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# Time Value of Money I: Single Payment Value
## Time Value of Money (TVM) Basics
### Learning Outcomes
By the end of this section, you will be able to:
1. Define future value and provide examples.
2. Explain how future dollar amounts are calculated using a single-period scenario.
3. Describe the impact of compounding.
Because we can invest our money in interest-bearing accounts and investments, its value can grow over time as interest income accrues or returns are realized on our investments. This concept is referred to as future value (FV). In short, future value refers to how a specific amount of money today can have greater value tomorrow.
### Single-Period Scenario
Let us start with the following example. Your friend is considering putting money in a bank account that will pay 4% interest per year and is particularly interested in knowing how much money they will have one year from now if they deposit $1,000 in this account. Your friend understands that you are studying finance and turns to you for help. By using the TVM principle of future value (FV), you can tell your friend that the answer is $1,040. The additional $40 that will be in the account after one year will be due to interest earned over that time. You can calculate this amount relatively easily by taking the original deposit (also referred to as the principal) of $1,000 and multiplying it by the annual interest rate of 4% for one period (in this case, one year).
By taking the interest earned amount of $40 and adding it to the original principal of $1,000, you will arrive at a total value of $1,040 in the bank account at the end of the year. So, the $1,040 one year from today is equal to $1,000 today when working with a 4% earning rate. Therefore, based on the concept of TVM, we can say that $1,040 represents the future value of $1,000 one year from today and at a 4% rate of interest. We will discuss interest rates and their importance in TVM decisions in more detail later in this chapter; for now, we can consider interest rate as a percentage of the principal amount that is earned by the original lender of funds and/or charged to the borrower of these same funds. Following are a few more examples of the single-period scenario.
If a person deposits $300 in an account that pays 5% per year, at the end of one year, they will have
If a company has earnings of $2.50 per share and experiences a 10% increase in the following year, the earnings per share in year two are
If a retail store decides on a 3% price increase for the following year on an item that is currently selling for $50, the new price in the following year will be
### The Impact of Compounding
What would happen if your friend were willing to wait one more year to receive their lump sum payment? What would the future dollar value in their account be after a two-year period? Returning to our earlier example, assume that during the second year, your friend leaves the principal ($1,000) and the earned interest ($40) in the account, thereby reinvesting the entire account balance for another year. The quoted interest rate of 4% reflects the interest the account would earn each year, not over the entire two-year savings period. So, during the second year of savings, the $1,000 deposit and the $40 interest earned during the first year would both earn 4%:
The additional $1.60 is interest on the first year’s interest and reflects the compounding of interest. Compound interest is the term we use to refer to interest income earned in subsequent periods that is based on interest income earned in prior periods. To put it simply, compound interest refers to interest that is earned on interest. Here, it refers to the $1.60 of interest earned in the second year on the $40.00 of interest earned in the first year. Therefore, at the end of two years, the account would have a total value of $1,081.60. This consists of the original principal of $1,000 plus the $40.00 interest income earned in year one and the $41.60 interest income earned in year two.
The amount of money your friend would have in the account at the end of two years, $1,081.60, is referred to as the future value of the original $1,000 amount deposited today in an account that will earn 4% interest every year.
Simple interest applies to year 1 while compound interest or “interest on interest” applies to year 2. This is calculated using the following method:
So, the total amount that would be in the account after two years, at 4% annual interest, would be .
To determine any future value of money in an interest-bearing account, we multiply the principal amount by 1 plus the interest rate for each year the money remains in the account. From this, we can develop the future value formula:
In this formula, the number of times we multiply by depends entirely on the number of years the money will remain in the bank account, earning interest, before it is withdrawn in a final lump sum distribution paid out from the account at the end of the chosen savings period. The 1 in the formula represents the principal amount, or the original $1,000 deposit, which will be included in the final total lump sum payment when the account is closed and all money is withdrawn at the end of the predetermined savings period.
We can write the above equation in a more condensed mathematical form using time value of money notation, as follows:
Using these inputs, we have the following formula:
With this equation, we can calculate the value of the savings account after any number of years. For example, suppose we are considering 3, 10, and 50 years from the original deposit date at the annual 4% interest rate:
How can this savings account have grown to be so large after 50 years? This question is answered by the impact of compounding interest. Every year, the interest earned in previous years will also earn interest along with the initial deposit. This will have the effect of accelerating the growth of the total dollar value of the account.
This is the important effect of the compounding of interest: money grows in larger and larger increments the longer you leave it in an interest-bearing account. In effect, the compounding of interest over time accelerates the growth of money.
In order to determine the FV of any amount of money, it will always be necessary to know the following pieces of information: (1) the principal, initial deposit, or present value (PV); (2) the rate of interest, usually expressed on an annual basis as r; and (3) the number of time periods that the money will remain in the account (n). The interest rate is often referred to as the growth rate, or the annual percentage increase on savings or on an investment. When the rate is raised to the power of the number of periods, the formula will yield a number that is commonly referred to as the future value interest factor (FVIF). As a result of this process, as n (time, or the number of periods) increases, the future value interest factor will increase. Also, as r (interest rate) increases, the FVIF will increases. For these reasons, the future value calculation is directly determined by both the interest rate being used and the total amount of time—specifically, the number of periods—being considered.
### How Time Impacts Compounding
We have just seen that time will lead to the growth of our money. As long as the prevailing growth or interest rate of any account we have our money in is positive, the passage of time will have the effect of growing the value of our money. The longer the period of time, the greater the growth and the larger the future value of the money will be. This can be reinforced very clearly with the following example.
Melvin is saving money in an account at a local bank that earns 5% per year. He begins with a deposit in his account of $100 and decides to save his money for exactly one year. He will not be making any further deposits into the account during the year. Melvin will earn or $5, in interest income. Adding this to the original deposit balance of $100 will give him a total of or $105, in the account at the end of one year.
Melvin likes this idea and believes he may be able to keep his money in the account for a longer period of time. How much money will he have in his account, without any further deposits, at the end of years two, three, four, and five?
Using the future value formula, the calculation is as follows:
### How the Interest Rate Impacts Compounding
Melvin likes the idea of earning more money over time, but he also believes that what he would earn in interest may not be enough for some of the things he plans to buy in the future. His friend suggests finding an account or some form of investment with a greater interest rate than the 5% he can get at his local bank.
Melvin thinks he can leave his money in an account or investment for a total of five years. He found investments that will provide annual returns of 6%, 7%, 10%, and 12%. Using the formula, we can complete the following calculations for him:
Again, Melvin likes this information, and he states that he will try to find the highest interest rate available. This makes sense, but it’s important to remember that investments are usually not guaranteed to earn you specific interest rates, or rates of return. Most investments, other than Treasury investments such as Treasury bonds, carry some form of financial risk, either small or large, and the greater the rate of return, the more likely it is that the risk associated with the investment will also be greater. This risk does not have any effect on the future calculations we have just completed, but it an important factor to bear in mind and consider well before moving ahead and putting your money in any investment or financial instrument.
###
Future value refers to the value that a current amount will eventually grow into at a given interest rate over a specific period of time. The single-period scenario is one way in which future amounts are calculated. Compounding, which is interest earned on interest, also affects the future value of money.
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# Time Value of Money I: Single Payment Value
## Methods for Solving Time Value of Money Problems
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain how future dollar amounts are calculated.
2. Explain how present dollar amounts are calculated.
3. Describe how discount rates are calculated.
4. Describe how growth rates are calculated.
5. Illustrate how periods of time for specified growth are calculated.
6. Use a financial calculator and Excel to solve TVM problems.
We can determine future value by using any of four methods: (1) mathematical equations, (2) calculators with financial functions, (3) spreadsheets, and (4) FVIF tables. With the advent and wide acceptance and use of financial calculators and spreadsheet software, FVIF (and other such time value of money tables and factors) have become obsolete, and we will not discuss them in this text. Nevertheless, they are often still published in other finance textbooks and are also available on the internet to use if you so choose.
### Using Timelines to Organize TVM Information
A useful tool for conceptualizing present value and future value problems is a timeline. A timeline is a visual, linear representation of periods and cash flows over a set amount of time. Each timeline shows today at the left and a desired ending, or future point (maturity date), at the right.
Now, let us take an example of a future value problem that has a time frame of five years. Before we begin to solve for any answers, it would be a good approach to lay out a timeline like that shown in :
The timeline provides a visual reference for us and puts the problem into perspective.
Now, let’s say that we are interested in knowing what today’s balance of $100 in our saving account, earning 5% annually, will be worth at the end of each of the next five years. Using the future value formula
that we covered earlier, we would arrive at the following values: $105 at the end of year one, $110.25 at the end of year two, $115.76 at the end of year three, $121.55 at the end of year four, and $127.63 at the end of year five.
With the numerical information, the timeline (at a 5% interest or growth rate) would look like :
Using timelines to lay out TVM problems becomes more and more valuable as problems become more complex. You should get into the habit of using a timeline to set up these problems prior to using the equation, a calculator, or a spreadsheet to help minimize input errors. Now we will move on to the different methods available that will help you solve specific TVM problems. These are the financial calculator and the Excel spreadsheet.
### Using a Financial Calculator to Solve TVM Problems
An extremely popular method of solving TVM problems is through the use of a financial calculator. Financial calculators such as the will typically have five keys that represent the critical variables used in most common TVM problems: N, I/Y, PV, FV, and PMT. These represent the following:
These are the only keys on a financial calculator that are necessary to solve TVM problems involving a single payment or lump sum.
### Example 1: Future Value of a Single Payment or Lump Sum
Let’s start with a simple example that will provide you with most of the skills needed to perform TVM functions involving a single lump sum payment with a financial calculator.
Suppose that you have $1,000 and that you deposit this in a savings account earning 3% annually for a period of four years. You will naturally be interested in knowing how much money you will have in your account at the end of this four-year time period (assuming you make no other deposits and withdraw no cash).
To answer this question, you will need to work with factors of $1,000, the present value (PV); four periods or years, represented by N; and the 3% interest rate, or I/Y. Make sure that the calculator register information is cleared, or you may end up with numbers from previous uses that will interfere with the solution. The register-clearing process will depend on what type of calculator you are using, but for the TI BA II Plus™ Professional calculator, clearing can be accomplished by pressing the keys 2ND and FV [CLR TVM].
Once you have cleared any old data, you can enter the values in the appropriate key areas: 4 for N, 3 for I/Y, and 1000 for PV. Now you have entered enough information to calculate the future value. Continue by pressing the CPT (compute) key, followed by the FV key. The answer you end up with should be displayed as 1,125.51 (see ).
### Important Notes for Using a Calculator and the Cash Flow Sign Convention
Please note that the PV was entered as negative $1,000 (or -$1000). This is because most financial calculators (and spreadsheets) follow something called the cash flow sign convention, which is a way for calculators and spreadsheets to keep the relative direction of the cash flow straight. Positive numbers are used to represent cash inflows, and negative numbers should always be used for cash outflows.
In this example, the $1,000 is an investment that requires a cash outflow. For this reason, -1000 is entered as the present value, as you will be essentially handing this $1,000 to a bank or to someone else to initiate the transaction. Conversely, the future value represents a cash inflow in four years’ time. This is why the calculator generates a positive 1,125.51 as the end result of this calculation.
Had you entered the present value of $1,000 as a positive number, there would have been no real concern, but the ending future value answer would have been returned expressed as a negative number. This would be correct had you borrowed $1,000 today (cash inflow) and agreed to repay $1,125.51 (cash outflow) four years from now. Also, it is important that you do not change the sign of any input value by using the - (minus) key). For example, on the TI BA II Plus™ Professional, you must use the +|- key instead of the minus key. If you enter 1000 and then hit the +|- key, you will get a negative 1,000 amount showing in the calculator display.
An important feature of most financial calculators is that it is possible to change any of the variables in a problem without needing to reenter all of the other data. For example, suppose that we wanted to find out the future value in our bank account if we left the money from our previous example invested for 20 years instead of 4. Before clearing any of the data, simply enter 20 for N and then press the CPT key and then the FV key. After this is done, all other inputs will remain the same, and you will arrive at an answer of $1,806.11.
### Example 2: Present Value of Lump Sums
Solving for the present value (discounted value) of a lump sum is the exact opposite of solving for a future value. Once again, if we enter a negative value for the FV, then the calculated PV will be a positive amount.
Taking the reverse of what we did in our example of future value above, we can enter -1,125.51 for FV, 3 for I/Y, and 4 for N. Hit the CPT and PV keys in succession, and you should arrive at a displayed answer of 1,000.
An important constant within the time value of money framework is that the present value will always be less than the future value unless the interest rate is negative. It is important to keep this in mind because it can help you spot incorrect answers that may arise from errors with your input.
### Example 3: Calculating the Number of Periods
There will be times when you will know both the value of the money you have now and how much money you will need to have at some unknown point in the future. If you also know the interest rate your money will be earning for the foreseeable future, then you can solve for N, or the exact amount of time periods that it will take for the present value of your money to grow into the future value that you will require for your eventual use.
Now, suppose that you have $100 today and you would like to know how long it will take for you to be able to purchase a product that costs $133.82.
After making sure your calculator is clear, you will enter 5 for I/Y, -100 for PV, and 133.82 for FV. Now press CPT N, and you will see that it will take 5.97 years for your money to grow to the desired amount of $133.82.
Again, an important thing to note when using a financial calculator to solve TVM problems is that you must enter your numbers according to the cash flow sign convention discussed above. If you do not make either the PV or the FV a negative number (with the other being a positive number), then you will end up getting an error message on the screen instead of the answer to the problem. The reason for this is that if both numbers you enter for the PV and FV are positive, the calculator will operate under the assumption that you are receiving a financial benefit without making any cash outlay as an initial investment. If you get such an error message in your calculations, you can simply press the CE/C key. This will clear the error, and you can reenter your data correctly by changing the sign of either PV or FV (but not both of these, of course).
### Example 4: Solving for the Interest Rate
Solving for an interest rate is a common TVM problem that can be easily addressed with a financial calculator. Let’s return to our earlier example, but in this case, we know that we have $1,000 at the present time and that we will need to have a total of $1,125.51 four years from now. Let’s also say that the only way we can add to the current value of our savings is through interest income. We will not be able to make any further deposits in addition to our initial $1,000 account balance.
What interest rate should we be sure to get on our savings account in order to have a total savings account value of $1,125.51 four years from now?
Once again, clear the calculator, and then enter 4 for N, -1,000 for PV, and 1,125.51 for FV. Then, press the CPT and I/Y keys and you will find that you need to earn an average 3% interest per year in order to grow your savings balance to the desired amount of $1,125.51. Again, if you end up with an error message, you probably failed to follow the sign convention relating to cash inflow and outflow that we discussed earlier. To correct this, you will need to clear the calculator and reenter the information correctly.
After you believe you are done and have arrived at a final answer, always make sure you give it a quick review. You can ask yourself questions such as “Does this make any sense?” “How does this compare to other answers I have arrived at?” or “Is this logical based on everything I know about the scenario?” Knowing how to go about such a review will require you to understand the concepts you are attempting to apply and what you are trying to make the calculator do. Further, it is critical to understand the relationships among the different inputs and variables of the problem. If you do not fully understand these relationships, you may end up with an incorrect answer. In the end, it is important to realize that any calculator is simply a tool. It will only do what you direct it to do and has no idea what your objective is or what it is that you really wish to accomplish.
### Using Excel to Solve TVM Problems
Excel spreadsheets can be excellent tools to use when solving time value of money problems. There are dozens of financial functions available in Excel, but a student who can use a few of these functions can solve almost any TVM problem. Special functions that relate to TVM calculations are as follows:
Excel also includes a function called Payment (PMT) that is used in calculations involving multiple payments or deposits (annuities). These will be covered in Time Value of Money II: Equal Multiple Payments.
### Future Value (FV)
The Future Value function in Excel is also referred to as FV and can be used to calculate the value of a single lump sum amount carried to any point in the future. The FV function syntax is similar to that of the other four basic time-value functions and has the following inputs (referred to as arguments), similar to the functions listed above:
Lump sum problems do not involve payments, so the value of Pmt in such calculations is 0. Another argument, Type, refers to the timing of a payment and carries a default value of the end of the period, which is the most common timing (as opposed to the beginning of a period). This may be ignored in our current example, which means the default value of the end of the period will be used.
The spreadsheet in shows two examples of using the FV function in Excel to calculate the future value of $100 in five years at 5% interest.
In cell E1, the FV function references the values in cells B1 through B4 for each of the arguments. When a user begins to type a function into a spreadsheet, Excel provides helpful information in the form of on-screen tips showing the argument inputs that are required to complete the function. In our spreadsheet example, as the FV formula is being typed into cell E2, a banner showing the arguments necessary to complete the function appears directly below, hovering over cell E3.
Cells E1 and E2 show how the FV function appears in the spreadsheet as it is typed in with the required arguments. Cell E4 shows the calculated answer for cell E1 after hitting the enter key. Once the enter key is pressed, the hint banner hovering over cell E3 will disappear. The second example of the FV function in our example spreadsheet is in cell E6. Here, the actual numerical values are used in the FV function equation rather than cell references. The method in cell E8 is referred to as hard coding. In general, it is preferable to use the cell reference method, as this allows for copying formulas and provides the user with increased flexibility in accounting for changes to input data. This ability to accept cell references in formulas is one of the greatest strengths of Excel as a spreadsheet tool.
Determining Future Value When Other Variables Are Known. You have $2,000 invested in a money market account that is expected to earn 4% annually. What will be the total value in the account in five years?
Note: Be sure to follow the sign conventions. In this case, the PV should be entered as a negative value.
Note: In Excel, interest and growth rates must be entered as percentages, not as whole integers. So, 4 percent must be entered as 4% or 0.04—not 4, as you would enter in a financial calculator.
Note: It is always assumed that if not specifically stated, the compounding period of any given interest rate is annual, or based on years.
Note: The Excel command used to calculate future value is as follows:
You may simply type the values for the arguments in the above formula. Another option is to use the Excel insert function option. If you decide on this second method, below are several screenshots of dialog boxes you will encounter and will be required to complete.
1. First, go to Formulas in the upper menu bar, and select the Insert Function option. When you do so, a dialog box will appear that looks like what you see in .
This dialog box allows you to either search for a function or select a function that has been used recently. In this example, you can search for FV by typing this in the search box and selecting Go, or you can simply choose FV from the list of most recently used functions (as shown here with the highlighted FV option).
2. Once you select FV and click the OK button, a new dialog box will appear for you to enter the necessary details. See .
Additional notes:
1. The Pmt argument or variable can be ignored in this instance, or you can enter a placeholder value of zero. This example shows a blank or ignored entry, but either option may be used in problems such as this where the information is not relevant.
2. The Type argument does not apply to this problem. Type refers to the timing of cash flows and is usually used in multiple payment or annuity problems to indicate whether payments or deposits are made at the beginning of periods or at the end. In single lump sum problems, this is not relevant information, and the Type argument box is left empty.
3. When you use cell addresses as function argument inputs, the numerical values within the cells are displayed off to the right. This helps you ensure that you are identifying the correct cells in your function. The final answer generated by the function is also displayed for your preliminary review.
Once you are satisfied with the result, hit the OK button, and the dialog box will disappear, with only the final numerical result appearing in the cell where you have set up the function.
The FV of this present value has been calculated as approximately $2,433.31.
### Present Value (PV)
We have covered the idea that present value is the opposite of future value. As an example, in the spreadsheet shown in , we calculated that the future value of $100 five years from now at a 5% interest rate would be $127.63. By reversing this process, we can safely state that $127.63 received five years from now with a 5% interest (or discount) rate would have a value of just $100 today. Thus, $100 is its present value. In Excel, the PV function is used to determine present value (see ).
The formula in cell E1 uses cell references in a similar fashion to our FV example spreadsheet above. Also similar to our earlier example is the hard-coded formula for this calculation, which is shown in cell E6. In both cases, the answers we arrive at using the PV function are identical, but once again, using cell references is preferred over hard coding if possible.
### Periods of Time
The following discussion will show you how to use Excel to determine the amount of time a given present value will need to grow into a specified future value when the interest or growth rate is known.
You want to be able to contribute $25,000 to your child’s first year of college tuition and related expenses. You currently have $15,000 in a tuition savings account that is earning 6% interest every year. How long will it take for this account grow into the targeted amount of $25,000, assuming no additional deposits or withdrawals are made?
Notes:
1. As with our other examples, interest and growth rates must be entered as percentages, not as whole integers. So, 6 percent must be entered as 6% or 0.06—not 6, as you would enter in a financial calculator.
2. The present value needs to be entered as a negative value in accordance with the sign convention covered earlier.
3. The Excel command used to calculate the amount of time, or number of periods, is this:
As with our FV and PV examples, you may simply type the values of the arguments in the above formula, or we can again use the Insert Function option in Excel. If you do so, you will need to work with the various dialog boxes after you select Insert Function.
1. First, go to Formulas in the upper menu bar, and select the Insert Function option. When you do so, the Insert Function dialog box will appear (see ).
As discussed in our previous examples on FV and PV, this menu allows you to either search for a function or select a function that has been used recently. In this example, you can search for NPER by typing this into the search box and selecting Go, or you can simply choose NPER from the list of most recently used functions.
2. Once you have highlighted NPER, click the OK button, and a new dialog box will appear for you to enter the necessary details. As in our previous examples, it will look like .
shows the completed Function Arguments dialog box. Note that once again, we are using cell addresses in this example.
As in the previous function examples, values are shown off to the right of the data input area, and our final answer of approximately 8.77 is displayed at the bottom. Also, once again, the Pmt and Type boxes are not relevant to this single lump sum example.
Review your answer, and once you are satisfied with the result, click the OK button. The dialog box will disappear, with only the final numerical result appearing in the cell where you have set up the function.
The amount of time required for the desired growth to occur is calculated as approximately 8.77 years.
### Interest or Growth Rate
You can also use Excel to determine the required growth rate when the present value, future value, and total number of required periods are known.
Let’s discuss a similar example to the one we used to calculate periods of time. You still want to help your child with their first year of college tuition and related expenses, and you still have a starting amount of $15,000, but you have not yet decided which savings plan to use.
Instead, the information you now have is that your child is just under 10 years old and will begin college at age 18. For simplicity’s sake, let’s say that you have eight and a half years until you will need to meet your total savings target of $25,000. What rate of interest will you need to grow your saved money from $15,000 to $25,000 in this time, again with no other deposits or withdrawals?
Note: The present value needs to be entered as a negative value.
Note: The Excel command used to calculate interest or growth rate is as follows:
As with our other TVM function examples, you may simply type the values for the arguments into the above formula. We also again have the same alternative to use the Insert Function option in Excel. If you choose this option, you will again see the Insert Function dialog box after you click the Insert Function button.
1. First, go to Formulas in the upper menu bar, and select the Insert Function option. When you do so, the Insert Function dialog box will appear (see ).
2. This time, find and highlight RATE, and click the OK button once you have done so. The Function Arguments dialog box will look like .
Once we complete the input, again using cell addresses for the required argument values, we will see what is shown in
As in our other examples, cell values are shown as numerical values off to the right, and our answer of approximately 0.0619, or 6.19%, is shown at the bottom of the dialog box.
This answer also can be checked from a logic point of view because of the similar example we worked through when calculating periods of time. Our present value and future value are the same as in that example, and our time period is now 8.5 years, which is just under the result we arrived at (8.77 years) in the periods example.
So, if we are now working with a slightly shorter time frame for the savings to grow from $15,000 into $25,000, then we would expect to have a slightly greater growth rate. That is exactly how the answer turns out, as the calculated required interest rate of approximately 6.19% is just slightly greater than the growth rate of 6% used in the previous example. So, based on this, it looks like our answer here passes a simple “sanity check” review.
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Calculations can be used to determine future and present dollar amounts, discount and growth rates, and periods of time required for specific growth. Time value of money problems can be solved using mathematical equations, calculators with financial functions, and spreadsheets. A useful tool for conceptualizing present value and future value problems is a timeline. A timeline is a visual, linear representation of the timing of periods and cash flows over a set amount of time.
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### Time Value of Money Calculations with Financial Calculator
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### Time Value of Money Using Excel with 10 Examples
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# Time Value of Money I: Single Payment Value
## Applications of TVM in Finance
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain how the time value of money can impact your personal financial goals.
2. Explain how the time value of money is related to inflation.
3. Explain how the time value of money is related to financial risk.
4. Explain how compounding period frequency affects the time value of money.
### Single-Period Scenario
Let’s say you want to buy a new car next year, and the one you have your eye on should be selling for $20,000 a year from now. How much will you need to put away today at 5% interest to have $20,000 a year from now? Essentially, you are trying to determine how much $20,000 one year from now is worth today at 5% interest over the year. To find a present value, we reverse the growth concept and lower or discount the future value back to the current period.
The interest rate that we use to determine the present value of a future cash flow is referred to as the discount rate because it is bringing the money back in time in terms of its value. The discount rate refers to the annual rate of reduction on a future value and is the inverse of the growth rate. Once we know this discount rate, we can solve for the present value (PV), the value today of tomorrow’s cash flow. By changing the FV equation, we can turn
into
which is the present value equation. The fraction shown above is referred to as the present value interest factor (PVIF). The PVIF is simply the reciprocal of the FVIF, which makes sense because these factors are doing exactly opposite things. Therefore, the amount you need to deposit today to earn $20,000 in one year at 5% interest is
### The Multiple-Period Scenario
There will often be situations when you need to determine the present value of a cash flow that is scheduled to occur several years in the future (see ). We can again use the formula for present value to calculate a value today of future cash flows over multiple time periods.
An example of this would be if you wanted to buy a savings bond for Charlotte, the daughter of a close friend. The face value of the savings bond you have in mind is $1,000, which is the amount Charlotte would receive in 30 years (the future value). If the government is currently paying 5% per year on savings bonds, how much will it cost you today to buy this savings bond?
The $1,000 face value of the bond is the future value, and the number of years n that Charlotte must wait to get this face value is 30 years. The interest rate r is 5.0% and is the discount rate for the savings bond. Applying the present value equation, we calculate the current price of this savings bond as follows:
So, it would cost you $231.38 to purchase this 30-year, 5%, $1,000 face-valued bond.
What we have done in the above example is reduce, or discount, the future value of the bond to arrive at a value expressed in today’s dollars. Effectively, this discounting process is the exact opposite of compounding interest that we covered earlier in our discussion of future value.
An important concept to remember is that compounding is the process that takes a present valuation of money to some point in the future, while discounting takes a future value of money and equates it to present dollar value terms.
Common applications in which you might use the present value formula include determining how much money you would need to invest in an interest-bearing account today in order to finance a college education for your oldest child and how much you would need to invest today to meet your retirement plans 30 years from now.
### TVM, Inflation, Compounding Interest, Investing, Opportunity Costs, and Risk
The time value of money (TVM) is a critical concept in understanding the value of money relative to the amount of time it is held, saved, or invested. The TVM concept and its specific applications are frequently used by individuals and organizations that might wish to better understand the values of financial assets and to improve investing and saving strategies, whether these are personal or within business environments.
As we have discussed, the key element behind the concept of TVM is that a given amount of money is worth more today than that same amount of money will be at any point in the future. Again, this is because money can be saved or invested in interest-bearing accounts or investments that will generate interest income over time, thus resulting in increased savings and dollar values as time passes.
### Inflation
The entire concept of TVM exists largely due to the presence of inflation. Inflation is defined as a general increase in the prices of goods and services and/or a drop in the value of money and its purchasing power.
The purchasing power of the consumer dollar is a statistic tracked by the part of the US Bureau of Labor Statistics and is part of the consumer price index (CPI) data that is periodically published by that government agency. In a way, purchasing power can be viewed as a mirror image or exact opposite of inflation or increases in consumer prices, as measured by the CPI. demonstrates the decline in the purchasing power of the consumer dollar over the 13-year period from 2007 to 2020.
With this in mind, we can work with the TVM formula and use it to help determine the present value of money you have in hand today, as well as how this same amount of money may be valued at any specific point in the future and at any specific rate of interest.
### The Relationship between TVM and Inflation
As we have seen, the future value formula can be very helpful in calculating the value of a sum of cash (or any liquid asset) at some future point in time. One of the important ideas relating to the concept of TVM is that it is preferable to spend money today instead of at some point in the future (all other things being equal) when inflation is positive. However, in very rare instances in our economy when inflation is negative, spending money later is preferable to spending it now. This is because in cases of negative inflation, the purchasing power of a dollar is actually greater in the future, as the costs of goods and services are declining as we move into the future.
Most investors would be inclined to take a payment of money today rather than wait five years to receive a payment in the same amount. This is because inflation is almost always positive, which means that general prices of goods and services tend to increase over the passage of time. This is a direct function and result of normal economic growth. The crux of the concept of TVM is directly related to maintaining the present value of financial assets or increasing the value of these financial assets at different points in the future when they may be needed to obtain goods and services. If a consumer’s monetary assets grow at a greater rate than inflation over any period of time, then the consumer will realize an increase in their overall purchasing power. Conversely, if inflation exceeds savings or investment growth, then the consumer will lose purchasing power as time goes by under such conditions.
### The Impact of Inflation on TVM
The difference between present and future values of money can be easily seen when considered under the effects of inflation. As we discussed above, inflation is defined as a state of continuously rising prices for goods and services within an economy. In the study of economics, the laws of supply and demand state that increasing the amount of money within an economy without increasing the amount of goods and services available will give consumers and businesses more money to spend on those goods and services. When more money is created and made available to the consuming public, the value of each unit of currency will diminish. This will then have the effect of incentivizing consumers to spend their money now, or in the very near future, instead of saving cash for later use. Another concept in economics states that this relationship between money supply and monetary value is one of the primary reasons why the Federal Reserve might at times take steps to inject money into a stagnant, lethargic economy. Increasing the money supply will lead to increased economic activity and consumer spending, but it can also have the negative effect of increasing the costs of goods and services, furthering an increase in the rate of inflation.
Consumers who decide to save their money now and for the foreseeable future, as opposed to spending it now, are simply making the economic choice to have their cash on hand and available. So, this ends up being a decision that is made despite the risk of potential inflation and perhaps losing purchasing power. When inflationary risk is low, most people will save their money to have it available to spend later. Conversely, in times when inflationary risk is high, people are more likely to spend their money now, before its purchasing power erodes. This idea of inflationary risk is the primary reason why savers and investors who decide to save now in order to have their money available at some point in the future will insist they are paid, through interest or return on investment, for the future value of any savings or financial instrument.
Lower interest rates will usually lead to higher inflation. This is because, in a way, interest rates can be viewed as the cost of money. This allows for the idea that interest rates can be further viewed as a tax on holding on to sums of money instead of using it. If an economy is experiencing lower interest rates, this will make money less expensive to hold, thus incentivizing consumers to spend their money more frequently on the goods and services they may require. We have also seen that the more quickly inflation rates rise, the more quickly the general purchasing power of money will be eroded. Rational investors who set money aside for the future will demand higher interest rates to compensate them for such periods of inflation. However, investors who save for future consumption but leave their money uninvested or underinvested in low-interest-bearing accounts will essentially lose value from their financial assets because each of their future dollars will be worth less, carrying less purchasing power when they end up needing it for use. This relationship of saving and planning for the future is one of the most important reasons to understand the concept of the time value of money.
### Nominal versus Real Interest Rates
One of the main problems of allowing inflation to determine interest rates is that current interest rates are actually nominal interest rates. Nominal rates are “stated,” not adjusted for the effects of inflation. In order to determine more practical real interest rates, the original nominal rate must be adjusted using an inflation rate, such as those that are calculated and published by the Bureau of Labor Statistics within the consumer price index (CPI).
A concept referred to as the Fisher effect, named for economist Irving Fisher, describes the relationship between inflation and the nominal and real interest rates and is expressed using the following formula:
where i is the nominal interest rate, R is the real interest rate, and h is the expected inflation rate.
An example of the Fisher effect would be seen in the case of a bond investor who is expecting a real interest rate of return of 6% on the bond, in an economy that is experiencing an expected inflation rate of 2%. Using the above formula, we have
So, the nominal interest rate on the bond amounts to 8.12%, with a real interest rate of 6% within an economy that is experiencing a 2% inflation rate. This is a logical result because in a scenario of positive inflation, a real rate of return would always be expected to amount to less than the stated or nominal rate.
### Interest and Savings
Savings are adversely affected by negative real interest rates. A person who holds money in the form of cash is actually losing future purchasing value when real interest rates are negative. A saver who decides to hold $1,000 in the form of cash for one year at a negative real interest rate of −3.65% per year will lose or $36.50, in purchasing power by the end of that year.
Ordinarily, interest rates would rise to compensate for negative real rates, but this might not happen if the Federal Reserve takes steps to maintain low interest rates to help stimulate and stabilize the economy. When interest rates are at such low levels, investors are forced out of Treasury and money market investments due to their extremely poor returns.
It soon becomes obvious that the time value of money is a critical concept because of its tremendous and direct impact on the daily spending, saving, and investment decisions of the people in our society. It is therefore extremely important that we understand how TVM and government fiscal policy can affect our savings, investments, purchasing behavior, and our overall personal financial health.
### Compounding Interest
As we discussed earlier, compound interest can be defined as interest that is being earned on interest. In cases of compounding interest, the amount of money that is being accrued on previous amounts of earned interest income will continue to grow with each compounding period. So, for example, if you have $1,000 in a savings account and it is earning interest at a 10% annual rate and is compounded every year for a period of five years, the compounding will allow for growth after one year to an amount of $1,100. This comprises the original principal of $1,000 plus $100 in interest. In year two, you would actually be earning interest on the total amount from the previous compounding period—the $1,100 amount.
So, to continue with this example, by the end of year two, you would have earned $1,210 ($1,100 plus $110 in interest). If you continue on until the end of year five, that $1,000 will have grown to approximately $1,610. Now, if we consider that the highest annual inflation rate over the last 20 years has been 3%, then in this scenario, choosing to invest your present money in an account where interest is being compounded leaves you in a much better position than you would be in if you did not invest your money at all. The concept of the time value of money puts this entire idea into context for us, leading to more informed decisions on personal saving and investing.
It is important to understand that interest does not always compound annually, as assumed in the examples we have already covered. In some cases, interest can be compounded quarterly, monthly, daily, or even continuously. The general rule to apply is that the more frequent the compounding period, the greater the future value of a savings amount, a bond, or any other financial instrument. This is, of course, assuming that all other variables are constant.
The math for this remains the same, but it is important that you be careful with your treatment and usage of rate (r) and number of periods (n) in your calculations.
For example, $1,000 invested at 6% for a year compounded annually would be worth . But that same $1,000 invested for that same period of time—one year—and earning interest at the same annual rate but compounded monthly would grow to , because the interest paid each month is earning interest on interest at a 6% rate. Note that we represent r as the interest paid per period and n as the number of periods (12 months in a year; ) rather than the number of years, which is only one.
Continuing with our example, that same $1,000 in an account with interest compounded quarterly, or four times a year, would grow to in one year. Note that this final amount ends up being greater than the annually compounded future value of $1,060.00 and slightly less than the monthly compounded future value of $1.061.67, which would appear to make logical sense.
The total differences in future values among annual, monthly, and quarterly compounding in these examples are insignificant, amounting to less than $1.70 in total. However, when working with larger amounts, higher interest rates, more frequent compounding periods, and longer terms, compounding periods and frequency become far more important and can generate some exceptionally large differences in future values.
Ten million dollars at 12% growth for one year and compounded annually amounts to , while 10 million dollars on the same terms but compounded quarterly will produce . Most wealthy and rational investors and savers would be very pleased to earn that additional $55,088.10 by simply having their funds in an account that features quarterly compounding.
In another example, $200 at 60% interest, compounded annually for six years, becomes , while this same amount compounded quarterly grows to .
An amount of $1 at 3%, compounded annually for 100 years, will be worth . The same dollar at the same interest rate, compounded monthly over the course of a century, will grow to .
This would all seem to make sense due to the fact that in situations when compounding increases in frequency, interest income is being received during the year as opposed to at the end of the year and thus grows more rapidly to become a larger and more valuable sum of money. This is important because we know through the concept of TVM that having money now is more useful to us than having that same amount of money at some later point in time.
### The Rule of 72
The rule of 72 is a simple and often very useful mathematical shortcut that can help you estimate the impact of any interest or growth rate and can be used in situations ranging from financial calculations to projections of population growth. The formula for the rule of 72 is expressed as the unknown (the required amount of time to double a value) calculated by taking the number 72 and dividing it by the known interest rate or growth rate. When using this formula, it is important to note that the rate should be expressed as a whole integer, not as a percentage. So, as a result, we have
This formula can be extremely practical when working with financial estimates or projections and for understanding how compound interest can have a dramatic effect on an original amount or monetary balance.
Following are just a few examples of how the rule of 72 can help you solve problems very quickly and very easily, often enabling you to solve them “in your head,” without the need for a calculator or spreadsheet.
Let’s say you are interested in knowing how long it will take your savings account balance to double. If your account earns an interest rate of 9%, your money will take or 8, years to double. However, if you are earning only 6% on this same investment, your money will take , or 12, years to double.
Now let’s say you have a specific future purchasing need and you know that you will need to double your money in five years. In this case, you would be required to invest it at an interest rate of , or 14.4%. Through these sample examples, it is easy to see how relatively small changes in a growth or interest rate can have significant impact on the time required for a balance to double in size.
To further illustrate some uses of the rule of 72, let’s say we have a scenario in which we know that a country’s gross domestic product is growing at 4% a year. By using the rule of 72 formula, we can determine that it will take the economy 72/4, or 18, years to effectively double.
Now, if the economic growth slips to 2%, the economy will double in , or 36, years. However, if the rate of growth increases to 11%, the economy will effectively double in , or 6.55, years. By performing such calculations, it becomes obvious that reducing the time it takes to grow an economy, or increasing its rate of growth, could end up being very important to a population, given its current level of technological innovation and development.
It is also very easy to use the rule of 72 to express future costs being impacted by inflation or future savings amounts that are earning interest.
To apply another example, if the inflation rate in an economy were to increase from 2% to 3%, consumers would lose half of the purchasing power of their money. This is calculated as the value of their money doubling in , or 24, years as compared to , or 36, years—quite a substantial difference.
Now, let’s say that tuition costs at a certain college are increasing at a rate of 7% per year, which happens to be greater than current inflation rates. In this case, tuition costs would end up doubling in , or about 10.3, years.
In an example related to personal finance, we can say that if you happen to have an annual percentage rate of 24% interest on your credit card and you do not make any payments to reduce your balance, the total amount you owe to the credit card company will double in only , or 3, years.
So, as we have seen, the rule of 72 can clearly demonstrate how a relatively small difference of 1 percentage point in GDP growth or inflation rates can have significant effects on any short- or long-term economic forecasting models.
It is important to understand that the rule of 72 can be applied in any scenario where we have a quantity or an amount that is in the process of growing or is expected to grow for any period of time into the future. A good nonfinancial use of the rule of 72 might be to apply it to some population projections. For example, an increase in a country’s population growth rate from 2% to 3% could present a serious problem for the planning of facilities and infrastructure in that country. Instead of needing to double overall economic capacity in or 36, years, capacity would have to be expanded in only , or 24, years. It is easy to see how dramatic an effect this would be when we consider that the entire schedule for growth or infrastructure would be reduced by 12 years due to a simple and relatively small 1% increase in population growth.
### Investing and Risk
Investing is usually a sound financial strategy if you have the money to do so. When investing, however, there are certain risks you should always consider first when applying the concepts of the time value of money. For example, making the decision to take $1,000 and invest it in your favorite company, even if it is expected to provide a 5% return each year, is not a guarantee that you will earn that return—or any return at all, for that matter. Instead, as with any investment, you will be accepting the risk of losing some or even all of your money in exchange for the opportunity to beat inflation and increase your future overall wealth. Essentially, it is risk and return that are responsible for the entire idea of the time value of money.
Risk and return are the factors that will cause a rational person to believe that a dollar risked should end up earning more than that single dollar.
To summarize, the concept of the time value of money and the related TVM formulas are extremely important because they can be used in different circumstances to help investors and savers understand the value of their money today relative to its earning potential in the future. TVM is critical to understanding the effect that inflation has on your money and why saving your money early can help increase the value of your savings dollars by giving them time to grow and outpace the effects of inflation.
### Opportunity Costs
The concept of opportunity cost arises from the idea that there will always be possible options that are sacrificed with every option we decide on or for every choice that we make. For example, let’s consider the decision to go to college after you graduate from high school. This decision, as with just about any other, will involve evaluating opportunity costs. If you choose to go to college, this will result in your sacrificing four years of potential earnings that you could have had if you had decided to take a job instead of attending school. Also, in addition to the lost salary, you would be losing out on four years of work experience that could have had a positive impact on your résumé or your future earnings prospects.
Of course, the entire idea behind furthering one’s education is that you are hopeful that by choosing to go to college, you will increase the likelihood of earning a greater salary over the course of your lifetime than you would have if you had chosen to join the workforce directly out of high school. So, this ends up being a bit of a risk, but one that you have considered. The idea is that you are hoping for a more significant payoff down the road than if you had made the decision not to continue with your studies. When it comes to opportunity costs and the time value of money, it is obvious that there will always be costs associated with every forgone financial opportunity we pass on when we make a different choice. The logical individual can only hope that these choices produce a better end result than if we had made different choices and pursued any of our forgone alternatives. This also applies in situations where we may sit idly by and decide to take no action at all.
For example, if you are putting $1,000 in a savings account to save for a house, you may be giving up an opportunity to grow that money in an investment account that would earn a greater rate of return. In another example, being able to calculate the future value of your money will tell you that instead of investing, you probably should be paying down your 24% APR credit card debt that is costing you hundreds of dollars a month—hundreds of dollars more than you might earn from an investment account.
###
The idea of the time value of money is often considered to be the cornerstone concept of the study of finance. TVM can help investors and savers understand the value of money today relative to its earning potential in the future. TVM is critical to understanding the effect that inflation has on money and why saving your money early can help increase the value of your savings dollars by giving them time to grow and outpace the effects of inflation. Of course, it is important to remember that there will always be possible options that are sacrificed with every option you decide on and every choice you make.
###
### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Time Value of Money II: Equal Multiple Payments
## Why It Matters
Although this text is directed at business finance students, our daily decisions as consumers are largely based on money and finance to just as great an extent. An old adage in finance claims, “If you aren’t in control of your money, your money is controlling you.” Fortunately, learning to manage your money is not difficult if you’re disciplined and understand some simple techniques. For example, several years ago, the author was negotiating for a three-year auto loan from a well-known regional dealer, who was offering an interest rate of 2%. When the manager left the room for a few minutes, we pulled out a financial calculator and proved in less than a minute that the actual interest rate in the payments he was proposing was nearly double the quoted and advertised rate.
In addition to understanding how the loan process works, which improves your negotiation skills when borrowing, businesses and individuals can better control their investments by understanding basic rules of finance, particularly as seen in this and the preceding chapter. Assume you pledge to invest $1,000 per year at 5% return per year and are curious about how much you will have accumulated by age 60. If you begin at age 30, you will have $69,760; if you begin at age 20, you will have $126,840. Can the extra 10 years make that much of a difference? We’ll see that indeed they can, and the calculations required to prove it can take less than two minutes.
As another example, many professionals confuse income and wealth during their career growth. In their popular book The Millionaire Next Door: The Surprising Secrets of America’s Wealthy, authors Thomas Stanley and William Danko illustrate these terms with a flowing river. A river is in constant movement, and as the flow or depth increases, this is comparable to one’s income increasing through the promotions, salary increases, and bonuses one receives. Unfortunately, many individuals then increase their spending habits in response, justifying a better car, a second home, or more lavish vacations. Wealth, in contrast, is comparable to taking a bucket of water from the river and holding it aside in a tank for oneself. Financial professionals often call this “paying yourself first.” Stanley and Danko list this among the “secrets” referenced in the title of their book. |
# Time Value of Money II: Equal Multiple Payments
## Perpetuities
### Learning Outcomes
By the end of this section, you will be able to:
1. Define perpetuity.
2. Explain how perpetuities are valued.
In Time Value of Money I, we learned that the value of money changes with the passage of time. Decision-makers consider how investments, projects, and even opportunity costs gain value as we move forward into the future. They similarly consider how value in the future can be reduced to a value in present or past periods. We saw that these value projections are called determination of future value (compounding, moving forward on a timeline) or present value (discounting, moving backward on a timeline). The easiest way to visualize this movement through time, whether forward or backward, is by use of a timeline.
Throughout the first chapter on the time value of money, we were analyzing a single amount. In this chapter, we deal with a stream of payments made periodically—in other words, payments made or received regularly over a span of time. We begin with the illustration of a perpetuity.
### What Is a Perpetuity?
A perpetuity is a series of payments or receipts that continues forever, or perpetually. One of the best ways to analyze the basics of an annuity (the stream of payments to be paid or received in the future) is by starting with a perpetuity. The most common examples of perpetuities in the author’s experience are college chair endowments and preferred stock.
If you gift $1,500,000 to a college to name a professor’s chair for your family, you might specify that the money must be held in perpetuity and invested by the college to yield a fixed 3%. The college will take those proceeds of the investment, leaving your original $1,500,000 intact, and use the annual interest of $45,000 to fund a portion of the professor’s salary.
Another common example is preferred stock. Most preferred stock issues carry a fixed and predetermined rate of dividend. If we assume that the dividend will not change in future years, then preferred dividend shareholders will receive a fixed amount of money in future years—assuming, of course, that the company’s board of directors declares the dividends sufficient to fund these requirements. If we assume that the dividend is declared and paid and that it remains constant, this represents a perpetuity.
For example, Shaw Inc. has issued 100,000 shares of preferred stock with a stated value of $50 and a 4% dividend. Therefore, if they can fund and decide to declare dividends for the full amount, they will pay out $200,000, or $2.00 per preferred share. Because shares such as these are created with the intention of continuity, the owner of this preferred stock can theoretically expect this dividend income stream in perpetuity.
To place a current market value on this stream of future income, how much should an investor pay for one share of this preferred stock? The calculation is a present value. The amount the investor pays today for that one share is equal to the annual dividend (assuming it is declared and paid) divided by the rate of return. But be careful—it is not the rate on the face of the preferred stock but the required rate of return, the “market rate” that investors expect from a stock of this level of risk. We must also note an important fact affecting all investment valuations: the value of an investment generally represents our expectations of all future cash flows from that investment, discounted to today’s dollars.
Because a perpetuity is a stream of payments continuing indefinitely, determining the future value isn’t possible. Determining the present value, however, is possible, although one might wonder how. As we learned earlier, the greater the amount of time used in a present value calculation, the smaller the amount of dollars needed at the beginning, regardless of the interest rate involved. Therefore, when we discount each payment in an infinite series, remembering that we would then add them together once we discount them to the present, the infinite payments become negligible at some point and will no longer have a significant impact on today’s value. To grow to one dollar 70 years from now, even at a growth rate of 5% per year, we would only need $0.0329—not even four cents. Keeping all other facts the same, if we had 100 years to grow an investment to one dollar, we would only need 0.76 cents—not even a whole penny! There is no question that the effect of time is substantial and dramatic.
The study of perpetuities in corporate finance is a first step to understanding valuation models of certain investments, such as the dividend discount model and the constant growth model, to be addressed in other chapters. Our ability to discount future cash flows, even infinite cash flows, to a present value is a clue to the price at which a company’s stock might trade. From a personal financial planning perspective, the individual investor is also better able to be certain that they are paying a fair price for holdings in their portfolio. For purposes of long-term or retirement planning, the investor must consider that a fixed and unchanging dividend, such as from preferred stock, might not adequately protect the holder from inflation in times of rising prices.
### How to Value a Perpetuity
Given these facts, how do we place a value on a perpetuity? Let’s keep the preferred stock example for Shaw Inc. in mind. The holder of one share will expect to receive a $2.00 dividend for every share owned. Although a perpetuity may allow for growth of that dividend, we will hold that constant now. We must know one additional fact: the required rate of return. This is our random variable, which can cause fluctuation in the price of the preferred stock. Let’s assume that the required rate of return, which we’ll call RS, is 7%. This is the rate of return that the market expects in order to take on the risk of an investment such as Shaw.
Determination of the price of Shaw’s preferred stock becomes quite simple because the expected annual cash flow should not change, making it a constant perpetuity. The constant perpetuity formula is
where PV is the price of the preferred stock, C is the constant dividend, and Rs is the required rate of return.
By substitution,
The price one should pay for a share of Shaw’s preferred stock is $28.57.
Here’s another constant perpetuity to try. The preferred stock of Rooney Corporation pays an annual dividend of $1.75 per share. If the required rate of return in the market for shares such as Rooney’s is 5.8%, at what price should these preferred shares be trading? The answer is , or $30.17.
Some investments might involve a growing perpetuity. In this case, some degree of change in the amount of the dividend is expected. The formula is altered slightly to include a rate of growth in the denominator, noted as G, making the growing perpetuity formula
To illustrate a growing perpetuity, let’s revisit Rooney Corp.’s stock, with its annual dividend of $1.75 and a required rate of return in the market of 5.8%. If the expected dividend growth rate G is 1.2%, then the value changes to , or $38.04. The expectation of growth in the dividend provides incentive for the investor to pay a higher price.
###
A perpetuity is an investment that is intended to provide an expected return indefinitely, either remaining constant or growing by an incremental amount. Preferred stock is a common example with a preestablished dividend formula. An indefinite stream of payments cannot be compounded into a future value, but it can be discounted to a present value, providing an opportunity to determine the amount an investor should be willing to pay for a share of that stock.
###
###
Use four decimal places on time value of money factors unless otherwise specified. Approximations and minor differences because of rounding are acceptable. Ignore the effect of taxes. Assume that all percentages are annual rates and that compounding occurs annually unless indicated otherwise. |
# Time Value of Money II: Equal Multiple Payments
## Annuities
### Learning Outcomes
By the end of this section, you will be able to:
1. Define annuity.
2. Distinguish between an ordinary annuity and an annuity due.
3. Calculate the present value of an ordinary annuity and an annuity due.
4. Explain how annuities may be used in lotteries and structured settlements.
5. Explain how annuities might be used in retirement planning.
### Calculating the Present Value of an Annuity
An annuity is a stream of fixed periodic payments to be paid or received in the future. Present or future values of these streams of payments can be calculated by applying time value of money formulas to each of these payments. We’ll begin with determining the present value.
Before exploring present value, it’s helpful to analyze the behavior of a stream of payments over time. Assume that we commit to a program of investing $1,000 at the end of each year for five years, earning 7% compounded annually throughout. The high rate is locked in based partly on our commitment beginning today, even though we will invest no money until the end of the first year. Refer to the timeline shown in .
At the end of the first year, we deposit the first $1,000 in our fund. Therefore, it has not yet had an opportunity to earn us any interest. The “new balance” number beneath is the cumulative amount in our fund, which then carries to the top of the column for the next year. In year 2, that first amount will earn 7% interest, and at the end of year 2, we add our second $1,000. Our cumulative balance is therefore $2,070, which then carries up to the top of year 3 and becomes the basis of the interest calculation for that year. At the end of the fifth year, our investing arrangement ends, and we’ve accumulated $5,750.74, of which $5,000 represents the money we invested and the other $750.74 represents accrued interest on both our invested funds and the accumulated interest from past periods.
Notice two important aspects that might appear counterintuitive: (1) we’ve “wasted” the first year because we deposited no funds at the beginning of this plan, and our first $1,000 begins working for us only at the beginning of the second year; and (2) our fifth and final investment earns no interest because it’s deposited at the end of the last year. We will address these two issues from a practical application point of view shortly.
Keeping this illustration in mind, we will first focus on finding the present value of an annuity. Assume that you wish to receive $25,000 each year from an existing fund for five years, beginning one year from now. This stream of annual $25,000 payments represents an annuity. Because the first payment will be received one year from now, we specifically call this an ordinary annuity. We will look at an alternative to ordinary annuities later. How much money do we need in our fund today to accomplish this stream of payments if our remaining balance will always be earning 8% annually? Although we’ll gradually deplete the fund as we withdraw periodic payments of the same amount, whatever funds remain in the account will always be earning interest.
Before we investigate a formula to calculate this amount, we can illustrate the objective: determining the present value of this future stream of payments, either manually or using Microsoft Excel. We can take each of the five payments of $25,000 and discount them to today’s value using the simple present value formula:
where FV is the future value, PV is the present value, r is the interest rate, and n is the number of periods.
For example, the first $25,000 is discounted by the equation as follows:
Proof that $23,148.15 will grow to $25,000 in one year at 8% interest:
If we use this same method for each of the five years, increasing the exponent n for each year, we see the result in .
We begin with the amount calculated in our table, $99,818.76. Before any money is withdrawn, a year’s worth of interest at 8% is compounded and added to our balance. Then our first $25,000 is withdrawn, leaving us with $82,804.26. This process continues until the end of five years, when, aside from a minor rounding difference, the fund has “done its job” and is equal to zero. However, we can make this simpler. Because each payment withdrawn (or added, as we will see later) is the same, we can calculate the present value of an annuity in one step using an equation. Rather than the multiple steps above, we will use the following equation:
where PVa is the present value of the annuity and PYMT is the amount of one payment.
In this example, PYMT is $25,000 at the end of each of five years. Note that the greater the number of periods and/or the size of the amount borrowed, the greater the chances of large rounding errors. We have used six decimal places in our calculations, though the actual time value of money factor, combining interest and time, can be much longer. Therefore, our solutions will often use ≅ rather than the equal sign.
By substitution, and then following the proper order of operations:
In both cases, barring a rounding difference caused by decimal expansion, we come to the same result using the equation as when we calculate each of multiple years. It’s important to note that rounding differences can become significant when dealing with larger multipliers, as in the financing of a multimillion-dollar machine or facility. In this text, we will ignore them.
In conclusion, five payments of $25,000, or $125,000 in total, can be funded today with $99,817.81, with the difference being obtained from interest always accumulating on the remaining balance at 8%. The running balance is obtained by calculating the year’s interest on the previous balance, adding it to that balance, and subtracting the $25,000 that is withdrawn on the last day of the year. In the last (fifth) year, just enough interest will accrue to bring the balance to the $25,000 needed to complete the fifth payment.
A common use of the PVa is with large-money lotteries. Let’s assume you win the North Dakota Lottery for $1.2 million, and they offer you $120,000 per year for 10 years, beginning one year from today. We will ignore taxes and other nonmathematical considerations throughout these discussions and problems. The Lottery Commission will likely contact you with an alternative: would you like to accept that stream of payments … or would you like to accept a lump sum of $787,000 right now instead? Can you complete a money-based analysis of these alternatives? Based purely on the dollars, no, you cannot. The reason is that you can’t compare future amounts to present amounts without considering the effect of time—that is, the time value of money. Therefore, we need an interest rate that we can use as a discounting factor to place these alternatives on the same playing field by expressing them in terms of today’s dollars, the present value. Let’s use 9%. If we discount the future stream of fixed payments (an ordinary annuity, as the payments are identical and they begin one year from now), we can then compare that result to the cash lump sum that the Lottery Commission is offering you instead.
By substitution, and following the proper order of operations:
All things being equal, that expected future stream of ten $120,000 payments is worth approximately $770,119 today. Now you can compare like numbers, and the $787,000 cash lump sum is worth more than the discounted future payments. That is the choice one would accept without considering such aspects as taxation, desire, need, confidence in receiving the future payments, or other variables.
### Calculating the Present Value of an Annuity Due
Earlier, we defined an ordinary annuity. A variation is the annuity due. The difference between the two is one period. That’s all—just one additional period of interest. An ordinary annuity assumes that there is a one-period lag between the start of a stream of payments and the actual first payment. In contrast, an annuity due assumes that payments begin immediately, as in the lottery example above. We would assume that you would receive the first annual lottery check of $120,000 immediately, not a year from now. In summary, whether calculating future value (covered in the next section) or present value of an annuity due, the one-year lag is eliminated, and we begin immediately.
Since the difference is simply one additional period of time, we can adjust for this easily by taking the formula for an ordinary annuity and multiplying by one additional period. One more period, of course, is (1 + i). Recall from Time Value of Money I that the formula for compounding is (1 + i), where i is the interest rate and N is the number of periods. The superscript N does not apply because it represents 1, for one additional period, and the power of 1 can be ignored. Therefore, faced with an annuity due problem, we solve as if it were an ordinary annuity, but we multiply by (1 + i) one more time.
In our original example from this section, we wished to withdraw $25,000 each year for five years from a fund that we would establish now. We determined how much that fund should be worth today if we intend to receive our first payment one year from now. Throughout this fund’s life, it will earn 8% annually. This time, let’s assume we’ll withdraw our first payment immediately, at point zero, making this an annuity due. Because we’re trying to determine how much our starting balance should be, it makes sense that we must begin with a larger number. Why? Because we’re pulling our first payment out immediately, so less money will remain to start compounding to the amount we need to fund all five of our planned payments! Our rule can be stated as follows:
Whether one is calculating present value or future value, the result of an annuity due must always be larger than that of an ordinary annuity, all other facts remaining constant. Here is the stream of solutions for the example above, but please notice that we will multiply by (1 + i), one additional period, following the same order of operations:
That’s how much we must start our fund with today, before we earn any interest or draw out any money. Note that it’s larger than the $99,817.81 that would be required for an ordinary annuity. It must be, because we’re about to diminish our compounding power with an immediate withdrawal, so we have to begin with a larger amount.
We notice several things:
1. The formula must change because the annual payment is subtracted first, prior to the calculation of annual interest.
2. We accomplish the same result, aside from an insignificant rounding difference: the fund is depleted once the last payment is withdrawn.
3. The last payment is withdrawn on the first day of the final year, not the last. Therefore, no interest is earned during the fifth and final year.
To reinforce this, let’s use the same approach for our lottery example above. Reviewing the facts, you have a choice of receiving 10 annual payments of your $1.2 million winnings, each worth $120,000, and you discount at a rate of 9%. The only difference is that this time, you can receive your first $120,000 right away; you don’t have to wait a year. This is now an annuity due. We solve it just as before, except that we multiply by one additional period of interest, (1 + i):
Again, this result must be larger than the amount we determined when this was calculated as an ordinary annuity.
The calculations above, representing the present values of ordinary annuities and annuities due, have been presented on an annual basis. In Time Value of Money I, we saw that compounding and discounting calculations can be based on non-annual periods as well, such as quarterly or monthly compounding and discounting. This aspect, quite common in periodic payment calculations, will be explored in a later section of this chapter.
### Calculating Annuities Used in Structured Settlements
In addition to lottery payouts, annuity calculations are often used in structured settlements by attorneys at law. If you win a $450,000 settlement for an insurance claim, the opposing party may ask you to accept an annuity so that they can pay you in installments rather than a lump sum of cash. What would a fair cash distribution by year mean? If you have a preferred discount rate (the percentage we all must know to calculate the time value of money) of 6% and you expect equal distributions of $45,000 over 10 years, beginning one year from now, you can use the present value of an annuity formula to compare the alternatives:
By substitution:
If the opposing attorney offered you a lump sum of cash less than that, all things equal, you would refuse it; if the lump sum were greater than that, you would likely accept it.
What if you negotiate the first payment to be made to you immediately, turning this ordinary annuity into an annuity due? As noted above, we simply multiply by one additional period of interest, (1 + 0.06). Repeating the last step of the solution above and then multiplying by (1 + 0.06), we determine that
You would insist on that number as an absolute minimum before you would consider accepting the offered stream of payments.
To further verify that ordinary annuity can be converted into an annuity due by multiplying the solution by one additional period’s worth of interest before applying the annuity factor to the payment, we can divide the difference between the two results by the value of the original annuity. When the result is expressed as a percent, it must be the same as the rate of interest used in the annuity calculations. Using our example of an annuity with five payments of $25,000 at 8%, we compare the present values of the ordinary annuity of $99,817.81 and the annuity due of $107,803.24.
The result shows that the present value of the annuity due is 8% higher than the present value of the ordinary annuity.
### Calculating the Future Value of an Annuity
In the previous section, we addressed discounting a periodic stream of payments from the future to the present. We are also interested in how to project the future value of a series of payments. In this case, an investment may be made periodically. Keeping with the definition of an annuity, if the amount of periodic investment is always the same, we may take a one-step shortcut to calculate the future value of that stream by using the formula presented below:
where FVa is the future value of the annuity, PYMT is a one-time payment or receipt in the series, r is the interest rate, and n is the number of periods.
As we did in our section on present values of annuities, we will begin with an ordinary annuity and then proceed to an annuity due.
Let’s assume that you lock in a contract for an investment opportunity at 4% per year, but you cannot make the first investment until one year from now. This is counterintuitive for an investor, perhaps, but because it is the basis of the formula and procedures for ordinary annuities, we will accept this assumption. You plan to invest $3,000 at the end of each year. How much money will you have at the end of five years?
Let’s start by placing this on a timeline like the one appearing earlier in this chapter (see ):
As we explained earlier when describing ordinary annuities, the payment for year 1 is not invested until the last day of that year, so year 1 is wasted as a compounding opportunity. Therefore, the amount only compounds for four years rather than five. Also, our fifth payment is not made until the last day of our contract in year 5, so it has no chance to earn a compounded future value. The investor has lost on both ends. In the table above, we have made five calculations, and for a longer-term contract such as 10, 25, or 40 years, this would be tedious. Fortunately, as with present values, this ordinary annuity can be solved in one step because all payments are identical.
Repeating the formula, and then by substitution:
This proof emphasizes that year 1 is wasted, with no compounding because the payment is made on the last day of year 1 rather than immediately. We lose compounding through this ordinary annuity in another way: year 5’s investment is made on the last day of this five-year contract and has no chance to accumulate interest. A more intuitive method would be to enter a contract for an annuity due so that our first payment can be made immediately. In this way, we don’t waste the first year, and all five payments work in year 5 as well. As stated previously, this means that annuities due will yield larger results than ordinary annuities, whether one is discounting (PVa) or compounding (FVa).
Let’s hold all facts constant with the previous example, except that we will invest at the beginning of each year, starting immediately upon locking in this five-year contract. We follow the same technique as in the present value section: we multiply by one additional period to convert this ordinary annuity factor into a factor for an annuity due. Whether one is solving for a future value or a present value, the result of an annuity due must always be larger than an ordinary annuity. With future value, we begin investing immediately, so the result will be larger than if we waited for a period to elapse. With present value, we begin extracting funds immediately rather than letting them work for us during the first year, so logically we would have to start with more.
Continuing our example but converting it to an annuity due, we will multiply by one additional period, (1 + i). All else remains the same:
Let’s provide one additional example of each. Assume that you have a chance to invest $15,000 per year for 10 years, earning 8% compounded annually. What amount would you have after the 10 years? If we can only make our first payment at the end of each year, our ending value will be
However, if we can make our first payment immediately and then make subsequent payments at the start of each following year, we modify the formula above by multiplying the annual payment by one additional period:
### How Annuities Are Used for Retirement Planning
On a final note, how might annuities be used for retirement planning? A person might receive a lump-sum windfall from an investment, and rather than choosing to accept the proceeds, they might decide to invest the sum (ignoring taxes) in an annuity. Their intention is to let this invested sum produce annual distributions to supplement Social Security payments. Assume the recipient just received $75,000, again ignoring tax effects. They have the chance to invest in an annuity that will provide a distribution at the end of each of the next five years, and that annuity contract provides interest at 3% annually. Their first receipt will be one year from now. This is an ordinary annuity.
We can also solve for the payment given the other variables, an important aspect of financial analysis. If the person with the $75,000 windfall wants this fund to last five years and they can earn 3%, then how much can they withdraw from this fund each year? To solve this question, we can apply the present value of an annuity formula. This time, the payment (PYMT) is the unknown, and we know that the PVa, or the present value that they have at this moment, is $75,000:
The person can withdraw this amount every year beginning one year from now, and when the final payment is withdrawn, the fund will be depleted. Interest accrues each year on the beginning balance, and then $16,376.60 is withdrawn at the end of each year.
###
An annuity is a stream of fixed periodic payments that is expected to be paid or received. Calculations of future value or present value are commonly performed on these payment streams for a wide number of reasons in business and personal financial analysis, as seen in the chapter focusing on single amounts, particularly in loan repayment. Annuities may be ordinary annuities, in which the first cash flow of a series occurs at the end of the first period, or annuities due, if the first cash flow occurs at the beginning point of the first period.
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### Future Value of Ordinary Annuities
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### Practical Example of Annuities
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# Time Value of Money II: Equal Multiple Payments
## Loan Amortization
### Learning Outcomes
By the end of this section, you will be able to:
1. Distinguish between different types of loans.
2. Explain how amortization works.
3. Create an amortization schedule.
4. Calculate the cost of borrowing.
### Types of Loans
Funds can be loaned to businesses of any type, including corporations, partnerships, limited liability companies, and proprietorships. Bankers often refer to these lending structures as facilities, and they can be tailored to the specific needs of the borrower in a number of ways. Similarly, lenders develop loans and lines of credit for individuals. Whether for a business or an individual, the purpose of the loan, method of repayment, interest rate, specific terms, and time involved must all be tailored to the goals of the borrower and the lender. In this chapter, we will focus on fixed-rate loans, although other alternatives exist.
Typical business loans include the following:
1. Term loans generally bear a maturity date and a set rate of interest and are typically used to finance investments in assets such as equipment, buildings, and possibly other acquired firms. The length of the term loan is generally designed to match the useful life of the asset being financed, and it will usually be repaid on a monthly schedule. It’s common for a term loan to be backed by collateral, such as the asset itself or other assets of the business.
2. Revolving lines of credit (revolvers), are used to finance the short-term working capital needs of a business. Revolvers will have a specific maximum but no set schedule of monthly payments. Interest accrues on the amount of cash that a company has drawn down from the facility. These credit lines may be secured by accounts receivable, inventory, other assets of the business, or sometimes simply the good faith and credit of the company if the firm is strong, creditworthy, and established with the lender. Revolvers must often be fully repaid and unused for a short period of time to assure the lender that the borrower is not using this facility for longer-term needs.
Personal loans also come in several types, designed for the purpose the borrower (consumer) has in mind, with assistance from the lender in determining the appropriate structure:
1. Personal lines of credit are similar to lines of credit on bank cards, with interest being charged on the outstanding balance of the credit line. These are available on the basis of personal credit scores, with data being supplied by the three best-known credit reporting firms: Experian, Equifax, and TransUnion. Individuals should check their scores with each of these companies at least once per year, which they can do for no charge. Additional requests from the same company require a small fee.
2. An unsecured personal loan is an installment loan, initially drawn for a fixed amount and repaid on a periodic schedule with interest, as we have seen in our annuity examples. Unsecured means that the loan is not secured by collateral but is instead based on the strong credit history of the borrower.
3. In contrast, a secured personal loan has an asset backing up the unpaid amount, and if the consumer defaults on the debt, the asset can be seized by the lender to satisfy their claim. A common example is an auto loan, which is secured by the car being purchased; nonpayment or default on the loan can lead to the borrower’s car being repossessed.
4. A mortgage loan is another type of secured personal loan, but for a longer period, such as 20, 25, or even 30 years. The home being purchased or built is the collateral, and the home may be foreclosed upon if the borrower defaults. Full title to the home typically remains with the lender as long as an unpaid balance remains on the debt.
5. Student loans are borrowings intended to fund college or career education, and they can come from a financial institution or the federal government. Interest rates on these loans are generally low and advantageous, and repayment does not begin until after the borrower’s education is complete (or if they drop below a certain level of time status, such as becoming a half-time student).
### Calculating Loan Payments Using Simple Amortization
Loan amortization refers to a schedule of how and when a debt will be repaid with interest. As noted, we will focus on fixed-rate debts, such as auto loans, personal loans with installment payments, or mortgages. Before entering into a borrowing agreement, the borrower can use any of a number of tools to verify the terms being offered, such as the monthly payment on a car loan financed by the dealer. In many cases, this is accomplished by using the present value of an annuity formula:
We’ve already reviewed the present value of ordinary annuities in several examples. Before we look into business or consumer loans and their repayment, we must review an area of Time Value of Money I.
We’re not likely to make annual payments on a home mortgage or auto loan, as these are commonly paid on a monthly basis. Fortunately, our formulas are easily adjusted from annual to non-annual periods. You will recall that we solve for non-annual periods in the same way, with two adjustments: (1) we divide the annual interest rate by the number of periods in the year, and (2) we multiply the time periods by the number of those periods within a year. Therefore, in the case of monthly debt service, including interest and principal, we use 12 periods.
Given a three-year car loan at 6%, rather than using 6% and 3 periods in our formula, we would instead use 0.5% (6% ÷ 12) and 36 periods (3 years × 12), and then apply the present value of an annuity formula in the same way. Let’s say the three-year, 6% auto loan is for $32,000. You need to know if you can squeeze the monthly payment into your budget. For our examples, we will ignore any other charges, fees, taxes, or extras that your lender might include in these payments, and we will focus only on interest and principal repayment. You will make the first payment one month from now, making this an ordinary annuity. What is the amount of your monthly debt service? In this case, you would be solving for a different unknown: the payment amount.
By substitution into the present value of an annuity formula, adjusting for monthly payments as noted:
Dividing both sides by 32.781 to isolate the payment amount (PYMT) gives us
Solving for the payment, we find that it’s approximately $973.50 per month. You consult your monthly budget and find that you can cover this monthly payment, so you conclude the deal. Ask the salesperson for the amortization table on this debt to show how your 36 payments of $973.50 will cover your interest plus repayment of the principal amount of the debt. At this point, you know how to complete your own table. Using a financial calculator or Microsoft Excel simplifies the operation above to a few keystrokes, as presented later in this chapter.
Two extracts from an amortization table are shown in .
This table resembles proofs we have seen of annuities, but let’s focus on some details:
1. Each fixed payment contains both interest and principal repayment.
2. Because the payments are fixed and the amount of remaining debt is decreasing, the monthly interest portion is always decreasing, and the amount of principal payment therefore must be increasing.
We can conclude that the lender is making more of their revenue (interest) in the early months than in the later months. In addition, the debt is decreasing slowly in the early months and more rapidly in the later months. We can all agree that lenders are compensated for the risks they take earlier rather than later. Of the 36 payments of $973.50, $32,000 has been repaid as the principal borrowed. The remaining $3,046.08 is the lender’s revenue, the cost of credit.
For an additional example, one that drives home the point that more interest is paid in the early months of a long-term loan, we will consider a 20-year home mortgage. Home mortgage payments are typically made monthly, and again, we will ignore additional charges by the lender, such as real estate tax and homeowner’s insurance. Let’s assume you buy a $200,000 home, pay $60,000 as a cash deposit, and will finance the remaining $140,000 over 20 years. The bank offers you a 3.6% annual interest rate. What will the amount of your monthly payment be for the interest and principal repayment? The bank will tell you, of course, but let’s prove it for ourselves. We’ll do it in exactly the same fashion as the car loan above, using the present value of an annuity formula. Remember that you are not financing the entire $200,000 purchase; you pay $60,000 in cash, so you are only financing the remaining $140,000.
We modify the periods from years to months by multiplying by 12, and we modify the annual rate to a monthly rate by dividing by 12, resulting in
By substitution into the present value of an annuity formula:
We divide both sides by 170.907667 to isolate the payment amount (PYMT):
Your monthly mortgage payment is $819.16. As in our auto loan example, we’ll complete an amortization table of our own—though, of course, you’ll remember to ask your lender for their version. Extracts from a full 240-month table are shown in below. The front-end packing of interest revenue is more obvious here because of the longer time period.
As with your car loan, earlier payments contain more interest than loan repayment, so the lender’s revenue is at a significant peak in the early years. The length of the loan, coupled with the frequent compounding, emphasizes this. In month 10, the interest and principal amounts “pass” each other, and now the loan balance is dropping at a quicker rate. Finally, note that you will pay more than $56,000 to finance this $140,000 borrowing. If you pay off this mortgage over 240 months as planned, the interest cost represents an additional 28% of the full cost of the home!
If the borrower has the means to make an accelerated payment against this debt—for example, due to a bonus or other windfall—doing so can make a significant difference in the total cost of financing over the life of the loan. Assume that after three years (month 36), you receive a bonus of $2,000 and decide to apply the entire amount to prepay the remaining balance. Your loan agreement allows you to apply the entire amount to the remaining unpaid balance of the mortgage. While this might seem equal to just 2.5 months’ worth of payment, the debt is fully paid off almost 6 months ahead of schedule, and total interest is reduced from over $56,000 to $55,000. The ability to prepay long-term debts such as this is clearly worth negotiating initially.
###
Loans are contracts between a lender and a borrower. Failure to observe the rules of that contract, such as payment of interest or repayment of the amount owed, can subject the borrower to substantial penalties as well as damage to their credit. Loan agreements bearing a fixed rate of interest have a scheduled amortization, or rate and time of repayments with interest. Several types of business and personal loans were described.
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# Time Value of Money II: Equal Multiple Payments
## Stated versus Effective Rates
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain the difference between stated and effective rates.
2. Calculate the true cost of borrowing.
### The Difference between Stated and Effective Rates
If you look at the bottom of your monthly credit card statement, you could see language such as “The interest rate on unpaid balances is 1.5% per month.” You might think to yourself, “So, that’s 12 months times 1.5%, or 18% per year.” This is a fine example of the difference between stated and effective annual interest rates. The effective interest rate reflects compounding within a one-year period, an important distinction because we tend to focus on annual interest rates. Because compounding occurs more than once per year, the true annual rate is higher than appears. Please remember that if interest is calculated and compounded annually, the stated and effective interest rates will be the same. Keep in mind that the following principles work whether you are the debtor paying off an obligation or an investor hoping for more frequent compounding. The dynamics of the time value of money apply in either direction.
### Effective Rates and Period of Compounding
Let’s remain with our example of a credit card statement that indicates an interest rate of 1.5% per month on unpaid balances. If you use this card only once, to make a $1,000 purchase in January, and then fail to pay the bill when it comes due, the issuer will bill you $15. Now you owe them $1,015. Assume you completely ignore this bill and never pay it throughout the rest of the year. The monthly calculation of interest starts to compound on past interest assessments in addition to the $1,000 initial purchase (see ).
Because interest compounds monthly rather than annually, the effective annual rate is 19.56%, not the intuitive rate of the stated 1.5% times 12 months, or 18%. Our basic compounding formula of (1+i)^n by substitution shows:
To isolate the effective annual rate, we then deduct 1 because our interest calculations are based on the value of $1:
Therefore, it falls to the consumer/borrower to understand the true cost of borrowing, especially when larger dollar amounts are involved. If we had been dealing with $10,000 rather than $1,000, the annual difference would be more than $156.
One example of the importance of understanding effective interest rates is an invention from the early 1990s: the payday advance loan (PAL). The practice of offering such loans can be controversial because it can lead to very high rates of interest, perhaps even illegally high, in an act known as usury. Although some states have outlawed PALs and others place limits on them, some do not. A PAL is a short-term loan in anticipation of a person’s next paycheck. A person in need of money for short-term needs will write a check on Thursday but date the check next Thursday, which is their normal payday; assume this transaction is for $200. The lender, typically operating from a storefront, will advance the $200 cash and hold the postdated check. The lender charges a fee—let’s say $14—as their compensation. The following Thursday, the borrower is expected to pay off the advance, and if they do not, the lender can deposit the postdated check. If that check has insufficient funds, more fees and penalties will likely be assessed.
One primary reason that arrangements such as these are controversial is the excessively high nominal (stated) interest rate that they can represent. For a one-week loan of $200, the borrower is paying $14, or 7% of the borrowed amount. If this is annualized, with 52 seven-day periods in a year, the stated rate is 364%! While a PAL might seem to be an effective immediate solution to a cash shortfall, the mathematics behind the true cost of borrowing simply do not make sense, and a person who uses such arrangements regularly is placing themselves at a dreadful financial disadvantage.
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For a borrower to understand the true cost of financing, they must be familiar with interannual compounding, which can cause a stated interest rate that appears to be annual to actually be higher. The effective rate of interest was demonstrated to understand that true cost. |
# Time Value of Money II: Equal Multiple Payments
## Equal Payments with a Financial Calculator and Excel
### Learning Outcomes
By the end of this section, you will be able to:
1. Use a financial calculator and Excel to solve perpetuity problems.
2. Use a financial calculator and Excel to solve annuity problems.
3. Calculate an effective rate of interest.
4. Schedule the amortization of a loan repayment.
### Solving Time Value of Money Problems Using a Financial Calculator
Since the 1980s, many convenient and inexpensive tools have become available to simplify business and personal calculations, including personal computers with financial applications and handheld/desktop or online calculators with many of the functions we’ve studied already. This section will explore examples of both, beginning with financial calculators. While understanding and mastery of the use of time value of money equations are part of a solid foundation in the study of business and personal finance, calculators are rapid and efficient.
We’ll begin with the constant perpetuity that we used to illustrate the constant perpetuity formula. A share of preferred stock of Shaw Inc., pays an annual $2.00 dividend, and the required rate of return that investors in this stock expect is 7%. The simple technique to solve this problem using the calculator is shown in .
Earlier we solved for the present value of a 5-year ordinary annuity of $25,000 earning 8% annually. We then solved for an annuity due, all other facts remaining the same. The two solutions were $99,817.50 and $107,802.50, respectively. We enter our variables as shown in to solve for an ordinary annuity:
Note that the default setting on the financial calculator is END to indicate that payment is made at the end of a period, as in our ordinary annuity. In addition, we follow the payment amount of $25,000 with the +/- keystroke—an optional step to see the final present value result as a positive value.
To perform the same calculation as an annuity due, we can perform the same procedures as above, but with two additional steps after Step 1 to change the default from payments at the end of each period to payments at the beginning of each period (see ).
The procedures to find future values of both ordinary annuities and annuities due are comparable to the two procedures above. We begin with the ordinary annuity, with reminders that this is the default for the financial calculator and that entering the payment as a negative number produces a positive result (see ).
Solving for an annuity due with the same details requires the keystrokes listed in .
Earlier in the chapter, we explored the effect of interannual compounding on the true cost of money, recalling the basic compounding formula:
We saw that when modified for monthly compounding at a stated rate of 1.5%, the actual (effective) rate of interest per year was 19.56%. One simple way to prove this is by using the calculator keystrokes listed in .
Had we assumed that the stated monthly interest rate of 1.5% could be simply multiplied by 12 months for an annual rate of 18%, we would be ignoring the effect of more frequent compounding. As indicated above, the annual interest on the money that we spent initially, accumulating at a rate of 1.5% per month, is 19.56%, not 18%:
The final example in this chapter will represent the amortization of a loan. Using a 36-month auto loan for $32,000 at 6% per year compounded monthly, we can easily find the monthly payment and the amortization of this loan on our calculator using the following procedures and keystrokes.
First, we find the monthly payment (see ).
We’ve verified the amount of our monthly debt service, including both the interest and repayment of the principal, as $973.50. The next step with our calculator is to verify our amortization at any point (see ).
Without resetting the calculator, we will try a second example, this time reviewing the second full year of amortization at the end of 24 months (see ).
### Solving Time Value of Money Problems Using Excel
Microsoft’s popular spreadsheet program Excel is arguably one of the most common and powerful numeric and data analysis products available. Yet while mastery of Excel requires extensive study and practice, enough basics can be learned in two or three hours to provide the user with the ability to solve problems quickly and conveniently, including extensive financial capability. Most of the calculations in this chapter were prepared with Excel.
The boxes in the Excel gridwork, known individually as cells (located at the intersection of a column and a row), can contain numbers, text, and very powerful formulas (or functions) for calculations and data analytics. Cells, rows, columns, and groups of cells (ranges) are easily moved, formatted, and replicated. In the mortgage amortization table for 240 months seen in Section 8.3.2, only the formulas for month 1 were typed in. With one simple command, that row of formulas was replicated 239 more times, with each line updating itself with relevant number adjustments automatically. With some practice, a long table such as that can be constructed by even a relatively new user in less than 10 minutes.
In this section, we will illustrate how to use Excel to solve problems from earlier in the chapter, including perpetuities, ordinary annuities, effective interest rates, and loan amortization. We will omit the basic dynamics of an Excel spreadsheet because they were presented sufficiently in preceding chapters.
Revisiting the constant perpetuity from Section 8.1, in which our shares of Shaw Inc., preferred stock pay an annual fixed dividend of $2.00 and the required rate of return is 7%, we do not use an Excel function for this simple operation. The two values are entered in cells B3 and B4, respectively.
We enter a formula in cell B6 to perform the division and display the result in that cell. The actual contents of cell B6 are typed below it for your reference, in cell B8 (see ).
To find the present value of an ordinary annuity, we revisit Section 8.2.1. You will draw $25,000 at the end of each year for five years from a fund earning 8% annually, and you want to know how much you need in that fund today to accomplish this. We accomplish this in Excel easily with the PV function. The format of the PV command is
Only the first three arguments inside the parentheses are used. We’ll place them in cells and refer to those cells in our PV function. As an option, you could also type the numbers into the parentheses directly. Notice the slight rounding error because of decimal expansion. Also, the payment must be entered as a negative number for your result to be positive; this can be accomplished either by making the $25,000 in cell B5 a negative amount or by placing a minus sign in front of the B5 in the formula’s arguments. In cell B3, you must enter the percent either as 0.08 or as 8% (with the percent sign). We repeated the formula syntax and the actual formula inputs in column A near the result, for your reference (see ).
We also found the present value of an annuity due. We use the same information from the ordinary annuity problem above, but you will recall that the first of five payments happens immediately at the start of year 1, not at the end. We follow the same procedures and inputs as in the previous example, but with one change to the PV function: the last argument in the parentheses will change from 0 to 1. This is a toggle switch that commands the PV function to treat this as an annuity due instead of an ordinary annuity (see ).
Section 8.2 introduced us to future values. Comparable to the PV function above, Excel provides the FV function. Using the same information—$3,000 invested annually for five years, starting one year from now, at 4%—we’ll solve using Excel (see ). The format of the command is
As with present values, using the same data but solving for an annuity due requires the fifth argument inside the parentheses to be changed from 0 to 1; all other values remain the same (see ).
In Section 8.4, we explained the difference between stated and effective rates of interest to show the true cost of borrowing, in this case for a one-year period, if interest is compounded for periods within a year. The syntax for the Excel effect function to calculate this rate is
where rate is the nominal rate and periods represents the number of periods within a year.
Earlier, our example showed that 1.5% compounded monthly results in not 18% per year but actually over 19.56% (see ).
Note several things: First, the nominal interest rate is entered as a percent. Second, the actual effect function in C7 is typed as =EFFECT(rate,B7); we use the word rate because we actually assigned a name to cell B3, so Excel can use it in a function and replicate it without it changing. When cell C7 is replicated to C8 and C9, rate remains the same, but the formulas automatically adjust to use B8 and B9 for the periods.
To assign a name to a cell, keep in mind that every cell has column-row coordinates. We want cell B3 to be the anchor of our effective rate calculations. Rather than referring to cell B3, we can name it, and in this case, we use the name rate, which we can then use in formulas like any other Excel cell letter-number reference. Place the cursor in cell B3. Now, look at cell A1 on the grid: right above that cell, you see a box displaying B3, the current cursor location. If you click in that box and type “rate” (without the quotation marks), as we did, then hit the enter key, the value in that box will change to rate. Now, if you type “rate” (again, without quotation marks) into a formula, Excel knows to use the contents of cell B3.
Excel provides convenient tools for figuring out amortization. We’ll revisit our 36-month auto loan for $32,000 at 6% per year, compounded monthly. A loan amortization table for a fixed interest rate debt is usually formatted as follows, with the Interest and Principal columns interchangeable:
In Excel, a table is completed by using the function PMT. The individual steps follow.
1. List the information about the loan in the upper left of the worksheet, and create the column headings for the schedule of amortization. Type “B5” (without the quotation marks) in cell E9 to begin the schedule. Then enter 1 for the first month under the Payment # (or Month) column, in cell A10 (see ).
2. Next, in cell B10, the payment is derived from the formula =PMT(rate,periods,pv), with PV representing the present value, or the loan amount. Because we are compounding monthly, enter C$2 and C$3 for the rate and periods, respectively. Cell B5 is used for the loan amount, but notice the optional minus sign placed in front of the entry B$5; this causes the results in the schedule to be displayed as positive numbers. The dollar sign ($) inserted in the cell references forces Excel to “freeze” those locations so that they don’t attempt to update when we replicate them later; this is known in spreadsheet programs as an absolute reference (see ).
3. The next step is to calculate the interest. We take the remaining balance from the previous line, in this case cell E9, and multiply it by the monthly interest rate in cell C2, typing C$2 to lock in the reference. The remaining balance of the loan should always be multiplied by this monthly percentage (see ).
4. Because this is a fixed-rate loan, whatever is left from each payment after first deducting the interest represents principal, the amount by which the balance of the outstanding loan balance is reduced. Therefore, the contents of cell D10 represent B10, the total payment, minus C10, the interest portion (see ). No dollar signs are included because this cell reference can adjust to each row into which this formula is replicated, as will be seen in the following examples.
5. Because our principal portion of the last payment has reduced our outstanding balance, it is subtracted from the preceding balance in cell E9 (see ). The command therefore is =E9-D10.
Now that the first full row is defined, an amortization schedule is easily developed by Excel’s replication abilities. Place the cursor on cell A10, hold down the left mouse button, and drag the cursor to cell E10. Cells A10 through E10 in row 10 should now be highlighted. Release the mouse button. Then “grab” the tiny square symbol at the bottom right of cell E10 and drag it downward as far as you need; in this case, you’ll need 35 more rows because this is a 36-month loan, so it will end at row 45. We added a line for totals.
This is now a complete loan amortization schedule (see ). The first several periods display, followed by the last few periods, to prove that the schedule is complete (data rows for month 4 to month 22 are hidden).
This will look familiar; it’s the same amortization table used as a proof in Section 8.3 (see ). There is no rounding error because Excel uses the full decimal expansion in its calculations.
This chapter has explored the time value of money by expanding on the concepts discussed in Time Value of Money I with additional funds being periodically added to or subtracted from our investment, either compounding or discounting them according to the situation. In all cases, the payments in the stream were identical. If they had not been identical, a separate set of operators would be required, and these will be addressed in the next chapter.
###
The use of two tools for managing and understanding the time value of money and its many applications was discussed: a professional financial calculator and the popular Microsoft Office Suite spreadsheet application Excel.
### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Time Value of Money III: Unequal Multiple Payment Values
## Why It Matters
Baseball legend Ted Williams once said, “Baseball is the only field of endeavor where a man can succeed three times out of ten and be considered a good performer.”Pete Palmer and Gary Gillette, eds. On routine or unimportant decisions, business managers might aspire to do as well as Ted Williams, making the right decisions only 30% of the time. But a professional decision maker must “hit it out of the park” when making major capital investment choices and recommendations.
As a student in a course of business studies and career development, it is highly likely that you will be a decision maker about projects that are likely to generate future cash flows but will also require a large initial expense. When you ask your manager to invest $500,000 or more in a new piece of equipment that could help your department meet or exceed its goals, you must be prepared to defend your request. Competing managers and departments will be asking for similar funding, and there simply might not be enough for everyone. This decision process requires financial analysis.
Mark Cuban of fame enjoys citing the series’ catchphrase: “Know thy numbers.” As a business professional, you must be able to assess potential profit against expenditures to be successful. In most cases, this is based on our understanding of cash flow. A major capital investment might seem initially like a gamble, but it is a gamble that can be hedged in your favor with understanding, analysis, and knowledge of your numbers.
The purpose of this chapter is to give you information and instruction on how this is done. The techniques we will discuss in this chapter will clarify decisions that must be made in the process of investing in a business. We focus first on decisions we make about our own money as investors if uneven cash receipts or payments are involved. |
# Time Value of Money III: Unequal Multiple Payment Values
## Timing of Cash Flows
### Learning Outcomes
By the end of this section, you will be able to:
1. Describe how multiple payments of unequal value are present in everyday situations.
2. Calculate the future value of a series of multiple payments of unequal value.
3. Calculate the present value of a series of multiple payments of unequal value.
### Multiple Payments or Receipts of Unequal Value: The Mixed Stream
At this point, you are familiar with the time value of money of single amounts and annuities and how they must be managed and controlled for business as well as personal purposes. If a stream of payments occurs in which the amount of the payments changes at any point, the techniques for solving for annuities must be modified. Shortcuts that we have seen in earlier chapters cannot be taken. Fortunately, with tools such as financial or online calculators and Microsoft Excel, the method can be quite simple.
The ability to analyze and understand cash flow is essential. From a personal point of view, assume that you have an opportunity to invest $2,000 every year, beginning next year, to save for a down payment on the purchase of your first home seven years from now. In the third year, you also inherit $10,000 and put it all toward this goal. In the fifth year, you receive a large bonus of $3,000 and also dedicate this to your ongoing investment.
The stream of regular payments has been interrupted—which is, of course, good news for you. However, it does add a new complexity to the math involved in finding values related to time, whether compounding into the future or discounting to the present value. Analysts refer to such a series of payments as a mixed stream. If you make the first payment on the first day of next year and continue to do so on the first day of each following year, and if your investment will always be earning 7% interest, how much cash will you have accumulated—principal plus earned interest—at the end of the seven years?
This is a future value question, but because the stream of payments is mixed, we cannot use annuity formulas or approaches and the shortcuts they provide. As noted in previous chapters, when solving a problem involving the time value of money, a timeline and/or table is helpful. The cash flows described above are shown in
. Remember that all money is assumed to be deposited in your investment at the beginning of each year. The cumulative cash flows do not yet consider interest.
By the end of seven years, you have invested $27,000 of your own money before we consider interest:
1. Seven years times $2,000 each year, or $14,000
2. The extra $10,000 you received in year 3 (which is invested at the start of year 4)
3. The extra $3,000 you received in year 5 (which is invested at the start of year 6)
These funds were invested at different times, and time and interest rate will work for you on all accumulated balances as you proceed. Therefore, focus on the line in your table with the cumulative cash flows. How much cash will you have accumulated at the end of this investment program if you’re earning 7% compounded annually? You could use the future value of a single amount equation, but not for an annuity. Because the amount invested changes, you must calculate the future value of each amount invested and add them together for your result.
Recall that the formula for finding the future value of a single amount is
, where FV is the future value we are trying to determine, PV is the value invested at the start of each period, i is the interest rate, and n is the number of periods remaining for compounding to take effect.
Let us repeat the table with your cash flows above.
includes a line to show for how many periods (years, in this case) each investment will compound at 7%.
The $2,000 that you deposit at the start of year 1 will earn 7% interest for the entire seven years. When you make your second investment at the start of year 2, you will now have spent $4,000. However, the interest from your first $2,000 investment will have earned you
, so you will begin year 2 with $4,140 rather than $4,000.
Before we complicate the problem with a schedule that ties everything together, let’s focus on years 1 and 2 with the original formula for the future value of a single amount. What will your year 1 investment be worth at the end of seven years?
You need to address the year 2 investment separately at this point because you’ve calculated the year 1 investment and its compounding on its own. Now you need to know what your year 2 investment will be worth in the future, but it will only compound for six years. What will it be worth?
You can perform the same operation on each of the remaining five invested amounts, remembering that you invest $12,000 at the start of year 4 and $5,000 at the start of year 6, as per the table. Here are the five remaining calculations:
Notice how the exponent representing n decreases each year to reflect the decreasing number of years that each invested amount will compound until the end of your seven-year stream. For clarity, let us insert each of these amounts in a row of
:
The solution to the original question—the value of your seven different investments at the end of the seven-year period—is the total of each individual investment compounded over the remaining years. Adding the compounded values in the bottom row provides the answer: $35,062.26. This includes the $27,000 that you invested plus $8,062.26 in interest earned by compounding.
It’s important to note that throughout these sections on the time value of money and compounded or discounted values of mixed streams and their analysis, we are placing the valuation at the end or beginning of a period for simplicity in the examples. In reality, businesses might consider valuations happening within the period to allow for a degree of regularity in the revenue streams provided by the asset being considered. However, because this is a technique of forecasting, which is inherently uncertain, we will continue with analysis by period.
Let’s take the example above and review it from a different angle. Keeping in mind that we have not yet explored the use of Excel, is there another way to view our solution? The problem above takes each annual investment and compounds it into the future, then adds the results of each calculation to find the total future value of the stream of payments.
But when you break the problem down, another way to look at the problem is as a five-year annuity of $10,000 per year plus added payments in years 4 and 5. Can we solve for the future value of an annuity first and then perform two separate calculations on the additional amounts ($5,000 each in years 4 and 5)? Yes, we can.
Let’s summarize:
1. Future value of a $10,000 annuity due, 4%, 5 years, plus
2. Future value of a single payment of $5,000, 4%, 2 years, plus
3. Future value of a single payment of $5,000, 4%, 1 year
This must give us the same result. The formula for the future value of an annuity due is
This problem can be solved in the three steps of the summary above.
Step 1:
Step 2:
Step 3:
Combining the results from each of the three steps gives us
It works. Whether you view this problem as five separate periods that can be compounded separately and then combined or as a combination of one or more annuities and/or single payment problems, we always arrive at the same solution if we are diligent about the time, the interest, and the stream of payments.
### The Present Value of a Mixed Stream
Now that we’ve seen the calculation of a future value, consider a present value. We will begin with a personal example. You win a cash windfall through your state’s lottery. You would like to take a portion of the funds and place them in a fixed investment so that you can draw $17,000 per year starting one year from now and continue to do so for the next two years. At the end of year 4, you want to withdraw $17,500, and at the end of year 5, you will withdraw the last $18,000 to close the account. When you take your last payment of $18,000, your fund will be totally depleted. You will always be earning 6% annually. How much of your cash windfall should you set aside today to accomplish this?
Let us break down the problem, remembering that we are thinking in reverse from the earlier problems that involved future values. In this case, we’re bringing future values back in time to find their present values. You will recall that this process is called discounting rather than compounding.
Regardless of how we solve this, the question remains the same: How much money must we invest today (present value) to achieve this? And remember that we will always be earning 6% compounded annually on any invested balances.
We are calculating present values as we did in previous chapters, given a known future value “target,” in order to determine how much money you need today to achieve that goal. Let us break this down by first reviewing the relevant equations from previous chapters.
Present value of an ordinary annuity:
Present value of a single amount:
where PVa is the present value of an annuity, PYMT is one payment in a consistent stream (an annuity), i is the interest rate (annual unless otherwise specified), n is the number of periods, PV is the present value of a single amount, and FV is the future value of a single amount.
You want to find out how much money you need to set aside today to accomplish your goal. You can also find out how much money you need to set aside in each period to accomplish this goal. Therefore, we can address this problem in increments. Let us look at potential solutions.
First, we will break this down into the cash flows of each year.
shows the timing of the future cash flows you’re expecting:
One method is to take each year’s cash flows, which happen at the end of the year, and discount them to today using the present value formula for a single amount:
Because year 1’s withdrawal from your fund only has one year to earn interest, we discounted it for one year. The second amount is discounted for two years:
The next three years are discounted in the same way, for three, four, and five years, respectively:
Notice how we reverse our thinking on the exponent n from our approach to future value. This time, it increases each period because we discount each future amount for a longer period to arrive at the value in today’s dollars.
When we add all five discounted present value amounts from above, we derive today’s value of $72,753.49. Expressed more simply, if you wanted to extract the specified stream of cash flows at the end of each year ($17,000 for three years, then $17,500, then $18,000), you would have to begin with $72,753.49. The thing to remember is that any amounts remaining in this fund, regardless of how you deplete it, will always be earning 6% annually. See .
Let us try another approach. Because the amount of cash withdrawn in the first three years remains constant at $17,000, it can be viewed as an annuity—specifically, a three-period annuity of $17,000 and two single payments of $17,500 and $18,000. Therefore, we could also discount (bring to present value) an annuity of $17,000 for three years (the first three) and then combine it with the year 4 discounted amount and the year 5 discounted amount. We can try it using the formulas for PVa and PVused above. In Step 1, we will discount the first three years as an annuity (ordinary, as the first withdrawal is not made until one year from now); in Step 2, we will discount the year 4 single payment amount; and in Step 3, we will do the same for the year 5 single payment amount. Then we can add them together.
Step 1: Find the present value of the annuity using the PVa formula:
Step 2: Discount the year 4 amount using the formula for the present value of a single amount:
Step 3: Perform the same operation as in Step 2 for the year 5 amount:
Now that all three amounts have been discounted to today’s value, we can add them:
Calculating the present value of cash flows is very common and critical in the analysis of capital investments in business for two compelling reasons: first, the investment is likely quite significant, and second, the risk will usually encompass a longer time frame. When the author of this chapter would purchase a large machine, it would likely take several years for that machine to justify its purchase with the revenues it would generate. This is one of the primary reasons that accountants require us to depreciate the cost of an asset over time: to assess the cost against the time it will take for that asset to produce profits and cash flow.
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To understand the true value and strength of cash, it is necessary to consider its timing. This is relevant for investments for the future and for analyses of the value of projects that require investment today to produce expected flows of cash later. These future cash flows could involve inflows or outflows of cash in unequal amounts. This section analyzed the determination of present and future value of these uneven or mixed cash flows.
###
General instructions: Approximations and minor differences because of rounding are acceptable. Ignore the effect of taxes. Please assume that all percentages are annual rates and compounding occurs annually unless otherwise indicated.
###
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### Calculate the Present Value for Multiple Cash Flows
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# Time Value of Money III: Unequal Multiple Payment Values
## Unequal Payments Using a Financial Calculator or Microsoft Excel
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate unequal payments using a financial calculator.
2. Calculate unequal payments using Microsoft Excel.
### Using a Financial Calculator
A financial calculator provides utilities to simplify the analysis of uneven mixed cash streams (see ).
Earlier, we explored the future value of a seven-year mixed stream, with $2,000 being saved each year, plus an additional $10,000 in year 4 and an additional $3,000 in year 6. All cash flows and balances earn 7% per year compounded annually, and the payments are made at the start of each year. We proved that this result totals approximately $35,062.26. We begin by clearing all memory functions and then entering each cash flow as follows:
At this point, we have found the net present value of this uneven stream of payments. You will recall, however, that we are not trying to calculate present values; we are looking for future values. The TI BA II Plus™ Professional calculator does not have a similar function for future value. This means that either we can find the future value for each payment in the stream and combine them, or we can take the net present value we just calculated and easily project it forward using the following keystrokes. Note the net present value solution in Step 20 above. We will use that and then use the simpler of the two approaches to calculate future value (see ).
This is consistent with the solution we found earlier, with a difference of one cent due to rounding error.
We may also use the calculator to solve for the present value of a mixed cash stream. Earlier in this chapter, we asked how much money you would need today to fund the following five annual withdrawals, with each withdrawal made at the end of the year, beginning one year from now, and all remaining money earning 6% compounded annually:
We determined these withdrawals to have a total present value of $72,753.30. Here is an approach to a solution using a financial calculator. In this example, we will store all cash flows in the calculator and perform an operation on them as a whole (see ). Because we will use the NPV function (to be explored in more detail in a later chapter), we enter our starting point as 0 because we do not withdraw any cash until one year after we begin.
This result, you will remember, was calculated earlier in the chapter by the formula approach.
### Using Microsoft Excel
Several of the exhibits already in this chapter have been prepared with Microsoft Excel. While full mastery of Excel requires extensive study and practice, enough basics can be learned in two or three hours to provide the user with the ability to quickly and conveniently solve problems, including extensive financial applications. Potential employers and internship hosts have come to expect basic Excel knowledge, something to which you are exposed in college.
We will demonstrate the same two problems using Excel rather than a calculator:
1. The future value of a mixed cash stream for a seven-year investment
2. The present value of a mixed cash stream of five withdrawals that you wish to make from a fund to be established today
Beginning with the future value problem, we created a simple matrix to lay out the mixed stream of future cash flows, starting on the first day of each year, with all funds earning 7% throughout. Our goal is to determine how much money you will have saved at the end of this seven-year period.
repeats the data from earlier in the chaper for your convenience.
is an Excel matrix that parallels above.
We begin by entering the cash flow as shown in . The assumed interest rate is 7%. The interest on the balance is calculated as the amount invested at the start of the year multiplied by the assumed interest rate. The cumulative cash flows of each year are calculated as follows: for year 1, the amount invested plus the interest on the balance; for years 2 through 7, the amount invested plus the interest on the balance plus the previous year’s running balance. By adding up the amount invested and the interest on the balance, you should arrive at a total of $35,062.27.
We can use Excel formulas to solve time value of money problems. For example, if we wanted to find the present value of the amount invested at 7% over the seven-year time period, we could use the NPV function in Excel. The dialog box for this function (Rate, Value 1, Value2) is shown in .
The function argument Rate is the interest rate; Value1, Value2, and so on are the cash flows; and “Formula result” is the answer.
We can apply the NPV function to our problem as shown in .
Please note that the Rate cell value (C1) and the Value1 cell range (C3:I3) will vary depending on how you set up your spreadsheet.
The non-Excel version of the problem, using an assumed interest rate of 7%, produces the same result.
We conclude with the second problem addressed earlier in this chapter: finding the present value of an uneven stream of payments. We can use Excel’s NPV function to solve this problem as well (see ).
Again, Rate is the interest rate; Value1, Value 2, and so on are the cash flows; and “Formula result” is the answer.
Let us apply the NPV function to our problem, as shown in and .
Please note that the Rate cell value (B3) and the Value1 cell range (C2:G2) will vary depending on how you set up your spreadsheet.
The non-Excel version of the problem produces the same result: an NPV of $72,753.49.
###
This section discussed the use of two tools for managing and understanding the time value of money and its many applications when the flows of cash are unequal: the TI BA II Plus™ Professional financial calculator and the spreadsheet application Excel.
###
### Future Value of Uneven Cash Flows
### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Bonds and Bond Valuation
## Why It Matters
When an investor purchases a bond, that person is, for all intents and purposes, making a loan to the bond issuer. Bonds issues are used to raise funds and can be issued by corporations, governments, or even subagencies of governments (including local municipalities).
As with any type of loan, the borrowing party is expected to offer something to the lender in exchange for their time and trouble. In this case, the bond-issuing entity will agree not only to repay the original face value of the loan on a specific date (the maturity of the bond) but also to pay the lender interest—or, in bond terminology, coupon payments.
Coupon payments are designed to make a bond purchase more acceptable for investors by helping compensate them for the time value of money. Because investors are parting with money that they have right now in order to make the initial bond purchase but will not see repayment of principal until the maturity date of the bond, they will experience the negative impact of time value over the bond term. When a bond issuer offers periodic coupon payments, this helps offset the negative effect of the delayed receipt of the principal amount for the investor. Also, because coupon payments will be coming to the investor throughout the term of the bond, essentially in installment payments (an annuity), the time value of money plays a critical role in bond transactions and in calculating bond valuation.
Bonds, along with stocks and mutual funds, are considered to be one of the most basic financial instruments available to any investor. It is quite common for investors to round out their portfolios by purchasing bonds, adding a degree of safety and diversity to their investment mix. |
# Bonds and Bond Valuation
## Characteristics of Bonds
### Learning Outcomes
By the end of this section, you will be able to:
1. List and define the basic characteristics of bonds.
2. List and describe the various types of bonds available.
3. Explain how a bond price is inversely related to its return (yield).
### Bonds as Investments
One way to look at bond investments is to consider the fact that any investor who purchases a bond is essentially buying a future cash flow stream that the bond issuer (or borrower) promises to make as per agreement.
Because bonds provide a set amount of cash inflow to their owners, they are often called fixed-income securities. Thus, future cash flows from the bond are clearly stated per agreement and fixed when the bond sale is completed.
Bonds are a basic form of investment that typically include a straightforward financial agreement between issuer and purchaser. Nevertheless, the terminology surrounding bonds is unique and rather extensive. Much of the specialized vocabulary surrounding bonds is designed to convey the concept that a bond is similar to other financial instruments in that it is an investment that can be bought and sold. Much of this unique terminology will be covered later in this chapter, but we can set out some of the basics here with an example.
Let’s say that you buy a $1,000 bond that was issued by Apple Inc. at 5% interest, paid annually, for 20 years. Here, you are the lender, and Apple Inc. is the borrower.
### Basic Terminology
We need to know the following basic bond terms and pricing in order to apply the necessary time value of money equation to value this Apple, Inc. bond issue:
1. Par value: A bond will always clearly state its par value, also called face amount or face value. This is equal to the principal amount that the issuer will repay at the end of the bond term or maturity date. In our example, the par value of the bond is $1,000.
2. Coupon rate: This is the interest rate that is used to calculate periodic interest, or coupon payments, on the bond. It is important to note that coupon rates are always expressed in annual terms, even if coupon payments are scheduled for different periods of time. The most common periods for coupon payments other than annual are semiannual and quarterly. Coupon rates will typically remain unchanged for the entire life of the bond. In our example, the coupon rate is the 5% interest rate.
3. Coupon payment: This refers to the regular interest payment on the bond. The coupon or periodic interest payment is determined by multiplying the par value of the bond by the coupon rate. It is important to note that no adjustment needs to be made to the coupon rate if the bond pays interest annually. However, if a bond pays interest on a semiannual or quarterly basis, the coupon rate will have to be divided by 2 or 4, respectively, to convert the stated annual rate to the correct periodic rate. In our example, coupons are paid annually, so the periodic or annual interest that is paid is equal to . You may notice that because these payments are the same amount and made at regular intervals, they constitute an annuity stream (refer to Time Value of Money II: Equal Multiple Payments.
4. Maturity date: The maturity date is the expiration date of the bond, or the point in time when the term of a bond comes to an end. On the maturity date, the issuer will make the final interest or coupon payment on the bond and will also pay off its principal, or face value. In our example, the maturity date is at the end of the 20-year period.
5. Yield to maturity (YTM): The YTM is essentially the discount rate used to bring the future cash flows of a bond into present value terms. It also equals the return that the investor will receive if the bond is held to maturity. The YTM helps quantify the overall investment value of a bond. We will explore how to compute this rate later in the chapter.
displays a selected listing of bonds available for purchase or sale. First, let’s review the columns so you can learn how to read this table.
1. Column 1: Issuer. The first column shows the company, city, or state issuing the bond. This bond listing includes two municipal issuers (City of Chicago and Tennessee Energy) as well as several corporate issuers.
2. Column 2: Bond Type. This describes the issue of the bond and indicates whether it is a corporation or a municipality.
3. Column 3: Current Price. The third column shows the price as a percent of par value. It is the price someone is willing to pay for the bond in today’s market. We quote the price in relation to $100. For example, the Nordstrom bond is selling for 112.905% of its par value, or $112.905 per $100.00 of par value. If this bond has a $1,000.00 par value, it will sell for . Note: Throughout this chapter, we use $1,000 as the par value of a bond because it is the most common par value for corporate bonds.
4. Column 4: Callable? This column states whether or not the bond has a call feature (if it can be retired or ended before its normal maturity date).
5. Column 5: Coupon Rate. The fifth column states the coupon rate, or annual interest rate, of each bond.
6. Column 6: Maturity Date. This column shows the maturity date of the issue—the date on which the corporation will pay the final interest installment and repay the principal.
7. Column 7: Yield. The seventh column indicates each bond’s yield to maturity—the yield or investment return that you would receive if you purchased the bond today at the price listed in column 3 and held the bond to maturity. We will use the YTM as the discount rate in the bond pricing formula.
8. Column 8: Rating. The final column gives the bond rating, a grade that indicates credit quality. As we progress through this chapter, we will examine prices, coupon rates, yields, and bond ratings in more detail
### Types of Bonds
There are three primary categories of bonds, though the specifics of these different types of bond can vary depending on their issuer, length until maturity, interest rate, and risk.
### Government Bonds
The safest category of bonds are short-term US Treasury bills (T-bills). These investments are considered safe because they have the full backing of the US government and the likelihood of default (nonpayment) is remote. However, T-bills also pay the least interest due to their safety and the economics of risk and return, which state that investors must be compensated for the assumption of risk. As risk increases, so should return on investment. Treasury notes are a form of government security that have maturities ranging from one to 10 years, while Treasury bonds are long-term investments that have maturities of 10 to 30 years from their date of issue.
Savings bonds are debt securities that investors purchase to pay for certain government programs. Essentially, the purchase of a US savings bond involves the buyer loaning money to the government with a guaranteed promise that they will earn back the face value of the bond plus a certain amount of interest in the future. Savings bonds are backed by the US government, meaning that there is virtually no possibility of the buyer losing their investment. For this reason, the return on savings bonds is relatively low compared to other forms of bonds and investments.
Municipal bonds (“munis”) are issued by cities, states, and localities or their agencies. Munis typically will return a little more than Treasury bills while being just a bit riskier.
### Corporate Bonds
Corporate bonds are issued by companies. They carry more risk than government bonds because corporations can’t raise taxes to pay for their bond issues. The risk and return of a corporate bond will depend on how creditworthy the company is. The highest-paying and highest-risk corporate bonds are often referred to as non-investment grade or, more commonly, junk bonds.
Corporate bonds that do not make regular coupon payments to their owners are referred to as zero-coupon bonds. These bonds are issued at a deep discount from their par values and will repay the full par value at their maturity date. The difference between what the investor spends on them in original purchase price and the par value paid at maturity will represent the investor’s total dollar value return.
Convertible bonds are similar to other types of corporate bonds but have a feature that allows for their conversion into a predetermined number of common stock shares. The conversion from the bond to stock can be done at certain times during the bond’s life, usually at the discretion of the bondholder.
### The Global Bond Market versus the Global Stock Market
Bonds have long been a trusted investment vehicle for many investors. Though the global fixed-income debt market remains considerably larger than the global stock market, this is not an entirely fair comparison. Bond markets include sovereign bonds, or bonds that are issues by governments, while stock markets do not. Some experts believe that a more relevant comparison is between the value of corporate bond markets only (excluding sovereign bonds) and total stock market value.
The chart in provides global market value information by category so that we may make our own conclusions about these markets.
While the total value of bond markets continues to exceed that of stocks, the prevailing trend over the past several years has been that stock markets are gaining in terms of total market size. The primary reason for this is that stocks have traditionally outperformed bonds in terms of return on investment over extended periods of time and are likely to continue to do so. This makes them more attractive to investors, despite the higher risk associated with stock.
### The Two Sides of a Bond Investment
There are essentially two sides to a bond investment, meaning the bondholder will receive two types of cash inflow from the bond investment over its term. These are the payment of the par value at maturity (often referred to as payment of the face value of the bond at term end), and the periodic coupon payments (also called interest income) from the bond. These coupon payments are contractually determined and clearly indicated in the bond issue documentation received by the bondholder upon purchase.
As a result of these two types of inflow, bond valuation requires two different time value of money techniques—specifically, present value calculations—to be computed separately and then added together.
### The Relationship between Bond Prices and Interest Rates
Bond price and interest rate have an inverse relationship. When interest rates fall, bond prices rise, and vice versa (see ). If interest rates increase, the value of bonds sold at lower interest rates will decline. Similarly, if interest rates decline, the value of fixed-rate bonds will increase. An exception to this general rule is floating-rate bonds (often referred to as “floaters,” floating-rate notes, or FRNs).
A floating-rate note is a form of debt instrument that is similar to a standard fixed-rate bond but has a variable interest rate. Rates for floating-rate bonds are typically tied to a benchmark interest rate that exists in the economy. Common benchmark rates include the US Treasury note rate, the Federal Reserve funds rate (federal funds rate), the London Interbank Offered Rate (LIBOR), and the prime rate.
So, investors who decide to purchase normal (fixed rate) bonds may not be thrilled to hear that the economy is signaling inflation and that interest rates are forecasted to rise. These bond investors are aware that when interest rates rise, their bond investments will lose value. This is not the case with floating- or variable-rate bonds.
Bonds with very low coupon rates are referred to as deep discount bonds. Of course, the bond that has the greatest discount is the zero-coupon bond, with a coupon rate of zero. The smaller the coupon rate, the greater the change in price when interest rates move.
###
Bonds are typically a basic form of investment that entails a straightforward financial agreement between issuer and purchaser. There are three primary categories of bonds: government bonds, corporate bonds, and convertible bonds. These different types of bond vary depending on their issuer, length until maturity, interest rate, and risk.
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# Bonds and Bond Valuation
## Bond Valuation
### Learning Outcomes
By the end of this section, you will be able to:
1. Determine the value (price) of a bond.
2. Understand the characteristics of and differences between discount and premium bonds.
3. Draw a timeline indicating bond cash flows.
4. Differentiate between fixed-rate and variable-rate bonds.
5. Determine bond yields.
### Pricing a Bond in Steps
Why do we want to learn how to price a bond? The answer goes to the heart of finance: the valuation of assets. We need to ascertain what a given bond is worth to a willing buyer and a willing seller. What is its value to these interested parties? Remember that a bond is a financial asset that a company sells to raise money from willing investors. Whether you are the company selling the bond or the investor buying the bond, you want to make sure that you are selling or buying at the best available price.
Let’s begin our pricing examples with the 3M Company corporate bond listed in above. The table information tells us that 3M issued a series of corporate bonds that promise to pay coupons annually on September 19 and to pay back the principal, or face value, on the maturity date of September 19, 2026. While this is not specified in the table, let’s say these are 15-year corporate bonds. In that case, we know that they were issued on September 20, 2011.
The 3M bonds have an annual coupon rate of 2.25%, which indicates that the annual interest payment on the bond will be the face value (assumed to be $1,000.00 multiplied by 2.25%), or $22.50. The appropriate discount rate to apply to these future payments is the yield to bond maturity, 1.24%.
Note that the 3M bond is selling at a premium (above par or face value) due to the fact that its coupon rate is greater than the YTM percentage. This means that the bond earns more value in interest than it loses due to discounting its cash flows to allow for the time value of money principle.
Finally, the table tells us some of the bond’s features. For example, Standard & Poor’s, an international rating agency, rates 3M Co. as A+ (high credit quality). Additionally, the bonds are designated as callable, meaning that 3M has the option of redeeming them before their maturity on September 19, 2026.
We can price a bond using the same methods from earlier chapters: an equation, a calculator, and a spreadsheet. Let’s start with the equation method (see ).
The first step is to identify the amounts and the timing of the two types of future cash flows to be received on the bond. Any bond that pays interest or coupon payments (coupon bonds) will have two sources of future cash flow to its bondholder/investor: the periodic coupon payments, which are a form of annuity, and the final lump sum payment of the face value amount at maturity.
As discussed above, the principal or face value is paid in a one-time lump sum payment at bond maturity. In our example with 3M Co., this is the $1,000 par value of the bond that will be paid on the maturity date of September 19, 2026. Step 1 is to lay out the timing and amount of the future cash flows. The first future cash flow we need to determine is the annual interest payment. Here, it is the coupon rate of 2.25% times the par value of the bond. As mentioned above, we will use $1,000 as the par value of this bond, so the annual coupon or interest payment will equal $22.50:
The next future cash flow that we need to determine is the payment of the par value or principal—in this case, the $1,000 par value of the bond—at the maturity date of September 19, 2026. We can set out the future cash flows for the bond as shown in :
Note that annual coupon payments are made each year on September 19, and the first annual coupon payment date is September 19, 2012. The annual payments continue for 15 years, with the last payment being made on September 19, 2026. At this point, we can apply previously learned concepts: the coupon payments constitute an annuity stream, or payments of the same amount at regular intervals.
The principal of $1,000 is also paid out at maturity. Here, we recognize another key concept: the final amount is a lump sum payment. So, we now have the promised set of future cash flows for the 3M Co. bond.
In Step 2, we will need to decide on a discount rate to use on these future bond cash payments. For now, we will jump to the answer and simply use the YTM of 1.24% from the bond data in . Later in the chapter, we will develop the concepts behind how an appropriate discount rate is determined.
For Step 3, we now apply two equations to the set of future cash flows from the bond. This will then provide us with the present values of these cash flows, or the expected present-day value of the bond. Because we know that the coupon payments constitute an annuity stream, we can use the equation for the present value of an annuity (discussed in Time Value of Money II: Equal Multiple Payments. To value the one-time par value payment, we use the equation for the present value of a lump sum payment. So, by combining these, we will have the present value of the coupon payment stream, or
So, for our example above, this becomes
Next, we need to determine the present value of the payment of the par or face value of the bond at maturity. This is calculated as follows:
Inserting our values into this formula gives us
Adding the present values of the two payment streams gives us
Our bond price is $1,137.47. This bond price represents the value of the financial asset to both a willing buyer and a willing seller.
In this example, the willing seller is 3M Company. The willing buyer is an investor who is demanding a 1.24% yield on the investment. As per above, the 3M bond sold for $1,051.20 in March 2021. However, we display the price as a percentage of the par value, so we have the displayed price as
Because we round the percent of par, we do not see the cents digit in the quoted price.
### Pricing a Bond Using a Financial Calculator
A financial calculator can also be used to solve common types of bond valuations. For example, what would be the current price (value) of a 4% coupon bond, paid semiannually, with a face value of $1,000 and a remaining term to maturity of 15 years, assuming a required YTM rate of 5%? The steps to solve this problem are shown in below.
The current price is $895.35.
### Time Value Connection
As we have briefly discussed, bond valuation is determined by time value of money techniques, most notably present value calculations. This makes logical sense when one considers that an investment in a bond involves a series of future cash inflows, or payments from the bond issuer to the bondholder over the term of the bond’s maturity.
To determine the value of a bond today, the two-step time value of money calculation we discussed earlier must be used, and the present value of a series of coupon payments (or an annuity) must be determined. This present value amount will then be added to the present value of a single lump sum payment (the principal or face value) that will come to the bondholder at the end of the bond’s term (maturity).
### Fixed Income
Because standard fixed-rate bonds have their coupon payments and maturity amounts locked in, they are often referred to as fixed-income investments. This is because their values are relatively straightforward to calculate. Bonds are generally viewed as stable investments that offer income and a lower amount of volatility compared to stocks.
While yields provided by corporate and government bonds such as US T-bills and municipal bonds are currently low because the Federal Reserve System (the Fed) has kept interest rates low for several years, investors may still consider adding bonds to their portfolios.Adam Hayes. “What Do Constantly Low Bond Yields Mean for the Stock Market?” This is especially true as investors enter their retirement years and seek to generate income while avoiding the volatility of the stock market. Such investors can add a mix of individual bonds, mutual funds, or exchange-traded funds to their portfolios, thus generating potential return while keeping risks at a minimum. Fixed-income investments such as intermediate- or longer-term bond funds are still providing good yields despite the low-interest-rate state of the economy.
It is important to note, however, that even though bonds are generally thought of as safer investments, they still are subject to a number of risks. Because income from most bonds is fixed, such instruments can have their values eroded by external factors such as interest rates and inflation. We will discuss some of these risks after the next section.
shows the cash inflow of a five-year, 9%, $100,000 corporate bond dated January 1, 2020. The bond will have coupon (interest) payment dates of June 30 and December 31 for each of the following five years. Because the bond was issued on January 1, 2020, the year 2020 is the first full year of the bond, followed by the years 2021, 2022, 2023, and 2024, with the bond maturing in December of the latter year.
Cash inflows will be (1) the coupon or interest payments of , paid to the bondholder every six months, and (2) the one-time principal or face-value payment of $100,000 upon maturity on December 31, 2024.
### Yields and Coupon Rates
The two interest rates that we associate with a bond are often confusing to students when they first begin to work with bonds. The coupon rate is the interest rate printed on the bond; this is only used to determine the interest or coupon payments. The yield to maturity (YTM) is an interest rate that is used to discount the bond’s future cash flow. The YTM is derived from the marketplace and is based on the riskiness of future cash flows.
As we have seen when pricing bonds, a bond’s YTM is the rate of return that the bondholder will receive at the current price if the investor holds the bond to maturity.
### Yield to Maturity
As noted above, the market sets this discount rate, or the yield to maturity. The YTM reflects the going rate in the bond market for this type of bond and the bond issuer’s perceived ability to make the future payments. Hence, we base the yield on a mutually agreeable price between seller and buyer. The bond market determines the YTM and the available supply of competing financial assets. By competing against other available financial assets, the YTM reflects the risk-free rate and inflation, plus such premiums as maturity and default specific to the issued bond.
The YTM is the expected return rate on the bond held to maturity. How do we determine the bond’s YTM? We can use our same three trusty methods: equations, a financial calculator, and Microsoft Excel (as shown at the end of the chapter).
### Determining Bond Yield Using an Equation
The solution, when solving for discount rates, requires us to revisit the bond pricing formula, which is
Of course, with one equation, we can solve for only one unknown, and here the variable of concern is r, which is the YTM. Unfortunately, it is difficult to isolate r on the left-hand side of the equation. Therefore, we need to use a calculator or spreadsheet to solve for the bond’s YTM.
Let’s take another bond, the Coca-Cola bond, from above and again back up our time to March 2021. If the Coca-Cola bond has just been issued in March 2021, then it would be a seven-year, semiannual bond with a coupon rate of 1.0% and an original price of $952.06 at the time of issue ().
### Determining Bond Yield Using a Calculator
For the Coca-Cola bond above, what was the bond’s YTM at its issue date? This is not an easy problem to solve with a mathematical formula. It is far more practical, not to mention easier, to use a financial calculator or an Excel spreadsheet to solve for bond prices, yields, and maturity periods.
We will cover Excel applications later, but we can jump into some calculator examples right now. So, to calculate the yield on the Coca-Cola bond, we’ll start by entering the values we have for this bond into a calculator. The values we know are as follows:
If the bond’s selling price was $952.06 at issue, we have all the information we need to determine the bond’s YTM at issue. shows the steps for using a calculator to come to an answer.
The calculated I/Y (interest rate or YTM) of 0.8651 is a semiannual figure because the periods and coupon payments we entered for the calculation are semiannual values. To covert the semiannual value into an annual rate, we will need multiply the calculated I/Y by 2. This gives us an amount of 1.73%.
So, the YTM of the Coca-Cola bond at issue date was 1.73%. It is important to know that unless otherwise indicated, bond yields are expressed in annual percentage terms.
We have just demonstrated how a calculator can be used to determine the YTM or interest rate of a bond. Let’s look at a few more examples that cover the most common types of bond problems. These are determining a YTM, calculating a bond’s current price (or value), and determining a bond’s maturity period.
First, let’s work through another example of calculating a YTM, but this time with a bond that has annual interest payments instead of semiannual coupons.
Let’s say you are considering buying a bond, but you want to calculate the YTM to determine if it will meet your overall return requirements. Some facts you have on the bond are that it has a $1,000 face value and that it matures in 12 years. Assume that the current price of the bond is $675 and it pays coupons annually at 3.5%. See for the steps to calculate the YTM.
By following the steps in the table above, you will arrive at a YTM of 7.76%.
Using a calculator is fast and accurate for finding bond yields. Thus, if you know the bond’s current price and all of the future cash flows, you can find the YTM, or the return rate that the bond buyer is receiving on the funds loaned to the bond issuer. As mentioned, Excel spreadsheets are as easy and accurate as a financial calculator for determining bond rates, and we will cover these later in the chapter.
### Determining Bond Price or Value Using a Calculator
Let’s say a friend recommends a 20-year bond that has a face value of $1,000 and a 6% annual coupon rate. If similar bonds are yielding 4% annually, what would be a fair price for this bond today? shows the steps to make this determination.
So, the bond should be priced today at $1,271.81.
### Determining Bond Maturity Using a Calculator
Imagine you are considering investing in a bond that is selling for $820, has a face value of $1,000, and has an annual coupon rate of 3%. If the YTM is 10%, how long would it take for the bond to mature? See for the steps to calculate the time to maturity.
So, the bond’s time to maturity would be 3.12 years.
### The Coupon Rate
The coupon rate is the rate that we use to determine the amount of a bond’s coupon payments. The issuer states the rate as an annual rate, even though payments may be made more frequently. Thus, for semiannual bonds, the most common type of corporate and government bond, the coupon payment is the par value of the bond multiplied by the annual coupon rate and then divided by the number of payments per year, 2.
We have already seen the coupon rate. The first bond we reviewed, the 3M Co. bond, was an annual coupon bond with a coupon rate of 2.25%. Using a par value of $1,000, we determined that the annual coupon payments would be .
For the Coca-Cola bond, we note from that it has a coupon rate of 1% and is paid semiannually. Using a par value of $1,000, we can determine that the coupon payments would be .
### The Relationship of Yield to Maturity and Coupon Rate to Bond Prices
The value or price of any bond has a direct relationship with the YTM and the coupon rate.
1. When the coupon rate of a bond exceeds the YTM, the bond sells at a premium compared to its par value. That is, market demand will push the price of the bond to an amount greater that than its face or par value. We call this kind of bond a premium bond.
2. When the coupon rate is less than the YTM, the bond sells at a discounted amount, or less than its par value. We refer to such a bond as a discount bond.
3. When the coupon rate and YTM are identical, a bond will sell at its par value. Bonds that experience this scenario in the market are referred to as par value bonds.
The interest or coupon payments of a bond are determined by its coupon rate and are calculated by multiplying the face value of the bond by this coupon rate.
The inverse relationship of interest rates and bond prices is an important concept for investors to know. Because interest rates fluctuate and can change significantly over time, it is important to understand how these changes will impact bond values.
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It is important to ascertain what a given bond is worth to a willing buyer and a willing seller. We can price a bond using an equation, a calculator, or a spreadsheet. The essential steps are (1) identify the amount and timing of the future cash flow; (2) determine the discount rate; (3) find the present values of the lump sum principal and the annuity stream of coupons; and (4) add the present value of the lump sum principal and the present value of the coupons.
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### Using TI BA II+ to Price a Bond
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# Bonds and Bond Valuation
## Using the Yield Curve
### Learning Outcomes
By the end of this section, you will be able to:
1. Use the yield curve to show the term structure of interest rates.
2. Describe and define changes in the yield curve shape.
3. Explain the importance of the yield curve shape.
### Term Structure of Interest Rates
The expected yields of various bonds across different maturity periods are referred to as the term structure of interest rates. This is because they represent interest rates for different periods of time, maturities, or terms.
When interest rate yields are plotted against their respective maturity periods and these plotted points are connected, the resulting line is called a yield curve. Essentially, the yield curve is a result of this plotting process and becomes a graphical representation of the term structure of interest rates. A yield curve will always be constructed by showing the value of yields (rates) on the y-axis and maturities or time periods on the x-axis (see ).
To create a useful graph of the yield curve, interest rate yields should be computed for all government bonds at all remaining times to maturity. For example, the yields on all government bonds with a single year remaining until maturity should be calculated. This value is then plotted on the y-axis against the one-year term on the x-axis. Similarly, yields on government bonds with two years remaining until maturity are calculated and plotted on the y-axis against two years on the x-axis, and so on, until a point of critical mass of information is reached and the resulting graph displays useful information.
The yield curve for government bonds is also known as the risk-free yield curve because these securities are thought of as safe investments that are not expected to fail or default and will in all likelihood repay or otherwise meet all financial obligations made through the bond issuance.
A normal yield curve slopes upward, with yield increasing as the term increases. This is because yields on fixed-income investments such as bonds will rise as maturity periods increase and produce greater levels of risk.
Corporate issuers of bonds will usually offer bond issues at higher yields that the government, which is understandable because they are potentially riskier for investors. Government securities are guaranteed by governments and have little to no chance of default or nonpayment. This is not the case for corporate bonds, where there is always a chance of default, though the likelihood of this occurring will vary by individual company or issuer as well as by bond type and term. We will discuss bond default and default risk next.
### Different Shapes of the Yield Curve
There are two important elements to any yield curve that will define its shape: its level and its slope. The level of a yield curve directly relates to the yield rates depicted on the y-axis of the graph (see ). The slope of the yield curve indicates the difference between yields on short-term and longer-term investments. The difference in yields is primarily due to investors’ expectations of the direction of interest rates in the economy and how the federal funds rate (referred to as cash rate in many countries) is uncertain and may differ significantly over time. As an example, yields on three-year bonds incorporate the expectations of investors on how bank rates might move over the next three years, combined with the uncertainty of those rates over the three-year period.
As we briefly discussed above, a positive or normal yield curve is indicative of the investment community’s requirements for higher rates of return as financial consideration for assuming the risk of entering into fixed-income investments, such as the purchase of bond issues. Typically, as a bond term increases, so will the potential interest rate risk to the bondholder. Therefore, bonds with longer terms will usually carry higher coupon rates to make returns greater for investors. Additionally, economists have come to believe that a steep positive yield curve is a sign that investors anticipate relatively high inflation in the future and thus higher interest rates accompanied by higher investment yields over shorter (inflationary) periods of time.
Normal yield curves are generally observed during periods of economic expansion, when growth and inflation are increasing. In any expansionary economy, there is a greater likelihood that future interest rates will be higher than current rates. This tends to occur because investors will anticipate the Fed or the central bank raising its short-term rates in response to higher inflation rates within the economy.
A flat shape for the yield curve occurs when there is not a great deal of difference between short-term and long-term yields (see ). A flat curve is usually not long lasting and is often observed when the curve is transitioning between a normal and an inverted shape, or vice versa.
A flat yield curve has also been observed as a result of low interest rate levels or some types of unconventional monetary policy.
### Why Is the Yield Curve Important?
Market technicians, brokers, and investment analysts will study the yield curve in great detail by keeping track of its many changes and movements. This is because of the overall importance of the yield curve as an economic indicator and how it can be representative of the ideas, attitudes, and bond market expectations of individuals as well as large institutional investors that exert significant influence on investment markets and the economy as a whole.
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When interest rate yields are plotted against their respective maturity periods and these plotted points are connected, the resulting line is called the yield curve. The yield curve is a graphical representation of the term structure of interest rates. A yield curve always shows the value of yields (rates) on the y-axis and maturities or time periods on the x-axis.
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# Bonds and Bond Valuation
## Risks of Interest Rates and Default
### Learning Outcomes
By the end of this section, you will be able to:
1. Define interest rate, default, and other common forms of bond risk.
2. Calculate the primary indicator of interest rate risk.
3. Determine factors impacting default risk.
4. Understand bond laddering as an investment strategy.
5. List major rating agencies and their indications of default.
6. Define and calculate the yield to maturity (YTM) on a bond.
### Bond Risks
As we touched on earlier, bonds are fixed-income investments, and because of this, they are subject to a number of risks that could have negative effects on their market value. The most common and best-known risks are interest rate risk and default risk, but other risks exist that should be understood. Among these risks are the following:
1. Credit risk. If investors believe that a bond issuer is unlikely to meet its payment commitments, they may demand a higher yield to purchase the bond issue in the first place. Due to the relative stability of governments compared to corporations, government bonds are considered to have low credit risk.
2. Liquidity risk. If investors believe that a bond may be difficult to sell, it will likely have a higher yield. This has the effect of compensating the bondholder for the lack of liquidity (the ability to cash out of the bond). Government bonds usually have the lowest yields of all investments available and are typically among the most liquid in any country where they are traded. Government securities will only face significant liquidity risk in times of great economic distress.
3. Duration risk. Duration risk is the risk associated with the sensitivity of a bond’s price to a single 1% change in interest rates. A bond’s duration is expressed in numerical measurements. The higher the duration number, the more sensitive a bond investment will be to changes in interest rates.
4. Call risk and reinvestment risk. Call risk is the risk of bonds being redeemed or called by the issuing firm before their maturity dates. Corporations may elect to call a bond issue (provided the bond issue has a call feature) when interest rates drop and companies are in a position to save a great deal of money by issuing new bonds with lower coupon rates. To investors, this is a risk in and of itself, but call risk also has the effect of potentially causing reinvestment risk. Reinvestment risk is defined as the risk to investors when they find themselves facing unfavorable alternatives for investing the proceeds from their called bonds in new, lower-paying investments. This can potentially lead to substantial financial loss for the original bond investors.
5. Term risk. Investors will generally demand higher returns for lending funds at fixed interest rates. This is because doing so exposes them to the risks presented by rising interest rates and the negative impact of these higher rates on their bond holdings. In a scenario of rising interest rates, investors will find that their return from lending money through a bond purchase just once, at a fixed interest rate, will be lower than the return they might have realized from making several different investments for much shorter periods of time. Term risk is usually measured by a special indicator referred to as the term premium.
As mentioned above, however, the most common forms of bond risk are interest rate risk and default risk.
### Interest Rate Risk
As we have discussed, when interest rates rise, bond values will fall. This is the general concept behind interest rate risk. Any investor in fixed-income securities (such as bonds) will have to contend with interest rate risk at one time or another. Interest rate risk is also referred to as market risk and usually increases the longer an investor maintains a bond investment.
### Default Risk
Any time a bond is purchased, the investor is taking a risk that the bond issuer may be late in making scheduled payments on a bond issue—or, in the worst case, may not be able to make payments at all. This is the underlying idea behind the concept of default risk.
Because US Treasury securities have the full backing of the government, they are generally considered free of default risk. However, most corporate bonds will face some possibility of default. Obviously, some bonds and their issuing companies are riskier in this respect than others.
To assist potential bond investors in understanding some of these risks, bond ratings are regularly published by a number of organizations to express their assessment of the risk quality of various bond issues. We will discuss these bond ratings and the companies that issue them next.
### Bond Ratings and Rating Providers
It is important for investors to know the risks they are assuming when investing in bonds. Many investors will take advantage of information provided by bond rating services to assess the likelihood of borrowers (bond issuers) defaulting on the financial obligations of their bond issues.
To help investors evaluate the default risks of bonds, rating agencies (bond rating services) were established to evaluate bonds and other fixed-income investments, taking into consideration and then analyzing any information that has been published or otherwise made available to the investing public. These services then apply a rating system that has been developed for measuring the quality of bonds and assign individual grades to each bond and its issuing company.
The three largest and best-known bond rating providers are Fitch Ratings, Moody’s Investors Service, and Standard & Poor’s (S&P) Global Ratings. The rating system used by these services identifies the very highest-quality bonds (the least likely to default) as triple-A (AAA or Aaa), followed next in quality level by double-A bonds (AA or Aa), and so on. Any bond that is rated BBB (S&P, Fitch) / Baa (Moody’s) or higher is referred to as investment grade and is considered strong and stable by the investment community (see ).
It is important to note that investment-grade bonds are among the most popular due to the fact that many commercial banks, as well as several pension funds, are only allowed to trade bonds that are investment grade.
Any bond that is below investment grade, or rated lower than BBB (S&P, Fitch) or Baa (Moody’s), is referred to as a high-yield bond or a junk bond. Junk bonds have had mixed levels of success for companies wishing to issue them to raise capital. In the early 1990s, the market for junk bonds collapsed, due in part to a political movement involving influential people who had been dominating corporate debt markets. This movement, combined with illegal insider trading activities conducted by investments banks, ultimately resulted in the bankruptcy of former financial giant Drexel Burnham Lambert.Lawrence Delevingne. “The Drexel Collapse, 25 Years Later.”
The market for junk bonds enjoyed a brief resurgence in popularity when the economy improved later in the 1990s. However, in 2001, the junk bond market shrank once again, resulting in 11% of US junk bond issues defaulting.
In general, it is important to understand that bond ratings are only judgments on corporations’ future ability to repay debt obligation and their growth prospects. There is no fixed methodology or basis for calculating a bond rating. However, some financial analysts can get a strong indication of how a bond will be rated by examining certain financial ratios of the issuing firm, such as company debt ratio, earnings-to-interest ratio, and their return on assets.
### Concepts of Bond Returns
Bond investors earn profits through two different means: collecting interest income and generating capital gains. These are important concepts for any investor who considers putting their money in fixed-income securities such as bonds.
### Collecting Interest Income
As we have covered, when investors buy bonds, they are lending money to bond issuers. The coupon rate of a bond is determined by the issuer and is generally tied to the overall level of interest rates in the economy at the time of issue as well as the maturity period of the bond and the credit rating of the issuer. The established coupon rate then governs how much periodic interest is paid to bondholders. For example, if an investor purchases a 5%, $1,000 bond with a 20-year maturity and annual coupon payments, that investor will receive 20 coupon payments equal to or for a total of $1,000.
Depending on interest and inflation rates over the 20-year period, this could be a very favorable situation resulting in significant realized return for the investor. However, if interest rates and inflation over the investment period are at high levels, the investment is not nearly as attractive.
### Generating Capital Gains
Many bonds are not held until their maturity dates. Should an investor require funds before maturity, they have the option to sell them through a broker in the secondary market. When this situation occurs, the investor may earn a capital gain or experience a capital loss, depending on whether the bond ends up being sold at a premium (above face value) or at a discount (below face value).
For example, if an investor bought a corporate bond yielding 7% and then the economy changed so that comparable bonds yielded 10%, the investor would have to lower their price on the original 7% bond until it also yielded the 10% market rate. Potential investors would not be very likely to buy the bond if they could simply buy a newly issued bond from an alternate issuer and receive a higher coupon rate.
It is equally possible that prevailing bond rates could fall and an investor could end up selling their bond at a higher price, thus earning a capital gain.
### Bond Laddering as an Investment Strategy
There are several successful strategies for successful bond investments, but perhaps one of the most common yet ingenious of these strategies is called . Bond ladders help investors achieve diversity in their portfolios and reduce risk while helping maintain regular cash inflows in the form of coupon payments or interest. In a bond ladder, an investor will divide their total investment dollars among various bonds that mature at regular intervals, thereby balancing risk and return. An example of a bond ladder would be to purchase 10 different bonds that have maturities of one year, two years, three years, and so on, all the way through to 10 years.
When the first bond matures, the investor will purchase a new bond that matures in 10 years to take its place in the ladder and continue the overall laddering strategy.
This strategy has several benefits. First, the shorter-term bonds in the ladder provide stability because they are less sensitive to risk than longer-term bonds. The longer-term bonds within the ladder will generally provide higher returns but with higher risk due to such factors as rising interest rates. So, by investing in bonds with different maturities and creating a bond ladder, investors can realize superior financial returns to what they would earn by only investing in short-term bonds. Also, the general level of risk from a bond ladder is reduced by the shorter-term component of the investment mix, making the bond ladder less risky than an investment that only included long-term bonds.
It is easy to see why bond laddering has become such a highly adopted bond investment strategy with investors ranging from novice to the most well-seasoned and experienced.
### Interest Rate Movements and Bond Prices
We now know that when investors buy bonds, either directly or through mutual funds, they are lending money to bond-issuing firms or governments. In turn, issuers promise to pay back the principal (par or face value) when the loan is due at the bond’s maturity date.
Issuers also promise to pay bondholders periodic interest or coupon payments to compensate them for the use of their money over the term of the bond. The rate at which issuers pay investors, or the bond’s stated coupon rate, is typically fixed at the time of issuance.
We have also covered the concept that bond values have an inverse relationship with interest rates. As interest rates rise, bond prices fall, and when interest rates fall, bond values increase. Movement of interest rates can have a dramatic effect on a bond’s value and presents the typical bondholder with a number of different financial risks that we have described in detail.
Also in this chapter, we have discussed how bond values can be estimated through the use of several different factors. Prevailing interest rates are among the most critical of these, but also important are factors such as maturity periods, the taxability of bond interest, the credit standing of bond issuers, and the likelihood of bond call, or issuers paying off their debt early.
When considering purchasing bonds or any such fixed-income investment, investors should remain aware that interest rates are always in a state of flux and can change at any time. The movements of bond values and bond yields will be significantly affected by these changes and can be favorable or unfavorable for any investor.
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Because bonds are fixed-income investments, they are subject to a number of risks that could have negative effects on their market value. The most common and best-known risks are interest rate risk and default risk, but there are some others risks that should be understood, such as credit risk, liquidity risk, duration risk, call risk, investment risk, and term risk. To assist potential bond investors in understanding some of these risks, bond ratings have been developed and are regularly published by a number of organizations to express their assessment of the risk quality of various bond issues.
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# Bonds and Bond Valuation
## Using Spreadsheets to Solve Bond Problems
### Learning Outcomes
By the end of this section, you will be able to:
1. Demonstrate bond valuations using Excel.
2. Demonstrate bond yield calculations using Excel.
### Calculating the Price (Present Value) of a Bond
The following examples illustrate how Microsoft Excel can be used to calculate common bond problems. Please be sure to refer to the chapters on the time value of money for examples of using spreadsheets to solve present value problems, as these same concepts are also used in solving bond problems.
You can use the following steps in Excel to determine the price or present value of a coupon bond. Suppose that a bond has a par or face value of $1,000, pays coupons semiannually at a 4% annual rate, and matures in 15 years. We can assume a YTM rate of 5%.
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select PV (see ).
2. When the PV function is selected, another dialog box will appear (see ). It is here that the function variables, or arguments, will be entered. It is preferable to use cell addresses to refer to these arguments so that the spreadsheet can be easily used again if inputs/arguments change.
3. Enter the function inputs or arguments (see ). We refer to the cell addresses as per our example spreadsheet.
Note that the result, the price or present value, will appear in the bottom left section of the Function Arguments box once the arguments are entered. It will appear as a negative value because of the sign convention and because the bond face value in cell F4 was entered as a positive value.
### Calculating the Yield to Maturity (Interest Rate) of a Bond
Use the following steps in Excel to determine the YTM (interest rate) of a bond. Assume that you want to find the YTM of a $1,000, 3.5% bond with annual coupon payments that is selling for $675.00 and will mature in 12 years.
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select Rate (see ).
2. After the dialog box appears, enter the variables or arguments. As with our earlier example, we will use the preferred method of identifying the arguments with cell addresses (see ).
3. Again, after all arguments are entered through their correct cell references, the answer will appear in the lower left corner of the box. Once satisfied with the result, you can hit Enter to insert this final calculated value in your spreadsheet. This has been set up in this sheet in cell H10.
### Calculating the Maturity Period (Term) of a Bond
You can use the following steps in Excel to determine the maturity period or term of a bond. Assume that you are considering investing in a bond that is selling for $820.00, has a face value of $1,000, and has an annual coupon rate of 3%. If the YTM is 10%, how long will it be until the bond matures?
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select Nper (see ).
2. When the dialog box appears, enter function arguments (see ). Once again, we will use the preferred method of using cell addresses as reference points.
3. When arguments have all been entered, the answer will appear in the lower left of the Function Arguments box, as per the above. We arrive at a final answer of 3.12 years until this bond matures.
### Calculating Coupon Rate and Interest (Coupon) Payments
Here is how you would determine the coupon or interest rate and coupon payment using Excel. Assume a $1,000 face value bond is selling for $595, has 20 years until it matures, and has a YTM of 6.5%. What are the coupon rate and the periodic coupon payment amount of the bond?
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select PMT (see ).
2. When the dialog box appears, enter function arguments. Once again, we will use the preferred method of using cell addresses as reference points (see ).
3. When arguments have all been entered, the answer will appear in the lower left of the Function Arguments box, as per the above. We arrive at a final answer of $28.24 as the coupon payment.
The coupon rate can be calculated by taking this coupon payment amount and dividing it by the face value:
So, the coupon rate is 2.824%.
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Microsoft Excel can be used to solve common bond problems. It can be used to calculate the value of a coupon bond, the yield to maturity (interest rate) of a bond, the maturity period of a bond, and the coupon rate and interest (coupon) payments of a bond.
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### Bond Pricing, Valuation, Formulas, and Functions in Excel
Review the examples included in this video, and practice setting up spreadsheets that solve for each of the five primary bond variables using the values in the videos (maturity 10 years, coupon rate 10%, coupon payment $100, yield to maturity 8%, and par value $1,000). Parallel the spreadsheets that are set up in the video, ensuring that you arrive at the same results for each bond variable amount.
### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Stocks and Stock Valuation
## Why It Matters
Similar to bonds, shares of common stock entitle investor owners to a portion of a company’s future earnings and cash flows. However, stocks differ significantly from bonds in how they are issued and managed by companies, the methodology used to calculate their values in public markets, and how they can generate income and eventual value for individual investors.
With common stock, there is no specific promise of how much cash investors will receive or when they will receive it. This differs from bond investments, which are valued entirely on the basis of their guaranteed timing of future cash flows to bondholders.
This means that with stocks, there are no maturity dates, face values, or coupon payment guarantees. It also means that stocks do not promise any specified cash flows in the form of coupons or a face value payment at some point in the future. Instead, stocks (only some, not all) may pay dividends. These dividends are declared after shares of stock have been issued by a company and then purchased by the investing public. Following a dividend declaration, the designated per-share amounts are paid to shareholders of record on a specified date, also determined by a company’s board.
Because stock investments carry no guarantee of payments to investors, they are far riskier than bonds and other forms of fixed-income investments.
While there are many reasons for an investor to choose to purchase common stock, three of the most common reasons are
1. to use stocks as instruments or repositories for maintaining value;
2. to accumulate wealth over the term of the stock investment; and
3. to earn income through capital gains and dividend payments.
As with any financial instrument, common stock purchases offer advantages and disadvantages to investors. Important advantages include the following:
1. Returns through dividends and price appreciation of shares can be substantial.
2. Stocks are a liquid form of investment and can be bought or sold within secondary markets relatively easily.
3. Information about companies, markets, and important trends are widely published and readily available to the investing public.
These advantages are significant and lead many individuals to move into stock investments. Yet it is important to realize that stock has some significant disadvantages, which can include the following:
1. General risk levels are greater than with bonds or other fixed-income investments.
2. Timing the buy-and-sell transactions of stock can be tricky and may lead to losses or not taking full advantage of share price opportunities.
3. Dividends (provided that the stock does indeed pay them, as not all do) are uncertain and subject to change based on decisions of company management.
We will discuss these topics in this chapter and cover many of the details regarding why corporations issue common stock and why investors purchase that stock. |
# Stocks and Stock Valuation
## Multiple Approaches to Stock Valuation
### Learning Outcomes
By the end of this section, you will be able to:
1. Define and calculate a P/E (price-to-earnings) ratio given company data.
2. Determine relative under- or overvaluation indicated by a P/E (price-to-earnings) ratio.
3. Define and calculate a P/B (price-to-book) ratio given company data.
4. Determine relative under- or overvaluation indicated by a P/B (price-to-book) ratio.
5. Define and detail alternative valuation multipliers, including P/S (price-to-sales) ratio, P/CF (price-to-cash-flow) ratio, and dividend yield.
### The Price-to Earnings (P/E) Ratio
Experienced investors use a number of different methods to evaluate information on companies and their common stock before deciding on any potential purchase. One of the most popular techniques used by investors and analysts is to study a company’s financial statements in order to uncover basic fundamental information on the company. This involves calculating a number of financial ratios that help identify trends, bringing elements of operational performance to light and allowing for clearer analysis and evaluation.
A well-proven analytical approach for investors to use in evaluating common stock is to review the overall market value of the company that issues a stock. One of the most consistently used calculations in this analysis, which has important applications in company and common stock evaluation, is the price-to-earnings (P/E) ratio.
The P/E ratio is computed using the following formula:
The P/E ratio is extremely useful to analysts in that it shows the expectations of the market. Essentially, the P/E ratio is representative of the price an investor must pay for every unit of current (or future) corporate earnings.
Bottom-line earnings are a critical factor in valuing common stock. Investors will always want to know how profitable a company is now as well as how profitable it will be in the future. When a company’s bottom line remains relatively flat over a period of time, leaving earnings per share (EPS) relatively unchanged, the P/E ratio can be interpreted as the payback period for the original amount paid for each share of common stock.
For example, the common stock price of Cameo Corp. is currently at $24.00 a share, and its EPS for the year is $4.00. Cameo’s P/E ratio is calculated as
This ratio would typically be expressed in the form . Essentially, this means that investors are willing to pay up to six dollars for every one dollar of earnings. It can also be stated that Cameo stock is currently trading at a multiple of six.
The P/E ratio is typically expressed in two primary ways. The first is as a metric listed by most finance websites and often carries the notation P/E (ttm). This refers to the Wall Street acronym for “trailing 12 months” and signals the company’s operating performance over the past 12 months.
Another form of the P/E ratio is known as the forward (or leading) P/E. This uses future earnings projections rather than actual trailing amounts. The leading P/E, sometimes called the estimated price to earnings, is useful for comparing current earnings to future earnings and helps provide a clearer picture of what earnings may look like, assuming there are no major changes in the company’s operations or accounting treatments.
Referring back to our calculation for Cameo Corp. above, because the current EPS was used in the calculation, this ratio would be classified as a trailing P/E ratio. If we had used an estimated or projected EPS as the denominator in the calculation, it would then be considered a leading P/E ratio.
Analyzing a company’s P/E ratio alone or within a vacuum will actually tell an analyst very little. It is only when a company’s P/E is compared to historical P/E ratios or the P/E ratios of other companies in the same industry that it becomes a useful tool for analysis. One of the most important benefits of using comparative P/E ratios is that they can standardize stocks with different prices and various earnings levels.
Generally speaking, it is very difficult to make any conclusions about a stand-alone stock value, such as whether a stock that has a ratio of is a good buy at its current price or if a stock with a P/E ratio of is too expensive, without performing any relevant comparisons or further analysis.
Analysts have many different ways to interpret P/E ratio data. One of the most common interpretations is that firms with high P/E ratios should be growth companies. Also, a high P/E ratio could mean that a stock’s price is high relative to earnings and possibly overvalued. This could signal a possible undesired downward adjustment in market price in the future.
We can extrapolate from the argument above to put forward the idea that stocks with low P/E ratios should be stabler, more mature organizations. A low P/E might indicate that the current stock price is low relative to earnings and that there may be an opportunity to take advantage of upward price movements and potential investment gains through stock price appreciation.
While this information is often very useful for evaluating stocks and making investment decisions, caution must always be used, as a current stock price may simply be out of line with the company’s earning potential, which would mean that price adjustments are likely to occur in the short term. This is why experienced analysts and investors will use multiple evaluation techniques when conducting stock analysis and evaluation and not rely solely on insights provided by a single set of facts or one form of statistical measurement.
### The Price-to-Book (P/B) Ratio
Another financial ratio commonly used by investors and analysts is the price-to-book (P/B) ratio, also called the market-to-book (M/B) ratio. This is a financial metric used to evaluate a company’s current market value relative to its book value.
The market value, or market capitalization, of a company is defined as the current price of all its outstanding shares of common stock. This is essentially equal to the total value of the company as perceived by the market. For all intents and purposes, the book value is representative of the residual of a company after it has liquidated all assets and paid off all of its liabilities.
Book value can be determined by performing some financial analysis on a company’s balance sheet. Essentially, analysts will use the P/B ratio to compare a business’s available net assets relative to the current sales price of its stock. The price-to-book-value ratio formula is
One of the primary uses of the P/B ratio is to understand market perceptions of a particular stock’s value. It is often the metric of choice for evaluating financial services firms such as real estate firms, insurance companies, and investment trusts. The P/B ratio has a notable shortcoming, however, in that it does not evaluate companies that have a high level of intangible assets such as patents, trademarks, and copyrights.
Ultimately, this ratio will tell an analyst exactly how much potential investors are willing to pay for each dollar of asset value. The PB(M/B) ratio is computed by dividing the current closing price of the stock by the company’s current book value per share, which is calculated by either of the following two equations:
Net book value is equal to net assets, or the total assets minus the total liabilities of the company.
Analysts often consider a low P/B ratio (less than 1) to indicate that a stock is undervalued and a higher ratio (greater than 1) to mean that a stock is overvalued. A low ratio may be an indication that something is wrong with the company or that an investor may be paying too much for any residual value should the company be liquidated.
However, many market experts will argue the exact opposite of the above interpretations. Because of these discrepancies in interpretation and overall variance of opinion, the use of alternate stock valuation metrics, either in addition to or in place of the P/B ratio, is always worth exploring.
In conclusion, the P/B ratio can help a company understand if its net assets are comparable to the market price of its stock. However, as with the P/E ratio, it is always a good idea to compare P/B ratios of companies within the same industry and use them in conjunction with other metrics and analytical methodologies.
### Alternative Multipliers
There are two main types of valuation metrics multiples used to value common stock. These are equity multiples and enterprise value (EV) multiples. Additionally, there are two primary methods by which to perform analysis using these multiples. These methods are comparable company analysis (comps) and precedent transaction analysis (precedents).
Experienced financial analysts advocate the use of multiples in valuation analysis for a number of reasons, the most important being that they help generate realistic and sound judgments of enterprise values (total company values), they are relatively easy to use and interpret, and they can provide helpful information on a company’s overall financial condition when used appropriately.
However, it should be noted that simplicity may have some important disadvantages. When such complex information is reduced to a single equation or final value, it can easily be misunderstood, and the influence of important factors may be masked or lost in the evaluation process.
What’s more, the calculation of multiples represents a snapshot in time for a firm and cannot easily show how a company grows or progresses. Thus, these calculations are only applicable to short-term analysis, not to long-term scenarios.
### Equity Multiples
Equity multiples are especially useful for investment decisions when an investor aspires to minority positions in companies. Below are some common equity multiples used in valuation analyses.
The price-to-earnings (P/E) ratio, which we discussed earlier, is probably the most common equity multiple used in stock valuation because it is relatively simple to calculate and all necessary data are easily accessible by analysts and investors. The market-to-book (M/B), or price-to-book (P/B), ratio is also useful if assets primarily drive a company’s earnings. Again, it is computed as the proportion of share price to book value per share.
Dividend yield is another form of equity multiple and is primarily used when conducting comparisons between cash returns and investment types. Dividend yield is computed as the proportion of dividend per share to share price. The price-to-sales (P/S) ratio is an additional metric used for firms that are experiencing financial losses. The P/S ratio is often used for quick estimates and is computed as the proportion of share price to sales (or revenue) per share.
Another useful metric is the price-to-cash-flow (P/CF) ratio. The P/CF ratio is used to compare a company’s market value to its operating cash flow (or the company’s stock price per share to its operating cash flow per share). This measurement is suitable only in certain cases, such as when a company has substantial noncash expenses (e.g., depreciation or amortization). In some situations, companies may have positive cash flows but still show a bottom-line loss due to large noncash expenses. The P/CF ratio is helpful for arriving at a less distorted view of such a company’s value.
While these various metrics are important, a financial analyst must always consider that companies often operate under their own unique sets of circumstances that ultimately will influence many of these equity multiples.
### Enterprise Value (EV) Multiples
The following are some common EV multiples used in valuation analyses:
Gordon growth model is as follows
EV multiples take an increasingly important role when value decisions surround recent mergers and acquisitions. Enterprise value (EV) is a measurement of the total value of a company. Companies often believe that EV offers a more accurate representation of a firm’s total value than a basic market capitalization method. Generally, EV is perceived to offer an aggregate value of the firm as an enterprise, which is a more comprehensive measurement (see ).
Following are two of the most common enterprise values metrics used in valuing companies and their common stock:
EV/Revenue (EV/R). Also called EV/Sales, EV/R is a valuation metric used to understand a company’s total valuation compared to its annual sales levels. EV/R can help provide an analyst with an idea of exactly how much investors pay for every dollar of a company’s sales revenue.
EV/R is considered a relatively crude metric but can be useful when analyzing companies that have different methods of revenue recognition. P/E ratios, for example, can be significantly affected by changes in the accounting policies of companies being evaluated. This is another reason why multiple metrics should be used in valuations.
Additionally, EV/R is a useful measure for companies that are consuming cash or experiencing financial losses. Such companies may be start-ups or emerging technology firms that have not fully matured and are still in a growth stage of development.
EV/EBIT. A firm’s earnings before interest and taxes (EBIT) is an indicator of its profitability before the effects of interest or taxes. EBIT is also referred to as operating earnings, operating profit, and profit before interest and tax.
1. EV/EBITDA. EV/EBITDA is a ratio that compares a company’s enterprise value (EV) to its earnings before interest, taxes, depreciation, and amortization (EBITDA). EBITDA is often used by analysts as a substitute for cash flow and can be applied to capital analysis using tools such as net present value and internal rate of return. It is relatively easy to calculate, as all information required to compete the calculation is available from any publicly traded company’s financial statements. Because of this, the EV/EBITDA ratio is a commonly used metric to compare the relative values of different businesses.
2. EV/EBITDAR. Another form of valuation based on enterprise value is EV/EBITDAR. This metric divides enterprise value by earnings before interest, tax, depreciation, amortization, and rental costs (EBITDAR). This multiple is used in businesses that have substantial rental and lease expenses, such as hotel chains and airlines. Capital investment can differ significantly for these firms, and when assets are leased, these companies tend to have artificially lower debt and operating income compared to firms that actually own their assets.
EV/Capital Employed. The EV-to-capital employed ratio is a measure of enterprise value compared to the level of capital used by a business. For example, a business with a large capital basis is bound to carry a large enterprise value simply due to its large capital holdings.
### Final Thoughts on Valuation Ratios and Multiples
There are many equity and enterprise value multiples used in company valuation, but the discussions above cover those that are most commonly used. In any case, gaining a thorough understanding of each multiple and its related concepts can help analysts make better use of these metrics in their stock analysis and valuation efforts. Also, as discussed, it is important that analysts and technicians use multiple ratios and alternate measures for any evaluation of a company and its common stock. Not doing so will limit the ultimate interpretation of the results, can lead to incorrect conclusions, and may cause fundamental mistakes in overall investment strategy.
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This section introduced common stock and some of the models and calculation methods used by investors and financial analysts to determine the prices or values of common shares. The most evaluative ratios that can be computed from a company’s financial statements include the price-to-earnings (P/E), price-to book (P/B), price-to-sales (P/S), and price-to cash-flow (P/CF) ratios.
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# Stocks and Stock Valuation
## Dividend Discount Models (DDMs)
### Learning Outcomes
By the end of this section, you will be able to:
1. Identify and use DDMs (dividend discount models).
2. Define the constant growth DDM.
3. List the assumptions and limitations of the Gordon growth model.
4. Understand and be able to use the various forms of DDM.
5. Explain the advantages and limitations of DDMs.
The dividend discount model (DDM) is a method used to value a stock based on the concept that its worth is the present value of all of its future dividends. Using the stock’s price, a required rate of return, and the value of the next year’s dividend, investors can determine a stock’s value based on the total present value of future dividends.
This means that if an investor is buying a stock primarily based on its dividend, the DDM can be a useful tool to determine exactly how much of the stock’s price is supported by future dividends. However, it is important to understand that the DDM is not without flaws and that using it requires assumptions to be made that, in the end, may not prove to be true.
### The Gordon Growth Model
The most common DDM is the Gordon growth model, which uses the dividend for the next year (D1), the required return (r), and the estimated future dividend growth rate (g) to arrive at a final price or value of the stock. The formula for the Gordon growth model is as follows:
This calculation values the stock entirely on expected future dividends. You can then compare the calculated price to the actual market price in order to determine whether purchasing the stock at market will meet your requirements.
Now that we have been introduced to the basic idea behind the dividend discount model, we can move on to cover other forms of DDM.
### Zero Growth Dividend Discount Model
The zero growth DDM assumes that all future dividends of a stock will be fixed at essentially the same dollar value forever, or at least for as long as an individual investor holds the shares of stock. In such a case, the stock’s intrinsic value is determined by dividing the annual dividend amount by the required rate of return:
When examined closely, it can be seen that this is the exact same formula that is used to calculate the present value of a perpetuity, which is
For the purpose of using this formula in stock valuation, we can express this as
where PV is equal to the price or value of the stock, D represents the dividend payment, and r represents the required rate of return.
This makes perfect sense because a stock that pays the exact same dividend amount forever is no different from a perpetuity—a continuous, never-ending annuity—and for this reason, the same formula can be used to price preferred stock. The only factor that might alter the value of a stock based on the zero-growth model would be a change in the required rate of return due to fluctuations in perceived risk levels.
Example:
What is the intrinsic value of a stock that pays $2.00 in dividends every year if the required rate of return on similar investments in the market is 6%?
Solution:
We can apply the zero growth DDM formula to get
While this model is relatively easy to understand and to calculate, it has one significant flaw: it is highly unlikely that a firm’s stock would pay the exact same dollar amount in dividends forever, or even for an extended period of time. As companies change and grow, dividend policies will change, and it naturally follows that the payout of dividends will also change. This is why it is important to become familiar with other DDMs that may be more practical in their use.
### Constant Growth Dividend Discount Model
As indicated by its name, the constant growth DDM assumes that a stock’s dividend payments will grow at a fixed annual percentage that will remain the same throughout the period of time they are held by an investor. While the constant growth DDM may be more realistic than the zero growth DDM in allowing for dividend growth, it assumes that dividends grow by the same specific percentage each year. This is also an unrealistic assumption that can present problems when attempting to evaluate companies such as Amazon, Facebook, Google, or other organizations that do not pay dividends. Constant growth models are most often used to value mature companies whose dividend payments have steadily increased over a significant period of time. When applied, the constant growth DDM will generate the present value of an infinite stream of dividends that are growing at a constant rate.
The constant growth DDM formula is
where D0 is the value of the dividend received this year, D1 is the value of the dividend to be received next year, g is the growth rate of the dividend, and r is the required rate of return.
As can be seen above, after simplification, the constant growth DDM formula becomes the Gordon growth model formula and works in the same way. Let’s look at some examples.
### Variable or Nonconstant Growth Dividend Discount Model
Many experienced analysts prefer to use the variable (nonconstant) growth DDM because it is a much closer approximation of businesses’ actual dividend payment policies, making it much closer to reality than other forms of DDM. The variable growth model is based on the real-life assumption that a company and its stock value will progress through different stages of growth.
The variable growth model is estimated by extending the constant growth model to include a separate calculation for each growth period. Determine present values for each of these periods, and then add them all together to arrive at the intrinsic value of the stock. The variable growth model is more involved than other DDM methods, but it is not overly complex and will often provide a more realistic and accurate picture of a stock’s true value.
As an example of the variable growth model, let’s say that Maddox Inc. paid $2.00 per share in common stock dividends last year. The company’s policy is to increase its dividends at a rate of 5% for four years, and then the growth rate will change to 3% per year from the fifth year forward. What is the present value of the stock if the required rate of return is 8%? The calculation is shown in .
Note:
The value of Maddox stock in this example would be $43.25 per share.
### Two-Stage Dividend Discount Model
The two-stage DDM is a methodology used to value a dividend-paying stock and is based on the assumption of two primary stages of dividend growth: an initial period of higher growth and a subsequent period of lower, more stable growth.
The two-stage DDM is often used with mature companies that have an established track record of making residual cash dividend payments while experiencing moderate rates of growth. Many analysts like to use the two-stage model because it is reasonably grounded in reality. For example, it is probably a more reasonable assumption that a firm that had an initial growth rate of 10% might see its growth drop to a more modest level of, say, 5% as the company becomes more established and mature, rather than assuming that the firm will maintain the initial growth rate of 10%. Experts tend to agree that firms that have higher payout ratios of dividends may be well suited to the two-stage DDM.
As we have seen, the assumptions of the two-stage model are as follows:
1. The first period analyzed will be one of high initial growth.
2. This stage of higher growth will eventually transition into a period of more mature, stable, and sustainable growth at a lower rate than the initial high-growth period.
3. The dividend payout ratio will be based on company performance and the expected growth rate of its operations.
Let’s use an example. Lore Ltd. estimates that its dividend growth will be 13% per year for the next five years. It will then settle to a sustainable, constant, and continuing rate of 5%. Let’s say that the current year’s dividend is $14 and the required rate of return (or discount rate) is 12%. What is the current value of Lore Ltd. stock?
Step 1:
First, we will need to calculate the dividends for each year until the second, stable growth rate phase is reached. Based on the current dividend value of $14 and the anticipated growth rate of 13%, the values of dividends (D1, D2, D3, D4, D5) can be determined for each year of the first phase. Because the stable growth rate is achieved in the second phase, after five years have passed, if we assume that the current year is 2021, we can lay out the profile for this stock’s dividends through the year 2026, as per .
Step 2:
Next, we apply the DDM to determine the terminal value, or the value of the stock at the end of the five-year high-growth phase and the beginning of the second, lower growth-phase.
We can apply the DDM formula at any point in time, but in this example, we are working with a stock that has constant growth in dividends for five years and then decreases to a lower growth rate in its secondary phase. Because of this timing and dividend structure, we calculate the value of the stock five years from now, or the terminal value. Again, this is calculated at the end of the high-growth phase, in 2026. By applying the constant growth DDM formula, we arrive at the following:
The terminal value can be calculated by applying the DDM formula in Excel, as seen in and . The terminal value, or the value at the end of 2026, is $386.91.
Step 3:
Next, we find the PV of all paid dividends that occur during the high-growth period of 2022–2026. This is shown in . Our required rate of return (discount rate) is 12%.
Step 4:
Next, we calculate the PV of the single lump-sum terminal value:
Remember that due to the sign convention, either the FV must be entered as a negative value or, if entered as a positive value, the resulting PV will be negative. This example shows the former.
Step 5:
Our next step is to find the current fair (intrinsic) value of the stock, which comprises the PV of all future dividends plus the PV of the terminal value. This is represented in the following formula, with all factors shown in :
So, we end up with a total current fair value of Lore Ltd. stock of $291.44 (due to Excel’s rounding), although the sum can also be calculated as shown below:
### Advantages and Limitations of DDMs
Some of the primary advantages of DDMs are their basis in the sound logic of present value concepts, their consistency, and the implication that companies that pay dividends tend to be mature and stable entities. Also, because the model is essentially a mathematical formula, there is little room for misinterpretation or subjectivity. As a result of these advantages, DDMs are a very popular form of stock evaluation that most analysts show faith in.
Because dividends are paid in cash, companies may keep making their dividend payments even when doing so is not in their best long-term interests. They may not want to manipulate dividend payments, as this can directly lead to stock price volatility. Rather, they may manipulate dividend payments in the interest of buoying up their stock price.
To further illustrate limitations of DDMs, let’s examine the Concepts in Practice case.
### Stock Valuation with Changing Growth Rates and Time Horizons
Before we move on from our discussion of dividend discount models, let’s work through some more examples of how the DDM can be used with a number of different scenarios, changing growth rates, and time horizons.
As we have seen, the value or price of a financial asset is equal to the present value of the expected future cash flows received while maintaining ownership of the asset. In the case of stock, investors receive cash flows in the form of dividends from the company, plus a final payout when they decide to relinquish their ownership rights or sell the stock.
Let’s look at a simple illustration of the price of a single share of common stock when we know the future dividends and final selling price.
Problem:
Steve wants to purchase shares of Old Peak Construction Company and hold these common shares for five years. The company will pay $5.00 annual cash dividends per share for the next five years.
At the end of the five years, Steve will sell the stock. He believes that he will be able to sell the stock for $25.00 per share. If Steve wants to earn 10% on this investment, what price should he pay today for this stock?
Solution:
The current price of the stock is the discounted cash flow that Steve will receive over the next five years while holding the stock. If we let the final price represent a lump-sum future value and treat the dividend payments as an annuity stream over the next five years, we can apply the time value of money concepts we covered in earlier chapters.
### Method 1: Using an Equation
### Method 2: Using a Financial Calculator
We can also use a calculator or spreadsheet to find the price of the stock (see ).
The stock price is calculated as $34.47.
Note that the value given is expressed as a negative value due to the sign convention used by financial calculators. We know the actual stock value is not negative, so we can just ignore the minus sign.
In cases such as the above, we find the present value of a dividend stream and the present value of the lump-sum future price. So, if we know the dividend stream, the future price of the stock, the future selling date of the stock, and the required return, it is possible to price stocks in the same manner that we price bonds.
### Method 3: Using Excel
shows a spreadsheet setup in Excel to reach a solution to this problem.
Due to the sign convention in Excel, we can ignore the parentheses around the solution, which indicate a negative value. Therefore, the price is $34.48. The Excel command used in cell F6 to calculate present value is as follows:
### Finding Stock Price with Constant Dividends
Example 1:
Four Seasons Resorts pays a $0.25 dividend every quarter and will maintain this policy forever. What price should you pay for one share of common stock if you want an annual return of 10% on your investment?
Solution:
You can restate your annual required rate of 10% as a quarterly rate of . Apply the quarterly dividend amount and the quarterly rate of return to determine the price:
Even though we anticipate that companies will be in business “forever,” we are not going to own a company’s stock forever. Therefore, the dividend stream to which we would have legal claim is only for that period of the company’s life during which we own the stock. We need to modify the dividend model to account for a finite period when we will sell the stock at some future time. This modification brings us from an infinite to a finite dividend pricing model, which we will use to price a finite amount of dividends and the future selling price of the stock. We will maintain a constant dividend assumption. Let’s assume we will hold a share in a company that pays a $1 dividend for 20 years and then sell the stock.
### Method 1: Using an Equation
The dividend pricing model under a finite horizon is a concept we have seen earlier. It is a simple present value annuity stream application:
We now need to determine the selling price that we would get in 20 years if we were to sell the stock to someone else at that time. What would a willing buyer give us for the stock 20 years from now? This price is difficult to estimate, so for the sake of this exercise, we will assume that the price in 20 years will be $30. So, what is the present value of the price in 20 years with a 10% discount rate? Again, this is just a simple application of the PV formula we covered earlier in the text:
We can now price the stock as if it were a bond with a dividend stream of 20 years, a sales price in 20 years, and a required return of 10%:
1. The dividend stream is analogous to the coupon payments.
2. The sales price is analogous to the bond’s principal.
3. The 20-year investment horizon is analogous to the bond’s maturity date.
4. The required return is analogous to the bond’s yield.
Carrying on with the PV calculations, we have
### Method 2: Using a Financial Calculator
We can also use a calculator or spreadsheet to find the price of the stock using constant dividends (see ).
The stock price resulting from the calculation is $12.97.
### Method 3: Using Excel
This same problem can be solved using Excel with a setup similar to that shown in .
Once again, we can ignore the negative indicator that is generated by the Excel sign convention because we know that the stock will not have a negative value 20 years from now. Therefore, the price is $12.97. The Excel command used in cell F13 to calculate present value is as follows:
Example 2:
Let’s look at an example and estimate current stock price given a 10.44% constant growth rate of dividends forever and a desired return on the stock of 13.5%. We will assume that the current stock owner has just received the most recent dividend, D0, and the new buyer will receive all future cash dividends, beginning with D1. This part of the setup of the model is important because the price reflects all future dividends, starting with D1, discounted back to today. (Price0 refers to the price at time zero, or today.) The first dividend the buyer would receive is one full period away. Using the discounted cash flow approach, we have
where g is the annual growth rate of the dividends and r is the required rate of return on the stock. We can simplify the equation above into the following:
As we discussed above, this classic model of constant dividend growth, known as the Gordon growth model, is a fundamental method of stock pricing. The Gordon growth model determines a stock’s value based on a future stream of dividends that grows at a constant rate. Again, we assume that this constantly growing dividend stream will pay forever. To see how the constant growth model works, let’s use our example from above once again as a test case. The most recent dividend (D0) is $1.76, the growth rate (g) is 10.44%, and the required rate of return (r) is 13.5%, so applying our PV equation, we have
Our estimated price for this example is $63.52. Notice that the formula requires the return rate r to be greater than the growth rate g of the dividend stream. If g were greater than r, we would be dividing by a negative number and producing a negative price, which would be meaningless.
Let’s pick another company and see if we can apply the dividend growth model and price the company’s stock with a different dividend history. In addition, our earlier example will provide a shortcut method to estimate g, although you could still calculate each year’s percentage change and then average the changes over the 10 years.
### Estimating a Stock Price from a Past Dividend Pattern
Problem:
Phased Solutions Inc. has paid the following dividends per share from 2011 to 2020:
If you plan to hold this stock for 10 years, believe Phased Solutions will continue this dividend pattern forever, and you want to earn 17% on your investment, what would you be willing to pay per share of Phased Solutions stock as of January 1, 2021?
Solution:
First, we need to estimate the annual growth rate of this dividend stream. We can use a shortcut to determine the average growth rate by using the first and last dividends in the stream and the time value of money equation. We want to find the average growth rate given an initial dividend (present value) of $0.70, the most recent dividend (future value) of $2.11, and the number of years (n) between the two dividends, or the number of dividend changes, which is 9. So, we calculate the average growth rate as follows:
shows the step-by-step process of using a financial calculator to solve for the growth rate.
The calculated growth rate is 13.04%.
We can also use Excel to set up a spreadsheet similar to the one in that will calculate this growth rate.
The Excel command used in cell G22 to calculate the growth rate is as follows:
We now have two methods to estimate g, the growth rate of the dividends. The first method, calculating the change in dividend each year and then averaging these changes, is the arithmetic approach. The second method, using the first and last dividends only, is the geometric approach. The arithmetic approach is equivalent to a simple interest approach, and the geometric approach is equivalent to a compound interest approach.
To apply our PV formula above, we had to assume that the company would pay dividends forever and that we would hold on to our stock forever. If we assume that we will sell the stock at some point in the future, however, can we use this formula to estimate the value of a stock held for a finite period of time? The answer is a qualified yes. We can adjust this model for a finite horizon to estimate the present value of the dividend stream that we will receive while holding the stock. We will still have a problem estimating the stock’s selling price at the end of this finite dividend stream, and we will address this issue shortly. For the finite growing dividend stream, we adjust the infinite stream in our earlier equation to the following:
where n is the number of future dividends.
This equation may look very complicated, but just focus on the far right part of the model. This part calculates the percentage of the finite dividend stream that you will receive if you sell the stock at the end of the nth year. Say you will sell Johnson & Johnson after 10 years. What percentage of the $60.23 (the finite dividend stream) will you get? Begin with the following:
Now, multiply the result by the price for your portion of the infinite stream:
The next step is to discount the selling price of Johnson & Johnson in 10 years at 17% and then add the two values to get the stock’s price. So, how do we estimate the stock’s price at the end of 10 years? If we elect to sell the stock after 10 years and the company will continue to pay dividends at the same growth rate, what would a buyer be willing to pay? How could we estimate the selling price (value) of the stock at that time?
We need to estimate the dividend in 10 years and assume a growth rate and the required return of the new owner at that point in time. Let’s assume that the new owner also wants a 17% return and that the dividend growth rate will remain at 13.04%. We calculate the dividend in 10 years by taking the current growth rate plus one raised to the tenth power times the current dividend:
We then use the dividend growth model with infinite horizon to determine the price in 10 years as follows:
Your price for the stock today—given that you will receive the growing dividend stream for 10 years and sell for $258.10 in 10 years, and also given that you want a 17% return over the 10 years—is as shown below:
Why did you get the same price of $60.23 for your stock with both the infinite growth model and the finite model? The reason is that the required rate of return of the stock remained at 17% (your rate) and the growth rate of the dividends remained at 13.04%. The infinite growth model gives the same price as the finite model with a future selling price as long as the required return and the growth rate are the same for all future sales of the stock.
Although this point may be subtle, what we have just shown is that a stock’s price is the present value of its future dividend stream. When you sell the stock, the buyer purchases the remaining dividend stream. If that individual should sell the stock in the future, the new owner would buy the remaining dividends. That will always be the case; a stock’s buyer is always buying the future dividend stream.
### Some Final Thoughts on Dividend Discount Models
The dividend used to calculate a price is the expected future payout and expected future dividend growth. This means the DDM is most useful when valuing companies that have long, consistent dividend records.
If the DDM formula is applied to a company with a limited dividend history, or in an industry exposed to significant risks that could affect a company’s ability to maintain its payout, the resulting derived value may not be entirely accurate.
In most cases, dividend models, whether constant growth or constant dividend, appeal to a fundamental concept of asset pricing: the future cash flow to which the owner is entitled while holding the asset and the required rate of return for that cash flow determine the value of a financial asset. However, problems can arise when using these models because the timing and amounts of future cash flows may be difficult to predict.
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The dividend discount model, or DDM, is a method used to value a stock based on the concept that its worth is the present value of all of its future dividends. The most common DDM is the Gordon growth model, which values stock entirely on expected future dividends. Other techniques include the zero growth DDM, which depends on fixed dividends; the constant growth DDM, which assumes that dividends will grow at a constant rate; and the variable growth or nonconstant growth DDM, which is based on the assumption that stock value will progress through different stages of growth. There is also the two-stage DDM, which is based on the assumption of two stages of dividend growth: an initial period of higher growth and a subsequent period of lower, more stable growth.
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### Efficient Markets
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# Stocks and Stock Valuation
## Discounted Cash Flow (DCF) Model
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain how the DCF model differs from DDMs.
2. Apply the DCF model.
3. Explain the advantages and disadvantages of the DCF model.
When investors buy stock, they do so in order to receive cash inflows at different points in time in the future. These inflows come in the form of cash dividends (provided the stock does indeed pay dividends, because not all do) and also in the form of the final cash inflow that will occur when the investor decides to sell the stock.
The investor hopes that the final sale price of the stock will be higher than the purchase price, resulting in a capital gain. The hope for capital gains is even stronger in the case of stocks that do not pay dividends. When securities have been held for at least one year, the seller is eligible for long-term capital gains tax rates, which are lower than short-term rates for most investors. This makes non-dividend-paying stocks even more attractive, provided that they do indeed appreciate in value over the investor’s holding period. Meanwhile, short-term gains, or gains made on securities held for less than one year, are taxed at ordinary income tax rates, which are usually higher and offer no particular advantage to an investor in terms of reducing their taxes.
### Understanding How the DCF Model Differs from DDMs
The valuation of an asset is typically based on the present value of future cash flows that are generated by the asset. It is no different with common stock, which brings us to another form of stock valuation: the discounted cash flow (DCF) model. The DCF model is usually used to evaluate firms that are relatively young and do not pay dividends to their shareholders. Examples of such companies include Facebook, Amazon, Google, Biogen, and Monster Beverage. The DCF model differs from the dividend discount models we covered earlier, as DDM methodologies are almost entirely based on a stock’s periodic dividends.
The DCF model is an absolute valuation model, meaning that it does not involve comparisons with other firms within any specific industry but instead uses objective data to evaluate a company on a stand-alone basis. The DCF model focuses on a company’s cash flows, determining the present value of the entire organization and then working this down to the share-value level based on total shares outstanding of the subject organization. This highly regarded methodology is the evaluation tool of choice for experienced financial analysts when evaluating companies and their common stock. Many analysts prefer DCF methods of valuation because these are based on a company’s cash flows, which are far less easily manipulated through accounting treatments than revenues or bottom-line earnings.
The DCF model formula in its mathematical form is presented below:
where CF1 is the estimated cash flow in year 1, CF2 is the estimated cash flow in year 2, and so on; TCF is the terminal cash flow, or expected cash flow from the ending asset sale; r is the discount rate or required rate of return; g is the anticipated growth rate of the cash flow; and n is the number of years covered in the model.
### Applying the DCF Model
We can apply the DCF model to an example to demonstrate this methodology and how the formula works. Calculate the value of Mayweather Inc. and its common stock based on the next six years of cash flow results. Assume that the discount rate (required rate of return) is 8%, Mayweather’s growth rate is 3%, and the terminal value (TCF) will be two and one-half times the discounted value of the cash flow in year 6.
Mayweather has a cash flow of $2.0 million in year 1, so its discounted cash flow after one year (CF1) is $1,851,851.85. We arrive at this amount by applying the discount rate of 8% for a one-year period to determine the present value.
In subsequent years, Mayweather’s cash flow will be increasing by 3%. These future cash flows also must be discounted back to present values at an 8% rate, so the discounted cash flow amounts over the next six years will be as follows:
Our earlier assumption that the terminal value will be 2.5 times the value in the sixth year gives us a total terminal cash flow (TCF) of , or $3,652,697. Now, if we take all these future discounted cash flows and add them together, we arrive at a grand total of $13,554,477. So, based on our DCF model analysis, the total value of Mayweather Inc. is just over $13.5 million.
At this point, we have the estimated value of the entire company, but we need to work this down to the level of per-share value of common stock.
Let’s say that Mayweather is currently trading at $12 per share, and it has 1,000,000 common shares outstanding. This tells us that the market capitalization of the company is , or $12 million, and that a $12 share price may be considered relatively low. The reason for this is that based on our DCF model analysis, investors would theoretically be willing to pay $13,554,477 divided by 1,000,000 shares, or $13.55 per share, for Mayweather. The overall conclusion would be that at $12.00 per share, Mayweather common stock would be a good buy at the present time. shows the Excel spreadsheet approach for arriving at the total value of Mayweather.
Cell E6 displays the present value formula that is active in cell D6.
### Advantages and Limitations of the DCF Model
Due to several corporate accounting scandals in recent years, many analysts have given increasing credence to the use of cash flow as a metric for determining accurate corporate valuations. However, it should be noted that cash flow is not always the best means of measuring financial health. A company can always sell a large portion of its assets to generate a positive cash flow, even if it is operating at a loss or experiencing other financial difficulties. Additionally, investors prefer to see companies reinvesting their cash back into their businesses rather than sitting on excessive balances of idle cash.
Similar to other models, the discounted cash flow model is only as good as the information entered. As the common expression goes, “garbage in, garbage out.” This can often be the case if reasonably accurate cash flow estimates are not available or if an unrealistic discount rate or required rate of return is used in the calculations. It is always best to use several different methods when valuing companies and their common stock.
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Investors buy stock to receive cash inflows at different points in the future. These inflows may come in the form of dividends or a final cash inflow. If the investor chooses to wait for a final cash flow, the hope is that capital gains will be even stronger. The DCF model is usually used to evaluate firms that are relatively young and do not pay dividends to their shareholders. The DCF model focuses on a company’s cash flows, determining the present value of an entire organization using objective data and then working this down to the share-value level based on total shares outstanding of the subject organization.
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# Stocks and Stock Valuation
## Preferred Stock
### Learning Outcomes
By the end of this section, you will be able to:
1. Define preferred stock.
2. Calculate the intrinsic value of preferred stock.
3. Understand the difference between common stock and preferred stock.
### Features of Preferred Stock
Preferred stock is a unique form of equity sold by some firms that offers preferential claims in ownership. Preferred stock will often feature a dividend that a company is obligated to pay out before it makes any dividend payments to common stockholders. In cases of bankruptcy and liquidation of the issuing company, preferred stockholders have priority claim to assets before common stockholders. Additionally, preferred stockholders are usually entitled to a set (or constant) dividend every period.
Preferred stock carries a stated par value, but unlike bonds, they have no maturity date, and consequently, there is no final payment of the par value. The only time a company would pay this par value to the shareholder would be if the company ceased operations or retired the preferred stock. Many preferred stock issues are cumulative in nature, meaning that if a company skips or is otherwise unable to pay a cash dividend, it becomes a liability to the company and must eventually be paid out to preferred shareholders at some point in the future. Other preferred stocks may be noncumulative, in which case if the company skips dividends, they are forever lost to the shareholder.
The term preferred comes from preferred shareholders receiving all past (if cumulative) and present dividends before common shareholders receive any cash dividends. In other words, preferred shareholders’ dividend claims are given preferential treatment over those of common shareholders. Preferred stock is usually a form of permanent funding, but there are circumstances or covenants that could alter the payoff stream.
For example, a company may convert preferred stock into common stock at a preset point in the future. It is not uncommon for companies to issue preferred stock that has a conversion feature. Such conversion features give preferred shareholders the right to convert to common shares after a predetermined period.
A review of the characteristics of preferred stock will lead to the conclusion that the constant growth dividend model is an excellent approach for valuing such stock. Because shares of preferred stock provide a constant cash dividend based on original par value and the stated dividend rate, these may be considered a form of perpetuity.
It is this constant, preferred dividend stream that makes preferred stock seem more like bonds or another form of debt than like stock. In addition, the constant dividend stream leads nicely to the pricing of preferred stock with the four dividend models we presented earlier in this chapter.
### Determining the Intrinsic Value of Preferred Stock
We can apply a version of the present value of a perpetuity formula to value preferred stock, as in the following example. Oh-Well Heath Services Inc. has issued preferred stock that has a par value of $1,000 and pays an annual dividend rate of 5%. If the market considers the risk of Oh-Well to warrant a 10% discount rate, what would be a fair market price for Oh-Well preferred stock?
First, we find the dividend value of Oh-Well:
We then use the constant dividend model with infinite horizon because we have g equal to zero and n equal to infinity:
We can also rearrange the formula to determine the required return on this stock, given its annual dividend and current price.
We have introduced the concept of return here, which should be thought of as both the anticipated return for the preferred stockholder and the company’s cost of borrowing money for this particular type of capital.
### Differences between Preferred and Common Stock
As we have discussed, preferred stock has important differences from common stock that apply to issuing firms and to investors. Some of the most important of these differences are listed in .
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Preferred stock is a unique form of equity sold by some firms that offers preferential claims in ownership. Preferred stock carries a stated par value, but unlike bonds, there is no maturity date, and consequently, there is no final payment of the par value. The term preferred comes from preferred shareholders receiving all past (if cumulative) and present dividends before common shareholders receive any cash dividends.
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### What Is Preferred Stock?
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# Stocks and Stock Valuation
## Efficient Markets
### Learning Outcomes
By the end of this section, you will be able to:
1. Understand what is meant by the term efficient markets.
2. Understand the term operational efficiency when referring to markets.
3. Understand the term informational efficiency when referring to markets.
4. Distinguish between strong, semi-strong, and weak levels of efficiency in markets.
### Efficient Markets
For the public, the real concern when buying and selling of stock through the stock market is the question, “How do I know if I’m getting the best available price for my transaction?” We might ask an even broader question: Do these markets provide the best prices and the quickest possible execution of a trade? In other words, we want to know whether markets are efficient. By efficient markets, we mean markets in which costs are minimal and prices are current and fair to all traders. To answer our questions, we will look at two forms of efficiency: operational efficiency and informational efficiency.
### Operational Efficiency
Operational efficiency concerns the speed and accuracy of processing a buy or sell order at the best available price. Through the years, the competitive nature of the market has promoted operational efficiency.
In the past, the NYSE (New York Stock Exchange) used a designated-order turnaround computer system known as SuperDOT to manage orders. SuperDOT was designed to match buyers and sellers and execute trades with confirmation to both parties in a matter of seconds, giving both buyers and sellers the best available prices. SuperDOT was replaced by a system known as the Super Display Book (SDBK) in 2009 and subsequently replaced by the Universal Trading Platform in 2012.
NASDAQ used a process referred to as the small-order execution system (SOES) to process orders. The practice for registered dealers had been for SOES to publicly display all limit orders (orders awaiting execution at specified price), the best dealer quotes, and the best customer limit order sizes. The SOES system has now been largely phased out with the emergence of all-electronic trading that increased transaction speed at ever higher trading volumes.
Public access to the best available prices promotes operational efficiency. This speed in matching buyers and sellers at the best available price is strong evidence that the stock markets are operationally efficient.
### Informational Efficiency
A second measure of efficiency is informational efficiency, or how quickly a source reflects comprehensive information in the available trading prices. A price is efficient if the market has used all available information to set it, which implies that stocks always trade at their fair value (see ). If an investor does not receive the most current information, the prices are “stale”; therefore, they are at a trading disadvantage.
### Forms of Market Efficiency
Financial economists have devised three forms of market efficiency from an information perspective: weak form, semi-strong form, and strong form. These three forms constitute the efficient market hypothesis. Believers in these three forms of efficient markets maintain, in varying degrees, that it is pointless to search for undervalued stocks, sell stocks at inflated prices, or predict market trends.
In weak form efficient markets, current prices reflect the stock’s price history and trading volume. It is useless to chart historical stock prices to predict future stock prices such that you can identify mispriced stocks and routinely outperform the market. In other words, technical analysis cannot beat the market. The market itself is the best technical analyst out there.
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Efficient markets are markets in which costs are minimal and prices are current and fair to all traders. There are two forms of efficiency: operational efficiency and informational efficiency. Operational efficiency concerns the speed and accuracy of processing a buy or sell order at the best available price. Informational efficiency concerns how quickly a source reflects comprehensive information in the available trading prices. Financial economists have devised three forms of efficient markets from an information perspective: weak form, semi-strong form, and strong form.
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### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Historical Performance of US Markets
## Why It Matters
Author Mark Twain spoke for many when he wrote, “October—this is one of the peculiarly dangerous months to speculate in stocks. The others are July, January, September, April, November, May, March, June, December, August, and February.” Twain’s comment, though humorous, reflects the serious risks associated with investing in the stock market. So why should we study the history of the US financial markets? Financial experts regularly remind us that past performance is no guarantee of future results. However, past performance can provide targets or benchmarks around which to build expectations. We can learn about the past to prepare for future possibilities, or we can suffer what Winston Churchill warned, “Those that fail to learn from history are doomed to repeat it.”
Carlos Slim Helu, a Mexican businessman and the richest person in the world from 2010 to 2013, once said, “With a good perspective on history, we can have a better understanding of the past and present, and thus a clear vision of the future.”Dan Western. “45 Carlos Slim Helu Quotes About Wealth and Success.” In this chapter, we examine current and historical performance in money, bond, and stock markets. Studying past market risk and return also allows investors to understand what is reasonable and what is not. |
# Historical Performance of US Markets
## Overview of US Financial Markets
### Learning Outcomes
By the end of this section, you will be able to:
1. Identify the various aspects of the US money markets.
2. Characterize government and corporate bond markets.
3. Detail the structure and operations of US equity markets.
### Money Markets
The money market is a multitrillion-dollar market. Features of money market securities include being short-term (with maturities of less than one year) and very low risk (rarely failing to make their required payments). Further, money market securities are also liquid, which means that they trade easily without losing value.
Financial institutions, corporations, and governments that have short-term borrowing and/or lending needs issue securities in the money market. Most of the transactions are quite large, with typical amounts of $10,000, $100,000, $1 million, or more. Money market securities are available in smaller amounts if you choose to invest in money market mutual funds (MMMFs) or certain types of exchange-traded funds (ETFs).
Treasury bills (T-bills) are short-term debt instruments issued by the federal government. T-bills are auctioned weekly by the Treasury Department through the trading window of the Federal Reserve Bank of New York, with maturities of 4, 8, 13, or 26 weeks. The Treasury also auctions 52-week T-bills once every four weeks. The federal government uses T-bills to meet short-term liquidity needs. T-bills have very short maturities and a broad secondary market and are default-risk free. T-bills are also exempt from state and local income taxes. As a result, they carry some of the lowest effective interest rates on publicly traded debt securities.
The volume of T-bills auctioned depends upon government borrowing needs. Much of the money raised at weekly T-bill auctions goes to repay the money borrowed 4, 8, 13, 26, or 52 weeks earlier. The gross amount of new T-bills issued in December 2020 was $1,591.1 billion, and the amount of T-bills retired in the same month was $1,570.6 billion, resulting in net new borrowing of “only” $20.5 billion.
In addition to the regular auction of new T-bills, there is also an active secondary market where investors can trade used or previously issued T-bills. Since 2001, the average daily trading volume for T-bills has exceeded $75 billion.
Commercial paper (CP) is a short-term, unsecured debt security issued by corporations and financial institutions to meet short-term financing needs such as inventory and receivables. For example, credit card companies use commercial paper to finance credit card payments. Commercial paper is a short-term debt instrument, with a typical maturity of 30 days and up to 270 days. The short maturity reduces US Securities and Exchange Commission (SEC) oversight. The lesser oversight and the unsecured nature of CP means that only highly rated firms are able to issue the uninsured paper.
Commercial paper typically carries a minimum face value of $100,000 and sells at a discount, with the face value as the repayment amount. Corporations and financial institutions, not the government, issue CP; thus, returns are taxable. Further, unlike T-bills, there is not a robust secondary market for CP. Most purchasers are large, such as mutual fund investment companies, and they tend to hold commercial paper until maturity. The default rate on commercial paper is typically low, but default rates did increase into the double-digit range during the financial crisis of 2008.
Negotiable certificates of deposit (NCDs) are very large CDs issued by financial institutions. They are redeemable only at maturity, but they can and often do trade prior to maturity in a broad secondary market. NCDs, or jumbo CDs, are so called because they sell in increments of $100,000 or more. However, typical amounts are $1 million, with a maturity of two weeks to six months.
NCDs differ in some important ways from the typical CD you may be familiar with from your local bank or credit union. The typical CD has a maturity date, interest rate, and face amount and has FDIC insurance. However, if an investor wishes to cash out prior to maturity, they will incur a substantial penalty from the issuer (bank or credit union). An NCD also has a maturity date and amount, but it is much larger than a regular CD and appeals to institutional investors. The principal is not insured. When the investor wishes to cash out early, there is a robust secondary market for trading the NCD. The issuing institution can offer higher rates on NCDs compared to CDs because it knows it will have use of the purchase amount for the entire maturity of the NCD and because the reserve requirements on NCDs by the Federal Reserve is lower than for other types of deposits.
Investment companies such as Vanguard and Fidelity, among many others, sell shares in money market mutual funds (MMMFs). The investment company purchases money market instruments, such as T-bills, CP, or NCDs; pools them; and then sells shares of ownership to investors (see ). Generally, MMMFs invest only in taxable securities, such as commercial paper and negotiable certificates of deposit, or only in tax-exempt government securities, such as T-bills. Investors can then choose which type of short-term liquid securities they would like to hold, taxable or nontaxable. MMMFs provide smaller firms and investors the opportunity to participate in the money market by facilitating smaller individual investment amounts.
The market for federal funds is notable because the Federal Reserve (Fed) targets the equilibrium interest rate on federal funds as one of its most important monetary policy tools. The federal funds market traditionally consists of the overnight borrowing and lending of immediately available funds among depository financial institutions, notably domestic commercial banks. Financial institutions such as banks are required to keep a fraction of their deposits on reserve with the Fed. When banks find they are short of reserves and immediately need cash to meet reserve requirements, they can borrow directly from the Fed through the so-called “discount window” or purchase excess reserves from other banks in the federal funds market. Often, the maturity of a federal funds contract is as short as a single day or overnight. The participants in the market negotiate the federal funds interest rate. However, the Federal Reserve effectively sets the target interest rate range in the federal funds market by controlling the supply of funds available for use in the market.
Since the financial crisis of 2008, the activities and functioning of the federal funds market has changed. The federal funds rate is still the rate targeted by the Fed for monetary policy, but the participants have evolved for several reasons. The market now includes foreign banks and non-depository financial institutions, such as the Federal Home Loan Banks. These institutions do not need to meet Fed reserve requirements and are not required to keep reserves with the Fed. In addition, the Fed now pays interest to commercial banks for reserves held at the Federal Reserve banks. Paying interest on reserves reduces the incentive for domestic commercial banks to enter the federal funds market since they can already earn interest on their excess reserves.
Daily trading volume in the federal funds market from 2016 through 2020 ranged from a high of $115 billion in March of 2018 to a low of only $34 billion on December 31, 2020.Federal Reserve Bank of New York. “Effective Federal Funds Rate.” The volume of federal funds activity is lower in periods of slower economic growth because banks have fewer good opportunities to issue loans and are less likely to be short of required reserves.
### Bond Markets
Bond markets are financial markets that make payments to investors for a specific period of time. Investors decide how much to pay for a bond depending on how much they expect inflation to affect the value of the fixed payment. There are several types of bonds: government bonds, corporate bonds, and municipal bonds.
### Government Bond Markets
In the section on money markets, we discussed T-bills, and we now discuss longer-term government securities in the form of Treasury notes and Treasury bonds.
We learned in Bonds and Bond Valuation that the federal government issues Treasury notes and bonds to raise money for current spending and to repay past borrowing. The size of the Treasury market is quite large, as the US federal government over the years has accumulated a total indebtedness of over $28 trillion dollars. The debt has grown so large we even have a real-time debt calculator online at https://www.usdebtclock.org/.
Treasury notes (T-notes) are US government debt instruments with maturities of 2, 3, 5, 7, or 10 years. The Treasury auctions notes on a regular basis, and investors may purchase new notes from TreasuryDirect.gov in the same way they would a T-bill. T-notes differ from T-bills in that they are longer term and pay semiannual coupon interest payments, as well as the par or face value of the note at maturity. T-bills, as you will recall, sell at a discount and pay the face value at maturity with no explicit interest payments. Upon issue of a note, the size, number, and timing of note payments is fixed. However, prices do change in the secondary market as interest rates change. Like T-bills, T-notes are generally exempt from state and local taxes.
Economists and investors keep a close eye on the 10-year T-note for several reasons. Mortgage lenders use it as a basis for setting and adjusting mortgage interest rates. In general, the rate on the 10-year T-note is a reliable market indicator of investor confidence.
There is an active secondary market for Treasury notes. From 2001 to 2020, the daily trading volume for Treasury notes has averaged $395 billion, or roughly five times the daily trading volume of T-bills. Treasury notes are the largest single type of government debt instrument, with over $11 trillion outstanding. As you can see from , the Treasury dramatically increased borrowing by issuing notes following the 2008 financial crisis.
Brokers, dealers, and investment companies provide secondary market opportunities for individual and institutional investors. Exchange-traded funds (ETFs) are popular investment vehicles for many types of government T-bill, T-note, and T-bond portfolios. An ETF is a basket of securities that can trade like stocks on a stock exchange. For example, IEI is an iShares ETF managed by BlackRock that invests in Treasury securities with three to seven years to maturity. When investors buy this ETF, they purchase a small bundle of Treasury notes that they can buy or sell, just as if they owned an individual share of stock. ETFs are a convenient way for investors to own broad portfolios of securities while still being able to trade the whole group in a single transaction if they choose.
Longer-term Treasury issues, Treasury bonds, have maturities of 20 or 30 years. T-bonds are like T-notes in that they pay semiannual coupon interest payments for the life of the security and pay the face value at maturity. They are longer term than notes and typically have higher coupon rates. T-bonds with maturities of 20 and 30 years are each auctioned only once per month. At the end of 2020, there were approximately $2.8 trillion of T-bonds outstanding, compared to approximately $11.1 trillion and $5 trillion of T-notes and T-bills.
In 1997, the Treasury began offering a new type of longer-term debt instrument, Treasury Inflation-Protected Securities, or TIPS. TIPS currently have maturities of 5, 10, or 30 years and are auctioned by the Treasury once per month. Like T-notes and T-bonds, they offer semiannual coupon interest payments for the life of the security and pay face value at maturity. The coupon interest rates are fixed, but the principal value adjusts monthly in response to changes in the consumer price index (CPI). Inflation and deflation cause the value of the principal to increase or decrease, which results in a larger or smaller semiannual coupon payment. With a total outstanding value of approximately $1.6 trillion at the end of 2020, TIPS are the smallest form of Treasury borrowing we have discussed.Nick Lioudis. “Where Can I Buy Government Bonds?”
State and local governments and taxing districts can issue debt in the form of municipal bonds (. Local borrowing carries more risk than Treasury securities, and default or bankruptcy is atypical but possible. Thus, munis have ratings that run a spectrum similar to corporate bonds in that they receive a bond rating based on the perceived default risk. The defining feature of municipal bonds is that some interest payments are tax-free. Interest on munis (municipal bonds) is always exempt from federal taxes and sometimes exempt from state and local taxes. This makes them very attractive to investors in high income brackets.
There are two primary types of municipal bonds: revenue bonds and general obligation (GO) bonds. GO bonds generate cash flows to repay the bonds by taxing a project. For instance, a local school district may tax residents to pay for capital construction, or a city may tax citizens to pay for a new public works building. Revenue bonds, on the other hand, may apply to projects that generate sufficient cash flows to repay the bond—perhaps a utility or local toll road.
### Corporate Bond Markets
Just as governments borrow money in the long-term from investors, so do corporations. A corporation often uses bank loans, commercial paper, or supplier credit for short-term borrowing needs and issues bonds for longer-term financing. Bond contracts identify very specific terms of agreement and outline the rules for the order, timing, and amount of contractual payments, as well as processes for when one or more of the required activities lapse. Indenture is the legal term for a bond contract. The indenture also includes limitations on the corporation for how they may use the bond proceeds.
A bond indenture includes both standard boilerplate contract language and specific conditions unique to a particular issue. Because of these non-standardized features of a bond contract, the secondary market for trading used bonds typically requires a broker, dealer, or investment company to facilitate a trade.
When a corporation uses a real asset, such as property or buildings, to guarantee a bond, the firm has issued a mortgage bond. However, it is more common for a corporation to issue an unsecured bond known as a debenture. The risk of a debenture reflects the risk of the entire corporation and does not rely on the value of a specific underlying asset, as is the case with a mortgage bond.
The risk a bondholder bears for buying a bond depends in part on the terms of the bond indenture, market conditions over the life of the bond contract, and the ability or inability of the firm to generate sufficient cash flows to meet its bond obligations. Fortunately, investors do not have to make these determinations about risk on their own. They can rely on bond rating services such as Moody’s, Standard and Poor’s, or Fitch to generate evaluations of the creditworthiness of bond issuers.
Ratings firms must adhere to rigorous standards when evaluating client creditworthiness. For example, Standard and Poor’s begins the explanation of its evaluation process with these paragraphs:
is a summary of how the three major credit rating agencies identify their ratings. Bond ratings are important for many reasons. The higher a firm’s rating, the lower the expected default risk and the lower the cost of borrowing for the firm. Pension funds may be restricted to investing in only medium- or higher-grade bonds. This could limit the number of investors who can participate in the market for lower-grade bonds, thereby reducing the liquidity, price, and tradability of those debt securities.
There are only two US companies with AAA credit ratings: Microsoft and Johnson & Johnson.The Wall Street Insider. “Why Only Two Companies Are Left with the AAA Rating.” Over the past 40 years, there has been a steady decline of AAA-rated companies (from sixty in 1980). Many institutions have found that this rating requires a more conservative approach to debt that can inhibit growth and revenue. So, in today’s market, credit ratings have begun to lose their importance. It seems that the ability to pay debts has become secondary to the potential for growth.
### Equity Markets
An important goal of firm managers is to maximize owners’ wealth. For corporations, shares of stock represent ownership. A corporation could have 100 shares, one million shares, or even several billion shares of stock. Stocks are difficult to value compared to bonds. Bonds typically provide periodic interest payments and a principal payment at maturity. The bond indenture specifies the timing and the amount of payments. Stocks might have periodic dividend payments, and an investor can plan to sell the stock at some point in the future. However, no contract guarantees the size of the dividends or the time and resale price of the stock. Thus, the cash flows from stock ownership are more uncertain and risky.
Corporations are the dominant form of business enterprise in the United States because of the ability to raise capital, the ease of transfer of ownership, and the benefit of limited liability to the owners. There are generally two types of stock, preferred and common. Preferred stock is a hybrid between common stock and bonds. Preferred stock has a higher claim to cash flows than common stockholders have (thus the term preferred), but it is lower than that of bondholders. In addition, preferred stock has fixed cash flows as bonds have and typically has no or few voting rights. Preferred stock dividends are after-tax payments by the corporation, as are common stock dividends, but bond interest payments, paid prior to taxes, are tax-deductible for firms. Of the three, preferred stock is the least used form of capital financing for corporations.
Common stockholders are the residual claimants and owners of the corporation. After all others who have a claim against the firm are paid, the common stockholders own all that remains. Common stockholders have voting rights, typically one vote per share, and choose the board of directors.
One popular way to rank the size of companies is to determine the value of their market capitalization, or market cap. Market cap is equal to the current stock price multiplied by the number of shares outstanding. According to the World Bank, the total market cap of US firms at the end of 2020 was $50.8 trillion, making up over half of the world’s total value of equity, estimated at $90 trillion.Siblis Research. “Total Market Value of US Stock Market.” The largest US company at that time was Apple, followed by Microsoft, Amazon, Alphabet (Google), and Facebook. The largest company by sales volume in 2020 was Walmart.Companies Market Cap. “Largest American Companies by Market Capitalization.”
### Geographical Location of Exchanges
Ownership is easily transferable for stocks that trade in one of the organized stock exchanges or in an over-the-counter (OTC) market. Definitions of a stock exchange and an OTC market blur as financial markets quickly adapt to innovations. However, stock markets have a centralized trading location, transactions require a broker to connect buyers and sellers, and the exchanges guarantee a basic level of liquidity so that investors are always able to buy or sell their stocks. An OTC market is an electronic market conducted on computer screens and consists of direct transactions among buyers and sellers, with no broker to bring the two together. Because there is no formal exchange present, it is possible that investors will have trouble finding buyers or sellers for their stocks.
Most of the trading consists of used or previously issued stocks in over-the-counter markets and organized exchanges. The two largest stock exchanges in the world, as measured by the market capitalization of the companies listed on the exchange, are the New York Stock Exchange (NYSE) and the NASDAQ. Both exchanges are located in the United States. Other large stock exchanges are located in Japan, China, Hong Kong, continental Europe, London, and Saudi Arabia.
### Process of Offering Equities
The primary market is the market for new securities, and the secondary market is the market for used securities. When issuing new equity, the issuing firm receives the proceeds of the sale. Having an active secondary market makes it easier for corporations to issue stock, as investors know they can resell if desired. Most of the trading of equity securities is for used securities on the secondary market.
An initial public offering, or IPO, occurs when a firm offers stock to the public for the first time. With a typical IPO, a private company decides to raise capital and go public with the help of an investment banker. The investment banker agrees to provide financial advice, recommend the price and number of shares to issue, and establish a syndicate of underwriters to finance and ultimately distribute the new shares to investors (see ). An IPO is expensive for the issuing firm, and it can expect to incur costs of 5% to 8% or more of the value of the IPO. As of the end of 2020, the largest successful IPO belongs to Saudi Aramco, a petroleum company, valued at $25.6 billion at issue in December 2019.Jennifer Rudden. “Largest IPOs Worldwide as of January 21, 2021.” The Ant Group had planned an IPO valued at over $34 billion dollars in 2020, but as of the end of 2020 that issue was put on hold by the Chinese government.Deborah D’Souza. “Ant Group Set to Be World’s Largest IPO Ever.”
Institutional and preferred individual investors are typically the initial purchasers of IPOs. Smaller investors rarely have the opportunity to purchase. However, any investor can buy the new shares once available for public trading. Investment author and financial expert Professor Burton Malkiel cautions that buying IPOs immediately after issue can be a money-losing investment. He cites research showing that, historically, IPOs have underperformed the market by an average of 4% per year.Burton Malkiel.
Another way for a corporation to raise capital in the equity market is through a seasoned equity offering (SEO). An IPO occurs when a firm transitions from a private to a public company. An SEO takes place when a corporation that is already publicly traded issues additional shares of stock to the public. An SEO is often part of a SEC Rule 415 offering, or so-called shelf registration. Shelf registrations allow a company to register with the SEC to issue new shares but wait up to two years before issuing the shares. This gives companies the ability to register their intent to issue new shares and to “set them on the shelf” until market conditions are most favorable for issuance to the public.
### Alternative Methods of Raising Capital
Special purpose acquisition companies (SPACs) were born in the 1990s and came of age in 2019.Nicholas Jasinksi. “Blank Check Companies Are Hot on Wall Street. Investors Can’t Ignore Them.” SPACs are a special form of IPO. We know that firms with products or services to sell and a documented operational and financial history often initiate an IPO to raise money by going public. A SPAC, however, is like an IPO that puts the cart before the horse. By this, we mean that rather than having a company ready to go public to raise capital, with a SPAC, a sponsor raises capital in anticipation of finding a firm ready to go public. This is why we sometimes refer to SPACs as “blank check” companies. Investors are providing capital to a firm that has no assets with the expectation that the sponsor will find a good investment.
Forming a SPAC shifts the risk and expenses associated with a firm going public. Because the money raised by the SPAC sponsor is the only asset, the process of filing with the SEC is less complicated, less expensive, and less time-consuming than filing an IPO. Often, when formed, a SPAC has a target company in mind, but this is not a requirement. Once the SPAC identifies a target firm, the sponsor can negotiate a purchase price and essentially merge with the target. Underpricing of IPOs is well documented,R. G. Ibbotson, J. L. Sindelar, and J. R. Ritter. “The Market’s Problems with the Pricing of Initial Public Offerings.” and the owners of a private company going public do not capture the significant increase in stock price that frequently occurs in the months following an IPO.
A SPAC offering allows the private firm owners to negotiate for a better price. A July 2020 study from Renaissance Capital reports that “of 223 SPAC IPOs conducted from the start of 2015 through July 2020, 89 have completed mergers and taken a company public.” According to the study, of those 89 mergers, “the common shares have delivered an average loss of 18.8% and a median return of minus 36.1%. That compares with the average after-market return from traditional IPOs of 37.2%” over the same time period.Ciara Linnane. “2020 Is the Year of the SPAC—Yet Traditional IPOs Offer Better Returns, Report Finds.”
### Sequence of Trade Execution
Investors who wish to trade stocks (buy or sell) execute trades via a broker. Many online brokers today will execute your trades at low to no cost once you have established a brokerage account. When trading, it is most common to make a market order or a limit order. A market order executes a trade at the current price, while a limit order specifies the price at which the investor is willing to buy or sell. provides a visual representation of how payment for an order flow works.
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One way to parse financial markets is by the maturity of financial instruments. With this dichotomy, we explored the money market and the capital market. The money market consists of short-term securities and the capital market of longer-term securities. The capital market discussion focused on debt and equity as financial instruments used to finance longer-term capital financing needs. IPOs or SPACs are vehicles for raising new equity. Most trading on organized exchanges or over-the-counter markets is for used, or secondary, securities.
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### How Private Companies Are Bypassing the IPO Process
### A Secret Meeting and the Birth of the Federal Reserve
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# Historical Performance of US Markets
## Historical Picture of Inflation
### Learning Outcomes
By the end of this section, you will be able to:
1. Define inflation with concrete examples.
2. Describe the practical impact of inflation on consumption and salary.
3. Explain how expected and actual inflation are measured.
4. Detail the behavior of inflation over various historical periods.
Over time, returns in the stock market have easily outperformed returns in the bond market. However, not everyone is comfortable investing in stocks. Taking a more informed and opposing view, financial economist Jeremy Siegel queries, “You have never lost money in stocks over any 20-year period, but you have wiped out half your portfolio in bonds (after inflation). So which is the riskier asset?”Jeremy J. Siegel. Siegel brings up a valid point about inflation. Returns adjusted for changes in prices provide a better measure of value or wealth. Baseball pundit Sam Ewing noted, “Inflation is when you pay fifteen dollars for the ten-dollar haircut you used to get for five dollars when you had hair.”Sam Ewing. “Sam Ewing Quotes.” If what you have earned on your investments fails to keep up with changing prices, you may have a larger portfolio with less purchasing power.
### Expected versus Actual Inflation
Inflation occurs when things cost more and your money buys less today than it did yesterday. It is understandable to dislike inflation and to be concerned over the prospect of rising prices. In fact, two of the primary policy objectives of the Federal Reserve are to work for price stability and moderate interest rates. However, is inflation necessarily good or bad? In practice, inflation can benefit some people and harm others at the same time. Consider the impact of differences in expected and actual realized inflation. Empirical evidence suggests that, on average, economists do a good job of developing inflation rate forecasts that match the actual rate of inflation. Estimated inflation is built into the interest rates investors require or are willing to pay for financial products, such as fixed-rate loans or bonds. When the actual rate of inflation exceeds the estimated rate on a product, such as a mortgage loan, this means borrowers are repaying the loan with less-valuable dollars and benefit from the increased inflation rate. Lenders, however, receive those inflation-impacted dollars and are harmed due to their unexpected decrease in purchasing power.
Deflation, or falling prices, is associated with economic recessions or even depressions and is thought to be an even more serious problem than inflation. Generally, policy makers tend to support moderate inflation, being careful to stay away from zero or negative price changes.
### Inflation Impacts
Ultimately, inflation redistributes wealth. Lenders providing fixed-rate loans receive less-valuable dollars in return. Borrowers repay with those same less-valuable dollars. Workers receive less-valuable dollars, especially when their wage increases lag behind changes in prices. Modest inflation can benefit a consumer-driven economy like the United States if consumers are motivated to spend money before prices increase. However, too much inflation can cause frenzied buying, drive prices even higher, and outpace the rate of wage increases. Higher inflation raises the rates on new borrowing instruments and can slow the rate of business investment and economic growth. Inflation raises overall prices and may cause hardship for consumers on a fixed income.
Finally, inflation does not have an equal impact on all goods and services. As shows, consumer prices did not increase at the same rate for the selected items shown. From 1980 to 2020, inflation, as measured by the consumer price index (CPI), grew at an average annual rate of 2.90%. However, the price of college tuition and fees increased at more than double that rate. Rent, another large expense for most college students, increased at an annual rate of 3.67%, also well above the average increase in the CPI. The price increases for ground beef and butter were slightly less than the CPI average. For the selected items presented here, only the average price for used cars and trucks rose at a rate significantly lower than the average rate of inflation.
### Using Graphs and Charts to Plot Inflation Behavior
The CPI is a measure of how prices have changed for a basket of goods across the United States. shows the eight major categories of expenditures included in the CPI.US Bureau of Labor Statistics. “Consumer Price Index.” Different regions of the country and population subgroups may experience different rates of inflation. Retired people may spend a greater portion of their income on healthcare than someone in their twenties.Chris Farrell. “The Truth About Health Care Costs in Retirement.” Residents on the East and West Coasts spend a higher percentage of their earnings on housing than those living in the Midwest.US Department of Housing and Urban Development, Office of Policy Development and Research. “Rental Burdens: Rethinking Affordability Measures.” PD&R Edge. September 22, 2014. https://www.huduser.gov/portal/pdredge/pdr_edge_featd_article_092214.html As such, there are several different measures of inflation based on geographic region or other factors.Khan Academy. “How Changes in the Cost of Living Are Measured.” It is worth noting that a basket of goods and services that cost $100 in 1984 would cost approximately $260 by the end of 2020, an increase of 160% (see )! Over that same period, the average annual rate of inflation as measured by the CPI was 2.55%. These numbers make it easy to see that even a relatively modest annual inflation rate, lower than the long-run annual US inflation rate of 3%, results in significant price increases.
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The Federal Reserve considers moderate inflation rates optimal in their oversight of the US economy. We measure inflation by comparing the price of a bundle or basket of goods over time and documenting how prices change. Since not everyone consumes similar baskets of goods, we calculate several different measures of inflation. The most commonly quoted measure of inflation uses changes in the Consumer Price Index (CPI).
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# Historical Performance of US Markets
## Historical Picture of Returns to Bonds
### Learning Outcomes
By the end of this section, you will be able to:
1. Detail the behavior of sovereign (government) bonds over various historical periods.
2. Detail the behavior of corporate bonds over various historical periods.
3. Extract sovereign and corporate bond investment results from plots and charts.
### Federal Government Bond Behavior, Discussion, Charts, and Graphs
United States Treasury (sovereign) bonds are among the safest investments available. In fact, T-bonds often serve as the proxy for a risk-free investment in financial modeling. However, even though T-bonds may be essentially default-risk free, they do change value as interest rates change. Look at , showing annual T-bond returns from 1980 to 1999. None of these bonds defaulted, yet investors realized very different returns on their investments from year to year. The total returns reflect interest payments plus the change in price because of changes in interest rates.
The late 1970s experienced high rates of inflation and interest rates. As those rates began to fall in late 1981, bond prices rose. Investors holding T-bond portfolios realized very large returns on their risk-free investments in 1982. That year provided the highest annual return in the 20-year period, with an annual return of 32.81%. The lowest annual return was in 1999, as the Federal Reserve began to raise rates to temper an overheating stock market caused by dot-com speculation gaining momentum. When the Fed raised interest rates, the prices on existing bonds fell, and investors realized a -8.25% return on their bond portfolios. Thus, the range in returns on these “low risk” investment securities was over 41% from the highest to the lowest annual return in this particular two-decade span. Overall, the average annual return on T-bonds from 1980 to 1999 was a robust 10.21%, boosted in part by the above average annual inflation rate of 4.28%.
Inflation slowed from 2000 to 2020, and the average annual rate of return on T-bonds fell accordingly (see ). With reduced variability of interest rates in the new century and interest rates in general being lower, the returns on bond portfolios were also lower on average. T-bonds in the first two decades of the twenty-first century averaged an annual return of 5.77%, very close to the long-run average return in the previous century. The range of returns was also smaller than the previous two decades, with the highest annual return topping out at 20.10% in 2008 and the lower end dipping down to -11.12% in 2009 as the Fed made a significant effort to reduce interest rates in an attempt to stimulate the economy following the Great Recession.
### Corporate Bond Behavior, Discussion, Charts, and Graphs
The performance of Baa bonds is very similar to T-Bonds over the four decades spanning 1980 to 2020 (see ). These bonds are not default-risk free and require a risk premium for investors. The average annual return of 12.07% from 1980 to 1999 topped T-bonds by 1.86%. The margin was greater in the period from 2000 to 2020 (see ), with Baa bonds (mid-tier corporate bonds) realizing an average annual premium of 2.30% over the default-risk free T-bonds.
These premiums translate to a substantial increase in investment performance. For example, had an investor placed $100 into a T-bond portfolio in 1980, the value of the investment would have been $1,931 by year-end 2020. This would have easily outpaced the rate of inflation but significantly lagged the ending value of $4,506 on a similar Baa bond portfolio investment (see ).
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Historical bond yields are published going back hundreds of years but are only reliably available for the last 100 years or so. In large part, the returns realized on portfolios of bonds have been smaller and less variable than the returns realized for equities.
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# Historical Performance of US Markets
## Historical Picture of Returns to Stocks
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain overall equity market behavior over various historical periods.
2. Explain different equity style and size behavior over various historical periods.
3. Extract various equity market performance results from plots and charts.
### Using Graphs and Charts to Plot Equity Market Behavior Stock Size Considerations
The Dow Jones Industrial Average (DJIA), also known as the Dow 30) and the S&P 500 Index are the most frequently quoted stock market indices among scholars, businesses, and the public in general. Both indices track the change in value of a group of large capitalization stocks. The changes in the two indices are highly correlated.
It may be fair to question if either index is a good representation of the value of equity and the changes in value in the market because there are over 6,000 publicly traded companies listed on organized exchanges and thousands of additional companies that trade only over the counter. As of year-end 2020, the S&P 500 firms had a combined market capitalization of $33.4 trillion, about 66% of the estimated US equity market capitalization of $50.8 trillion.Spencer Israel. “The Number of Companies Publicly Traded in the US Is Shrinking—Or Is It?” It is widely agreed that the performance of the S&P 500 is a good representation of the broader market and more specifically of large capitalization firms.
provides a visualization of how S&P 500 stock returns have stacked up since 1900. This figure makes it clear that equity returns roughly follow a bell curve, or normal distribution. Thus, we are able to measure risk with standard deviation. A lower standard deviation of returns suggests less uncertainty of returns and therefore less risk.
Capital market history demonstrates that the average return to stocks has significantly outperformed other financial security classes, such as government bonds, corporate bonds, or the money market. provides the return and standard deviation of several US investment classes over the 40-year period 1981–2020. As you can see, stocks outperformed bonds, bills, and inflation. This has led many investment advisers to emphasize asset allocation first and individual security selection second. The intuition is that the decision to invest in stocks rather than bonds has a greater long-run payoff than the change in performance resulting from the selection of any individual or group of stocks.
demonstrates the growth of a $100 investment at the start of 1928. Note that the value of the large company portfolio is more than 50 times greater than the equal investment in long-term US government bonds. This supports the importance of thoughtful asset allocation.
Still, the size of a firm has a significant impact on how investors choose equity securities. Capital market history also shows that a portfolio of small company stocks has realized larger average annual returns, as well as greater variability, than a portfolio of large companies as represented by the S&P 500. Small-cap stock total returns ranged from a high of 142.9% in 1933 to a low of -58.0% in 1937.
More recently, the differential return between small and large capital stocks has not been as pronounced. From 1980 through 2020, the Wilshire US Small-Cap Index has averaged an annual compound return of 12.13% compared to the Wilshire US Large Cap Index average of 11.82% over the same period. The 31-basis point premium is much smaller than that realized in the 1926–2019 period, which saw a small-cap average annual compounded return of 11.90% versus 10.14% for the large-cap portfolio.
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Stocks have produced the greatest average annual rates of return of the money and capital market assets discussed in this chapter. Stockholders bear more risk than bondholders or money market investors and receive on average higher average annual returns. Despite the relatively high average annual rate of return for portfolios of stock, history shows that the equity markets earn negative annual returns about 25% of the time. The negative returns realized by equities occur far more often than the negative results realized by money market or debt market instruments.
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# Statistical Analysis in Finance
## Why It Matters
Statistical analysis is used extensively in finance, with applications ranging from consumer concerns such as credit scores, retirement planning, and insurance to business concerns such as assessing stock market volatility and predicting inflation rates. As a consumer, you will make many financial decisions throughout your life, and many of these decisions will be guided by statistical analysis. For example, what is the probability that interest rates will rise over the next year, and how will that affect your decision on whether to refinance a mortgage? In your retirement planning, how should the investment mix be allocated among stocks and bonds to minimize volatility and ensure a high probability for a secure retirement? When running a business, how can statistical quality control methods be used to maintain high quality levels and minimize waste? Should a business make use of consumer focus groups or customer surveys to obtain business intelligence data to improve service levels? These questions and more can benefit from the use and application of statistical methods.
Running a business and tracking its finances is a complex process. From day-to-day activities such as managing inventory levels to longer-range activities such as developing new products or expanding a customer base, statistical methods are a key to business success. For finance considerations, a business must manage risk versus return and optimize investments to ensure shareholder value. Business managers employ a wide range of statistical processes and tools to accomplish these goals. Increasingly, companies are also interested in data analytics to optimize the value gleaned from business- and consumer-related data, and statistical analysis forms the core of such analytics. |
# Statistical Analysis in Finance
## Measures of Center
By the end of this section, you will be able to:
1. Calculate various measures of the average of a data set, such as mean, median, mode, and geometric mean.
2. Recognize when a certain measure of center is more appropriate to use, such as weighted mean.
3. Distinguish among arithmetic mean, geometric mean, and weighted mean.
### Arithmetic Mean
The average of a data set is a way of describing location. The most widely used measures of the center of a data set are the mean (average), median, and mode. The arithmetic mean is the most common measure of the average. We will discuss the geometric mean later.
Note that the words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term is arithmetic mean, and average technically refers only to a center location. Formally, the arithmetic mean is called the first moment of the distribution by mathematicians. However, in practice among non-statisticians, average is commonly accepted as a synonym for arithmetic mean.
To calculate the arithmetic mean value of 50 stock portfolios, add the 50 portfolio dollar values together and divide the sum by 50. To calculate the arithmetic mean for a set of numbers, add the numbers together and then divide by the number of data values.
In statistical analysis, you will encounter two types of data sets: sample data and population data. Population data represents all the outcomes or measurements that are of interest. Sample data represents outcomes or measurements collected from a subset, or part, of the population of interest.
The notation is used to indicate the sample mean, where the arithmetic mean is calculated based on data taken from a sample. The notation is used to denote the sum of the data values, and is used to indicate the number of data values in the sample, also known as the sample size.
The sample mean can be calculated using the following formula:
Finance professionals often rely on averages of Treasury bill auction amounts to determine their value. lists the Treasury bill auction amounts for a sample of auctions from December 2020.
To calculate the arithmetic mean of the amount paid for Treasury bills at auction, in billions of dollars, we use the following formula:
### Median
To determine the median of a data set, order the data from smallest to largest, and then find the middle value in the ordered data set. For example, to find the median value of 50 portfolios, find the number that splits the data into two equal parts. The portfolio values owned by 25 people will be below the median, and 25 people will have portfolio values above the median. The median is generally a better measure of the average when there are extreme values or outliers in the data set.
An outlier or extreme value is a data value that is significantly different from the other data values in a data set. The median is preferred when outliers are present because the median is not affected by the numerical values of the outliers.
The ordered data set from appears as follows:
The middle value in this ordered data set is the third data value, which is 39.7. Thus, the median is $39.7 billion.
You can quickly find the location of the median by using the expression . The variable n represents the total number of data values in the sample. If n is an odd number, the median is the middle value of the data values when ordered from smallest to largest. If n is an even number, the median is equal to the two middle values of the ordered data values added together and divided by 2. In the example from , there are five data values, so n = 5. To identify the position of the median, calculate , which is , or 3. This indicates that the median is located in the third data position, which corresponds to the value 39.7.
As mentioned earlier, when outliers are present in a data set, the mean can be nonrepresentative of the center of the data set, and the median will provide a better measure of center. The following Think It Through example illustrates this point.
### Mode
Another measure of center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal. For example, assume that the weekly closing stock price for a technology stock, in dollars, is recorded for 20 consecutive weeks as follows:
To find the mode, determine the most frequent score, which is 72. It occurs five times. Thus, the mode of this data set is 72. It is helpful to know that the most common closing price of this particular stock over the past 20 weeks has been $72.00.
### Geometric Mean
The arithmetic mean, median, and mode are all measures of the center of a data set, or the average. They are all, in their own way, trying to measure the common point within the data—that which is “normal.” In the case of the arithmetic mean, this is accomplished by finding the value from which all points are equal linear distances. We can imagine that all the data values are combined through addition and then distributed back to each data point in equal amounts.
The geometric mean redistributes not the sum of the values but their product. It is calculated by multiplying all the individual values and then redistributing them in equal portions such that the total product remains the same. This can be seen from the formula for the geometric mean, x̃ (pronounced x-tilde):
The geometric mean is relevant in economics and finance for dealing with growth—of markets, in investments, and so on. For an example of a finance application, assume we would like to know the equivalent percentage growth rate over a five-year period, given the yearly growth rates for the investment.
For a five-year period, the annual rate of return for a certificate of deposit (CD) investment is as follows: 3.21%, 2.79%, 1.88%, 1.42%, 1.17%. Find the single percentage growth rate that is equivalent to these five annual consecutive rates of return. The geometric mean of these five rates of return will provide the solution. To calculate the geometric mean for these values (which must all be positive), first multiplyIn this chapter, the interpunct dot will be used to indicate the multiplication operation in formulas. the rates of return together—after adding 1 to the decimal equivalent of each interest rate—and then take the nth root of the product. We are interested in calculating the equivalent overall rate of return for the yearly rates of return, which can be expressed as 1.0321, 1.0279, 1.0188, 1.0142, and 1.0117:
Based on the geometric mean, the equivalent annual rate of return for this time period is 2.09%.
### Weighted Mean
A weighted mean is a measure of the center, or average, of a data set where each data value is assigned a corresponding weight. A common financial application of a weighted mean is in determining the average price per share for a certain stock when the stock has been purchased at different points in time and at different share prices.
To calculate a weighted mean, create a table with the data values in one column and the weights in a second column. Then create a third column in which each data value is multiplied by each weight on a row-by-row basis. Then, the weighted mean is calculated as the sum of the results from the third column divided by the sum of the weights.
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Several measurements are used to provide the average of a data set, including mean, median, and mode. The terms mean and average are often used interchangeably. To calculate the mean for a set of numbers, add the numbers together and then divide the sum by the number of data values. The geometric mean redistributes not the sum of the values but the product by multiplying all of the individual values and then redistributing them in equal portions such that the total product remains the same. To calculate the median for a set of numbers, order the data from smallest to largest and identify the middle data value in the ordered data set.
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# Statistical Analysis in Finance
## Measures of Spread
By the end of this section, you will be able to:
1. Define and calculate standard deviation for a data set.
2. Define and calculate variance for a data set.
3. Explain the relationship between standard deviation and variance.
### Standard Deviation
An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated close to the mean; in other data sets, the data values are more widely spread out. For example, an investor might examine the yearly returns for Stock A, which are 1%, 2%, -1%, 0%, and 3%, and compare them to the yearly returns for Stock B, which are -9%, 2%, 15%, -5%, and 0%.
Notice that Stock B exhibits more volatility in yearly returns than Stock A. The investor may want to quantify this variation in order to make the best investment decisions for a particular investment objective.
The most common measure of variation, or spread, is standard deviation. The standard deviation of a data set is a measure of how far the data values are from their mean. A standard deviation
1. provides a numerical measure of the overall amount of variation in a data set; and
2. can be used to determine whether a particular data value is close to or far from the mean.
The standard deviation provides a measure of the overall variation in a data set. The standard deviation is always positive or zero. It is small when the data values are all concentrated close to the mean, exhibiting little variation or spread. It is larger when the data values are more spread out from the mean, exhibiting more variation.
Suppose that we are studying the variability of two different stocks, Stock A and Stock B. The average stock price for both stocks is $5. For Stock A, the standard deviation of the stock price is 2, whereas the standard deviation for Stock B is 4. Because Stock B has a higher standard deviation, we know that there is more variation in the stock price for Stock B than in the price for Stock A.
There are two different formulas for calculating standard deviation. Which formula to use depends on whether the data represents a sample or a population. The notation s is used to represent the sample standard deviation, and the notation is used to represent the population standard deviation. In the formulas shown below, x̄ is the sample mean, is the population mean, n is the sample size, and N is the population size.
Formula for the sample standard deviation:
Formula for the population standard deviation:
### Variance
Variance also provides a measure of the spread of data values. The variance of a data set measures the extent to which each data value differs from the mean. The more the individual data values differ from the mean, the larger the variance. Both the standard deviation and the variance provide similar information.
In a finance application, variance can be used to determine the volatility of an investment and therefore to help guide financial decisions. For example, a more cautious investor might opt for investments with low volatility.
Similar to standard deviation, the formula used to calculate variance also depends on whether the data is collected from a sample or a population. The notation is used to represent the sample variance, and the notation σ2 is used to represent the population variance.
Formula for the sample variance:
Formula for the population variance:
This is the method to calculate standard deviation and variance for a sample:
1. First, find the mean of the data set by adding the data values and dividing the sum by the number of data values.
2. Set up a table with three columns, and in the first column, list the data values in the data set.
3. For each row, subtract the mean from the data value , and enter the difference in the second column. Note that the values in this column may be positive or negative. The sum of the values in this column will be zero.
4. In the third column, for each row, square the value in the second column. So this third column will contain the quantity (Data Value – Mean)2 for each row. We can write this quantity as . Note that the values in this third column will always be positive because they represent a squared quantity.
5. Add up all the values in the third column. This sum can be written as .
6. Divide this sum by the quantity (n – 1), where n is the number of data points. We can write this as .
7. This result is called the sample variance, denoted by s2. Thus, the formula for the sample variance is .
8. Now take the square root of the sample variance. This value is the sample standard deviation, called s. Thus, the formula for the sample standard deviation is .
9. Round-off rule: The sample variance and sample standard deviation are typically rounded to one more decimal place than the data values themselves.
As the above example illustrates, calculating the variance and standard deviation is a tedious process. A financial calculator can calculate statistical measurements such as mean and standard deviation quickly and efficiently.
There are two steps needed to perform statistical calculations on the calculator:
1. Enter the data in the calculator using the [DATA] function, which is located above the 7 key.
2. Access the statistical results provided by the calculator using the [STAT] function, which is located above the 8 key.
Follow the steps in to calculate mean and standard deviation using the financial calculator. The ages data set from the Think It Through example above is used again here: 40, 36, 44, 51, 54, 55, 39, 47, 44, 50.
From the statistical results, the mean is shown as 46, and the sample standard deviation is shown as 6.50.
Excel provides a similar analysis using the built-in functions =AVERAGE (for the mean) and =STDEV.S (for the sample standard deviation). To calculate these statistical results in Excel, enter the data values in a column. Let’s assume the data values are placed in cells A2 through A11. In any cell, type the Excel command =AVERAGE(A2:A11) and press enter. Excel will calculate the arithmetic mean in this cell. Then, in any other cell, type the Excel command =STDEV.S(A2:A11) and press enter. Excel will calculate the sample standard deviation in this cell. shows the mean and standard deviation for the 10 ages.
### Relationship between Standard Deviation and Variance
In the formulas shown above for variance and standard deviation, notice that the variance is the square of the standard deviation, and the standard deviation is the square root of the variance.
Once you have calculated one of these values, you can directly calculate the other value. For example, if you know the standard deviation of a data set is 12.5, you can calculate the variance by squaring this standard deviation. The variance is then 12.52, which is 156.25.
In the same way, if you know the value of the variance, you can determine the standard deviation by calculating the square root of the variance. For example, if the variance of a data set is known to be 31.36, then the standard deviation can be calculated as the square root of 31.36, which is 5.6.
One disadvantage of using the variance is that the variance is measured in square units, which are different from the units in the data set. For example, if the data set consists of ages measured in years, then the variance would be measured in years squared, which can be confusing. The standard deviation is measured in the same units as the original data set, and thus the standard deviation is used more commonly than the variance to measure the spread of a data set.
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The standard deviation and variance are measures of the spread of a data set. The standard deviation is small when the data values are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation. The formula used to calculate the standard deviation depends on whether the data represents a sample or a population, as the formulas for the sample standard deviation and the population standard deviation are slightly different.
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# Statistical Analysis in Finance
## Measures of Position
By the end of this section, you will be able to:
1. Define and calculate z-scores for a measurement.
2. Define and calculate quartiles and percentiles for a data set.
3. Use quartiles as a method to detect outliers in a data set.
### z-Scores
A , also called a , is a measure of the position of an entry in a data set. It represents the number of standard deviations by which a data value differs from the mean. For example, suppose that in a certain year, the rates of return for various technology-focused mutual funds are examined, and the mean return is 7.8% with a standard deviation of 2.3%. A certain mutual fund publishes its rate of return as 12.4%. Based on this rate of return of 12.4%, we can calculate the relative standing of this mutual fund compared to the other technology-focused mutual funds. The corresponding z-score of a measurement considers the given measurement in relation to the mean and standard deviation for the entire population.
The formula for a z-score calculation is as follows:
where x is the measurement, is the mean, and is the standard deviation.
### Quartiles and Percentiles
If a person takes an IQ test, their resulting score might be reported as in the 87th percentile. This percentile indicates the person’s relative performance compared to others taking the IQ test. A person scoring in the 87th percentile has an IQ score higher than 87% of all others taking the test. This is the same as saying that the person is in the top 13% of all people taking the IQ test.
Common measures of location are quartiles and percentiles. Quartiles are special percentiles. The first quartile, Q1, is the same as the 25th percentile, and the third quartile, Q3, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile.
To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. If you score in the 90th percentile of an exam, that does not necessarily mean that you receive 90% on the test. It means that 90% of the test scores are the same as or less than your score and the remaining 10% of the scores are the same as or greater than your score.
Percentiles are useful for comparing values. In a finance example, a mutual fund might report that the performance for the fund over the past year was in the 80th percentile of all mutual funds in the peer group. This indicates that the fund performed better than 80% of all other funds in the peer group. This also indicates that 20% of the funds performed better than this particular fund.
Quartiles are values that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second quartile. The first quartile, Q1, is the middle value, or median, of the lower half of the data, and the third quartile, Q3, is the middle value of the upper half of the data. As an example, consider the following ordered data set, which represents the rates of return for a group of technology-based mutual funds in a certain year:
The median, or second quartile, is the middle value in this data set, which is 7.2. Notice that 50% of the data values are below the median, and 50% of the data values are above the median. The lower half of the data values are 5.4, 6.0, 6.3, 6.8, 7.1 Notice that these are the data values below the median. The upper half of the data values are 7.4, 7.5, 7.9, 8.2, 8.7, which are the data values above the median.)
To find the first quartile, Q1, locate the middle value of the lower half of the data. The middle value of the lower half of the data set is 6.3. Notice that one-fourth, or 25%, of the data values are below this first quartile, and 75% of the data values are above this first quartile.
To find the third quartile, Q3, locate the middle value of the upper half of the data. The middle value of the upper half of the data set is 7.9. Notice that one-fourth, or 25%, of the data values are above this third quartile, and 75% of the data values are below this third quartile.
The interquartile range (IQR) is a number that indicates the spread of the middle half, or the middle 50%, of the data. It is the difference between the third quartile, Q3, and the first quartile, Q1.
In the above example, the IQR can be calculated as
### Outlier Detection
Quartiles and the IQR can be used to flag possible outliers in a data set. For example, if most employees at a company earn about $50,000 and the CEO of the company earns $2.5 million, then we consider the CEO’s salary to be an outlier data value because is significantly different from all the other salaries in the data set. An outlier data value can also be a value much lower than the other data values, so if one employee only makes $15,000, then this employee’s low salary might also be considered an outlier.
To detect outliers, use the quartiles and the IQR to calculate a lower and an upper bound for outliers. Then any data values below the lower bound or above the upper bound will be flagged as outliers. These data values should be further investigated to determine the nature of the outlier condition.
To calculate the lower and upper bounds for outliers, use the following formulas:
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Several measures are used to indicate the position of a data value in a data set. One measure of position is the z-score for a particular measurement. The z-score indicates how many standard deviations a particular measurement is above or below the mean. Other measures of position include quartiles and percentiles. Quartiles are special percentiles. The first quartile, Q1, is the same as the 25th percentile, and the third quartile, Q3, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile. To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths.
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# Statistical Analysis in Finance
## Statistical Distributions
By the end of this section, you will be able to:
1. Construct and interpret a frequency distribution.
2. Apply and evaluate probabilities using the normal distribution.
3. Apply and evaluate probabilities using the exponential distribution.
### Frequency Distributions
A frequency distribution provides a method to organize and summarize a data set. For example, we might be interested in the spread, center, and shape of the data set’s distribution. When a data set has many data values, it can be difficult to see patterns and come to conclusions about important characteristics of the data. A frequency distribution allows us to organize and tabulate the data in a summarized way and also to create graphs to help facilitate an interpretation of the data set.
To create a basic frequency distribution, set up a table with three columns. The first column will show the intervals for the data, and the second column will show the frequency of the data values, or the count of how many data values fall within each interval. A third column can be added to include the relative frequency for each row, which is calculated by taking the frequency for that row and dividing it by the sum of all the frequencies in the table.
### Normal Distribution
The normal probability density function, a continuous distribution, is the most important of all the distributions. The normal distribution is applicable when the frequency of data values decreases with each class above and below the mean. The normal distribution can be applied to many examples from the finance industry, including average returns for mutual funds over a certain time period, portfolio values, and others. The normal distribution has two parameters, or numerical descriptive measures: the mean, , and the standard deviation, . The variable x represents the quantity being measured whose data values have a normal distribution.
The curve in is symmetric about a vertical line drawn through the mean, . The mean is the same as the median, which is the same as the mode, because the graph is symmetric about . As the notation indicates, the normal distribution depends only on the mean and the standard deviation. Because the area under the curve must equal 1, a change in the standard deviation, , causes a change in the shape of the normal curve; the curve becomes fatter and wider or skinnier and taller depending on . A change in causes the graph to shift to the left or right. This means there are an infinite number of normal probability distributions.
To determine probabilities associated with the normal distribution, we find specific areas under the normal curve, and this is further discussed in Apply the Normal Distribution in Financial Contexts. For example, suppose that at a financial consulting company, the mean employee salary is $60,000 with a standard deviation of $7,500. A normal curve can be drawn to represent this scenario, in which the mean of $60,000 would be plotted on the horizontal axis, corresponding to the peak of the curve. Then, to find the probability that an employee earns more than $75,000, you would calculate the area under the normal curve to the right of the data value $75,000.
Excel uses the following command to find the area under the normal curve to the left of a specified value:
For example, at the financial consulting company mentioned above, the mean employee salary is $60,000 with a standard deviation of $7,500. To find the probability that a random employee’s salary is less than $55,000 using Excel, this is the command you would use:
Thus, there is a probability of about 25% that a random employee has a salary less than $55,000.
### Exponential Distribution
The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, a finance professional might want to model the time to default on payments for company debt holders.
An exponential distribution is one in which there are fewer large values and more small values. For example, marketing studies have shown that the amount of money customers spend in a store follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The random variable for the exponential distribution is continuous and often measures a passage of time, although it can be used in other applications. Typical questions may be, What is the probability that some event will occur between x1 hours and x2 hours? or What is the probability that the event will take more than x1 hours to perform? In these examples, the random variable x equals either the time between events or the passage of time to complete an action (e.g., wait on a customer). The probability density function is given by
where is the historical average of the values of the random variable (e.g., the historical average waiting time). This probability density function has a mean and standard deviation of .
To determine probabilities associated with the exponential distribution, we find specific areas under the exponential distribution curve. The following formula can be used to calculate the area under the exponential curve to the left of a certain value:
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A frequency distribution provides a method of organizing and summarizing a data set and allows us to organize and tabulate the data in a summarized way. Once a frequency distribution is generated, it can be used to create graphs to help facilitate an interpretation of the data set. The normal distribution has two parameters, or numerical descriptive measures: the mean, , and the standard deviation, . The exponential distribution is often concerned with the amount of time until some specific event occurs.
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### Normal Distribution Stock Return Calculations
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# Statistical Analysis in Finance
## Probability Distributions
By the end of this section, you will be able to:
1. Calculate portfolio weights in an investment.
2. Calculate and interpret the expected values.
3. Apply the normal distribution to characterize average and standard deviation in financial contexts.
### Calculate Portfolio Weights
In many financial analyses, the weightings by asset category in a portfolio are a key index used to assess if the portfolio is meeting allocation metrics. For example, an investor approaching retirement age may wish to shift assets in a portfolio to more conservative and lower-volatility investments. Weightings can be calculated in several different ways—for example, based on individual stocks in a portfolio or on various sectors in a portfolio. Weightings can also be calculated based on number of shares or the value of shares of a stock.
To calculate a weighting in a portfolio based on value, take the value of the particular investment and divide it by the total value of the overall portfolio. As an example, consider an individual’s retirement account for which the desired portfolio weighting is determined to be 40% stocks, 50% bonds, and 10% cash equivalents. shows the current assets in the individual’s portfolio, broken out according to stocks, bonds, and cash equivalents.
To determine the weighting in this portfolio for stocks, bonds, and cash, take the total value for each category and divide it by the total value of the entire portfolio. These results are summarized in . Notice that the portfolio weightings shown in the table do not match the target, or desired, allocation weightings of 40% stocks, 50% bonds, and 10% cash equivalents.
Portfolio rebalancing is a process whereby the investor buys or sells assets to achieve the desired portfolio weightings. In this example, the investor could sell approximately 10% of the stock assets and purchase bonds with the proceeds to align the asset categories to the desired portfolio weightings.
### Calculate and Interpret Expected Values
A probability distribution is a mathematical function that assigns probabilities to various outcomes. For example, we can assign a probability to the outcome of a certain stock increasing in value or decreasing in value. One application of a probability distribution function is determining expected value.
In many financial situations, we are interested in determining the expected value of the return on a particular investment or the expected return on a portfolio of multiple investments. To calculate expected returns, we formulate a probability distribution and then use the following formula to calculate expected value:
where P1, P2, P3, ⋯ P are the probabilities of the various returns and R1, R2, R3, ⋯ R are the various rates of return.
In essence, expected value is a weighted mean where the probabilities form the weights. Typically, these values for P and R are derived from historical data. As an example, consider a probability distribution for potential returns for United Airlines common stock. Assume that from historical data gathered over a certain time period, there is a 15% probability of generating a 12% return on investment for this stock, a 35% probability of generating a 5% return, a 25% probability of generating a 2% return, a 14% probability of generating a 5% loss, and an 11% probability of resulting in a 10% loss. This data can be organized into a probability distribution table as seen in .
Using the expected value formula, the expected return of United Airlines stock over an extended period of time
follows:
Based on the probability distribution, the expected value of the rate of return for United Airlines common stock over an extended period of time is 2.25%.
We can extend this analysis to evaluate the expected return for an investment portfolio consisting of various asset categories, such as stocks, bonds, and cash equivalents, where the probabilities are associated with the weighting of each category relative to the total value of the portfolio. Using historical return data for each of the asset categories, the expected return of the overall portfolio can be calculated using the expected value formula.
Assume an investor has assets in stocks, bonds, and cash equivalents as shown in .
Based on the probability distribution, the expected value of the rate of return for this portfolio over an extended period of time is 8.44%.
### Apply the Normal Distribution in Financial Contexts
The normal, or bell-shaped, distribution can be utilized in many applications, including financial contexts. Remember that the normal distribution has two parameters: the mean, which is the center of the distribution, and the standard deviation, which measures the spread of the distribution. Here are several examples of applications of the normal distribution:
1. IQ scores follow a normal distribution, with a mean IQ score of 100 and a standard deviation of 15.
2. Salaries at a certain company follow a normal distribution, with a mean salary of $52,000 and a standard deviation of $4,800.
3. Grade point averages (GPAs) at a certain college follow a normal distribution, with a mean GPA of 3.27 and a standard deviation of 0.24.
4. The average annual gain of the Dow Jones Industrial Average (DJIA) over a 40-year time period follows a normal distribution, with a mean gain of 485 points and a standard deviation of 1,065 points.
5. The average annual return on the S&P 500 over a 50-year time period follows a normal distribution, with a mean rate of return of 10.5% and a standard deviation of 14.3%.
6. The average annual return on mid-cap stock funds over the five-year period from 2010 to 2015 follows a normal distribution, with a mean rate of return of 8.9% and a standard deviation of 3.7%.
When analyzing data sets that follow a normal distribution, probabilities can be calculated by finding areas under the normal curve. To find the probability that a measurement is within a specific interval, we can compute the area under the normal curve corresponding to the interval of interest.
Areas under the normal curve are available in tables, and Excel also provides a method to find these areas. The empirical rule is one method for determining areas under the normal curve that fall within a certain number of standard deviations of the mean (see ).
If x is a random variable and has a normal distribution with mean µ and standard deviation , then the empirical rule states the following:
1. About 68% of the x-values lie between and units from the mean (within one standard deviation of the mean).
2. About 95% of the x-values lie between and units from the mean (within two standard deviations of the mean).
3. About 99.7% of the x-values lie between and units from the mean (within three standard deviations of the mean). Notice that almost all the x-values lie within three standard deviations of the mean.
4. The z-scores for and are and , respectively.
5. The z-scores for and are and , respectively.
6. The z-scores for and are and , respectively.
As an example of using the empirical rule, suppose we know that the average annual return for mid-cap stock funds over the five-year period from 2010 to 2015 follows a normal distribution, with a mean rate of return of 8.9% and a standard deviation of 3.7%. We are interested in knowing the likelihood that a randomly selected mid-cap stock fund provides a rate of return that falls within one standard deviation of the mean, which implies a rate of return between 5.2% and 12.6%. Using the empirical rule, the area under the normal curve within one standard deviation of the mean is 68%. Thus, there is a probability, or likelihood, of 0.68 that a mid-cap stock fund will provide a rate of return between 5.2% and 12.6%.
If the interval of interest is extended to two standard deviations from the mean (a rate of return between 1.5% and 16.3%), using the empirical rule, we can determine that the area under the normal curve within two standard deviations of the mean is 95%. Thus, there is a probability, or likelihood, of 0.95 that a mid-cap stock fund will provide a rate of return between 1.5% and 16.3%.
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A probability distribution is a mathematical function that assigns probabilities to various outcomes. In many financial situations, we are interested in determining the expected value of the return on a particular investment or the expected return on a portfolio of multiple investments. When analyzing distributions that follow a normal distribution, probabilities can be calculated by finding the area under the graph of the normal curve.
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### Portfolio Weights
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# Statistical Analysis in Finance
## Data Visualization and Graphical Displays
By the end of this section, you will be able to:
1. Determine appropriate graphs to use for various types of data.
2. Create and interpret univariate graphs such as bar graphs and histograms.
3. Create and interpret bivariate graphs such as time series graphs and scatter plot graphs.
### Graphing Univariate Data
Data visualization refers to the use of graphical displays to summarize data to help to interpret patterns and trends in the data. Univariate data refers to observations recorded for a single characteristic or attribute, such as salaries or blood pressure measurements. When graphing univariate data, we can choose from among several types of graphs, such as bar graphs, time series graphs, and so on.
The most effective type of graph to use for a certain data set will depend on the nature of the data and the purpose of the graph. For example, a time series graph is typically used to show how a measurement is changing over time and to identify patterns or trends over time.
Below are some examples of typical applications for various graphs and displays.
Graphs used to show the distribution of data:
1. Bar chart: used to show frequency or relative frequency distributions for categorical data
2. Histogram: used to show frequency or relative frequency distributions for continuous data
Graphs used to show relationships between data points:
1. Time series graph: used to show measurement data plotted against time, where time is displayed on the horizontal axis
2. Scatter plot: used to show the relationship between a dependent variable and an independent variable
### Bar Graphs
A bar graph consists of bars that are separated from each other and compare percentages. The bars can be rectangles, or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graph shown in the example below has age groups represented on the and proportions on the .
By the end of 2021, a certain social media site had over 146 million users in the United States. shows three age groups, the number of users in each age group, and the proportion (%) of users in each age group. A bar graph using this data is shown in .
### Histograms
A histogram is a bar graph that is used for continuous numeric data, such as salaries, blood pressures, heights, and so on. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.
A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either Frequency or Relative Frequency (or Percent Frequency or Probability). The graph will have the same shape regardless of the label on the vertical axis. A histogram, like a stem-and-leaf plot, can give you the shape of the data, the center, and the spread of the data.
The relative frequency is equal to the frequency of an observed data value divided by the total number of data values in the sample. Remember, frequency is defined as the number of times a solution occurs. Relative frequency is calculated using the formula
where f = frequency, n = the total number of data values (or the sum of the individual frequencies), and RF = relative frequency.
To construct a histogram, first decide how many bars or intervals, also called classes, will represent the data. Many histograms consist of 5 to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval that is less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1, and if this is the smallest value, a convenient starting point is 6.05 (because ). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is . If all the data values happen to be integers and the smallest value is 2, then a convenient starting point is . Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.
Example: The following data values are the portfolio values, in thousands of dollars, for 100 investors.
60, 60.5, 61, 61, 61.5
63.5, 63.5, 63.5
64, 64, 64, 64, 64, 64, 64, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5
66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5
68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69.5, 69.5, 69.5, 69.5, 69.5
70, 70, 70, 70, 70, 70, 70.5, 70.5, 70.5, 71, 71, 71
72, 72, 72, 72.5, 72.5, 73, 73.5
74
The smallest data value is 60. Because the data values with the most decimal places have one decimal place (for instance, 61.5), we want our starting point to have two decimal places. Because the numbers 0.5, 0.05, 0.005, and so on are convenient numbers, use 0.05 and subtract it from 60, the smallest value, to get a convenient starting point: , which is more precise than, say, 61.5 by one decimal place. Thus, the starting point is 59.95. The largest value is 74, and , so 74.05 is the ending value.
Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide the result by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars. The interval width is calculated as follows:
We will round up to 2 and make each bar or class interval 2 units wide. Rounding up to 2 is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary, even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number if necessary. For example, if there are 150 data values, take the square root of 150 and round to 12 bars or intervals. The boundaries are as follows:
The data values 60 through 61.5 are in the interval 59.95–61.95. The data values of 63.5 are in the interval 61.95–63.95. The data values of 64 and 64.5 are in the interval 63.95–65.95. The data values 66 through 67.5 are in the interval 65.95–67.95. The data values 68 through 69.5 are in the interval 67.95–69.95. The data values 70 through 71 are in the interval 69.95–71.95. The data values 72 through 73.5 are in the interval 71.95–73.95. The data value 74 is in the interval 73.95–75.95. The histogram shown in displays the portfolio values on the x-axis and relative frequency on the y-axis.
### Graphing Bivariate Data
Bivariate data refers to paired data, where each value of one variable is paired with a value of a second variable. An example of paired data would be if data were collected on employees’ years of experience and their corresponding salaries. Typically, it is of interest to investigate possible associations or correlations between the two variables under analysis.
### Time Series Graphs
Suppose that we want to track the consumer price index (CPI) over the past 10 years. One feature of the data that we may want to consider is the element of time. Because each year is paired with the CPI value for that year, we do not have to think of the data as being random. We can instead use the years given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing CPI value as the decade progresses is called a time series graph.
To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a point in time and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.
Example: The following data set shows the annual CPI for 10 years. We need to construct a time series graph for the (rounded) annual CPI data (see ). The time series graph is shown in .
### Scatter Plots
A scatter plot, or scatter diagram, is a graphical display intended to show the relationship between two variables. The setup of the scatter plot is that one variable is plotted on the horizontal axis and the other variable is plotted on the vertical axis. Then each pair of data values is considered as an (x, y) point, and the various points are plotted on the diagram. A visual inspection of the plot is then made to detect any patterns or trends. Additional statistical analysis can be conducted to determine if there is a correlation or other statistically significant relationship between the two variables.
Assume we are interested in tracking the closing price of Nike stock over the one-year time period from April 2020 to March 2021. We would also like to know if there is a correlation or relationship between the price of Nike stock and the value of the S&P 500 over the same time period. To visualize this relationship, we can create a scatter plot based on the (x, y) data shown in . The resulting scatter plot is shown in .
Note the linear pattern of the points on the scatter plot. Because the data points generally align along a straight line, this provides an indication of a linear correlation between the price of Nike stock and the value of the S&P 500 over this one-year time period.
The scatter plot can be generated using Excel as follows:
1. Enter the x-data in column A of a spreadsheet.
2. Enter the y-data in column B.
3. Highlight the data with your mouse.
4. Go to the Insert menu and select the icon for a scatter plot, as shown in .
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Data visualization refers to the use of graphical displays to summarize a data set to help to interpret patterns and trends in the data. Univariate data refers to observations recorded for a single characteristic or attribute, such as salaries or blood pressure measurements. When graphing univariate data, we can choose from among several types of graphs. The type of graph to be used for a certain data set will depend on the nature of the data and the purpose of the graph. Examples of graphs for univariate data include line graphs, bar graphs, and histograms. Bivariate data refers to paired data where each value of one variable is paired with a value of a second variable. Examples of graphs for bivariate data include time series graphs and scatter plots.
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# Statistical Analysis in Finance
## The R Statistical Analysis Tool
By the end of this section, you will be able to:
1. Create a vector of data values for the R statistical analysis tool.
2. Write basic statistical commands using the R statistical analysis tool.
### Commands and Vectors in R
R is a statistical analysis tool that is widely used in the finance industry. It is available as a free program and provides an integrated suite of functions for data analysis, graphing, and statistical programming. R is increasingly being used as a data analysis and statistical tool as it is an open-source language and additional features are constantly being added by the user community. The tool can be used on many different computing platforms and can be downloaded at the R Project website.
Once you have installed and started R on your computer, at the bottom of the R console, you should see the symbol >, which indicates that R is ready to accept commands.
R is a command-driven language, meaning that the user enters commands at the prompt, which R then executes one at a time. R can also execute a program containing multiple commands. There are ways to add a graphic user interface (GUI) to R. An example of a GUI tool for R is RStudio.
The R command line can be used to perform any numeric calculation, similar to a handheld calculator. For example, to evaluate the expression enter the following expression at the command line prompt and hit return:
Most calculations in R are handled via functions. For statistical analysis, there are many preestablished functions in R to calculate mean, median, standard deviation, quartiles, and so on. Variables can be named and assigned values using the assignment operator <-. For example, the following R commands assign the value of 20 to the variable named x and assign the value of 30 to the variable named y:
These variable names can be used in any calculation, such as multiplying x by y to produce the result 600:
The typical method for using functions in statistical applications is to first create a vector of data values. There are several ways to create vectors in R. For example, the c function is often used to combine values into a vector. The following R command will generate a vector called salaries that contains the data values 40,000, 50,000, 75,000, and 92,000:
This vector salaries can then be used in statistical functions such as mean, median, min, max, and so on, as shown:
Another option for generating a vector in R is to use the seq function, which will automatically generate a sequence of numbers. For example, we can generate a sequence of numbers from 1 to 5, incremented by 0.5, and call this vector example1, as follows:
If we then type the name of the vector and hit enter, R will provide a listing of numeric values for that vector name.
Often, we are interested in generating a quick statistical summary of a data set in the form of its mean, median, quartiles, min, and max. The R command called summary provides these results.
For measures of spread, R includes a command for standard deviation, called sd, and a command for variance, called var. The standard deviation and variance are calculated with the assumption that the data set was collected from a sample.
To calculate a weighted mean in R, create two vectors, one of which contains the data values and the other of which contains the associated weights. Then enter the R command weighted.mean(values, weights).
The following is an example of a weighted mean calculation in R:Assume your portfolio contains 1,000 shares of XYZ Corporation, purchased on three different dates, as shown in . Calculate the weighted mean of the purchase price, where the weights are based on the number of shares in the portfolio.
Here is how you would create two vectors in R: the price vector will contain the purchase price, and the shares vector will contain the number of shares. Then execute the R command weighted.mean(price, shares), as follows:
A list of common R statistical commands appears in .
### Graphing in R
There are many statistical applications in R, and many graphical representations are possible, such as bar graphs, histograms, time series plots, scatter plots, and others. The basic command to create a plot in R is the plot command, plot(x, y), where x is a vector containing the x-values of the data set and y is a vector containing the y-values of the data set.
The general format of the command is as follows:
For example, we are interested in creating a scatter plot to examine the correlation between the value of the S&P 500 and Nike stock prices. Assume we have the data shown in , collected over a one-year time period.
Note that data can be read into R from a text file or Excel file or from the clipboard by using various R commands. Assume the values of the S&P 500 have been loaded into the vector SP500 and the values of Nike stock prices have been loaded into the vector Nike. Then, to generate the scatter plot, we can use the following R command:
As a result of these commands, R provides the scatter plot shown in . This is the same data that was used to generate the scatter plot in in Excel.
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R is an open-source statistical analysis tool that is widely used in the finance industry. It provides an integrated suite of functions for data analysis, graphing, and statistical programming. R is increasingly being used as a data analysis and statistical tool as it is an open-source language, and additional features are constantly being added by the user community. This tool can be used on many different computing platforms and can be downloaded at The R Project for Statistical Computing.
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### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Regression Analysis in Finance
## Why It Matters
Correlation and regression analysis are used extensively in finance applications. Correlation analysis allows the determination of a statistical relationship between two numeric quantities. Regression analysis can be used to predict one quantity based on a second quantity, assuming there is a significant correlation between the two quantities. For example, in finance, we use regression analysis to calculate the beta coefficient of a stock, which represents the volatility of the stock versus overall market volatility, with volatility being a measure of risk.
A business may want to establish a correlation between the amount the company spent on advertising versus its recorded sales. If a strong enough correlation is established, then the business manager can predict sales based on the amount spent on advertising for a given time period.
Finance professionals often use correlation analysis to predict future trends and mitigate risk in a stock portfolio. For example, if two investments are strongly correlated, an investor might not want to have both investments in a certain portfolio since the two investments would tend to move in the same directions during up markets or down markets. To diversify a portfolio, an investor might seek investments that are not strongly correlated with one another.
Regression analysis can be used to establish a mathematical equation that relates a dependent variable (such as sales) to an independent variable (such as advertising expenditure). In this discussion, the focus will be on analyzing the relationship between one dependent variable and one independent variable, where the relationship can be modeled using a linear equation. This type of analysis is called . |
# Regression Analysis in Finance
## Correlation Analysis
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate a correlation coefficient.
2. Interpret a correlation coefficient.
3. Test for the significance of a correlation coefficient.
### Calculate a Correlation Coefficient
In correlation analysis, we study the relationship between bivariate data, which is data collected on two variables where the data values are paired with one another.
Correlation is the measure of association between two numeric variables. For example, we may be interested to know if there is a correlation between bond prices and interest rates or between the age of a car and the value of the car. To investigate the correlation between two numeric quantities, the first step is to create a scatter plot that will graph the (x, y) ordered pairs. The independent, or explanatory, quantity is labeled as the x-variable, and the dependent, or response, quantity is labeled as the y-variable.
For example, we may be interested to know if the price of Nike stock is correlated with the value of the S&P 500 (Standard & Poor’s 500 stock market index). To investigate this, monthly data can be collected for Nike stock prices and value of the S&P 500 for a period of time, and a scatter plot can be created and examined. A scatter plot, or scatter diagram, is a graphical display intended to show the relationship between two variables. The setup of the scatter plot is that one variable is plotted on the horizontal axis and the other variable is plotted on the vertical axis. Each pair of data values is considered as an (x, y) point, and the various points are plotted on the diagram. A visual inspection of the plot is then made to detect any patterns or trends on the scatter diagram. shows the relationship between the Nike stock price and its S&P value over a one-year time period.
To assess linear correlation, the graphical trend of the data points is examined on the scatter plot to determine if a straight-line pattern exists. If a linear pattern exists, the correlation may indicate either a positive or a negative correlation. A positive correlation indicates that as the independent variable increases, the dependent variable tends to increase as well, or, as the independent variable decreases, the dependent variable tends to decrease (the two quantities move in the same direction). A negative correlation indicates that as the independent variable increases, the dependent variable decreases, or, as the independent variable decreases, the dependent variable increases (the two quantities move in opposite directions). If there is no relationship or association between the two quantities, where one quantity changing does not affect the other quantity, we conclude that there is no correlation between the two variables.
From the scatter plot in the Nike stock versus S&P 500 example (see ), we note that the trend reflects a positive correlation in that as the value of the S&P 500 increases, the price of Nike stock tends to increase as well.
When inspecting a scatter plot, it may be difficult to assess a correlation based on a visual inspection of the graph alone. A more precise assessment of the correlation between the two quantities can be obtained by calculating the numeric correlation coefficient (referred to using the symbol r).
The correlation coefficient, which was developed by statistician Karl Pearson in the early 1900s, is a measure of the strength and direction of the correlation between the independent variable x and the dependent variable y.
The formula for r is shown below; however, technology, such as Excel or the statistical analysis program R, is typically used to calculate the correlation coefficient.
where n refers to the number of data pairs and the symbol indicates to sum the x-values.
provides a step-by-step procedure on how to calculate the correlation coefficient r.
Note that since r is calculated using sample data, r is considered a sample statistic used to measure the strength of the correlation for the two population variables. Sample data indicates data based on a subset of the entire population.
Given the complexity of this calculation, Excel or other software is typically used to calculate the correlation coefficient.
The Excel command to calculate the correlation coefficient uses the following format:
where A1:A10 are the cells containing the x-values and B1:B10 are the cells containing the y-values.
### Interpret a Correlation Coefficient
Once the value of r is calculated, this measurement provides two indicators for the correlation:
1. the strength of the correlation based on the value of r
2. the direction of the correlation based on the sign of r
The value of r gives us this information:
1. The value of r is always between and : .
2. The size of the correlation r indicates the strength of the linear relationship between the two variables. Values of r close to or to indicate a stronger linear relationship.
3. If , there is no linear relationship between the two variables (no linear correlation).
4. If , there is perfect positive correlation.
5. If there is perfect negative correlation. In both of these cases, all the original data points lie on a straight line.
The sign of r gives us this information:
1. A positive value of r means that when x increases, y tends to increase, and when x decreases, y tends to decrease (positive correlation).
2. A negative value of r means that when x increases, y tends to decrease, and when x decreases, y tends to increase (negative correlation).
The Excel command used to find the value of the correlation coefficient for the Nike stock versus S&P 500 example (refer back to ) is
In this example, the value of is calculated by Excel to be .
Since this is a positive value close to 1, we conclude that the relationship between Nike stock and the value of the S&P 500 over this time period represents a strong, positive correlation.
The correlation coefficient r can also be determined using the statistical capability on the financial calculator:
1. Step 1 is to enter the data in the calculator (using the [DATA] function that is located above the 7 key).
2. Step 2 is to access the statistical results provided by the calculator (using the [STAT] function that is located above the 8 key) and scroll to the correlation coefficient results.
Follow the steps in for calculating the correlation data for the data set of Nike stock price and value of the S&P 500 shown previously.
From the statistical results shown on the calculator display, the correlation coefficient r is 0.93, which indicates that the relationship between Nike stock and the value of the S&P 500 over this time period represents a strong, positive correlation.
Note: A strong correlation does not suggest that x causes y or y causes x. We must remember that correlation does not imply causation.
### Test a Correlation Coefficient for Significance
The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y. The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population (that is, all measurements of interest), we could find the population correlation coefficient, which is labeled as the Greek letter ρ (pronounced “rho”). But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
1. ρ = population correlation coefficient (unknown)
2. r = sample correlation coefficient (known; calculated from sample data)
An important step in the correlation analysis is to determine if the correlation is significant. By this, we are asking if the correlation is strong enough to allow meaningful predictions for y based on values of x. One method to test the significance of the correlation is to employ a hypothesis test. The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is close to zero or significantly different from zero. We decide this based on the sample correlation coefficient r and the sample size n.
If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is significant.
1. Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y variables because the correlation coefficient is significantly different from zero.
2. What the conclusion means: There is a significant linear relationship between the x and y variables. If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that the correlation coefficient is not significant.
A hypothesis test can be performed to test if the correlation is significant. A hypothesis test is a statistical method that uses sample data to test a claim regarding the value of a population parameter. In this case, the hypothesis test will be used to test the claim that the population correlation coefficient ρ is equal to zero.
Use these hypotheses when performing the hypothesis test:
1. Null hypothesis:
2. Alternate hypothesis:
The hypotheses can be stated in words as follows:
1. Null hypothesis : The population correlation coefficient is not significantly different from zero. There is not a significant linear relationship (correlation) between x and y in the population.
2. Alternate hypothesis : The population correlation coefficient is significantly different from zero. There is a significant linear relationship (correlation) between x and y in the population.
A quick shorthand way to test correlations is the relationship between the sample size and the correlation. If then this implies that the correlation between the two variables demonstrates that a linear relationship exists and is statistically significant at approximately the 0.05 level of significance. As the formula indicates, there is an inverse relationship between the sample size and the required correlation for significance of a linear relationship. With only 10 observations, the required correlation for significance is 0.6325; for 30 observations, the required correlation for significance decreases to 0.3651; and at 100 observations, the required level is only 0.2000.
NOTE:
1. If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x-values.
2. If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
3. If r is significant and the scatter plot shows a linear trend, the line may not be appropriate or reliable for prediction outside the domain of observed x-values in the data.
Correlations may be helpful in visualizing the data, but they are not appropriately used to explain a relationship between two variables. Perhaps no single statistic is more misused than the correlation coefficient. Citing correlations between health conditions and everything from place of residence to eye color have the effect of implying a cause-and-effect relationship. This simply cannot be accomplished with a correlation coefficient. The correlation coefficient is, of course, innocent of this misinterpretation. It is the duty of analysts to use a statistic that is designed to test for cause-and-effect relationships and to report only those results, if they are intending to make such a claim. The problem is that passing this more rigorous test is difficult, therefore lazy and/or unscrupulous researchers fall back on correlations when they cannot make their case legitimately.
###
Correlation is the measure of association between two numeric variables. A correlation coefficient called r is used to assess the strength and direction of the correlation. The value of r is always between and . The size of the correlation r indicates the strength of the linear relationship between the two variables. Values of r close to or to indicate a stronger linear relationship. A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation). A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation). |
# Regression Analysis in Finance
## Linear Regression Analysis
### Learning Outcomes
By the end of this section, you will be able to:
1. Analyze a regression using the method of least squares and residuals.
2. Test the assumptions for linear regression.
### Method of Least Squares and Residuals
Once the correlation coefficient has been calculated and a determination has been made that the correlation is significant, typically a regression model is then developed. In this discussion we will focus on linear regression, where a straight line is used to model the relationship between the two variables. Once a straight-line model is developed, this model can then be used to predict the value of the dependent variable for a specific value of the independent variable.
Recall from algebra that the equation of a straight line is given by
where m is the slope of the line and b is the y-intercept of the line.
The slope measures the steepness of the line, and the y-intercept is that point on the y-axis where the graph crosses, or intercepts, the y-axis.
In linear regression analysis, the equation of the straight line is written in a slightly different way using the model
In this format, b is the slope of the line, and a is the y-intercept. The notation is called y-hat and is used to indicate a predicted value of the dependent variable y for a certain value of the independent variable x.
If a line extends uphill from left to right, the slope is a positive value, and if the line extends downhill from left to right, the slope is a negative value. Refer to .
When generating the equation of a line in algebra using , two (x, y) points were required to generate the equation. However, in regression analysis, all (x, y) points in the data set will be utilized to develop the linear regression model.
The first step in any regression analysis is to create the scatter plot. Then proceed to calculate the correlation coefficient r, and check this value for significance. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If x is the independent variable and y the dependent variable, then we can use a regression line to predict y for a given value of x.
As an example of a regression equation, assume that a correlation exists between the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company. After collecting (x, y) data for a certain time period, the company determines the regression equation is of the form
where x represents the monthly amount spent on advertising (in thousands of dollars) and represents the monthly revenues for the company (in thousands of dollars).
A scatter plot of the (x, y) data is shown in .
The Fortune 500 company would like to predict the monthly revenue if its executives decide to spend $150,000 in advertising next month. To determine the estimate of monthly revenue, let in the regression equation and calculate a corresponding value for :
This predicted value of y indicates that the anticipated revenue would be $18,646,700, given the advertising spend of $150,000.
Notice that from past data, there may have been a month where the company actually did spend $150,000 on advertising, and thus the company may have an actual result for the monthly revenue. This actual, or observed, amount can be compared to the prediction from the linear regression model to calculate a residual.
A residual is the difference between an observed y-value and the predicted y-value obtained from the linear regression equation. As an example, assume that in a previous month, the actual monthly revenue for an advertising spend of $150,000 was $19,200,000, and thus . The residual for this data point can be calculated as follows:
Notice that residuals can be positive, negative, or zero. If the observed y-value exactly matches the predicted y-value, then the residual will be zero. If the observed y-value is greater than the predicted y-value, then the residual will be a positive value. If the observed y-value is less than the predicted y-value, then the residual will be a negative value.
When formulating the linear regression line of best fit to the points on the scatter plot, the mathematical analysis generates a linear equation where the sum of the squared residuals is minimized. This analysis is referred to as the method of least squares. The result is that the analysis generates a linear equation that is the “best fit” to the points on the scatter plot, in the sense that the line minimizes the differences between the predicted values and observed values for y.
The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
Once the (x, y) has been collected, the slope (b) and y-intercept (a) can be calculated using the following formulas:
where n refers to the number of data pairs and indicates sum of the x-values.
Notice that the formula for the y-intercept requires the use of the slope result (b), and thus the slope should be calculated first and the y-intercept should be calculated second.
When making predictions for y, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best-fit line to make predictions for y, given x within the domain of x-values in the sample data, but not necessarily for x-values outside that domain.
Note: Computer spreadsheets, statistical software, and many calculators can quickly calculate the best-fit line and create the graphs. The calculations tend to be tedious if done by hand.
### Assumptions for Linear Regression
Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence that we can conclude that there is a linear relationship between x and y in the population.
The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population (). Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.
These are the assumptions underlying the test of significance:
1. There is a linear relationship in the population that models the average value of y for varying values of x. In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
2. The y-values for any particular x-value are normally distributed about the line. This implies that there are more y-values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y-values lie on the line.
3. The standard deviations of the population y-values about the line are equal for each value of x. In other words, each of these normal distributions of y-values has the same shape and spread about the line.
4. The residual errors are mutually independent (no pattern).
5. The data are produced from a well-designed, random sample or randomized experiment.
###
Linear regression analysis uses a straight-line fit to model the relationship between the two variables. Once a straight-line model is developed, this model can then be used to predict the value of the dependent variable for a specific value of the independent variable. Two parameters are calculated for the linear model, the slope of the best-fit line and the y-intercept of the best-fit line. The method of least squares is used to generate these parameters; this method is based on minimizing the squared differences between the predicted values and observed values for y. |
# Regression Analysis in Finance
## Best-Fit Linear Model
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate the slope and y-intercept for a linear regression model using technology.
2. Interpret and apply the slope and y-intercepts.
### Calculate the Slope and y-Intercept for a Linear Regression Model Using Technology
Once a correlation has been deemed as significant, a best-fit linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
The slope (b) and y-intercept (a) can be calculated using the following formulas:
These formulas can be quite cumbersome, especially for a significant number of data pairs, and thus technology is often used (such as Excel, a calculator, R statistical software, etc.).
Using Excel: To calculate the slope and y-intercept of the linear model, start by entering the (x, y) data in two columns in Excel. Then the Excel commands =SLOPE and =INTERCEPT can be used to calculate the slope and intercept, respectively.
The following data set will be used as an example: the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months (data is shown in ).
To calculate the slope of the regression model, use the Excel command
It’s important to note that this Excel command expects that the y-data range is entered first and the x-data range is entered second. Since revenue depends on amount spent on advertising, revenue is considered the y-variable and amount spent on advertising is considered the x-variable. Notice the y-data is contained in cells C2 through C13 and the x-data is contained in cells B2 through B13. Thus the Excel command for slope would be entered as
In the same way, the Excel command to calculate the y-intercept of the regression model is
For the data set shown in the above table, the Excel command would be
The results are shown in , where
Based on this, the regression equation can be written as
where x represents the amount spent on advertising (in thousands of dollars) and y represents the amount of revenue (in thousands of dollars).
### Using a Financial Calculator
The financial calculator provides the slope and y-intercept for the linear regression model once the (x, y) data is inputted into the calculator.
Follow the steps in for calculating the slope and y-intercept for the data set of amounts spent on advertising and revenue shown previously.
From the statistical results shown on the calculator display, the slope b is 61.8 and the y-intercept a is 9,367.7.
Based on this, the regression equation can be written as
### Interpret and Apply the Slope and y-Intercept
The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.
### Interpretation of the Slope
The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.
In the previous example, the linear regression model for the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months was generated as follows:
Since the slope was determined to be 61.8, the company can interpret this to mean that for every $1,000 dollars spent on advertising, on average, this will result in an increase in revenues of $61,800.
The intercept of the regression equation is the corresponding y-value when.
### Interpretation of the Intercept
The intercept of the best-fit line tells us the expected mean value of y in the case where the x-variable is equal to zero.
However, in many scenarios it may not make sense to have the x-variable equal zero, and in these cases, the intercept does not have any meaning in the context of the problem. In other examples, the x-value of zero is outside the range of the x-data that was collected. In this case, we should not assign any interpretation to the y-intercept.
In the previous example, the range of data collected for the x-variable was from $49 to $153 spent per month on advertising. Since this interval does not include an x-value of zero, we would not provide any interpretation for the intercept.
###
Once a correlation has been deemed significant, a linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation: . Typically some technology, such as Excel, R statistical tool, or a calculator, is used to generate the coefficients a and b since manual calculations are cumbersome. |
# Regression Analysis in Finance
## Regression Applications in Finance
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate the regression model for a single independent variable as applied to financial forecasting.
2. Extract measures of slope and intercept from regression analysis in financial applications.
### Regression Model for a Single Independent Variable
Regression analysis is used extensively in finance-related applications. Many typical applications involve determining if there is a correlation between various stock market indices such as the S&P 500, the Dow Jones Industrial Average (DJIA), and the Russell 2000 index.
As an example, suppose we would like to determine if there is a correlation between the Russell 2000 index and the DJIA. Does the value of the Russell 2000 index depend on the value of the DJIA? Is it possible to predict the value of the Russell 2000 index for a certain value of the DJIA? We can explore these questions using regression analysis.
shows a summary of monthly closing prices of the DJIA and the Russell 2000 for a 12-month time period. We consider the DJIA to be the independent variable and the Russell 2000 index to be the dependent variable.
The first step is to create a scatter plot to determine if the data points appear to follow a linear pattern. The scatter plot is shown in . The scatter plot clearly shows a linear pattern; the next step is to calculate the correlation coefficient and determine if the correlation is significant.
1. Using the Excel command =CORREL, the correlation coefficient is calculated to be 0.947. This value of the correlation coefficient is significant using the test for significance referenced earlier in Correlation Analysis.
2. Using the Excel commands =SLOPE and =INTERCEPT, the value of the slope and y-intercept are calculated as 0.11 and , respectively, when rounded to two decimal places.
The Excel output is shown below:
Based on these results, the corresponding linear regression model is
Assume the DJIA has reached a value of 32,000. Predict the corresponding value of the Russell 2000 index. To determine this, substitute the value of the independent variable, (this is the given value of the DJIA), and calculate the corresponding value for the dependent variable, which is the predicted value for the Russell 2000 index:
Thus the predicted value for the Russell 2000 index is approximately 2,024 when the DJIA reached a value of 32,000.
### Measures of Slope and Intercept from Regression Analysis
An important application of regression analysis is to determine the systematic risk for a particular stock, which is referred to as beta. A stock’s beta is a measure of the volatility of the stock compared to a benchmark such as the S&P 500 index. If a stock has more volatility compared to the benchmark, then the stock will have a beta greater than 1.0. If a stock has less volatility compared to the benchmark, then the stock will have a beta less than 1.0.
Beta can be determined as the slope of the regression line when the stock returns are plotted versus the returns for the benchmark, such as the S&P 500. As an example, consider the calculation for beta of Nike stock based on monthly returns of Nike stock versus monthly returns for the S&P 500 over the time period from May 2020 to March 2021. The monthly return data is shown in .
The scatter plot that graphs S&P monthly return versus Nike monthly return is shown in .
The slope of the regression line is 0.83, obtained by using the =SLOPE command in Excel.
This indicates the value of beta for Nike stock is 0.83, which indicates that Nike stock had lower volatility versus the S&P 500 for the time period of interest.
###
Regression analysis is used extensively in finance-related applications. Many typical applications involve determining if there is a correlation between various stock market indices such as the S&P 500, the DJIA, and the Russell 2000 index. The procedure is to first generate a scatter plot to determine if a visual trend is observed, then calculate a correlation coefficient and check for significance. If the correlation coefficient is significant, a linear model can then be generated and used for predictions. |
# Regression Analysis in Finance
## Predictions and Prediction Intervals
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate predictions for the dependent variable using the regression model.
2. Generate prediction intervals based on a prediction for the dependent variable.
### Predicting the Dependent Variable Using the Regression Model
A key aspect of generating the linear regression model is to use the model for predictions, provided the correlation is significant. To generate predictions or forecasts using the linear regression model, substitute the value of the independent variable (x) in the regression equation and solve the equation for the dependent variable (y).
In a previous example, the linear regression equation was generated to relate the amount of monthly revenue for a Fortune 500 company to the amount of monthly advertising spend. From the previous example, it was determined that the regression equation can be written as
where x represents the amount spent on advertising (in thousands of dollars) and y represents the amount of revenue (in thousands of dollars).
Let’s assume the Fortune 500 company would like to predict the monthly revenue for a month where it plans to spend $80,000 for advertising. To determine the estimate of monthly revenue, let in the regression equation and calculate a corresponding value for ŷ:
This predicted value of y indicates that the forecasted revenue would be $14,320,700, assuming an advertising spend of $80,000.
1. Excel can provide this forecasted value directly using the =FORECAST command.
2. To use this command, enter the value of the independent variable
3. Using this Excel command, the forecasted value for the revenue is $14,320.52 when the advertising spend is $80 (in thousands of dollars) (see ). (Note: The discrepancy in the more precise Excel result and the formula result is due to rounding in interim calculations.)
A word of caution when predicting values for y: it is generally recommended to only predict values for y using values of x that are in the original range of the data collection.
As an example, assume we have developed a linear model to predict the height of male children based on their age. We have collected data for the age range from years old to years old, and we have confirmed that the scatter plot shows a linear trend and that the correlation is significant.
It would be erroneous to use this model to predict the height of a 25-year-old male since is outside the range of the x-data, which was from 3 to 10 years old. The reason this is not recommended is that a linear pattern cannot be assumed to continue beyond the x-value of 10 years old unless some data collection has occurred at ages greater than 10 to confirm that the linear pattern is consistent for x-values beyond 10 years old.
### Generating Prediction Intervals
One important value of an estimated regression equation is its ability to predict the effects on y of a change in one or more values of the independent variables. The value of this is obvious. Careful policy cannot be made without estimates of the effects that may result. Indeed, it is the desire for particular results that drive the formation of most policy. Regression models can be, and have been, invaluable aids in forming such policies.
Remember that point estimates do not carry a particular level of probability, or level of confidence, because points have no “width” above which there is an area to measure. There are actually two different approaches to the issue of developing estimates of changes in the independent variable (or variables) on the dependent variable. The first approach wishes to measure the expected mean value of y from a specific change in the value of x.
The second approach to estimate the effect of a specific value of x on y treats the event as a single experiment: you choose x and multiply it times the coefficient, and that provides a single estimate of y. Because this approach acts as if there were a single experiment, the variance that exists in the parameter estimate is larger than the variance associated with the expected value approach.
The conclusion is that we have two different ways to predict the effect of values of the independent variable(s) on the dependent variable, and thus we have two different intervals. Both are correct answers to the question being asked, but there are two different questions. To avoid confusion, the first case where we are asking for the expected value of the mean of the estimated y is called a confidence interval. The second case, where we are asking for the estimate of the impact on the dependent variable y of a single experiment using a value of x, is called the prediction interval.
The prediction interval for an individual y for can be calculated as
where s is the standard deviation of the error term, s is the standard deviation of the x-variable, and is the critical value of the t-distribution at the confidence level.
Tabulated values of the t-distribution are available in online references such as the Engineering Statistics Handbook. The mathematical computations for prediction intervals are complex, and usually the calculations are performed using software. The formula above can be implemented in Excel to create a 95% prediction interval for the forecast for monthly revenue when is spent on monthly advertising. shows the detailed calculations in Excel to arrive at a 95% prediction interval of (13,270.95, 15,370.09) for the monthly revenue. (The commands refer to the Excel data table shown in .)
This prediction interval can be interpreted as follows: there is 95% confidence that when the amount spent on monthly advertising is $80,000, the corresponding monthly revenue will be between $13,270.95 and $15,370.09.
Various computer regression software packages provide programs within the regression functions to provide answers to inquiries of estimated predicted values of y given various values chosen for the x-variable(s). For example, the statistical program R provides these prediction intervals directly. It is important to know just which interval is being tested in the computer package because the difference in the size of the standard deviations will change the size of the interval estimated. This is shown in .
shows visually the difference the standard deviation makes in the size of the estimated intervals. The confidence interval, measuring the expected value of the dependent variable, is smaller than the prediction interval for the same level of confidence. The expected value method assumes that the experiment is conducted multiple times rather than just once, as in the other method. The logic here is similar, although not identical, to that discussed when developing the relationship between the sample size and the confidence interval using the central limit theorem. There, as the number of experiments increased, the distribution narrowed, and the confidence interval became tighter around the expected value of the mean.
It is also important to note that the intervals around a point estimate are highly dependent upon the range of data used to estimate the equation, regardless of which approach is being used for prediction. Remember that all regression equations go through the point of means—that is, the mean value of y and the mean values of all independent variables in the equation. As the value of x gets further and further from the (x, y) point corresponding to the mean value of x and the mean value of y, the width of the estimated interval around the point estimate increases. Choosing values of x beyond the range of the data used to estimate the equation poses an even greater danger of creating estimates with little use, very large intervals, and risk of error. shows this relationship.
demonstrates the concern for the quality of the estimated interval, whether it is a prediction interval or a confidence interval. As the value chosen to predict y, in the graph, is further from the central weight of the data, , we see the interval expand in width even while holding constant the level of confidence. This shows that the precision of any estimate will diminish as one tries to predict beyond the largest weight of the data and most certainly will degrade rapidly for predictions beyond the range of the data. Unfortunately, this is just where most predictions are desired. They can be made, but the width of the confidence interval may be so large as to render the prediction useless.
###
A key aspect of generating the linear regression model is to then use the model for predictions, provided that the correlation is significant. To generate predictions or forecasts using the linear regression model, substitute the value of the independent variable (x) in the regression equation and solve the equation for the dependent variable (y). When making predictions using the linear model, it is generally recommended to only predict values for y using values of x that are in the original range of the data collection. |
# Regression Analysis in Finance
## Use of R Statistical Analysis Tool for Regression Analysis
### Learning Outcomes
By the end of this section, you will be able to:
1. Generate correlation coefficients using the R statistical tool.
2. Generate linear regression models using the R statistical tool.
### Generate Correlation Coefficients Using the R Statistical Tool
R is an open-source statistical analysis tool that is widely used in the finance industry. R is available as a free program and provides an integrated suite of functions for data analysis, graphing, and statistical programming. R provides many functions and capabilities for regression analysis.
Recall that most calculations in R are handled via functions.
The typical method for using functions in statistical applications is to first create a vector of data values. There are several ways to create vectors in R. For example, the c function is often used to combine values into a vector. For example, this R command will generate a vector called salaries, containing the data values 40,000, 50,000, 75,000, and 92,000:
To calculate the correlation coefficient r, we use the R command called cor.
As an example, consider the data set in , which tracks the return on the S&P 500 versus return on Coca-Cola stock for a seven-month time period.
Create two vectors in R, one vector for the S&P 500 returns and a second vector for Coca-Cola returns:
The R command called cor returns the correlation coefficient for the x-data vector and y-data vector:
### Generate Linear Regression Models Using the R Statistical Tool
To create a linear model in R, assuming the correlation is significant, the command lm (for linear model) will provide the slope and y-intercept for the linear regression equation.
The format of the R command is
Notice the use of the tilde symbol as the separator between the dependent variable vector and the independent variable vector.
We use the returns on Coca-Cola stock as the dependent variable and the returns on the S&P 500 as the independent variable, and thus the R command would be
The R output provides the value of the y-intercept as and the value of the slope as 0.8641. Based on this, the linear model would be
where x represents the value of S&P 500 return and y represents the value of Coca-Cola stock return.
The results can also be saved as a formula and called “model” using the following R command. To obtain more detailed results for the linear regression, the summary command can be used, as follows:
In this output, the y-intercept and slope is given, as well as the residuals for each x-value. The output includes additional statistical details regarding the regression analysis.
Predicted values and prediction intervals can also be generated within R.
First, we can create a structure in R called a data frame to hold the values of the independent variable for which we want to generate a prediction. For example, we would like to generate the predicted return for Coca-Cola stock, given that the return for the S&P 500 is 6.
We use the R command called predict.
To generate a prediction for the linear regression equation called model, using the data frame where the value of the S&P 500 is 6, the R commands will be
The output from the predict command indicates that the predicted return for Coca-Cola stock will be 4.8% when the return for the S&P 500 is 6%.
We can extend this analysis to generate a 95% prediction interval for this result by using the following R command, which adds an option to the predict command to generate a prediction interval:
Thus the 95% prediction interval for Coca-Cola return is (0.05%, 9.62%) when the return for the S&P 500 is 6%.
###
R is an open-source statistical analysis tool that is widely used in the finance industry and can be found online. R provides an integrated suite of functions for data analysis, graphing, and correlation and regression analysis. R is increasingly being used as a data analysis and statistical tool because it is an open-source language and additional features are constantly being added by the user community. The tool can be used on many different computing platforms.
### Multiple Choice
### Questions
### Problems
###
### Simple Linear Regression
### How to Calculate Correlation for Stocks, Bonds, and Funds
|
# How to Think about Investing
## Why It Matters
Having finished her college degree and embarked on her career, Maria is now contemplating her financial future. She is considering how she might invest some of her hard-earned money. As a short-term goal, she wants to build an emergency fund so that she could cover her expenses for six months if she became ill or injured and had to take time off of work. She would also like to save money for a down payment on a home and to purchase new furniture. Although she is not yet 30 years old, Maria also knows that it is prudent to begin saving for retirement.
What should she do with her savings? Maria has some friends who have told her how successful they have been investing in stocks. Bart bragged about doubling his money in just over a year when he purchased Facebook stock, and Tiffany quickly tripled her money when she purchased shares in Netflix. But Maria also knows that her uncle lost a significant amount of money when his Boeing stock dropped from over $300 per share to under $150 within a couple of months at the beginning of 2020. Just how risky would it be to invest in stocks? What type of return might Maria expect? Are there strategies she could follow that would allow her to avoid her uncle’s fate? |
# How to Think about Investing
## Risk and Return to an Individual Asset
### Learning Outcomes
By the end of this section, you will be able to:
1. Compute the realized return from an individual investment.
2. Compute the average return and volatility of returns from historical data.
3. Describe firm-specific risk.
### Measuring Historical Returns
Risk and return are often referred to as the two Rs of finance. Investors are interested in both risk and return because understanding one without the other is really meaningless. In terms of investment, the concept of return is fairly straightforward; return is the benefit, or profit, the investor expects from an expenditure. It is the reward for investing—the reason an investment is made in the first place. However, no investment is a sure thing. The return may not be what the investor was expecting. This uncertainty about what the return will be is referred to as risk.
We begin by looking at how to measure both risk and return when considering an individual asset, such as one stock. If your grandparents bought 100 shares of Apple, Inc. stock for you when you were born, you are interested in knowing how well that investment has done. You may even want to compare how that investment has fared to how an investment in a different stock, perhaps Disney, would have done. You are interested in measuring the historical return.
### Individual Investment Realized Return
The realized return of an investment is the total return that occurs over a particular time period. Suppose that you purchased a share of Target (TGT) at the beginning of January 2020 for $128.74. At the end of the year, you sold the stock for $176.53, which was $47.79 more than you paid for it. This increase in value is known as a capital gain. As the owner of the stock, you also received $2.68 in dividends during 2020. The total dollar return from your investment is calculated as
It is common to express investment returns in percentage terms rather than dollar terms. This allows you to answer the question “How much do I receive for each dollar invested?” so that you can compare investments of different sizes. The total percent return from your investment is
The dividend yield is calculated by dividing the dividends you received by the initial stock price. This calculation says that for each dollar invested in TGT in 2020, you received $0.0208 in dividends. The capital gain yield is the change in the stock price divided by the initial stock price. This calculation says that for each dollar invested in TGT in 2020, you received $0.3712 in capital gains. Your total percent return of 39.20% means that you made $0.392 for every dollar invested when your gains from both dividends and stock price appreciation are totaled together.
Of course, investors seldom purchase a stock and then sell it exactly one year later. Assume that you purchased shares of Facebook (FB) on June 1, 2020, for $228.50 per share and sold the shares three months later for $261.90. You received no dividends. In this case, your holding period percentage return is calculated as
This 14.62% is your return for a three-month holding period. To compare them to other investment opportunities, you need to express returns on a per-year, or annualized, basis. The holding period returned is converted to an effective annual rate (EAR) using the formula
where m is the number of holding periods in a year.
There are four three-month periods in a year. So, the EAR for this investment is
What happens if you own a stock for more than one year? Your holding period return would have occurred over a period longer than a year, but the process to calculate the EAR is the same. Suppose you purchased shares of FB in May 2015, when it was selling for $79.30 per share. You held the stock until May 2020, when you sold it for $224.59. Your holding period percentage return would be . You more than tripled your money, but it took you five years to do so. Your EAR, which will be smaller than this five-year holding period return rate, is calculated as
### Average Annual Returns
Suppose that you purchased shares of Delta Airlines (DAL) at the beginning of 2011 for $11.19 and held the stock for 10 years before selling it for $40.21. You made on your investment over a 10-year period. This is a 259.34% holding period return. The EAR for this investment is
To calculate the EAR using the above formula, the holding period return must first be calculated. The holding period return represents the percentage return earned over the entire time the investment is held. Then the holding period return is converted to an annual percentage rate using the formula.
You can also use the basic time value of money formula to calculate the EAR on an investment. In time value of money language, the initial price paid for the investment, $11.19, is the present value. The price the stock is sold for, $40.21, is the future value. It takes 10 years for the $11.19 to grow to $40.21. Using the time value of money will result in a calculation of
The EAR formula and the time value of money both result in a 13.65% annual return. Mathematically, the two formulas are the same; one is simply an algebraic rearrangement of the other.
If you earned 13.65% each year, compounded for 10 years, you would have converted your $11.19 per share investment to $40.21 per share. Of course, DAL stock did not increase by exactly 13.65% each year. The returns for DAL for each year are shown in . Some years, the return was much higher than 13.65%. In 2013, the return was almost 133%! Other years, the return was much lower than 13.65%; in fact, in the return was negative in four of the years.
Although an investment in DAL of $11.19 at the beginning of 2011 grew to $40.20 by the end of 2020, this growth was not consistent each year. The amount that the stock was worth at the end of each year is also shown in . During 2011, the return for DAL was −35.79%, resulting in the value of the investment falling to . The following year, 2012, the return for DAL was 46.72%. Therefore, the value of the investment was at the end of 2012. This process continues each year that the stock is held.
The compounded annual return derived from the EAR and time value of money formulas is also known as a geometric average return. A geometric average return is calculated using the formula
where R is the return for each year in the time period for which the average is calculated.
The calculation of the geometric average return for DAL is shown in the right column of . (The slight difference in the geometric average return of 13.64% from the 13.65% derived from the EAR and time value of money calculations is due to rounding errors.)
Looking at , you will notice that the geometric average return differs from the mean return. Adding each of the annual returns and dividing the sum by 10 results in a 22.4% average annual return. This 22.4% is called the arithmetic average return.
The geometric average return will be smaller than the arithmetic average return (unless the returns for all years are identical). This is due to the basic arithmetic of compounding. Think of a very simple example in which you invest $100 for two years. If you have a positive return of 50% the first year and a negative 50% return the second year, you will have an arithmetic average return of , but you will have a geometric average return of . With a 50% positive return the first year, you ended the year with $150. The second year, you lost 50% of that balance and were left with only $75.
Another important fact when studying average returns is that the order in which you earn the returns is not important. Consider what would have occurred if the returns in the two years were reversed, so that you faced a loss of 50% in the first year of your investment and a gain of 50% in the second year of your investment. With a −50% return in the first year, you would have ended that year with only $50. Then, if that $50 earned a positive 50% return the second year, you would have a $75 balance at the end of the two-year period. A negative return of 50% followed by a positive return of 50% still results in an arithmetic average return of 0% and a geometric average return of .
Both the arithmetic average return and the geometric average return are “correct” calculations. They simply answer different questions. The geometric average tells you what you actually earned per year on average, compounded annually. It is useful for calculating how much a particular investment grows over a period of time. The arithmetic average tells you what you earned in a typical year. When we are looking at the historical description of the distribution of returns and want to predict what to expect in a particular year, the arithmetic average is the relevant calculation.
### Measuring Risk
Although the arithmetic average return for Delta Airlines (DAL) for 2011–2020 was 22.4%, there is not a year in which the return was exactly 22.4%. In fact, in some years, the return was much higher than the average, such as in 2013, when it was 132.61%. In other years, the return was negative, such as 2011, when it was −35.79%. Looking at the yearly returns in , the return for DAL varies widely from year to year. In finance, this volatility of returns is considered risk.
### Volatility of Returns
The most commonly used measure of volatility of returns in finance is the standard deviation of the returns. The standard deviation of returns for DAL for the sample period 2011–2020 is 51.9%. Remember that if the normal distribution (a bell−shaped curve) describes returns, then 68% (or about two-thirds) of the time, the return in a particular year will be within one standard deviation above and one standard deviation below the arithmetic average return. Given DAL’s average return of 22.4%, the actual yearly return will be somewhere between −29.5% and 74.29% in two out of three years. A very high return of greater than 74.29% would occur 16% of the time; a very large loss of more than 29.5% would also occur 16% of the time.
As you can see, there is a wide range of what can be considered a “typical” year for DAL. Although we can calculate an average return, the return in any particular year is likely to vary from that average. The larger the standard deviation, the greater this range of returns is. Thus, a larger standard deviation indicates a greater volatility of returns and, hence, more risk.
### Firm-Specific Risk
Investors purchase a share of stock hoping that the stock will increase in value and they will receive a positive return. You can see, however, that even with well-established companies such as ExxonMobil and CVS, returns are highly volatile. Investors can never perfectly predict what the return on a stock will be, or even if it will be positive.
The yearly returns for four companies—Delta Airlines (DAL), Southwest Airlines (LUV), ExxonMobil (XOM), and CVS Health Corp. (CVS)—are shown in . Each of these stocks had years in which the performance was much better or much worse than the arithmetic average. In fact, none of the stocks appear to have a typical return that occurs year after year.
contains a graph of the returns for each of these four stocks by year. In this graph, it is easy to see that DAL and LUV both have more volatility, or returns that vary more from year to year, than do XOM or CVS. This higher volatility leads to DAL and LUV having higher standard deviations of returns than XOM or CVS.
Standard deviation is considered a measure of the risk of owning a stock. The larger the standard deviation of a stock’s annual returns, the further from the average that stock’s return is likely to be in any given year. In other words, the return for the stock is highly unpredictable. Although the return for CVS varies from year to year, it is not subject to the wide swings of the returns for DAL or LUV.
Why are stock returns so volatile? The value of the stock of a company changes as the expectations of the future revenues and expenses of the company change. These expectations may change due to a number of events and new information. Good news about a company will tend to result in an increase in the stock price. For example, DAL announcing that it will be opening new routes and flying to cities it has not previously serviced suggests that DAL will have more customers and more revenue in future years. Or if CVS announces that it has negotiated lower rent for many of its locations, investors will expect the expenses of the company to fall, leading to more profits. Those types of announcements will often be associated with a higher stock price. Conversely, if the pilots and flight attendants for DAL negotiate higher salaries, the expenses for DAL will increase, putting downward pressure on profits and the stock price.
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Investors are interested in both the return they can expect to receive when making an investment and the risk associated with that investment. In finance, risk is considered the volatility of the return from time period to time period. Historical returns are measured by the arithmetic average, and the risk is measured by the standard deviation of returns.
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### How to Double Your Money in Seven Years
In this video, Jim Cramer explains how compounding can help investors build and preserve wealth. He provides suggestions for how young people can use the stock market to build financial independence. |
# How to Think about Investing
## Risk and Return to Multiple Assets
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain the benefits of diversification.
2. Describe the relationship between risk and return for large portfolios.
3. Compare firm-specific and systematic risk.
4. Discuss how portfolio size impacts risk.
### Diversification
So far, we have looked at the return and the volatility of an individual stock. Most investors, however, own shares of stock in multiple companies. This collection of stocks is known as a portfolio. Let’s explore why it is wise for investors to hold a portfolio of stocks rather than to pick just one favorite stock to own.
We saw that investors who owned DAL experienced an average annual return of 20.87% but also a large standard deviation of 51.16%. Investors who used all their funds to purchase DAL stock did exceptionally well during 2012–2014. But in 2020, those investors lost almost one-third of their money as COVID-19 caused a sharp reduction in air travel worldwide. To protect against these extreme outcomes, investors practice what is called diversification, or owning a variety of stocks in their portfolios.
Suppose, for example, you have saved $50,000 that you want to invest. If you purchased $50,000 of DAL stock, you would not be diversified. Your return would depend solely on the return on DAL stock. If, instead, you used $5,000 to purchase DAL stock and used the remaining $45,000 to purchase nine other stocks, you would be diversifying. Your return would depend not only on DAL’s return but also on the returns of the other nine stocks in your portfolio. Investors practice diversification to manage risk.
It is akin to the saying “Don’t put all of your eggs in one basket.” If you place all of your eggs in one basket and that basket breaks, all of your eggs will fall and crack. If you spread your eggs out across a number of baskets, it is unlikely that all of the baskets will break and all of your eggs will crack. One basket may break, and you will lose the eggs in that basket, but you will still have your other eggs. The same idea holds true for investing. If you own stock in a company that does poorly, perhaps even goes out of business, you will lose the money you placed in that particular investment. However, with a diversified portfolio, you do not lose all your money because your money is spread out across a number of different companies.
shows the returns of investors who placed 50% of their money in DAL and the remaining 50% in LUV, XOM, or CVS. Notice that the standard deviation of returns is lower for the two-stock portfolios than for DAL as an individual investment.
As investors diversify their portfolios, the volatility of one particular stock becomes less important. XOM has good years with above-average returns and bad years with below-average (and even negative) returns, just like DAL. But the years in which those above-average and below-average returns occur are not always the same for the two companies. In 2014, for example, the return for DAL was greater than 80%, while the return for XOM was negative. On the other hand, in 2011, when DAL had a return of −35.15%, XOM had a positive return. When more than one stock is held, the gains in one stock can offset the losses in another stock, washing away some of the volatility.
When an investor holds only one stock, that one stock’s volatility contributes 100% to the portfolio’s volatility. When two stocks are held, the volatility of each stock contributes to the volatility of the portfolio. However, the volatility of the portfolio is not simply the average of the volatility of each stock held independently. How correlated the two stocks are, or how much they move together, will impact the volatility of the portfolio.
You will recall from our study of correlation in Regression Analysis in Finance that a correlation coefficient describes how two variables move relative to each other. A correlation coefficient of 1 means that there is a perfect, positive correlation between the two variables, while a correlation coefficient of −1 means that the two variables move exactly opposite of each other. Stocks that are in the same industry will tend to be more strongly correlated than stocks that are in much different industries. During the 2011–2020 time period, the correlation coefficient for DAL and LUV was 0.87, the correlation coefficient for DAL and XOM was 0.35, and the correlation coefficient for DAL and CVS was 0.79. Combining stocks that are not perfectly positively correlated in a portfolio decreases risk.
Notice that investors who owned DAL and LUV from 2011 to 2020 would have had a lower portfolio standard deviation, but not much lower, than investors who just owned DAL. Because the correlation coefficient is less than one, the standard deviation fell. However, because the two stocks are in the same industry and exposed to many of the same economic issues, the correlation coefficient is relatively high, and combining those two stocks provides only a small decrease in risk.
This is because, as airlines, DAL and LUV face many of the same market conditions. In years when the economy is strong, the weather is good, fuel prices are low, and people are traveling a lot, both companies will do well. When something such as bad weather conditions reduces the amount of air travel for several weeks, both companies are harmed. By holding LUV in addition to DAL, investors can reduce exposure to risk that is specific to DAL (perhaps a problem that DAL has with its reservation system), but they do not reduce exposure to the risk associated with the airline industry (perhaps rising jet fuel prices). DAL and LUV tend to experience positive returns in the same years and negative returns in the same years.
On the other hand, investors who added XOM to their portfolio saw a significantly lower standard deviation than those who held just DAL. In years when jet fuel prices rise, harming the profits of both DAL and LUV, XOM is likely to see high profits. Diversifying a portfolio across firms that are less correlated will reduce the standard deviation of the portfolio more.
### Portfolio Size and Risk
As you add more stocks to a portfolio, the volatility, or standard deviation, of the portfolio decreases. The volatility of individual assets becomes less and less important. As we discussed earlier, the risk that is associated with events related to a particular company is called firm-specific risk, or unsystematic, risk. Examples of unsystematic risk would include a company facing a product liability lawsuit, a company inventing a new product, or accounting irregularities being detected. Holding a portfolio of stocks means that if one company you have invested in goes out of business because of poor management, you do not lose all your savings because some of your money is invested in other companies. Portfolio diversification protects you from being significantly impacted by unsystematic risk.
However, there is a level below which the portfolio risk does not drop, no matter how diversified the portfolio becomes. The risk that never goes away is known as systematic risk. Systematic risk is the risk of holding the market portfolio.
We have talked about reasons why a firm’s returns might be volatile; for example, the firm discovering a new technology or having a product liability lawsuit brought against it will impact that firm specifically. There are also events that broadly impact the stock market. Changes in the Federal Reserve Bank’s monetary policy and interest rates impact all companies. Geopolitical events, major storms, and pandemics can also impact the entire market. Investors in stocks cannot avoid this type of risk. This unavoidable risk is the systematic risk that investors in stocks have. This systematic risk cannot be eliminated through diversification.
In addition, as per research conducted by Meir Statman,Meir Statman. “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis 22, no. 3 (September 1987): 353–363. https://doi.org/10.2307/2330969 the standard deviation of a portfolio drops quickly as the number of stocks in the portfolio increases from one to two or three (see Figure 2 illustration in this subsequent article by Statman for context). Increasing the size of the portfolio decreases the standard deviation, and thus the risk, of the portfolio. However, as the portfolio increases in size, the amount of risk reduced by adding one more stock to the portfolio will decrease. How many stocks does an investor need for a portfolio to be well-diversified? There is not an exact number that all financial managers agree on. A portfolio of 15 highly correlated stocks will offer less benefits of diversification than a portfolio of 10 stocks with lower correlation coefficients. A portfolio that consists of American Airlines, Spirit Airlines, United Airlines, Southwest Airlines, Delta Airlines, and Jet Blue, along with a few other stocks, is not very diversified because of the heavy concentration in the airline industry. The term diversified portfolio is a relative concept, but the average investor can create a reasonably diversified portfolio with approximately a dozen stocks.
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As investors hold multiple assets in a portfolio, they are able to eliminate firm-specific risk. However, systematic or market risk remains, even if an investor holds the market portfolio. The return to a portfolio is measured by the arithmetic average, and the risk is measured by the standard deviation of the returns of the portfolio. The risk of the portfolio will be lower than the weighted average of the risk of the individual securities because the returns of the securities are not perfectly correlated. A low or negative return for one stock in a period can be offset by a high return for another stock in the same period.
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### John Bogle and the Buy-and-Hold Strategy
In this video, the legendary investor Jack Bogle, founder and former CEO of Vanguard, discusses strategies for investors. |
# How to Think about Investing
## The Capital Asset Pricing Model (CAPM)
### Learning Outcomes
By the end of this section, you will be able to:
1. Define risk premium.
2. Explain the concept of beta.
3. Compute the required return of a security using the CAPM.
### Risk-Free Rate
The capital asset pricing model (CAPM) is a financial theory based on the idea that investors who are willing to hold stocks that have higher systematic risk should be rewarded more for taking on this market risk. The CAPM focuses on systematic risk, rather than a stock’s individual risk, because firm-specific risk can be eliminated through diversification.
Suppose that your grandparents have given you a gift of $20,000. After you graduate from college, you plan to work for a few years and then apply to law school. You want to use the $20,000 your grandparents gave you to pay for part of your law school tuition. It will be several years before you are ready to spend the money, and you want to keep the money safe. At the same time, you would like to invest the money and have it grow until you are ready to start law school.
Although you would like to earn a return on the money so that you have more than $20,000 by the time you start law school, your primary objective is to keep the money safe. You are looking for a risk-free investment. Lending money to the US government is considered the lowest-risk investment that you can make. You can purchase a US Treasury security. The chances of the US government not paying its debts is close to zero. Although, in theory, no investment is 100% risk-free, investing in US government securities is generally considered a risk-free investment because the risk is so miniscule.
The rate that you can earn by purchasing US Treasury securities is a proxy for the risk-free rate. It is used as an investing benchmark. The average rate of return for the three-month US Treasury security from 1928 to 2020 is 3.36%.“Historical Return on Stocks, Bonds and Bills: 1928–2020.” Damodaran Online. Stern School of Business, New York University, January 2021. http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html You can see that you will not become immensely wealthy by investing in US Treasury bills. Another characteristic of US Treasury securities, however, is that their volatility tends to be much lower than that of stocks. In fact, the standard deviation of returns for the US Treasury bills is 3.0%. Unlike the returns for stocks, the return on US Treasury bills has never been negative. The lowest annual return was 0.03%, which occurred in 2014.Ibid.
### Risk Premium
You know that if you use your $20,000 to invest in stock rather than in US Treasury bills, the outcome of the investment will be uncertain. Your investments may do well, but there is also a risk of losing money. You will only be willing to take on this risk if you are rewarded for doing so. In other words, you will only be willing to take the risk of investing in stocks if you think that doing so will make you more than you would make investing in US Treasury securities.
From 1928 to 2020, the average return for the S&P 500 stock index has been 11.64%, which is much higher than the 3.36% average return for US Treasury bills.Ibid. Stock returns, with a standard deviation of 19.49%, however, have also been much more volatile. In fact, there were 25 years in which the return for the S&P 500 index was negative.
You may not be willing to take the risk of losing some of the money your grandparents gave you because you have been setting it aside for law school. If that’s the case, you will want to invest in US Treasury securities. You may have money that you are saving for other long-term goals, such as retirement, with which you are willing to take some risk. The extra return that you will earn for taking on risk is known as the risk premium. The risk premium can be thought of as your reward for being willing to bear risk.
The risk premium is calculated as the difference between the return you receive for taking on risk and what you would have returned if you did not take on risk. Using the average return of the S&P 500 (to measure what investors who bear the risk earn) and the US Treasury bill rate (to measure what investors who do not bear risk earn), the risk premium is calculated as
### Beta
The risk premium represents how much an investor who takes on the market portfolio is rewarded for risk. Investors who purchase one stock—DAL, for example—experience volatility, which is measured by the standard deviation of that stock’s returns. Remember that some of that volatility, the volatility caused by firm-specific risk, can be diversified away. Because investors can eliminate firm-specific risk through diversification, they will not be rewarded for that risk. Investors are rewarded for the amount of systematic risk they incur.
### Interpreting Beta
The relevant risk for investors is the systematic risk they incur. The systematic risk of a particular stock is measured by how much the stock moves with the market. The measure of how much a stock moves with the market is known as its beta. A stock that tends to move in sync with the market will have a beta of 1. For these stocks, if the market goes up 10%, the stock generally also goes up 10%; if the market goes down 5%, stocks with a beta of 1 also tend to go down 5%.
If a company has a beta greater than 1, then the stock tends to have a more pronounced move in the same direction as a market move. For example, if a stock has a beta of 2, the stock will tend to increase by 20% when the market goes up by 10%. If the market falls by 5%, that same stock will tend to fall by twice as much, or 10%. Thus, stocks with a beta greater than 1 experience greater swings than the overall market and are considered to be riskier than the average stock.
On the other hand, stocks with a beta less than 1 experience smaller swings than the overall market. A beta of 0.5, for example, means that a stock tends to experience moves that are only 50% of overall market moves. So, if the market increases by 10%, a stock with a beta of 0.5 would tend to rise by only 5%. A market decline of 5% would tend to be associated with a 2.5% decrease in the stock.
### Calculating Betas
The calculation of beta for DAL is demonstrated in . Monthly returns for DAL and for the S&P 500 are plotted in the diagram. Each dot in the scatter plot corresponds to a month from 2018 to 2020; for example, the dot that lies furthest in the upper right-hand corner represents November 2020. The return for the S&P 500 was 10.88% that month; this return is plotted along the horizontal axis. The return for DAL during November 2020 was 31.36%; this return is plotted along the vertical axis.
You can see that generally, when the overall stock market as measured by the S&P 500 is positive, the return for DAL is also positive. Likewise, in months in which the return for the S&P 500 is negative, the return for DAL is also usually negative. Drawing a line that best fits the data, also known as a regression line, summarizes the relationship between the returns for DAL and the S&P 500. The slope of this line, 1.39, is DAL’s beta. Beta measures the amount of systematic risk that DAL has.
### CAPM Equation
Because DAL’s beta of 1.39 is greater than 1, DAL is riskier than the average stock in the market. Finance theory suggests that investors who purchase DAL will expect a higher rate of return to compensate them for this risk. DAL has 139% of the average stock’s systematic risk; therefore, investors in the stock should receive 139% of the market risk premium.
The capital asset pricing model (CAPM) equation is
where Re is the expected return of the asset, Rf is the risk-free rate of return, and Rm is the expected return of the market. Given the average S&P 500 return of 11.64% and the average US Treasury bill return of 3.36%, the expected return of DAL would be calculated as
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The capital asset pricing model (CAPM) relates the expected return of an asset to the systematic risk of that asset. Investors will be rewarded for taking on systematic risk. They will not be rewarded for taking on firm-specific risk, however, because that risk can be diversified away.
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# How to Think about Investing
## Applications in Performance Measurement
### Learning Outcomes
By the end of this section, you will be able to:
1. Interpret a Sharpe ratio.
2. Interpret a Treynor measurement.
3. Interpret Jensen’s alpha.
### Sharpe Ratio
Investors want a measure of how good a professional money manager is before they entrust their hard-earned funds to that professional for investing. Suppose that you see an advertisement in which McKinley Investment Management claims that the portfolios of its clients have an average return of 20% per year. You know that this average annual return is meaningless without also knowing something about the riskiness of the firm’s strategy. In this section, we consider some ways to evaluate the riskiness of an investment strategy.
A basic measure of investment performance that includes an adjustment for risk is the Sharpe ratio. The Sharpe ratio is computed as a portfolio’s risk premium divided by the standard deviation of the portfolio’s return, using the formula
The portfolio risk premium is the portfolio return RP minus the risk-free return Rf; this is the basic reward for bearing risk. If the risk-free return is 3%, McKinley Investment Management’s clients who are earning 20% on their portfolios have an excess return of 17%.
The standard deviation of the portfolio’s return, , is a measure of risk. Although you see that McKinley’s clients earn a nice 20% return on average, you find that the returns are highly volatile. In some years, the clients earn much more than 20%, and in other years, the return is much lower, even negative. That volatility leads to a standard deviation of returns of 26%. The Sharpe ratio would be , or 0.65.
Thus, the Sharpe ratio can be thought of as a reward-to-risk ratio. The standard deviation in the denominator can be thought of as the units of risk the investor has. The numerator is the reward the investor is receiving for taking on that risk.
### Treynor Measurement of Performance
Another reward-to-risk ratio measurement of investment performance is the Treynor ratio. The Treynor ratio is calculated as
Just as with the Sharpe ratio, the numerator of the Treynor ratio is a portfolio’s risk premium; the difference is that the Treynor ratio focus focuses on systematic risk, using the beta of the portfolio in the denominator, while the Shape ratio focuses on total risk, using the standard deviation of the portfolio’s returns in the denominator.
If McKinley Investment Management has a portfolio with a 20% return over the past five years, with a beta of 1.2 and a risk-free rate of 3%, the Treynor ratio would be
Both the Sharpe and Treynor ratios are relative measures of investment performance, meaning that there is not an absolute number that indicates whether an investment performance is good or bad. An investment manager’s performance must be considered in relation to that of other managers or to a benchmark index.
### Jensen’s Alpha
Jensen’s alpha is another common measure of investment performance. It is computed as the raw portfolio return minus the expected portfolio return predicted by the CAPM:
Suppose that the average market return has been 12%. What would Jensen’s alpha be for McKinley Investment Management’s portfolio with a 20% average return and a beta of 1.2?
Unlike the Sharpe and Treynor ratios, which are meaningful in a relative sense, Jensen’s alpha is meaningful in an absolute sense. An alpha of 0.062 indicates that the McKinley Investment Management portfolio provided a return that was 6.2% higher than would be expected given the riskiness of the portfolio. A positive alpha indicates that the portfolio had an abnormal return. If Jensen’s alpha equals zero, the portfolio return was exactly what was expected given the riskiness of the portfolio as measured by beta.
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Because investors are not simply interested in returns but are also interested in risk, the success of a portfolio cannot be measured simply by considering the portfolio’s return. In order to compare investment portfolios, risk and return must both be taken into consideration. The Sharpe ratio and the Treynor ratio are two measures that provide a reward-to-risk measure of a portfolio. Jensen’s alpha provides a measure of the abnormal return of a portfolio, considering the portfolio’s risk level.
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# How to Think about Investing
## Using Excel to Make Investment Decisions
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate the average return and standard deviation for a stock.
2. Calculate the average return and standard deviation for a portfolio.
3. Calculate the beta of a stock.
### Average Return and Standard Deviation for a Single Stock
Excel can be used to calculate the average returns and the standard deviation of returns for both a single stock and a portfolio of stocks. It can also be used to calculate the beta for a stock. Historic stock price data for stocks you are interested in analyzing can easily be downloaded from sites such as Yahoo! Finance into Excel. The examples in this section use monthly stock data from December 2017 to December 2020 from Yahoo! Finance.
Monthly price data for AMZN (Amazon) is shown in column B of . To begin, monthly returns must be calculated from the price data using the formula
The ending prices shown in are the last price the stock traded for each month. Each month, the return is calculated under the assumption that you purchased the stock at the last trading price of the previous month and sold at the last price of the current month. Thus, the return for January 2018 is calculated as
This is accomplished in Excel by placing the formula =(B3-B2)/B2 in cell C3. This formula can then be copied down the spreadsheet through row C38. Now that each monthly return is in column C, you can calculate the average of the monthly returns in cell C39 and the standard deviation of returns in cell C40.
Over the three-year period, the average monthly return for AMZN was 3.3%. However, this return was highly volatile, with a standard deviation of 9.33%. Remember that this means that approximately two-thirds of the time, the monthly return from AMZN was between −6.03% and 12.63%.
### Average Return and Standard Deviation for a Portfolio
The Excel screenshot in shows the return and standard deviation calculation for a portfolio. This sample four-stock portfolio contains AMZN, CVS, AAPL (Apple), and NFLX (Netflix). This portfolio is constructed as an equally weighted portfolio; because there are four stocks in this portfolio, each has a weight of 25%.
The monthly returns for each stock are recorded in their respective columns. The portfolio return for each month is calculated as the weighted average of the four monthly individual stock returns. The formula for the portfolio return is
The formula =$B$1*B3+$C$1*C3+$D$1*D3+$E$1*E3 is placed in cell F3. The formula is then copied down column F to calculate the portfolio return for each month. After the monthly portfolio return is calculated, then the average monthly portfolio return is calculated in cell F39. The average monthly portfolio return is 2.69%.
Because this is an equally weighted portfolio, with each of the four stocks impacting the portfolio return in the same way, the average monthly portfolio return of 2.69% is the same as the sum of the average monthly returns of the four stocks divided by four, or .
The standard deviation of the monthly portfolio returns is calculated in cell F40. This four-stock portfolio has a standard deviation of 7.10%. Unlike the average return, this standard deviation is not equal to the average of the standard deviations of returns of the four stocks. In fact, the standard deviation for the portfolio is less than the standard deviation for any one of the four stocks. Remember that this occurs because the stock returns are not perfectly positively correlated. The high return of one of the stocks in one month is dampened by a lower return in another stock during the same month. Likewise, a negative return in one stock during a month might be offset by a positive return in one of the other three stocks during the same month. This is the risk reduction benefit of holding a portfolio of stocks.
### Calculating Beta
The standard deviation of a stock’s returns indicates the stock’s volatility. Remember that the volatility is caused by both firm-specific and systematic risk. Investors will not be rewarded for firm-specific risk because they can diversify away from it. Investors are, however, rewarded for systematic risk. To determine how much of a firm’s risk is due to systematic risk, you can use Excel to calculate the stock’s beta.
To calculate a stock’s beta, you need the monthly return for the market in addition to the monthly market return for the stock. In column B in , the monthly return for SPY, the SPDR S&P 500 Trust, is recorded. SPY is an ETF that was created to mimic the performance of the S&P 500 index by State Street Global Advisors and is often used as a proxy for the overall market performance. The monthly returns for AMZN are visible in column C. It is important that these returns be lined up so that the returns for a particular month for both securities appear in the same row number. Also, you want to place the returns for SPY in the column to the left of the returns for AMZN so that when you create your graph, SPY will automatically appear on the horizontal axis.
You will use a scatter plot to create a graph. In Excel, go to the Insert tab, and then from the Chart menu, choose the first scatter plot option.
Selecting the scatter plot option will result in a chart being inserted that looks like the chart in . Each dot represents one month’s combination of returns, with the return for SPY measured on the horizontal axis and the return for AMZN measured on the vertical axis. Consider, for example, the dot in the furthest upper right-hand section of the figure. This dot is the plot of returns for the month of April 2020, when the return for SPY was 13.36% (measured on the horizontal axis) and the return for AMZN was 26.89% (measured on the vertical axis).
Hover your mouse over one of the dots, and right-click the dot to pull up a chart formatting menu. This menu will allow you to add labels to your axis and polish your chart in additional ways if you would like. Select the option Add Trendline.
When the trendline is inserted, a formatting box will appear on the right of your screen (see ). If it is not already selected, choose the Linear trendline option. Scroll down and select the “Display Equation on chart” option. You will see the equation appear on the screen. This is the equation for the best-fit line that shows how AMZN moves when the market moves. The slope of this line, 1.1477, is the beta for AMZN. This tells you that for every 10% move the overall market makes, AMZN tends to move 11.477%. Because AMZN tends to move a little more than the broader market, it has a little more systematic risk than the average stock in the market.
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Using Excel to manipulate publicly available stock data makes calculating the average return of a stock and the standard deviation of returns easy. The average return for a portfolio and the standard deviation of the portfolio returns can also be calculated easily. By comparing the returns of a stock with the returns of the overall market using Excel charting tools, the beta for a stock, which measures systematic risk, can be determined.
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### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# How Companies Think about Investing
## Why It Matters
One of the most important decisions a company faces is choosing which investments it should make. Should an automobile manufacturer purchase a new robot for its assembly line? Should an airline purchase a new plane to add to its fleet? Should a hotel chain build a new hotel in Atlanta? Should a bakery purchase tables and chairs to provide places for customers to eat? Should a pharmaceutical company spend money on research for a new vaccine? All of these questions involve spending money today to make money in the future.
The process of making these decisions is often referred to as capital budgeting. In order to grow and remain competitive, a firm relies on developing new products, improving existing products, and entering new markets. These new ventures require investments in fixed assets. The company must decide whether the project will generate enough cash to cover the costs of these initial expenditures once the project is up and running.
For example, Sam’s Sporting Goods sells sporting equipment and uniforms to players on local recreational and school teams. Customers have been inquiring about customizing items such as baseball caps and equipment bags with logos and other designs. Sam’s is considering purchasing an embroidery machine so that it can provide these customized items in-house. The machine will cost $16,000. Purchasing the embroidery machine would be an investment in a fixed asset. If it purchases the machine, Sam’s will be able to charge customers for customization.
The managers think that selling customized items will allow the company to increase its cash flow by $2,000 next year. They predict that as customers become more aware of this service, the ability to customize products in-house will increase the company’s cash flow by $4,000 the following year. The managers expect the machine will be used for five years, with the embroidery products increasing cash flows by $5,000 during each of the last three years the machine is used. Should Sam’s Sporting Goods invest in the embroidery machine? In this chapter, we consider the main capital budgeting techniques Sam’s and other companies can use to evaluate these types of decisions. |
# How Companies Think about Investing
## Payback Period Method
### Learning Outcomes
By the end of this section, you will be able to:
1. Define payback period.
2. Calculate payback period.
3. List the advantages and disadvantages of using the payback period method.
The payback period method provides a simple calculation that the managers at Sam’s Sporting Goods can use to evaluate whether to invest in the embroidery machine. The payback period calculation focuses on how long it will take for a company to make enough free cash flow from the investment to recover the initial cost of the investment.
### Payback Period Calculation
In order to purchase the embroidery machine, Sam’s Sporting Goods must spend $16,000. During the first year, Sam’s expects to see a $2,000 benefit from purchasing the machine, but this means that after one year, the company will have spent $14,000 more than it has made from the project. During the second year that it uses the machine, Sam’s expects that its cash inflow will be $4,000 greater than it would have been if it had not had the machine. Thus, after two years, the company will have spent $10,000 more than it has benefited from the machine. This process is continued year after year until the accumulated increase in cash flow is $16,000, or equal to the original investment. The process is summarized in .
Sam’s Sporting Goods is expecting its cash inflow to increase by $16,000 over the first four years of using the embroidery machine. Thus, the payback period for the embroidery machine is four years. In other words, it takes four years to accumulate $16,000 in cash inflow from the embroidery machine and recover the cost of the machine.
### Advantages
The principal advantage of the payback period method is its simplicity. It can be calculated quickly and easily. It is easy for managers who have little finance training to understand. The payback measure provides information about how long funds will be tied up in a project. The shorter the payback period of a project, the greater the project’s liquidity.
### Disadvantages
Although it is simple to calculate, the payback period method has several shortcomings. First, the payback period calculation ignores the time value of money. Suppose that in addition to the embroidery machine, Sam’s is considering several other projects. The cash flows from these projects are shown in . Both Project B and Project C have a payback period of five years. For both of these projects, Sam’s estimates that it will take five years for cash inflows to add up to $16,000. The payback period method does not differentiate between these two projects.
However, we know that money has a time value, and receiving $6,000 in year 1 (as occurs in Project C) is preferable to receiving $6,000 in year 5 (as in Projects B and D). From what we learned about the time value of money, Projects B and C are not identical projects. The payback period method breaks the important finance rule of not adding or comparing cash flows that occur in different time periods.
A second disadvantage of using the payback period method is that there is not a clearly defined acceptance or rejection criterion. When the payback period method is used, a company will set a length of time in which a project must recover the initial investment for the project to be accepted. Projects with longer payback periods than the length of time the company has chosen will be rejected. If Sam’s were to set a payback period of four years, Project A would be accepted, but Projects B, C, and D have payback periods of five years and so would be rejected. Sam’s choice of a payback period of four years would be arbitrary; it is not grounded in any financial reasoning or theory. No argument exists for a company to use a payback period of three, four, five, or any other number of years as its criterion for accepting projects.
A third drawback of this method is that cash flows after the payback period are ignored. Projects B, C, and D all have payback periods of five years. However, Projects B and C end after year 5, while Project D has a large cash flow that occurs in year 6, which is excluded from the analysis. The payback method is shortsighted in that it favors projects that generate cash flows quickly while possibly rejecting projects that create much larger cash flows after the arbitrary payback time criterion.
Fourth, no risk adjustment is made for uncertain cash flows. No matter how careful the planning and analysis, a business is seldom sure what future cash flows will be. Some projects are riskier than others, with less certain cash flows, but the payback period method treats high-risk cash flows the same way as low-risk cash flows.
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The payback period is the simplest project evaluation method. It is the time it takes the company to recoup its initial investment. Its usefulness is limited, however, because it ignores the time value of money.
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# How Companies Think about Investing
## Net Present Value (NPV) Method
### Learning Outcomes
By the end of this section, you will be able to:
1. Define net present value.
2. Calculate net present value.
3. List the advantages and disadvantages of using the net present value method.
4. Graph an NPV profile.
### Net Present Value (NPV) Calculation
Sam’s purchasing of the embroidery machine involves spending money today in the hopes of making more money in the future. Because the cash inflows and outflows occur in different time periods, they cannot be directly compared to each other. Instead, they must be translated into a common time period using time value of money techniques. By converting all of the cash flows that will occur from a project into present value, or current dollars, the cash inflows from the project can be compared to the cash outflows. If the cash inflows exceed the cash outflows in present value terms, the project will add value and should be accepted. The difference between the present value of the cash inflows and the present value of cash outflows is known as net present value (NPV).
The equation for NPV can be written as
Consider Sam’s Sporting Goods’ decision of whether to purchase the embroidery machine. If we assume that after six years the embroidery machine will be obsolete and the project will end, when placed on a timeline, the project’s expected cash flow is shown in :
Calculating NPV is simply a time value of money problem in which each cash flow is discounted back to the present value. If we assume that the cost of funds for Sam’s is 9%, then the NPV can be calculated as
Because the NPV is positive, Sam’s Sporting Goods should purchase the embroidery machine. The value of the firm will increase by $2,835.63 as a result of accepting the project.
Calculating NPV involves computing the present value of each cash flow and then summing the present values of all cash flows from the project. This project has six future cash flows, so six present values must be computed. Although this is not difficult, it is tedious.
A financial calculator is able to calculate a series of present values in the background for you, automating much of the process. You simply have to provide the calculator with each cash flow, the time period in which each cash flow occurs, and the discount rate that you want to use to discount the future cash flows to the present.
Follow the steps in for calculating NPV:
### Advantages
The NPV method solves several of the listed problems with the payback period approach. First, the NPV method uses the time value of money concept. All of the cash flows are discounted back to their present value to be compared. Second, the NPV method provides a clear decision criterion. Projects with a positive NPV should be accepted, and projects with a negative NPV should be rejected. Third, the discount rate used to discount future cash flows to the present can be increased or decreased to adjust for the riskiness of the project’s cash flows.
### Disadvantages
The NPV method can be difficult for someone without a finance background to understand. Also, the NPV method can be problematic when available capital resources are limited. The NPV method provides a criterion for whether or not a project is a good project. It does not always provide a good solution when a company must make a choice between several acceptable projects because funds are not available to pursue them all.
### NPV Profile
The NPV of a project depends on the expected cash flows from the project and the discount rate used to translate those expected cash flows to the present value. When we used a 9% discount rate, the NPV of the embroidery machine project was $2,836. If a higher discount rate is used, the present value of future cash flows falls, and the NPV of the project falls.
Theoretically, we should use the firm’s cost to attract capital as the discount rate when calculating NPV. In reality, it is difficult to estimate this cost of capital accurately and confidently. Because the discount rate is an approximate value, we want to determine whether a small error in our estimate is important to our overall conclusion. We can do this by creating an NPV profile, which graphs the NPV at a variety of discount rates and allows us to determine how sensitive the NPV is to changes in the discount rate.
To construct an NPV profile for Sam’s, select several discount rates and compute the NPV for the embroidery machine project using each of those discount rates. below shows the NPV for several discount rates. Notice that if the discount rate is zero, the NPV is simply the sum of the cash flows. As the discount rate becomes larger, the NPV falls and eventually becomes negative.
The information in is presented in a graph in . We can see that the graph crosses the horizontal axis at about 14%. To the left, or at lower discount rates, the NPV is positive. If you are confident that the firm’s cost of attracting funds is less than 14%, the company should accept the project. If the cost of capital is more than 14%, however, the NPV is negative, and the company should reject the project.
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Net present value (NPV) is calculated by subtracting the present value of a project’s cash outflows from the present value of the project’s cash inflows. A project should be accepted if its NPV is positive and rejected if its NPV is negative.
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### Calculating NPV and IRR
Businesses use NPV and IRR to determine whether or not a project will add value for shareholders. Watch this CFA® Level I Corporate Finance video to learn more. Working along with the video, you will gain practice in using your financial calculator to calculate IRR. |
# How Companies Think about Investing
## Internal Rate of Return (IRR) Method
### Learning Outcomes
By the end of this section, you will be able to:
1. Define internal rate of return (IRR).
2. Calculate internal rate of return.
3. List advantages and disadvantages of using the internal rate of return method.
### Internal Rate of Return (IRR) Calculation
The internal rate of return (IRR) is the discount rate that sets the present value of the cash inflows equal to the present value of the cash outflows. In considering whether Sam’s Sporting Goods should purchase the embroidery machine, the IRR method approaches the time value of money problem from a slightly different angle. Instead of using the company’s cost of attracting funds for the discount rate and solving for NPV, as we did in the first NPV equation, we set NPV equal to zero and solve for the discount rate to find the IRR:
The IRR is the discount rate at which the NPV profile graph crosses the horizontal axis. If the IRR is greater than the cost of capital, a project should be accepted. If the IRR is less than the cost of capital, a project should be rejected. The NPV profile graph for the embroidery machine crossed the horizontal axis at 14%. Therefore, if Sam’s Sporting Goods can attract capital for less than 14%, the IRR exceeds the cost of capital and the embroidery machine should be purchased. However, if it costs Sam’s more than 14% to attract capital, the embroidery machine should not be purchased.
In other words, a company wants to accept projects that have an IRR that exceed the company’s cost of attracting funds. The cash flow from these projects will be great enough to cover the cost of attracting money from investors in addition to the other costs of the project. A company should reject any project that has an IRR less than the company’s cost of attracting funds; the cash flows from such a project are not enough to compensate the investors for the use of their funds.
Calculating IRR without a financial calculator is an arduous, time-consuming process that requires trial and error to find the discount rate that makes NPV exactly equal zero. Your calculator uses the same type of trial-and-error iterative process, but because it uses an automated process, it can do so much more quickly than you can. A problem that might require 30 minutes of detailed mathematical calculations by hand can be completed in a matter of seconds with the assistance of a financial calculator.
All the information your calculator needs to calculate IRR is the value of each cash flow and the time period in which it occurs. To calculate IRR, begin by entering the cash flows for the project, just as you do for the NPV calculation (see ). After these cash flows are entered, simply compute IRR in the final step.
### Advantages
The primary advantage of using the IRR method is that it is easy to interpret and explain. Investors like to speak in terms of annual percentage returns when evaluating investment possibilities.
### Disadvantages
One disadvantage of using IRR is that it can be tedious to calculate. We knew the IRR was about 14% for the embroidery machine project because we had previously calculated the NPV for several discount rates. The IRR is about, but not exactly, 14%, because NPV is not exactly equal to zero (just very close to zero) when we use 14% as the discount rate. Before the prevalence of financial calculators and spreadsheets, calculating the exact IRR was difficult and time-consuming. With today’s technology, this is no longer a major consideration. Later in this chapter, we will look at how to use a spreadsheet to do these calculations.
No Single Mathematical Solution. Another disadvantage of using the IRR method is that there may not be a single mathematical solution to an IRR problem. This can happen when negative cash flows occur in more than one period in the project. Suppose your company is considering building a facility for an upcoming Olympic competition. The construction cost would be $350 million. The facility would be used for one year and generate cash inflows of $950 million. Then, the following year, your company would be required to convert the facility into a public park area for the city, which is expected to cost $620 million. Placing these cash flows in a timeline results in the following ():
The NPV profile for this project looks like . The NPV is negative at low interest rates, becomes positive at higher interest rates, and then turns negative again as the interest rate continues to rise. Because the NPV profile line crosses the horizontal axis twice, there are two IRRs. In other words, there are two interest rates at which NPV equals zero.
Reinvestment Rate Assumption. The IRR assumes that the cash flows are reinvested at the internal rate of return when they are received. This is a disadvantage of the IRR method. The firm may not be able to find any other projects with returns equal to a high-IRR project, so the company may not be able to reinvest at the IRR.
The reinvestment rate assumption becomes problematic when a company has several acceptable projects and is attempting to rank the projects. We will look more closely at the issues that can arise when considering mutually exclusive projects later in this chapter. If a company is simply deciding whether to accept a single project, the reinvestment assumption limitation is not relevant.
Overlooking Differences in Scale. Another disadvantage of using the IRR method to choose among various acceptable projects is that it ignores differences in scale. The IRR converts the cash flows to percentages and ignores differences in the size or scale of projects. Issues that occur when comparing projects of different scales are covered later in this chapter.
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The internal rate of return (IRR) of a project is the discount rate that sets the present value of a project’s cash inflows exactly equal to the present value of the project’s cash outflows. A project should be accepted if its IRR is greater than the firm’s cost of attracting capital.
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# How Companies Think about Investing
## Alternative Methods
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate profitability index.
2. Calculate discounted payback period.
3. Calculate modified internal rate of return.
### Profitability Index (PI)
The profitability index (PI) uses the same inputs as the NPV calculation, but it converts the results to a ratio. The numerator is the present value of the benefits of doing a project. The denominator is the present value of the cost of doing the project. The formula for calculating PI is
For the embroidery machine project that Sam’s Sporting Goods is considering, the PI would be calculated as
The numerator of the PI formula is the benefit of the project, and the denominator is the cost of the project. Thus, the PI is the benefit relative to the cost. When NPV is greater than zero, PI will be greater than 1. When NPV is less than zero, PI will be less than 1. Therefore, the decision criterion using the PI method is to accept a project if the PI is greater than 1 and reject a project if the PI is less than 1.
Note that the NPV method and the PI method of project evaluation will always provide the same answer to the accept-or-reject question. The advantage of using the PI method is that it is helpful in ranking projects from best to worst. Issues that arise when ranking projects are discussed later in this chapter.
### Discounted Payback Period
The payback period method provides a fast, simple approach to evaluating a project, but it suffers from the fact that it ignores the time value of money. The discounted payback period method addresses this flaw by discounting cash flows using the company’s cost of funds and then using these discounted values to determine the payback period.
Consider Sam’s Sporting Goods’ decision regarding whether to purchase an embroidery machine. The expected cash flows and their values when discounted using the company’s 9% cost of funds are shown in . Earlier, we calculated the project’s payback period as four years; that is how long it would take the company to recover all of the cash that it would spend on the project. Remember, however, that the payback period does not consider the company’s cost of funds, so it underestimates the true breakeven time period.
When the cash flows are appropriately discounted, the project still has not broken even by the end of year 5. The discounted payback period would be years. This adjusted calculation addresses the payback period method’s flaw of not considering the time value of money, but managers are still confronted with the other disadvantages. No objective criterion for acceptance or rejection exists because of the lack of a theoretical underpinning for what is an acceptable payback period length. The discounted payback period ignores any cash flows after breakeven occurs; this is a serious drawback, especially when comparing mutually exclusive projects.
### Modified Internal Rate of Return (MIRR)
Financial analysts have developed an alternative evaluation technique that is similar to the IRR but modified in an attempt to address some of the weakness of the IRR method. This modified internal rate of return (MIRR) is calculated using the following steps:
1. Find the present value of all of the cash outflows using the firm’s cost of attracting capital as the discount rate.
2. Find the future value of all cash inflows using the firm’s cost of attracting capital as the discount rate. All cash inflows are compounded to the point in time at which the last cash inflow will be received. The sum of the future value of cash inflows is known as the project terminal value.
3. Compute the yield that sets the future value of the inflows equal to the present value of the outflows. This yield is the modified internal rate of return.
For our embroidery machine project, the MIRR would be calculated as shown in :
1. The only cash outflow is the $16,000 at time period 0.
2. The future value of each of the six expected cash inflows is calculated using the company’s 9% cost of attracting capital. Each of the cash flows is translated to its value in time period 6, the time period of the final cash inflow. The sum of the future value of these six cash flows is $31,595.22. Thus, the terminal value is $31,595.22
3. The interest rate that equates the present value of the outflows, $16,000, to the terminal value of $31,595.22 six years later is found using the formula
The MIRR solves the reinvestment rate assumption problem of the IRR method because all cash flows are compounded at the cost of capital. In addition, solving for MIRR will result in only one solution, unlike the IRR, which may have multiple mathematical solutions. However, the MIRR method, like the IRR method, suffers from the limitation that it does not distinguish between large-scale and small-scale projects. Because of this limitation, the MIRR cannot be used to rank projects; it can only be used to make accept-or-reject decisions.
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The discounted payback period uses the time value of money to discount future cash flows to see how long it will be before the initial investment of a project is recovered. MIRR provides a variation on IRR in which all cash flows are compounded using the cost of capital, resolving the reinvestment rate assumption problem of the IRR method; unlike IRR, which may have multiple mathematical solutions, MIRR will result in one solution. The profitability index is calculated as the NPV of the project divided by the initial cost of the project.
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# How Companies Think about Investing
## Choosing between Projects
### Learning Outcomes
By the end of this section, you will be able to:
1. Choose between mutually exclusive projects.
2. Compare projects with different lives.
3. Compare projects of different scales.
4. Rank projects when resources are limited.
So far, we have considered methods for deciding to accept or to reject a single stand-alone project. Sometimes, managers must make decisions regarding which of two projects to accept, or a company might be faced with a number of good, acceptable projects and have to decide which of those projects to take on during the current year.
### Choosing between Mutually Exclusive Projects
Earlier in this chapter, we saw that the embroidery machine that Sam’s Sporting Goods was considering had a positive NPV, making it a project that Sam’s should accept. However, another, more expensive embroidery machine may be available that is able to make more stitches per minute. Although the initial cost of this heavy-duty machine is higher, it would allow Sam’s to embroider and sell more items each year, generating more revenue. The two embroidery machines are mutually exclusive projects. Mutually exclusive projects compete with one another; purchasing one embroidery machine excludes Sam’s from purchasing the other embroidery machine.
shows the cash outflow and inflows expected from the original embroidery machine considered as well as the heavy-duty machine. The heavy-duty machine costs $25,000, but it will generate more cash inflows in years 3 through 6. Both machines have a positive NPV, leading to decisions to accept the projects. Also, both machines have an IRR exceeding the company’s 9% cost of raising capital, also leading to decisions to accept the projects.
When considered by themselves, each of the machines is a good project for Sam’s to pursue. The question the managers face is which is the better of the two projects. When faced with this type of decision, the rule is to take the project with the highest NPV. Remember that the goal is to choose projects that add value to the company. Because the NPV of a project is the estimate of how much value it will create, choosing the project with the higher NPV is choosing the project that will create the greater value.
### Choosing between Projects with Different Lives
Suppose you are considering starting an ice-cream truck business. You find that you can purchase a used truck for $50,000. You estimate that the truck will last for three years, and you will be able to sell enough ice cream treats to generate a cash inflow of $40,000 during each of those years. Your cost of capital is 10%. The positive NPV of $49,474 for the project makes this an acceptable project.
Another ice-cream truck is also for sale for $50,000. This truck is smaller and will not be able to hold as many frozen treats. However, the truck is newer, with lower mileage, and you estimate that you can use it for six years. This newer truck will allow you to generate a cash inflow of $30,000 each year for the next six years. The NPV of the newer truck is $80,658.
Because both trucks are acceptable projects but you can only drive one truck at a time, you must choose which truck to purchase. At first, it may be tempting to purchase the newer, lower-mileage truck because of its higher NPV. Unfortunately, when comparing two projects that have different lives, a decision cannot be made simply by comparing the NPVs. Although the ice-cream truck with the six-year life span has a much higher NPV than the larger truck, it consumes your resources for a long time.
There are two methods for comparing projects with different lives. Both assume that when the short-life project concludes, another, similar project will be available.
### Replacement Chain Approach
With the replacement chain approach, as many short-life projects as necessary are strung together to equal the life of the long-life project. You can purchase the newer, lower-mileage ice-cream truck and run your business for six years. To make a comparison, you assume that if you purchase the larger truck that will last for three years, you will be able to repeat the same project, purchasing another larger truck that will last for the next three years. In essence, you are comparing a six-year project with two consecutive three-year projects so that both options will generate cash inflows for six years. Your timeline for the projects (comparing an older, larger truck with a newer, lower-mileage truck) will look like :
The present values of all of the cash inflows and outflows from purchasing two of the older, larger trucks consecutively are added together to find the NPV of that alternative. The NPV of this alternative is $86,645, which is higher than the NPV of $80,658 of the newer truck, as shown in :
When using the replacement chain approach, the short-term project is repeated any number of times to equal the length of the longer-term project. If one project is 5 years and another is 20 years, the short one is repeated four times. This method can become tedious when the length of the longer project is not a multiple of the shorter project. For example, when choosing between a five-year project and a seven-year project, the short one would have to be duplicated seven times and the long project would have to be repeated five times to get to a common length of 35 years for the two projects.
### Equal Annuity Approach
The equal annuity approach assumes that both the short-term and the long-term projects can be repeated forever. This approach involves the following steps:
Step 1: Find the NPV of each of the projects.
1. The NPV of the larger, older ice-cream truck is $49,474.
2. The NPV of the smaller, newer ice-cream truck is $80,658.
Step 2: Find the annuity that has the same present value as the NPV and the same number of periods as the project.
1. For the larger, older ice-cream truck, we want to find the three-year annuity that would have a present value of $49,474 when using a 10% discount rate. This is $19,894.
2. For the smaller, newer ice-cream truck, we want to find the six-year annuity that would have a present value of $80,658 when using a 10% discount rate. This is $18,520.
Step 3: Assume that these projects, or similar projects, can be repeated over and over and that these annuities will continue forever. Calculate the present value of these annuities continuing forever using the perpetuity formula.
We again find that the older, larger truck is preferred to the newer, smaller truck.
These methods correct for unequal lives, but managers need to be aware that some unavoidable issues come up when these adjustments are made. Both the replacement chain and equal annuity approaches assume that projects can be replicated with identical projects in the future. It is important to note that this is not always a reasonable assumption; these replacement projects may not exist. Estimating cash flows from potential projects is prone to errors, as we will discuss in Financial Forecasting these errors are compounded and become more significant as projects are expected to be repeated. Inflation and changing market conditions are likely to result in cash flows varying in the future from our predictions, and as we go further into the future, these changes are potentially greater.
### Choosing Projects When Resources Are Limited
Choosing positive NPV projects adds value to a company. Although we often assume that the company will choose to pursue all positive NPV projects, in reality, managers often face a budget that restricts the amount of capital that they may invest in a given time period. Thus, managers are forced to choose among several positive NPV projects. The goal is to maximize the total NPV of the firm’s projects while remaining within budget constraints.
For example, suppose Southwest Manufacturing is considering the seven projects displayed in . Each of the projects has a positive NPV and would add value to the company. The firm has a budget of $200 million to put toward new projects in the upcoming year. Doing all seven of the projects would require initial investments totaling $430 million. Thus, although all of the projects are good projects, Southwest Manufacturing cannot fund them all in the upcoming year and must choose among these projects. Southwest Manufacturing could choose the combination of Projects A and D; the combination of Projects B, C, and E; or several other combinations of projects and exhaust its $200 million investment budget.
To decide which combination results in the largest added NPV for the company, rank the projects based on their profitability index, as is done in . Projects A, E, and F should be chosen, as they have the highest profitability indexes. Because those three projects require a cumulative investment of $200 million, none of the remaining projects can be undertaken at the present time. Doing those three projects will add $78 million in NPV to the firm. Out of this set of choices, there is no combination of projects that is affordable given Southwest Manufacturing’s budget that would add more than $78 million in NPV.
Notice that when choices must be made among projects, the decision cannot be made by simply ranking the projects from highest to lowest NPV. Project D has an NPV of $15 million, which is higher than both the $11 million of Project E and the $7 million of Project F. However, Project D requires $50 million for an initial investment. For the same $50 million of investment funds, Southwest Manufacturer can accept both Projects E and F for a total NPV of $18 million. Investment capital is a scare resource for this company. By ranking projects based on their profitability index, the company is able to determine the best way to allocate its scarce capital for the largest potential increase in NPV.
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Firms may need to choose among a variety of good projects. The projects may have different lives or be differently sized projects that require different amounts of resources. By choosing projects with the highest profitability index, companies can take on the projects that will lead to the greatest increase in value for the company.
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# How Companies Think about Investing
## Using Excel to Make Company Investment Decisions
### Learning Outcomes
By the end of this section, you will be able to:
1. Calculate NPV using Excel.
2. Calculate IRR using Excel.
3. Create an NPV profile using Excel.
A Microsoft Excel spreadsheet provides an alternative to using a financial calculator to automate the arithmetic necessary to calculate NPV and IRR. An advantage of using Excel is that you can quickly change any assumptions or numbers in your problem and recalculate NPV or IRR based on that updated information. Excel is a versatile tool with more than one way to set up most problems. We will consider a couple of straightforward examples of using Excel to calculate NPV and IRR.
Suppose your company is considering a project that will cost $30,000 this year. The cash inflow from this project is expected to be $6,000 next year and $8,000 the following year. The cash inflow is expected to increase by $2,000 yearly, resulting in a cash inflow of $18,000 in year 7, the final year of the project. You know that your company’s cost of funds is 9%. Your company would like to evaluate this project.
### Calculating NPV Using Excel
To calculate NPV using Excel, you would begin by placing each year’s expected cash flows in a sheet, as in row 5 in . One approach to calculating NPV is to use the formula for discounting future cash flows, as is shown in row 6.
shows the present value of each year’s cash flow resulting from the formula. The NPV is then calculated by summing the present values of the cash flows.
Alternatively, Excel is programmed with financial functions, including a calculation for NPV. The NPV formula is shown in cell J7 in below. However, it is important to pay attention to how Excel defines NPV. The Excel NPV function calculates the sum of the present values of the cash flows occurring from period 1 through the end of the project using the designated discount rate, but it fails to include the initial investment at time period zero at the beginning of the project. The NPV function in cell J6 will return $56,947 for this project. You must subtract the initial cash outflow of $30,000 that occurs at time 0 to get the NPV of $26,947 for the project.
When entering the Excel-programmed NPV function, you must remember to include references only to the cells that contain cash flows from year 1 to the end of the project. Then, subtract the initial investment of year 0 to calculate NPV according to the standard definition of NPV—the present values of the cash inflows minus the present value of the cash outflow. Note: Because of the nonstandard use of the term NPV by Excel, many users prefer to use the method described above rather than this predefined function.
### Calculating IRR Using Excel
Excel also provide a function for calculating IRR. This function is shown in , cell J8. The IRR function properly uses all the project’s cash flows, including the initial cash outflow at time 0, in its calculation, unlike the NPV function. This function will correctly return the IRR of 27.7% for the project. shows the completed spreadsheet.
### Using Excel to Create an NPV Profile
Firms often do not know exactly what their cost of attracting capital is, so they must use estimates in their decision-making. Also, the cost of attracting capital can change with economic and market conditions. Especially if markets are volatile, a company may use an NPV profile to see how sensitive their decisions are to changes in financing costs. Excel simplifies the creation of an NPV profile.
Middleton Manufacturing is considering installing solar panels to heat water and provide lighting throughout its plant. To do so will cost the company $800,000 this year. However, this upgrade will save the company an estimated $150,000 in electrical costs each year for the next 10 years. Constructing an NPV profile of this project will allow Middleton to see how the NPV of the project changes with the cost of attracting funds.
First, the project cash flows must be placed in an Excel spreadsheet, as is shown in cells D2 through N2 in . The company’s cost of funds is placed in cell B1; begin by putting in 10% for this rate. Next, the formula for NPV is placed in cell B6; cell B6 shows the NPV of the cash flows in cells D2 through N2, using the rate that is in cell B1.
For reference, compute IRR in cell B4. Calculating IRR is not necessary for creating the NPV profile. However, it gives a good reference point. Remember that if the IRR of a project is greater than the firm’s cost of attracting capital, then the NPV will be positive; if the IRR of a project is less than the firm’s cost of attracting capital, then the NPV will be negative.
An NPV profile is created by calculating the NPV of the project for a variety of possible costs of attracting capital. In other words, you want to calculate NPV using the project cash flows in cells D2 through N2, using a variety of discount rates in cell B1. This is accomplished by using the Excel data table function. The data table function shows how the outcome of an Excel formula changes when one of the cells in the spreadsheet changes. In this instance, you want to determine how the value of the NPV formula (cell B6) changes when the discount rate (cell B1) changes.
To do this, enter the range of interest rates that you want to consider down a column, beginning in cell A7. This example shows rates from 1% to 20% entered in cells A7 through A26. Your Excel file should now look like the screenshot in .
Next, highlight the cells containing the NPV calculation and the range of discount rates. Thus, you will highlight cells A6 through A26 and B6 through B26 (see ). Click Data at the top of the Excel menu so that you see the What-If Analysis feature. Choose Data Table. Because the various discount rates you want to use are in a column, use the “Column input cell” option. Enter “B1” in this box. You are telling Excel to calculate NPV using each of the numbers in this column as the cost of attracting funds in cell B1. Click OK.
After clicking OK, the cells in column B next to the list of various discount rates will fill with the NPVs corresponding to each of the rates. This is shown in .
Now that the various NPVs are calculated, you can create the NPV profile graph. To create the graph, begin by highlighting the discount rates and NPVs that are in cells A7 through A26 and B7 through B26. Next, go to the Insert tab in the menu at the top of Excel. Several different chart options will be available; choose Scatter. You will end up with a chart that looks like the one in . You can customize the chart by renaming it, labeling the axes, and making other cosmetic changes if you like.
You will notice that the NPV profile crosses the x-axis between 13% and 14%; remember that the NPV will be zero when the discount rate that is used to calculate the NPV is equal to the project’s IRR, which we previously calculated to be 13.43%. If the firm’s cost of raising funds is lower than 13.43%, the NPV profile shows that the project has a positive NPV, and the project should be accepted. Conversely, if the firm’s cost of raising funds is greater than 13.43%, the NPV of this project will be negative, and the project should not be accepted.
Middleton Manufacturing can use this NPV profile to evaluate its solar panel installation project. If the managers think that the cost of attracting funds for the company is 10%, then the project has a positive NPV of $121,685 and the company should install the panels. The NPV profile shows that if the managers are underestimating the cost of funds even by 30% and it will really cost Middleton 13% to attract funds, the project is still a good project. The cost of attracting funds would have to be higher than 13.43% for the solar panel project to be rejected.
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Excel spreadsheets provide a way to easily calculate the NPV and IRR of a project. Using Excel to create an NPV profile allows a company to see how much its estimates of the cost of raising funds can err from the true cost and have the project still be an acceptable project.
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### The Tokyo Olympics
The capital investment a city must undertake to host the Olympic Games is massive. Learn more about the capital investments and expenses Tokyo faced as host of the 2020 Summer Olympics and how it was impacted by a global pandemic by watching this video, How the Tokyo Olympics Became the Most Expensive Summer Games Ever.
### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# How Firms Raise Capital
## Why It Matters
The most important job that company managers have is to maximize the value of the company. Some obvious things come to mind when you think of how managers would do this. For example, to maximize the value of American Airlines, the managers need to attract customers and sell seats on flights. They also need to keep costs as low as possible, which means keeping the costs of purchasing fuel and making plane repairs as low as possible. While the concept of keeping costs low is simple, the specific decisions a firm makes can be complex. If American Airlines wants to purchase a new airplane, it needs to consider not just the dollar cost of the initial purchase but also the passenger and cargo capacity of the plane as well as ongoing maintenance costs.
In addition to paying salaries to its pilots and flight attendants, American Airlines must pay to use investors’ money. If the company wants to purchase a new airplane, it may borrow money to pay for the plane. Even if American Airlines does not need to incur debt to buy the plane, the money it uses to buy the plane ultimately belongs to the owners or shareholders of the company. The company must consider the opportunity cost of this money and the return that shareholders are expecting on their investments.
Just as different planes have distinctive characteristics and costs, the different types of financing that American Airlines can use will have different characteristics and costs. One of the tasks of the financial manager is to consider the trade-offs of these sources of funding. In this chapter, we look at the basic principles that managers use to minimize the cost of funding and maximize the value of the firm. |
# How Firms Raise Capital
## The Concept of Capital Structure
By the end of this section, you will be able to:
1. Distinguish between the two major sources of capital appearing on a balance sheet.
2. Explain why there is a cost of capital.
3. Calculate the weights in a company’s capital structure.
### The Basic Balance Sheet
In order to produce and sell its products or services, a company needs assets. If a firm will produce shirts, for example, it will need equipment such as sewing machines, cutting boards, irons, and a building in which to store its equipment. The company will also need some raw materials such as fabric, buttons, and thread. These items the company needs to conduct its operations are assets. They appear on the left-hand side of the balance sheet.
The company has to pay for these assets. The sources of the money the company uses to pay for these assets appear on the right-hand side of the balance sheet. The company’s sources of financing represent its capital. There are two broad types of capital: debt (or borrowing) and equity (or ownership).
is a representation of a basic balance sheet. Remember that the two sides of the balance sheet must be . Companies typically finance their assets through equity (selling ownership shares to stockholders) and debt (borrowing money from lenders). The debt that a firm uses is often referred to as financial leverage. The relative proportions of debt and equity that a firm uses in financing its assets is referred to as its capital structure.
### Attracting Capital
When a company raises money from investors, those investors forgo the opportunity to invest that money elsewhere. In economics terms, there is an opportunity cost to those who buy a company’s bonds or stock.
Suppose, for example, that you have $5,000, and you purchase Tesla stock. You could have purchased Apple stock or Disney stock instead. There were many other options, but once you chose Tesla stock, you no longer had the money available for the other options. You would only purchase Tesla stock if you thought that you would receive a return as large as you would have for the same level of risk on the other investments.
From Tesla’s perspective, this means that the company can only attract your capital if it offers an expected return high enough for you to choose it as the company that will use your money. Providing a return equal to what potential investors could expect to earn elsewhere for a similar risk is the cost a company bears in exchange for obtaining funds from investors. Just as a firm must consider the costs of electricity, raw materials, and wages when it calculates the costs of doing business, it must also consider the cost of attracting capital so that it can purchase its assets.
### Weights in the Capital Structure
Most companies have multiple sources of capital. The firm’s overall cost of capital is a weighted average of its debt and equity costs of capital. The average of a firm’s debt and equity costs of capital, weighted by the fractions of the firm’s value that correspond to debt and equity, is known as the weighted average cost of capital (WACC).
The weights in the WACC are the proportions of debt and equity used in the firm’s capital structure. If, for example, a company is financed 25% by debt and 75% by equity, the weights in the WACC would be 25% on the debt cost of capital and 75% on the equity cost of capital. The balance sheet of the company would look like .
These weights can be derived from the right-hand side of a market-value-based balance sheet. Recall that accounting-based book values listed on traditional financial statements reflect historical costs. The market-value balance sheet is similar to the accounting balance sheet, but all values are current market values.
Just as the accounting balance sheet must balance, the market-value balance sheet must balance:
This equation reminds us that the values of a company’s debt and equity flow from the market value of the company’s assets.
Let’s look at an example of how a company would calculate the weights in its capital structure. Bluebonnet Industries has debt with a book (face) value of $5 million and equity with a book value of $3 million. Bluebonnet’s debt is trading at 97% of its face value. It has one million shares of stock, which are trading for $15 per share.
First, the market values of the company’s debt and equity must be determined. Bluebonnet’s debt is trading at a discount; its market value is . The market value of Bluebonnet’s equity equals . Thus, the total market value of the company’s capital is . The weight of debt in Bluebonnet’s capital structure is . The weight of equity in its capital structure is .
Capital structure refers to how a company finances its assets. The two main sources of capital are debt financing and equity financing. A cost of capital exists because investors want a return equivalent to what they would receive on an investment with an equivalent risk to persuade them to let the company use their funds. The market values of debt and equity are used to calculate the weights of the components of the capital structure. |
# How Firms Raise Capital
## The Costs of Debt and Equity Capital
By the end of this section, you will be able to:
1. Calculate the after-tax cost of debt capital.
2. Explain why the return to debt holders is not the same as the cost to the firm.
3. Calculate the cost of equity capital.
The costs of debt and equity capital are what company lenders (those who allow the firm to use their capital) expect in return for providing that capital. Just as current market values of debt and equity should be used in determining their weights in the capital structure, current market values of debt and equity should be used in determining the costs of those types of financing.
### Cost of Debt Capital
A company’s cost of debt is the interest rate it would have to pay to refinance its existing debt. Because a firm’s existing debt trades in the marketplace, its price changes according to market conditions. The overall credit environment can change due to changing macroeconomic conditions, causing a change in the price of debt securities. In addition, as there are changes in the overall riskiness of the firm and its ability to repay its creditors, the price of the debt securities issued by the firm will change.
The market price of a company’s existing bonds implies a yield to maturity. Recall that the yield to maturity is the return that current purchasers of the debt will earn if they hold the bond to maturity and receive all of the payments promised by the borrowing firm.
### Yield to Maturity and the Cost of Debt
Bluebonnet’s debt is selling for 97% of its face value. This means that for every $100 of face value, investors are currently paying $97 for an outstanding bond issued by Bluebonnet Industries. This debt has a coupon rate of 6%, paid semiannually, and the bonds mature in 15 years.
Because the bonds are selling at a discount, the yield that investors who purchase these bonds will receive if they hold the bond to maturity exceeds 6%. The purchasers of these bonds will receive a coupon payment of every six months for the next 15 years. They will also receive the $100 face value when the bonds mature in 15 years. To calculate the yield to maturity of these bonds using your financial calculator, input the information shown in .
The yield to maturity (YTM) of Bluebonnet Industries bonds is 6.312%. This YTM should be used in estimating the firm’s overall cost of capital, not the coupon rate of 6% that is stated on the outstanding bonds. The coupon rate on the existing bonds is a historical rate, set under economic conditions that may have been different from the current market conditions. The YTM of 6.312% represents what investors are currently requiring to purchase the debt issued by the company.
### After-Tax Cost of Debt
Although current debt holders demand to earn 6.312% to encourage them to lend to Bluebonnet Industries, the cost to the firm is less than 6.312%. This is because interest paid on debt is a tax-deductible expense. When a firm borrows money, the interest it pays is offset to some extent by the tax savings that occur because of this deductible expense.
The after-tax cost of debt is the net cost of interest on a company’s debt after taxes. This after-tax cost of debt is the firm’s effective cost of debt. The after-tax cost of debt is calculated as , where is the before-tax cost of debt, or the return that the lenders receive, and T is the company’s tax rate. If Bluebonnet Industries has a tax rate of 21%, then the firm’s after-tax cost of debt is
This means that for every $1,000 Bluebonnet borrows, the company will have to pay its lenders in interest every year. The company can deduct $63.12 from its income, so this interest payment reduces the taxes the company must pay to the government by . Thus, Bluebonnet’s effective cost of debt is , or .
### Cost of Equity Capital
Companies can raise money by selling stock, or ownership shares, of the company. Stock is known as equity capital. The cost of common stock capital cannot be directly observed in the market; it must be estimated. Two primary methods for estimating the cost of common stock capital are the capital asset pricing model (CAPM) and the constant dividend growth model.
### CAPM
The CAPM is based on using the firm’s systematic risk to estimate the expected returns that shareholders require to invest in the stock. According to the CAPM, the cost of equity (re) can be estimated using the formula
For example, suppose that Bluebonnet Industries has an equity beta of 1.3. Because the beta is greater than one, the stock has more systematic risk than the average stock in the market. Assume that the rate on 10-year US Treasury notes is 3% and serves as a proxy for the risk-free rate. If the long-run average return for the stock market is 11%, the market risk premium is this means that people who invest in the stock market are rewarded for the risk they are taking by being paid 8% more than they would have been paid if they had purchased US Treasury notes. Bluebonnet Industries cost of equity capital can be estimated as
### Constant Dividend Growth Model
The constant dividend growth model provides an alternative method of calculating a company’s cost of equity. The basic formula for the constant dividend growth model is
Thus, three things are needed to complete this calculation: the current stock price, what the dividend will be in one year, and the growth rate of the dividend. The current price of the stock is easy to obtain by looking at the financial news. The other two items, the dividend next year and the growth rate of the dividend, will occur in the future and at the current time are not known with certainty; these two items must be estimated.
Suppose Bluebonnet paid a dividend of $1.50 per share to its shareholders last year. Also suppose that this dividend has been growing at a rate of 2% each year for the past several years and that growth rate is expected to continue into the future. Then, the dividend in one year can be expected to be . If the current stock price is $12.50 per share, then that cost of equity is estimated as
The yield to maturity (YTM) on a company’s outstanding bonds represents the return that debt holders are requiring to lend money to the company. Because interest expenses are tax-deductible, the cost of debt to the company is less than the YTM. The cost of equity capital is not directly observed, so financial managers must estimate this cost. Two common methods for estimating the cost of equity capital are the constant dividend growth model and the capital asset pricing model (CAPM). |
# How Firms Raise Capital
## Calculating the Weighted Average Cost of Capital
By the end of this section, you will be able to:
1. Calculate the weighted average cost of capital (WACC).
2. Describe issues that arise from estimating the cost of equity capital.
3. Describe the use of net debt in calculating WACC.
Once you know the weights in a company’s capital structure and have estimated the costs of the different sources of its capital, you can calculate the company’s weighted average cost of capital (WACC).
### WACC Equation
WACC is calculated using the equation
D%, P%, and E% represent the weight of debt, preferred stock, and common equity, respectively, in the capital structure. Note that must equal 100% because the company must account for 100% of its financing. The after-tax cost of debt is . The cost of preferred stock capital is represented by rpfd, and the cost of common stock capital is represented by re.
For a company that does not issue preferred stock, P% is equal to zero, and the WACC equation is simply
Earlier in this chapter, we calculated the weights in Bluebonnet Industries’ capital structure to be and . We also calculated the after-tax cost of debt for Bluebonnet to be 4.99%. If we use the CAPM to estimate the cost of equity capital for the firm, Bluebonnet’s WACC is computed as
If we use the constant dividend discount model to estimate the cost of equity for Bluebonnet Industries, the WACC is computed as
### Calculating WACC in Practice
The equation for calculating WACC is straightforward. However, issues come up when financial managers calculate WACC in practice. Both the weights of the equity components and the cost of the equity components are needed to calculate the WACC. The WACC that financial managers derive will depend on the assumptions and models they use to determine what weights and capital costs to use.
### Issues in Estimating the Cost of Equity Capital
We have explored two ways of estimating the cost of equity capital: the CAPM and the constant dividend growth model. Often, these methods will produce similar estimates of the cost of capital; seldom will the two methods provide the same value.
In our example for Bluebonnet Industries, the CAPM estimated the cost of equity capital as 13.4%. The constant dividend growth model estimated the cost of capital as 14.24%. The exact value of the WACC calculation depends on which of these estimates is used. It is important to remember that the WACC is an estimate that is based on a number of assumptions that financial managers made.
For example, using the CAPM requires assumptions be made regarding the values of the risk-free interest rate, the market risk premium, and a firm’s beta. The risk-free interest rate is generally determined using US Treasury security yields. In theory, the yield on US Treasury securities that have a maturity equivalent to the length of the company’s investors’ investment horizon should be used. It is common for financial analysts to use yields on long-term US Treasury bonds to determine the risk-free rate.
To estimate the market risk premium, analysts turn to historical data. Because this historical data is used to estimate the future market risk premium, the question arises of how many years of historical data should be used. Using more years of historical data can lead to more accurate estimates of what the average past return has been, but very old data may have little relevance if today’s financial market environment is different from what it was in the past. Old data may have little relevance for investors’ expectations today. Typical market risk premiums used by financial managers range from 5% to 8%.
The same issue with how much historical data should be considered arises when calculating a company’s beta. Different financial managers can calculate significantly different betas even for well-established, stable companies. In April 2021, for example, the beta for IBM was reported as 0.97 by MarketWatch and as 1.25 by Yahoo! Finance.
The CAPM estimate of the cost of equity capital for IBM is significantly different depending on what source is used for the company’s beta and what value is used for the market risk premium. Using a market risk premium of 5%, the beta of 0.97 provided by MarketWatch, and a risk-free rate of 3% results in a cost of capital of
If, instead, a market risk premium of 8% and the beta of 1.25 provided by Yahoo! Finance are used, the cost of capital is estimated to be
The CAPM estimate depends on assumptions made, but issues also exist with the constant dividend growth model. First, the constant dividend growth model can be used only for companies that pay dividends. Second, the model assumes that the dividends will grow at a constant rate in the future, an assumption that is not always reasonable. It also assumes that the financial manager accurately forecasts the growth rate of dividends; any error in this forecast results in an error in estimating the cost of equity capital.
Given the differences in assumptions made when using the constant dividend growth model and the CAPM to estimate the equity cost of capital, it is not surprising that the numbers from the two models differ. When estimating the cost of equity capital for a particular firm, financial managers must examine the assumptions made for both approaches and decide which set of assumptions is more realistic for that particular company.
### Net Debt
Many practitioners use net debt rather than total debt when calculating the weights for WACC. Net debt is the amount of debt that would remain if a company used all of its liquid assets to pay off as much debt as possible. Net debt is calculated as the firm’s total debt, both short-term and long-term, minus the firm’s cash and cash equivalents. Cash equivalents are current assets that can quickly and easily be converted into cash, such as Treasury bills, commercial paper, and marketable securities.
Consider, for example, Apple, which had $112.436 billion in total debt in 2020. The company also had $38.016 billion in cash and cash equivalents. This meant that the net debt for Apple was only $74.420 billion. If Apple used all of its cash and cash equivalents to pay debt, it would be left with $74.420 billion in debt.“Historical Data.” Apple Inc. (AAPL). Yahoo! Finance, accessed October 29, 2021. https://finance.yahoo.com/quote/AAPL/history/
Cash and cash equivalents can be viewed as negative debt. For firms with relatively low levels of cash, this adjustment will not have a large impact on the overall WACC estimate. However, the adjustment can be important for firms that hold substantial cash reserves.
Calculate the weighted average cost of capital (WACC) using the formula
Remember that the WACC is an estimate; different methods of estimating the cost of equity capital can lead to different estimations of WACC.
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### Calculating the Weighted Average Cost of Capital
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# How Firms Raise Capital
## Capital Structure Choices
By the end of this section, you will be able to:
1. Distinguish between a levered and an unlevered firm.
2. Explain why the choice of capital structure does not impact the value of a firm in perfect financial markets.
3. Calculate the interest tax shield.
4. Explain how the interest tax shield encourages the use of leverage.
So far, we have taken the company’s capital structure as given. Each firm’s capital structure, however, is a result of intentional decisions made by the financial managers of the company. We now turn our attention to the issues that financial managers consider when making these decisions.
### The Unlevered Firm
Let’s begin our discussion of capital structure choices by exploring the financing decisions you would face if you were to start a T-shirt business. Suppose that your hometown will host an international cycling competition. The competition itself will last for a month; cyclists will arrive early to train in the local climate. News coverage will be significant, meaning a lot of media personnel will be visiting your area. In addition to fans attending the event, it is expected that tourism will increase over the next year as recreational cyclists will want to ride the route of the professional race. You decide to operate a business for one year that will sell T-shirts highlighting this event.
You will need to make an up-front investment of $40,000 to start the business. You estimate that you will generate a cash flow of $52,000, after you cover all of your operating costs, at the end of next year. You know that these profits are risky; you think a 10% risk premium is appropriate for the level of riskiness of the business. If the risk-free rate is 4%, this means that the appropriate discount rate for you to use is 14%. The value of this business opportunity is
This looks as if it will be a profitable business that should be undertaken. However, you do not have the $40,000 for the up-front investment and will need to raise it.
First, consider raising money solely by selling ownership shares to your family and friends. How much would those shares be worth? The value of the stock would be equal to the present value of the expected future cash flows. The potential stockholders would expect to receive $45,614 in one year. If they agree with you that the riskiness of this T-shirt business warrants a discount rate of 14%, then they will value the stock at
If you sell all of the equity in the company for $45,614 and purchase the equipment necessary for the project for $40,000, you have $5,614 to keep as the entrepreneur who created the business.
This business would be financed 100% by equity. The lack of any debt in the capital structure means the firm would have no financial leverage. The equity in a firm that has no financial leverage is called unlevered equity.
### The Levered Firm
Next, consider borrowing some of the money that you will need to start this T-shirt business. Although the cash flows from the business are uncertain, suppose you are certain that the business will generate at least $18,000. (Perhaps you have a guaranteed order from the cycling competition sponsors.) If you borrowed $17,000 at the risk-free interest rate of 4%, you would owe to the lenders at the end of the year. Because you are certain that you will generate at least $18,000 in cash, which is greater than $17,680, you can borrow the $17,000 without any risk of defaulting.
The $17,000 will not be enough to pay for all the start-up costs. You will also need to raise some capital by selling equity. Because your firm will have some debt, or financial leverage, the equity that you raise will be known as levered equity. The equity holders expect the firm to generate $52,000 in cash flows. Debt holders must be paid before equity holders, so this will leave for the shareholders.
The expected future cash flows generated by the business are determined by the productivity of its assets, not the manner in which those assets are financed. It is the present value of these expected future cash flows that determines the firm’s value. Thus, the firm’s value in perfect capital markets will not change as a result of the company taking on leverage.
The value of your T-shirt business remains at $45,614. You can calculate the value of the levered equity as
Now, shareholders are willing to pay $28,614 for ownership in this company. They expect to get $34,320 in one year in return for purchasing this equity. What discount rate does this imply?
Notice that the expected return to shareholders has risen from 14% for the unlevered firm to 19.94% for the levered firm. Recall that the expected return to shareholders equals the risk-free rate plus a risk premium. The risk-free rate has remained 4%. With leverage, the risk premium rises from 10% to 15.94%.
Why does this risk premium increase? Recall that debt holders are paid before equity holders. Equity holders are residual claimants; they will only receive payment if there is money left over after the debt holders are fully paid. The business is risky. You are certain that the company will have cash flow of at least $18,000 at the end of the year and that $17,680 will be paid to the debt holders. Therefore, if the company performs poorly (perhaps bad weather results in the cancellation of much of the cycling competition) and the cash flows fall way below what you are expecting, there may be only several hundred dollars left for the shareholders.
When the firm was unlevered, if the cash flow at the end of the year was only $18,000, the shareholders would receive $18,000. When leverage is used, the same cash flow would result in shareholders receiving only $320. The risk to the shareholders increases as leverage is used; thus, the risk premium that shareholders require also increases as leverage is used.
### Leverage and the WACC
What happens to the WACC as leverage is used? To figure this out, we must calculate the weights of debt and equity in the capital structure:
In perfect capital markets, an assumption we are making for now, there are no taxes. Because we are using only debt and common stock, the weight of preferred stock is zero, and our WACC can be calculated as
Notice that the use of leverage does not change the WACC. When only equity was used to finance the business, stockholders required a 14% expected return to encourage them to let the firm use their capital. When leverage was used, the debt holders only required a 4% return. However, the existence of debt holders, who stand in front of shareholders in the order of claimants, puts shareholders in a riskier position. There is a greater chance that the shareholders will not receive payment from this uncertain business. Thus, the shareholders require a higher rate of return to let the leveraged firm use their capital.
The cost-savings benefits of using lower-cost debt in your company’s capital structure are exactly offset by the higher return that shareholders require when leverage is used. Mathematically, the increase in the cost of equity when leverage is used will be proportional to the debt–equity ratio. Financial managers refer to this outcome as MM Proposition II. The relationship is expressed by the formula
where ru is the required return to equity holders of the unlevered firm.
shows how the cost of equity increases as the weight of debt in the capital structure increases. As the company uses more debt, the risk to equity holders increases. Because equity holders risk that there will be no residual money after bondholders are paid, the equity holders require a higher rate of return to invest in the company as its use of leverage increases. Although debt holders face less risk than equity holders, the risk that they face increases as the amount of debt the company takes on increases. Once the company’s debt exceeds its guaranteed cash flow, which is $18,000 in our example, debt holders face some risk that the company will not be able to pay them. At that point, the cost of debt rises above the risk-free rate. As the weight of debt approaches 100%, the cost of debt capital approaches the cost of equity of the unlevered firm. In other words, if you financed the T-shirt business solely through the use of debt, the debt holders would require a 14% return because they would be bearing the entire risk of the business and would demand to be rewarded for doing so.
As the leverage of the firm increases, both the cost of debt capital and the cost of equity capital increase. However, as the firm’s leverage increases, it is using proportionately more of the relatively cheaper source of capital—debt— and proportionately less of the relatively more expensive source of capital—equity. Thus, the WACC remains constant as leverage increases, despite the rising cost of each component.
### The Impact of Taxes
In perfect capital markets, the choice of capital structure will not impact the value of the firm or the cost of the firm’s financing. In the real world, however, capital markets are not perfect. One of the important market imperfections is the presence of corporate taxes. Because the choice of capital structure can impact the taxes that a company pays, in the real world, capital structure can impact the cost of capital and the firm’s value.
Assume that your T-shirt business venture will result in earnings before interest and taxes (EBIT) of $52,000 next year and that the corporate tax rate is 28%. If your company is unlevered, it has no interest expense, and its net income will be $37,440, as shown in .
If your company uses leverage, raising $7,000 of financing by issuing debt with a 4% interest rate, it will have an interest expense of $280. This lowers its taxable income to $51,720 and its taxes to $14,481.60. Because interest is a tax-deductible expense, using leverage lowers the company’s taxes.
shows that the company’s net income is lower with leverage than it would be without leverage. In other words, debt obligations will reduce the value of the equity. However, less equity is needed because some of the firm is financed through debt. The important consideration is how the use of leverage changes the total amount of dollars available to all investors. shows this impact.
Using leverage allows the firm to generate $37,518.40 to pay its investors, compared to only $37,440 that is available if the firm is unlevered. Where does the extra $78.40 to pay investors come from? It comes from the reduction in taxes that the firm pays due to leverage. If the company uses no debt, it pays $14,560 in taxes. The levered firm pays only $14,481.60 in taxes, a savings of $78.40.
The $280 that the levered company pays in interest is shielded from the corporate tax, resulting in tax savings of . The additional amount available to investors because of the tax deductibility of interest payments is known as the interest tax shield. The interest tax shield is calculated as
When interest is a tax-deductible expense, the total value of the levered firm will exceed the value of the unlevered firm by the amount of this interest tax shield. The tax-advantage status of debt financing impacts the WACC. The WACC with taxes is calculated as
This formula can be written as
Thus, the WACC with taxes is lower than the pretax WACC because of the interest tax shield. The more debt the firm has, the greater the dollar amount of this interest tax shield. The presence of the interest tax shield encourages firms to use debt financing in their capital structures.
An unlevered firm uses no debt in its capital structure. A levered firm uses both debt and equity in its capital structure. In perfect financial markets, the value of the firm will be the same regardless of the firm’s decision to use leverage. With the tax deductibility of interest expenses, however, the value of the firm can increase through the use of debt. As the level of debt increases, the value of the interest tax shield increases. |
# How Firms Raise Capital
## Optimal Capital Structure
By the end of this section, you will be able to:
1. Explain how increased use of leverage increases the possibility of financial distress.
2. Explain how the possibility of financial distress impacts the cost of capital.
3. Discuss the trade-offs a firm faces as it increases its leverage.
4. Explain the concept of an optimal capital structure.
### Debt and Financial Distress
The more debt a company uses in its capital structure, the larger the dollar value of the interest tax shield. Why, then, do we not see firms using a capital structure composed 100% of debt to maximize this interest tax shield?
The answer to this question lies in the fact that as a company increases its debt, there is a greater chance that the firm will be unable to make its required interest payments on the debt. If the firm has difficulty meeting its debt obligations, it is said to be in financial distress.
A firm in financial distress incurs both direct and indirect costs. The direct costs of financial distress include fees paid to lawyers, consultants, appraisers, and auctioneers. The indirect costs include loss of customers and suppliers.
### Trade-Off Theory
Trade-off theory weighs the advantages and disadvantages of using debt in the capital structure. The advantage of using debt is the interest tax shield. The disadvantage of using debt is that it increases the risk of financial distress and the costs associated with financial distress.
A company has an incentive to increase leverage to exploit the interest tax shield. However, too much debt will make it more likely that the company will default and incur financial distress costs. Calculating the precise balance between these two is difficult if not impossible.
For companies with a low level of debt, the risk of default is low, and the main impact of an increase in leverage will be an increase in the interest tax shield. At some point, however, the tax savings that result from increasing the amount of debt in the capital structure will be just offset by the increased probability of incurring the costs of financial distress. For firms that have higher costs of financing distress, this point will be reached sooner. Thus, firms that face higher costs of financial distress have a lower optimal level of leverage than firms that face lower costs of financial distress.
demonstrates how the value of a levered firm varies with the level of debt financing used. Vu is the value of the unlevered firm, or the firm with no debt. As the firm begins to add debt to its capital structure, the value of the firm increases due to the interest tax shield. The more debt the company takes on, the greater the tax benefit it receives, up until the point at which the company’s interest expense exceeds its earnings before interest and taxes (EBIT). Once the interest expense equals EBIT, the firm will have no taxable income. There is no tax benefit from paying more interest after that point.
As the firm increases debt and increases the value of the tax benefit of debt, it also increases the probability of facing financial distress. The magnitude of the costs of financial distress increases as the debt level of the company rises. To some degree, these costs offset the benefit of the interest tax shield.
The optimal debt level occurs at the point at which the value of the firm is maximized. A company will use this optimal debt level to determine what the weight of debt should be in its target capital structure. The optimal capital structure is the target. Recall that the market values of a company’s debt and equity are used to determine the costs of capital and the weights in the capital structure. Because market values change daily due to economic conditions, slight variations will occur in the calculations from one day to the next. It is neither practical nor desirable for a firm to recalculate its optimal capital structure each day.
Also, a company will not want to make adjustments for minor differences between its actual capital structure and its optimal capital structure. For example, if a company has determined that its optimal capital structure is 22.5% debt and 77.5% equity but finds that its current capital structure is 23.1% debt and 76.9% equity, it is close to its target. Reducing debt and increasing equity would require transaction costs that might be quite significant.
shows the average WACC for some common industries. The calculations are based on corporate information at the end of December 2020. A risk-free rate of 3% and a market-risk premium of 5% are assumed in the calculations. You can see that the capital structure used by firms varies widely by industry. Companies in the online retail industry are financed almost entirely through equity capital; on average, less than 7% of the capital comes from debt for those companies. On the other hand, companies in the rubber and tires industry tend to use a heavy amount of debt in their capital structure. With a debt weight of 63.62%, almost two-thirds of the capital for these companies comes from debt.
Industries that have high betas, such as hotels/gaming and air transport, have high equity costs of capital. More recession-proof industries, such as food processing and household products, have low betas and low equity costs of capital. The WACC for each industry ends up being influenced by the weights of equity and debt the company chooses, the riskiness of the industry, and the tax rates faced by companies in the industry.
A company wants to choose a capital structure that maximizes its value. Although increasing the level of financial leverage, or debt, in the capital structure increases the value of the interest tax shield, it also increases the probability of financial distress. As the weight of debt in the capital structure increases, the return that providers of both debt and equity capital require to entice them to provide money to the firm increases because their risk increases. Trade-off theory suggests that the value of a company that uses debt equals the value of the unlevered firm plus the value of the interest tax shield minus financial distress costs.
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### Capital Structure for Real Estate Companies
Click to view video content |
# How Firms Raise Capital
## Alternative Sources of Funds
By the end of this section, you will be able to:
1. Calculate the required return to preferred shareholders.
2. Calculate the WACC of a firm that issues preferred shares.
3. Discuss how issuing new equity impacts the cost of equity capital.
4. Explain the functionality of convertible debt.
A company can finance its assets in two ways: through debt financing and through equity financing. Thus far, we have treated these sources as two broad categories, each with a single cost of capital. In reality, a company may have different types of debt or equity, each with its own cost of capital. The same principle would apply: the WACC of the firm would be calculated using the weights of each of these types multiplied by the cost of that particular type of debt or equity capital.
### Preferred Shares
Although our calculations of WACC thus far have assumed that companies finance their assets only through debt and common equity, we saw at the beginning of the chapter that the basic WACC formula is
In addition to common stock, a company can raise equity capital by issuing preferred stock. Owners of preferred stock are promised a fixed dividend, which must be paid before any dividends can be paid to common stockholders.
In the order of claimants, preferred shareholders stand in line between bondholders and common shareholders. Bondholders are paid interest before preferred shareholders are paid annual dividends. Preferred shareholders are paid annual dividends before common shareholders are paid dividends. Should the company face bankruptcy, the same priority of claimants is followed in settling claims—first bondholders, then preferred stockholders, with common stockholders standing at the end of the line.
Preferred stock shares some characteristics with debt financing. It has a promised cash flow to its holders. Unlike common equity, the dividend on preferred stock is fixed and known. Also, there are consequences if those preferred dividends are not paid. Common shareholders cannot receive any dividends until preferred dividends are paid, and in some cases, preferred shareholders receive voting rights until they are paid the dividends that are due. However, preferred shareholders cannot force the company into bankruptcy as debt holders can. For tax and legal purposes, preferred stock is treated as equity.
The cost of the preferred equity capital is calculated using the formula
Suppose that Greene Building Company has issued preferred stock that pays a dividend of $2.00 each year. If this preferred stock is selling for $21.80 per share, then the company’s cost of preferred stock is
### Issuing New Common Stock
An existing firm can acquire equity capital to expand its assets in two ways: the retention of earnings or the sale of new shares of stock. Thus far in the chapter, the cost of equity capital calculations have assumed that the earnings were being retained for equity capital financing.
The net income that is left after all expenses are paid is the residual income that belongs to the shareholders. Instead of receiving a fixed payment for letting the firm use their capital (like bondholders who receive fixed interest payments), the reward to shareholders for letting the company use their capital varies from year to year. In a good year, net income and the reward to shareholders is high. In a poor year, net income is low or perhaps even negative.
The net income can either be paid immediately and directly to shareholders in the form of dividends or be retained within the company to fund growth. Shareholders are willing to allow the company to retain these earnings because they expect that the money will be used to fund profitable projects, leading to an even larger reward for shareholders in future years.
Although managers do not need to actively solicit the funds that are retained to fund the business, managers cannot view these funds as costless. The shareholders will require a return on those funds to entice them to allow the company to delay paying the dollars to them immediately in terms of a dividend.
Suppose a company has $1 million in net income one year. If it pays $250,000 in dividends and retains $750,000, then it can finance $750,000 more in assets. If the company has a capital structure of 25% debt and 75% equity and wants to maintain that capital structure, it must increase its debt by $250,000 to balance the increase in equity. Thus, the company would be increasing its total financing by $1 million. Of that financing, 25% would be debt financing, and 75% would be equity financing.
To increase its assets by more than $1 million, the company would need to decide to either change its capital structure or issue new stock. Consider the firm represented by the market-value balance sheet in . The firm has $900 million in assets. These assets are financed by $225 million in debt capital and $675 million in equity capital, resulting in a capital structure of 25% debt and 75% equity.
The retained earnings of $750,000 cause the equity on the balance sheet to increase to $675.75 million. The company could sell $250,000 in bonds, increasing its debt to $225.25 million. shows the impact on the balance sheet. The company has increased its financing by $1,000,000 and can expand assets by $1,000,000. The capital structure remains 25% debt and 75% equity.
What if the economy is in an expansionary period and this company thinks it has the opportunity to grow at a rate of 5%? The company knows that it will need more assets to be able to grow. If it needs 5% more assets, its assets will need to increase to $945 million. To increase the left-hand side of its balance sheet, the company will also need to increase the right-hand side of the balance sheet.
Where does the company get the $45 million in capital? With $750,000 in retained earnings, the company can increase its equity to $675.75 million, but if the remainder of $44.25 million was financed through debt, the company’s capital structure would change. Its weight of debt would increase to
If the company has determined that its optimal capital structure is 25% debt and 75% equity, financing the majority of the growth through debt would cause it to stray from these levels. Funding the growth while keeping the capital structure the same would require the firm to issue new shares. shows how the firm would need to finance $45 million in growth while maintaining its desirable capital structure. The firm would need to increase equity capital to $708.75 million; retained earnings could provide $750,000, but $33 million of new equity would need to be sold.
Investors who are providing common equity financing require a return to entice them to let the company use their money. If this company has paid $0.50 per share in dividends to shareholders and this dividend is expected to increase by 3% each year, we can use the constant dividend growth model to estimate how much common shareholders require. If the stock is trading for $8.00 per share, the cost of common equity financing is estimated as
If, however, the firm must issue more equity, its cost of equity for those additional shares will be higher than 9.44%. Even if shareholders are willing to pay $8.00 per share for the stock, the firm will incur flotation costs; this means the firm will not receive the entire $8.00 to use to finance new assets and generate a profit for shareholders. Flotation costs include the costs of filing with the Securities and Exchange Commission (SEC) as well as the fees paid to investment bankers to place the new shares.
When new equity must be issued to finance the company, the flotation costs must be subtracted from the price of the stock to determine the net proceeds the firm will receive. The cost of this new equity capital is calculated as
where F represents the flotation costs of the new stock issue. If, in this example, the flotation cost is $0.25 per share, then the cost of raising new equity capital is
Issuing new common equity is the most expensive form of raising capital. Equity capital is already expensive because the common shareholders are the residual claimants who will only be paid if all other claimants are paid. Because of this risk, they require a higher rate of return than providers of capital who have precedence in the order of claimants. Flotation costs must be added to this equity cost when new shares are issued to grow the company.
### Convertible Debt
Some companies issue convertible bonds. These corporate bonds have a provision that gives the bondholder the option of converting each bond held into a fixed number of shares of common stock. The number of common shares the bondholder would receive for each bond is known as the conversion ratio.
Suppose that you own a convertible bond issued by Sheridan Sodas with a face value of $1,000 and a conversion ratio of 20 shares that matures today. If you convert the bond today, you will receive 20 shares of Sheridan common stock. If you do not convert, you will receive $1,000. If you convert, you are basically paying $1,000 for 20 shares of Sheridan stock. The conversion price is If Sheridan is trading for more than $50 per share, you would want to convert. If Sheridan is trading for less than $50 per share, you would not want to convert; you would prefer the $1,000. In other words, you will choose to convert whenever the stock price exceeds the conversion price at maturity.
A convertible bond gives the holder an option; the bondholder is able to choose between the face value cash or receiving shares of stock. Options always have a positive value to holders. It is always preferable to be able to choose $1,000 or shares of stock than to simply be given $1,000. There is a possibility that the shares of stock will be more valuable, and there is no way the choice can put you in a worse position.
Because holders of convertible bonds have the valuable option of conversion that holders of nonconvertible bonds do not have, convertible debt can be offered with a lower interest rate. It might seem as if the firm could lower its weighted average cost of capital by issuing convertible debt rather than nonconvertible debt. However, this is not the case. Remember that holders of convertible bonds choose whether they would prefer to convert the bond and become a stockholder or receive the face value of the bond at maturity.
If a bond has a face value of $1,000, the convertible bond holders will consider whether the stock they can convert to is worth more than $1,000. Only when the price of the stock has increased enough that the value of the stock received is more than $1,000 will the bondholders convert. However, this means that instead of paying $1,000, the firm is paying the bondholder in stock worth more than $1,000. In essence, the firm (and the current shareholders) would be selling an equity position in the company for less than the market price of that equity position. The lower interest rate compensates for the possibility that conversion will occur.
Preferred stock is a type of equity capital; the owners of preferred stock receive preferential treatment over common stockholders in the order of claimants. A fixed dividend is paid to preferred shareholders and must be paid before common shareholders receive dividends. Equity capital can be raised through either retaining earnings or selling new shares of stock. Significant flotation costs are associated with issuing new shares of stock, making it the most expensive source of financing. Convertible debt allows the debt holders to convert their debt into a fixed number of common shares instead of receiving the face value of the stock at maturity.
### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# Financial Forecasting
## Why It Matters
Though no one in business has a crystal ball, managers must often do all they can to predict the future as accurately as possible. This is called forecasting. Accounting and finance professionals use past performance along with what they know about the business, its competitors, the economy, and the company’s plans for the future to assemble detailed financial forecasts. Forecasts are useful to many individuals for different reasons. A budget, a type of static forecast, helps accountants and managers see how their plans for the coming year can be achieved. It outlines sales targets and how much can be spent on cost of goods sold and expenses to achieve the company’s bottom-line (net income) targets. Investors use financial forecasts to help guide their decisions to buy, sell or hold stocks or to estimate future potential income through dividends. Perhaps most importantly, for our purposes in finance, forecasts are used to help predict and manage cash flows.
A business can have all the profit in the world at the end of the year, but if it doesn’t raise enough cash (liquidity) to pay the bills and pay its employees halfway through the year, it could still go bankrupt despite being profitable. Forecasting sales and expenses helps assemble a cash forecast—when sales will be collected and when expenses will be paid—so that financial managers can look forward far enough to have enough time to react accordingly and secure short- or long-term financing to meet gaps in cash flow. |
# Financial Forecasting
## The Importance of Forecasting
### Learning Outcomes
By the end of this section, you will be able to:
1. Discuss how to use financial statements in forecasting firm financials.
2. Explain why balance sheet items are important in forecasting a firm’s financial result.
3. Explain why income statement items are important in forecasting a firm’s financial result.
In this section, we will briefly review some of the basic elements of financial statements and how we can analyze historical statements to help assemble financial forecasts. Financial forecasting is important to short- and long-term firm success. It helps a firm plan for the resources it will need, ensuring it will have enough cash on hand at the right time to cover daily operations and capital expenditures. It helps the firm communicate its future potential and manage its shareholders’ expectations. It also helps management assess future risk and set plans in place to mitigate that risk.
Financial forecasting involves using historical data, analysis tools, and other information we can gather to make an educated guess about the future financial performance of the firm. Historical figures provide a reasonable starting point. We use tools such as ratios, common size, and trend analysis to fine-tune our forecast. And finally, we assess what we know about the firm, its competitors, the economy, and anything else that might impact performance and further fine-tune our forecast from there.
It’s important to take a moment to consider the role of ethics in forecasting. Ethics is a huge issue in the world of accounting and finance in general, and forecasting is no different. There can be tremendous pressure on management to perform, to deliver certain levels of profit, and to meet shareholder expectations.
Forecasting, as you will learn throughout this chapter, is not an exact science. There is a great deal of subjectivity that can come into play when forecasting sales and expenses. Ethical behavior is crucial in this area. Those who create forecasts must have a firm understanding of where their data comes from, how reliable it is, and whether or not their assumptions and projections are reasonably justified.
### Financial Statement Foundations
In Financial Statements, you were introduced to a firm called Clear Lake Sporting Goods. You learned about the four key financial statements: the income statement, balance sheet, statement of stockholders’ equity, and statement of cash flows. Each one provides a different view of the firm’s financial health and performance.
Clear Lake Sporting Goods is a small merchandising company (a company that buys finished goods and sells them to consumers) that sells hunting and fishing gear. It uses financial statements to understand its profitability and current financial position, to manage cash flow, and to communicate its finances to outside parties such as investors, governing bodies, and lenders. We will use Clear Lake’s company information and historical financial statements in this chapter as we explore its forecasting process. It’s important to note that in this chapter, we are focusing on just one firm and the one method its managers have chosen to forecast financial performance. There are a variety of types of firms in actual application, and they may choose to forecast their financial performance differently. We are demonstrating just one approach here.
The balance sheet shows all the firm’s assets, liabilities, and equity at one point in time. It also supports the accounting equation in a very clear and transparent way. We find one section of the balance sheet contains all current and noncurrent assets that must total the other section of the balance sheet: total liabilities and equity. In , we see that Clear Lake Sporting Goods has total assets of $250,000 in the current year, which balances with its total liabilities and equity of $250,000.
The income statement reflects the performance of the firm over a period of time. It includes net sales, cost of goods sold, operating expenses, and net income. In , we see that Clear Lake had $120,000 in net sales, $60,000 in cost of goods sold, and $35,000 in net income in the current year.
Finally, the statement of cash flows is used to reconcile net income to cash balances. The statement begins with net income, then reflects adjustments to balance sheet accounts and noncash expenses. The statement of cash flows is broken down into three key categories: operating, investing, and financing. This allows users to clearly see what elements of the business are generating or using cash. In , we see that Clear Lake had cash flow from operating activities of $53,600, cash used for investing activities of ($18,600), and cash used for financing activities of ($15,000).
Another key concept to remember about the financial statements is that the statement of cash flows is necessary to truly understand how the firm is using and generating cash. A common misconception is that if a firm reports net income on its income statement, then it must have plenty of cash, and if it reports a loss, it must be short on cash. Although this can be true, it’s not necessarily the case. Historically speaking, we need the statement of cash flows to get the full picture of how cash was used or generated in the past. Looking to the future, we need a cash flow forecast to plan for possible gaps in cash flow and, potentially, how to make the best use of any cash surplus. Throughout this chapter, we will see how to use historical financial statements to help develop the future cash forecast.
It’s also important to remember that the four financial statements are tied together. Net income from the income statement feeds into retained earnings, which live on the balance sheet. Equity balances on the balance sheet feed information to the statement of stockholders’ equity. And information from both the income statement (net income and noncash expenses) and the balance sheet (changes in working capital accounts) all feed into the statement of cash flows. These relationships will be helpful to understand when using historical statements and preparing forecasts.
### Balance Sheet Analysis
Fully understanding the items that are on the balance sheet and how they relate to one another and to other financial statements will help you create a financial forecast. In Financial Statements, you learned that on the classified balance sheet, both assets and liabilities are broken down into current and noncurrent categories. You also know that the balance sheet must live up to its name—it must balance. This means that total assets (what the company owns) must equal total liabilities and equity (what the company owes).
You continued your financial statement development in Measures of Financial Health, where you saw how to use elements of the balance sheet to assess financial health. Ratios based on balance sheet accounts can be useful for understanding relationships between balance sheet items—how they related in the past and then, in forecasting, how those relationships might change or remain the same in the future. Examples of balance sheet ratios include the current ratio, quick ratio, cash ratio, debt-to-assets ratio, and debt-to-equity ratio.
In Financial Statements, you also explored common-size analysis. To prepare a common-size analysis of the balance sheet, every item on the statement must be expressed as a percentage of total assets. Seeing each item as a percentage—that is, seeing its relationship to total assets—is also helpful for assessing historical statements and how those percentages or relationships can be used to predict future balances in the forecast. For example, in , you can see that Clear Lake’s current assets represented 80% of its total assets in both the current and prior years.
### Income Statement Analysis
Like balance sheet analysis, income statement analysis is also quite helpful in preparing for the forecasting process. In Financial Statements, you learned that the income statement is commonly broken down into a few sections. Cost of goods sold is deducted from net sales to arrive at gross margin. Gross margin refers to the profits earned solely on the sale of the product itself, without consideration for the expenses incurred to run the business. Next, operating expenses are deducted to reflect operating income. Operating income reflects the profits of the core business function. Finally, other items, such as interest expense, tax expense, and other gains and losses, are deducted to arrive at net income, a.k.a. the bottom line. Each segment of the income statement is helpful for assessing past performance and estimating future expenses for a forecast.
You continued your financial statement development in Measures of Financial Health, where you saw how to use elements of the income statement to assess historical financial performance. Ratios based on the income statement can be useful for understanding relationships between net sales and expenses—how they related in the past and then, in forecasting, how those relationships might change or remain the same. Examples of income statement ratios include gross margin, operating margin, and profit margin. Common ratios that incorporate items from both the balance sheet and the income statement include return on assets (ROA), return on equity (ROE), inventory turnover, accounts receivable turnover, and accounts payable turnover.
In Financial Statements, you also explored common-size analysis. To prepare a common-size analysis of the income statement, every item on the statement must be expressed as a percentage of net assets. Seeing each item as a percentage, in terms of its relationship to total sales, is also helpful for assessing historical statements and how those percentages or relationships can be used to predict future balances in the forecast. For example, in , you can see that Clear Lake’s cost of goods sold represented 50% of its net sales in both the current and prior years.
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Forecasting financial statements is important to different users for different reasons. In finance, it’s most important for assessing the value of future growth plans and planning for future cash flow needs.
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# Financial Forecasting
## Forecasting Sales
### Learning Outcomes
By the end of this section, you will be able to:
1. Explain how sales are the main driver for a financial forecast.
2. Determine a past time period to formulate the basis for a financial forecast.
3. Explain the advantages and disadvantages of using past data to forecast future financial performance.
4. Calculate past sales growth averages.
5. Justify adjusting relationships when forecasting future financial performance.
In this section of the chapter, you will begin to explore the first step of creating a forecast: forecasting sales. We will discuss common time frames for sales forecasts and why we use historical data in our forecasts (but only with caution), and we will work through the process of forecasting future sales. We will be using the percent-of-sales method to forecast some expenses for Clear Lake Sporting Goods, the example used throughout the chapter. This method relies on sales data, further highlighting why accuracy in forecasting sales is crucial.
### Sales as the Driver
A significant portion of a business’s costs are driven by how much it sells. Thus, the sales forecast is the necessary first step in preparing a financial forecast. Common costs driven by sales include direct product costs, direct labor costs, and other key variable costs (i.e., costs that vary proportionately to sales), such as sales commissions.
### Looking to the Past
Forecasting sales is not always an easy task, as no one knows the future. We can, however, use the information we do have to forecast future sales with the greatest accuracy possible. Most firms start by looking at the past. A firm may look at past sales from a variety of prior periods. It’s common to look at the past 12 months to estimate the coming 12 months. Looking at 12 consecutive months helps identify seasonality of sales trends, what time of year sales tend to drop off and when they increase, possible sales spikes that might reoccur, and any other trends that tend to appear over a 12-month period. In , we see Clear Lake’s sales by month for the past 12 months.
Past data is often used in conjunction with probabilities and weighted average calculations derived from probabilities. Though used in several areas of forecasting, this approach is particularly common in drafting the sales forecast. Using multiple scenarios and the probability of each scenario occurring is a common approach to estimating future sales.
We can see at first glance that sales remain fairly steady from January to April. Sales then goes up significantly in April and May, seem to peak in June, taper off a bit in July, then decline steeply from August to the end of the year, with the lowest sales being in November and December. Though not exact, it’s easy to quickly see that sales follow a seasonal pattern. We will focus on just one year of data here to keep things simple. However, it’s important to note that when a firm has a seasonal sales pattern, it normally uses more than one year of data to detect and evaluate the pattern. It’s not uncommon for firms to have a seasonal sales pattern that fluctuates based on an external factor such as weather patterns, patterns in business or demand, or other factors such as holidays. Common examples might include farm-based businesses that function on a weather pattern for harvesting and selling crops or a toy company that fluctuates around gift-giving holidays.
This knowledge is helpful when assembling a first pass at the next year’s sales forecast. Using common-size and horizontal (trend) analyses on sales is also helpful, as shown in . We can see the exact percentages that sales went up or down each month:
1. In January, the company had sales of $9,000, which was of the total annual sales.
2. In June, the company had $19,000 sales, which was of the total annual sales and of January sales.
Once a baseline in the 12-month period is assessed, it can also be helpful to look for trends in other ways. For example, the past several years might be assessed to see if there is a trend in total growth or decline for those years on a summary basis or by period. Clear Lake Sporting Goods had sales in the current year of $126,000, in the prior year of $105,000, and two years ago of $89,000. This reflects a 20% increase and an 18% increase, respectively. It might be reasonable to expect a roughly 18 to 20% increase in total sales in the future with only this information in mind. Keep in mind that we will learn about many other factors to consider in the forecast, so the 18 to 20% increase is a good general guideline to consider along with other factors.
Looking at , assume that Clear Lake Sporting Goods decides to take its first pass at a forecast using the more conservative estimate of 18% total sales growth. The company could consider last year’s sales of $126,000 and increase them by 18% to arrive at total forecasted sales for next year of . Next, to get the monthly sales, the company could use the same percent of the total for each month that it did for the previous year. For example, sales in January of last year were 7.1% of the full year’s sales. To find the forecast for the next year, the company would take the forecasted sales of $148,680 for the year and multiply that by 7.1% to get $10,620 for January. The process is repeated for each month to get the full year.
Keep in mind that this is only a starting point. These estimates will be reviewed, assessed, and updated as more information and other factors are taken into consideration.
It can also be helpful to look at a shorter period, perhaps just the last few months, on a more detailed basis (by department, by customer, etc.) to see if there are any possible new trends beginning to develop that might be an indicator of performance in the coming year. For example, Clear Lake Sporting Goods might look at detailed sales records for October, November, and December and see that it had an old product line that was discontinued in early October, which contributed to a 2% reduction in monthly sales. This reduction in monthly sales will likely continue into the new year until the new line the company has signed on begins arriving in stores. Thus, the management team feels they should reduce their first quarter monthly estimates by 2%, as reflected in . January is now , for example.
### Changes for the Future
It’s important to note that the past is not always a reliable predictor of the future. Circumstances can often change to make the future quite different from the past. The business itself may change, the economy can change, the customer base may undergo a shift in demographics or a change in buying habits, new competition may emerge, and so on. So while past performance is helpful, it is only one step in the process of forecasting sales.
Most firms first look to the past to target some form of baseline estimate for the coming year; then, managers begin making adjustments based on what they know about the future. Assume that Clear Lake Sporting Goods will be adding a new brand to its collection of fishing supplies in March. The manufacturer plans to begin running its commercials in late February, which managers anticipate will increase Clear Lake’s monthly sales by about $500 in March, $1,000 in April, $1,400 in May, and $2,000 per month in June, July, and August. We see the monthly adjustments to Clear Lake’s latest sales forecast in . March, for example, is now $10,908 ($10,408 prior estimate plus $500 increase from new brand).
What we have discussed here are only some brief examples of the myriad factors that might impact a sales budget for the coming year. It’s critical that all members of the team take the time and effort to research their customers and the factors that impact their business in order to effectively assess the impact of these factors on future sales. Though only two adjustments were made here, it’s likely that a large firm would have to consider many, many factors that would ultimately impact monthly sales figures before arriving at a conclusion.
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The sales forecast is the foundation on which much of the rest of the forecast is built. Thus, the sales forecast is completed first. Historical sales data and any other information on the firm, its products, the economy, its customers, and its competitors are all used to create the most accurate sales forecast possible.
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# Financial Forecasting
## Pro Forma Financials
### Learning Outcomes
By the end of this section, you will be able to:
1. Define pro forma in the context of a financial forecast.
2. Describe the factors that impact the length of a financial forecast.
3. Explain the risks associated with a financial forecast.
In this section of the chapter, we will move beyond the sales forecast and look at the general nature, length, and timeline of forecasts and the risks associated with using them. We’ll look at why we use them, how long they generally are, what the key variables in a forecast are, and how we pair those variables with common-size analysis to develop the forecast.
### Purpose of a Forecast
As mentioned earlier in the chapter, forecasts serve different purposes depending on who is using them. Our focus here, however, is the world of finance. In this realm, the key purpose of pro forma (future-looking) financial statements is to manage a firm’s cash flow and assess the overall value that the firm is generating through future sales growth. Growing just for the sake of growing doesn’t always yield favorable income for the firm. A larger top-line sales figure that results in lower net income doesn’t make sense in the grand scheme of things. The same is true of profitable sales that don’t generate enough cash flows at the right time. The firm may make a profit, but if it doesn’t manage the timing of its cash flows, it could be forced to shut down if it can’t cover the costs of payroll or keep the lights on. Forecasting helps assess both cash flow and the profitability of future growth. Managers can forecast cash flow using data from forecasted financial statements; this allows them to identify potential gaps in cash and plan ahead in order to either alter collection and payment policies or obtain funding to cover the gap in the timing of cash flows.
### Length of a Forecast
Forecasts can generally be for any length of time. The length generally depends on the user’s needs. A one-year forecast, broken down by month, is quite typical. A firm will often go through a formal budgeting process near the end of its calendar or fiscal year to project financial plans and goals for the coming year. Once that is done, a rolling financial forecast is then done monthly to adjust as time moves on, more information becomes available, and circumstances change.
To be useful, the future forecast for financial planning purposes is almost always calculated as monthly increments rather than one total figure for the next 12 months. Breaking the data down by month allows finance managers to more clearly see fluctuations in cash flows in and out, identify potential gaps in cash flow, and plan ahead for their cash needs.
Forecasts can also be done for several years into the future. In fact, they commonly are. However, once the firm is looking out beyond 12 months, it gets difficult to forecast items with a great degree of accuracy. Often, forecasts beyond a year will be completed only to quarterly or even annual figures rather than monthly. Forecasts that far into the future are often strategic in nature, made more to communicate future plans for the firm than for more detailed decision-making and cash flow planning.
### Common-Size Financials
As we saw earlier in the chapter, common-size analysis involves using historical financial statements as a basis for future forecasts. Financial statements provide a great starting point for analysis, as we can see the relationships between sales and costs on the income statement and the relationships between total assets and line items on the balance sheet.
For example, in , we saw that for the past two years, cost of goods sold has been 50% of sales. Thus, in the first draft of a forecast for Clear Lake, it’s likely that managers would estimate cost of goods sold at 50% of their forecasted sales. We can begin to see why forecasting sales first is crucial and why doing so as accurately as possible is also important.
### Select Variables to Use
A simple way to begin a full financial statement forecast might be to simply use the common-size statements and forecast every item using historical percentages. It’s a logical way to begin a very rough draft of the forecast. However, several variables should be taken into consideration. First, managers must address the cost of an account and determine if it’s a variable or fixed item. Variable costs tend to vary directly and proportionally with production or sales volume. Common examples include direct labor and direct materials. Fixed costs, on the other hand, do not change when production or sales volume increases or decreases within the relevant range. Granted, if production were to increase or decrease by a large amount, fixed costs would indeed change. However, in normal month-to-month changes, fixed costs often remain the same. Common examples of fixed costs include rent and managerial salaries.
So, if we were to approach our common-size income statement, for example, we would likely use the percentage of sales as a starting point to forecast variable items such as cost of goods sold. However, fixed costs may not be accurately forecast as a percentage of sales because they won’t actually change with sales. Thus, we would likely look at the history of the dollar values of fixed costs in order to forecast them.
### Determine Potential Changes in Variables
So far, we have focused on using historical common-size statements to create a draft (not a final version) of the forecast. This is because the past isn’t always a perfect indicator of the future, and our finances don’t always follow a linear pattern. We use the past as a good starting point; then, we must assess what else we know to fine-tune and make adjustments to the forecast.
Many items impact the forecast, and they will vary from one organization to another. The key is to do research, gather data, and look around at the market, the economy, the competition, and any other factors that have the potential to impact the future sales, costs, and financial health of the company. Though certainly not an exhaustive list, here are a few examples of items that may impact Clear Lake Sporting Goods.
1. It has an old product line that was discontinued in early October, contributing to a 2% reduction in monthly sales that will likely continue into the new year until a new line begins arriving in stores.
2. It will be adding a new brand to its collection of fishing supplies in March. The manufacturer plans to begin running commercials in late February. Managers anticipate that this will increase Clear Lake’s monthly sales by about $500 in March, $1,000 in April, $1,400 in May, and $2,000 per month in June, July, and August.
3. The company has just finished updating its employee compensation package. It goes into effect in January of the new year and will result in an overall 4% increase in the cost of labor.
4. The landlord indicated that rent will increase by $50 per month starting July 1.
5. Some fixed assets will be fully depreciated by the end of March. Thus, depreciation expense will go down by $25 per month beginning in April.
6. There are rumors of new regulations that will impact the costs of importing some of the more difficult-to-obtain hunting supplies. Managers aren’t entirely sure of the full impact of the new legislation at this time, but they anticipate that it could increase cost of goods sold for the affected product line when the new legislation goes into effect in the last quarter. Their best estimate is that it could increase the overall cost of goods sold by up to 2%.
We will use all of this data later in the chapter when we are ready to compile a complete forecast for Clear Lake.
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Pro forma financial statements are forward looking in nature. They use the sales forecast, historical data, financial statement analyses, relationships between accounts and statements, and any other information known about the firm, the environment, and the future to create the most accurate financial statement forecast possible.
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### What Is a Pro Forma?
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# Financial Forecasting
## Generating the Complete Forecast
### Learning Outcomes
By the end of this section, you will be able to:
1. Generate a forecasted income statement that incorporates pertinent sales, functional, and policy variables.
2. Generate a forecasted balance sheet.
3. Connect the balance sheet and income statement forecasts with appropriate feedback linkages.
In this section of the chapter, we will tie together what we have learned so far about forecasting sales, common-size analysis, and using what we know about the company and its environment to create a full set of pro forma (forward-looking or forecasted) financial statements.
### Forecast the Income Statement
To arrive at a fully forecasted income statement, we use historical income statements, common-size income statements, and any additional information we have about future sales and costs, such as the effects of the economy and competition. As we saw earlier in the chapter, we begin with forecasted sales because they are the basis for many of the forecasted costs.
Let’s begin with the sales forecast for Clear Lake Sporting Goods that we saw earlier in the chapter, in , and use it along with the prior year income statement by month shown in . We will consider other data we have about the business to begin creating a full income statement (see ).
The first two key points regarding product lines have already been built into the sales forecast. Notice that the cost of goods sold was 50% in the prior year. However, based on possible future legislation, to be conservative, we should increase the cost of goods sold by 2% in the last quarter of the year. Thus, we will forecast cost of goods sold at 50% of sales in the first nine months and increase it to 52% in the last three months of the year.
Rent is a fixed cost that historically amounts to $458 per month. However, we know that the landlord is increasing rent by $50 starting on July 1. Thus, we will forecast rent at the same fixed cost of $458 per month for the first six months and increase it to $508 per month for the second half of the year.
Depreciation, also a fixed cost, was historically $300 per month. However, we know that depreciation expense will go down by $25 beginning in April. Thus, we forecast depreciation at $300 for the first three months and at $275 for the last 9 months.
Salaries expense has historically been $450 per month. However, we know that the company is implementing a new compensation program on January 1 that will increase salaries expense by 4% ($18). Thus, we will forecast salaries for the whole year at $468.
Utilities expense seems to vary somewhat by sales from month to month, as shops are open longer hours during their busy season. However, the total utilities expense is not expected to change for the coming year. Thus, the forecast for utilities expense remains at $2,500, broken down by month as a percentage of sales.
Interest expense is a fixed cost and isn’t anticipated to change. Thus, the same $167 interest expense per month is forecast for the coming year.
Finally, income tax expense is forecasted as a percentage of operating income because tax liability is incurred as a direct result of operating income. shows the next 12 months’ forecast for Clear Lake Sporting Goods using all of this data.
### Forecast the Balance Sheet
Now that we have a reasonable income statement forecast, we can move on to the balance sheet. The balance sheet, however, is entirely different from the income statement. It requires a bit more research and additional assumptions. Just like the income statement, it’s often a work in progress. A first draft is a good starting point, but adjustments must be made once it is created, and all the interrelationships between the statements, cash flow in particular, are taken into consideration.
The balance sheet is a bit more difficult to forecast because the statement reflects balances at just a given point in time. Account balances change daily, so forecasting just one snapshot in time for each month can be a challenge. A good starting point is to assess general company financial policies or rules of thumb. For example, assume that Clear Lake pays most of its vendors on net 30-day terms. A good way to forecast accounts payable on the balance sheet might be to add up the cost of goods sold from the forecasted income statement for the prior month. For example, in , we see that Clear Lake has forecasted its accounts payable for March as the cost of goods sold in March from its forecasted income statement.
For accounts receivable, Clear Lake generally receives payment from customers within net 90-day terms. Thus, it uses the sum of the current and prior two months’ forecasted sales to estimate its accounts receivable balance.
Inventory will vary throughout the year. For the first six months, the company tries to build inventory for four months of sales. Once the busy season hits, inventory goes down to three months’ worth of future sales, then finally drops to only two months of sales in December. Thus, managers use their sales forecast by month to estimate their inventory ending balance each month.
The equipment balance is forecasted by reducing the prior month’s balance by the forecasted depreciation expense on the forecasted income statement.
Unearned revenue is historically around 50% of the current month’s sales. Thus, Clear Lake estimates its unearned revenue balance each month by taking the current month’s net sales from the forecasted income statement and multiplying it by 50%.
Short-term investments, notes payable, and common stock are not anticipated to change, so the current balance is forecasted to remain the same for the next 12 months.
To forecast the ending balance for retained earnings for each month, managers add the monthly net income from the forecasted balance sheet to the prior balance and subtract a quarterly $10,000 dividend.
Once all of these accounts are completed, the balance sheet is out of balance. Given that all of these events are somewhat related but are not tied together dollar for dollar, it’s not surprising when the forecasted balance sheet is finished and does not balance. To complete the first draft (see ), the cash account is used as a variable and plugged in to make the balance sheet balance. Notice that by the end of the year, the company has $59,905 in cash. However, look at what happens midyear—the cash account falls to only $8,782. In the next section, we will generate a cash flow forecast, which will allow Clear Lake to update its balance sheet forecast once it estimates what it will do to cover the cash flow gaps.
### Linkages between the Forecasted Balance Sheet and the Income Statement
Notice that in the discussion in the prior section on the balance sheet forecast, a lot of the information in the forecasted income statement was used to generate the forecasted balance sheet. The balance sheet accounts generally depend on activity reported in the income statement. For example, for many firms, the balance in their accounts receivable account is tied to their sales. Looking at historical balances in the accounts receivable account and how those relate to historical sales will help determine how to use the forecasted future sales to estimate the future balance of accounts receivable.
The same is true of accounts payable. Looking at past balances, past expenses (normally cost of goods sold), and the firm’s payment terms for its vendors allows managers to use forecasted cost of goods sold or other expenses to estimate the balance in the accounts payable account.
We learned in Financial Statements that net income flows into retained earnings. Thus, the net income from the forecasted income statement can be used to help estimate the ending balance in retained earnings. If the firm intends to issue any dividends in the coming year, managers should also estimate that reduction in their forecast.
It’s also common to find other general policies or procedures that help drive performance and aid in forecasting balances. For example, if the company has a goal of maintaining a certain level of inventory or a minimum balance in its cash account, that information can be used to guide the estimate for those accounts.
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Interrelationships among historical data, the forecasted income statement, and the forecasted balance sheet are all used to estimate each line item in the financial statements.
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# Financial Forecasting
## Forecasting Cash Flow and Assessing the Value of Growth
### Learning Outcomes
By the end of this section, you will be able to:
1. Generate a cash flow forecast.
2. Assess a cash flow forecast to determine future cash funding needs.
3. Use pro forma financial statements and cash flow forecasts to assess the value of growth to the firm.
In this section of the chapter, we will use the forecasted income statement, forecasted balance sheet, and other information we know about the firm’s policies and goals for the coming year to generate and assess a cash flow forecast.
### Create a Cash Flow Forecast
A cash flow forecast isn’t overly complex, yet it is not easy to assemble because it requires making many assumptions about the future. A cash forecast begins with the beginning cash balance, adds anticipated cash inflows, and deducts anticipated cash outflows. This identifies cash surpluses and shortages.
For Clear Lake Sporting Goods, for example, we see in that the company begins with cash of $42,581,000 in January of the new year. Next, it lists the cash inflows, or cash received from customers. Given the assumption that customers pay in 90-day terms, the cash flow is filled in by plugging in the sales forecast for the three prior months. For example, the cash flow from customers of $10,508 for June is the same as the net sales forecast for March (see ).
Next, Clear Lake identifies cash outflows, which include accounts payable, salaries, rent, utilities, dividends, and interest payments. Accounts payable are normally paid within 30 days, so the forecast for cost of goods sold for the prior month is used as an estimate of amount paid for payables. For example, in , we see that the accounts payable settled in June of $8,610 is the cost of goods sold for May from the forecasted income statement.
Salaries are paid monthly and thus represent the same recurring monthly cash outflow, as does rent. Utilities, like accounts payable, are assumed to be paid within 30 days. Thus, the cash outflow for utilities is the utilities expense for the prior month from the forecasted income statement.
Management intends to pay a quarterly dividend of $10,000. Thus, in , we see $10,000 cash outflows forecasted for March, June, September, and December. Interest on the long-term liability is paid quarterly. Thus, the $500 cash outflows in March, June, September, and December are simply the monthly interest expense of $167 from the income statement, summed for each quarter.
### Using a Cash Forecast to Determine Additional Funds Needed
Finally, at the end of the cash flow forecast, cash outflows are subtracted from the cash inflows. This identifies whether a cash surplus (extra) or cash deficit (not enough) exists for each month. For example, in , we see that in March, Clear Lake is forecasting $4,800 of cash inflows and $17,800 of total cash outflows, which results in a cash deficit of $13,000.
Clear Lake has a general policy to not let its cash balance fall below $35,000. Thus, managers need to assess their monthly balances and potential deficits and identify months when financing is necessary. For example, the deficit of $13,000 in March is enough to push the cash balance lower than $35,000. Thus, it’s estimated that the company will need $5,000 in short-term financing in March. It has an estimated surplus in April, so $3,000 of the borrowing is returned.
### Assessing the Value of Growth
It’s a fairly common assumption that most, if not all, businesses want to grow. While it certainly can be good as a firm to grow in size, growth just for the sake of growth isn’t necessarily a good goal. A firm can grow in size based on customers, employees, locations, or simply sales. However, that doesn’t mean that the growth will increase profits. Growth may increase profits, but this is not a safe assumption. Scaling up operations takes careful planning, which includes monitoring the profitability of the sales and, of course, the cash flow it would require. Growing a business can require more inventory, more locations, more equipment, and more manpower, all of which cost money. Even if the forecasted growth is profitable, it may pose problems from a cash flow perspective. It’s important that the firm review not only its forecasted income statement and balance sheet but also its cash forecast, as this can reveal some serious gaps in funding depending on the extent, timing, and nature of the planned growth.
For example, assume that Clear Lake Sporting Goods intends to run a large-scale ad campaign to boost sales in its busy season. Historically, the store relied primarily on its prime location for high volumes of retail foot traffic. Managers felt, however, that given the increase in competition, they could boost sales significantly by running the ad campaign in the first quarter. The campaign would cost $30,000. Forecasts already reflect a cash deficit at the end of the first quarter of $13,000, so the additional $30,000 ad campaign, which would require payment up front, would create a much larger need for funding. It’s also important that managers look at the increased cost of doing business along with the increased cost in advertising to ensure that the move would be profitable. Fortunately, Excel or other forecasting software can be used to create a forecast with formulas that tie together, making scenario analysis such as this a much easier process.
### Scenarios in Forecasting
Forecasting is almost never a linear process. In other words, we don’t do one forecast and call it good. The first draft is completed using historical data, and then changes are made a bit at a time as all potential variables are assessed for their impact on the forecast. It’s quite common to then use the work-in-progress forecast to complete scenario analysis. This is particularly true when the forecast is completed in Excel or other budgeting or forecasting software. Elements of the forecast can be changed to see what the overall impact would be to the firm. Assuming the forecast is set up using formulas in Excel or other software, a change to one figure or one variable would then “ripple” through the forecast to reflect the overall impact.
Often, a firm may complete an initial forecast (scenario) under the assumption that the economy is in a “normal state.” The firm can then alter the initial forecast for different scenarios, such as the economy in a recession or the economy in a state of expansion. This helps the firm understand different possible future states and highlights how changes in the economy such as inflation may cause revenue and expenses to increase.
Assume that Clear Lake’s initial forecast is created under the assumption that the economy will remain average. Management also wants to know the worst-case scenario. What will their financial results look like if the economy were in a recession, for example? If management assumes their sales would drop to only 60% of the prior year sales in a recessionary economy, they could alter the formula in Excel driving their sales and variable costs, resulting in a new pro forma income statement. In , we can see that net income would drop to $16,391 under this assumption, compared to the net income of $47,653 forecasted under average economy assumptions in .
Though creating a full forecast in Excel can be a bit complex, it is a powerful tool that is useful for analysis. Elements can be used to vary just about anything, from something small such as a 1% increase in the cost of a product to a company-wide increase in salaries, the introduction of an entire new product line, or the purchase of a new production machine, among other possibilities.
For example, assume that Clear Lake has completed a first pass at its forecast and is reviewing the forecasted profit for the next 12 months. Managers feel the profit is currently low, as they always want to target a certain percentage. They might tinker with variables in the forecast file to see the impact on profits of potential changes they are considering. They may reduce the new salaries package by a percentage point to see if it gets them closer to their goal. They may adjust cost of goods sold by a certain percentage if they feel they can negotiate with vendors to work down their costs. They may adjust rent and see if they can find a better retail location to either reduce costs or increase sales due to increased foot traffic in a new location. They may save an entirely new version of the forecast and change it drastically to see what investing in opening a second retail location would do.
As you can see, the list of possibilities is endless. Though the main goal of financial managers may be cash planning, the power of a well-developed forecast is tremendous. It can help assess potential growth, new opportunities, and even small changes in the business as well.
### Sensitivity Analysis in Forecasting
Sensitivity analysis will often look at the change in just one variable rather than the entire scenario. It examines how sensitive a particular output (commonly net income) will be to a change in a particular underlying input (sales or costs, for example). What if sales are 10% more or less than forecasted? What if the prices the firm can charge its customers are 10% more or less? What if the cost of goods sold increases by 10%? The purpose is to see which variables are crucial to “get right.” It isn’t worth spending a lot of research dollars to make sure you are accurately predicting a variable if that variable won’t notably change the outcome. However, a slight change in other variables may have significant impact.
Using pro forma financial statements created in Excel allows management to quickly generate new pro forma financials and see the impact that each possible variable might have on the overall financial results.
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Once the income statement and balance sheet forecasts are complete, data from those statements, information on company policies, and account relationships are used to generate a cash forecast. The cash forecast is important for identifying any gaps in cash flow so that financial managers can plan for cash needs. It’s also important to review not only the cash forecast but all forecasted financial statements to assess the overall impact and value of proposed firm growth.
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### Cash Flow Forecasting Explained: How to Complete a Cash Flow Forecast Example
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# Financial Forecasting
## Using Excel to Create the Long-Term Forecast
### Learning Outcomes
By the end of this section, you will be able to:
1. Generate a financial statement forecast using spreadsheet tools.
2. Connect the balance sheet and income statement using appropriate formula referencing.
3. Use spreadsheet functions to generate appropriate iterations that balance financial forecasts.
Throughout this chapter, we have seen forecasted financial statements for Clear Lake Sporting Goods along with its forecasted cash flow. These statements could all have been generated by hand, of course, but that wouldn’t be an effective use of time. As mentioned in prior sections, several different types of software can be quite effective in making the forecasting process faster and more flexible. In this section, we will review just one common option, Microsoft Excel.
### Using the “Sheet”
Creating a budget in Excel can be very simple or extremely complex, depending on the size and complexity of the business and the number of formulas and dependencies that are written into the Excel workbook.
Creating the forecast in Excel follows the same steps and flow we just explored in this chapter but with the power of a software program to do the math for you. We begin with the sales forecast, which uses several key formulas in Excel.
1. First, sales are projected to be 18% higher than the prior year. Thus, a total projection for the year is calculated using a simple link and multiplication function tied to last year’s total sales. In , you can see the formula in cell O4 is “='Figure 18.12'!N4*1.18”. This formula simply does the math to increase the prior year’s sales by 18%.
2. Next, the sales are distributed by month. In , we see in cell B5 that the forecasted income statement sheet is linked to the percent of annual sales from the Prior Year Income Statement () sheet. Then, in cell B4, January sales are estimated with a formula that multiplies the total forecasted sales in O4 by the percent of annual sales for January of the prior year. Notice that the formula then multiplies that product by 0.98. This is because Clear Lake discontinued a product line in the last quarter of the prior year, and management feels that this will reduce sales in the first quarter of the new year by roughly 2%.
3. As Clear Lake continues to fill out its forecasted income statement, the next formula we see is a simple sum formula to calculate net sales in B8 (see ). It’s a simple formula that subtracts sales returns and allowances in B7 from gross sales in B4. Similar formulas are also found in B10 for gross margin and B18 for net income.
4. In cell B9, we see a multiplication formula that multiplies sales from B4 by 0.5, or 50%. This is because management feels that cost of goods sold will remain the same as last year, in most quarters at least, and last year’s percentage was 50%.
5. Rent, depreciation, and salaries are all simply typed in, as they are fixed expenses that remain the same as last year.
6. The utilities calculation, found in cell B14, is somewhat similar to the sales calculation. The total utilities expense from O14 is multiplied by the current month’s sales in B4 divided by the total annual sales in O4. This spreads out the utility cost by month based on the percentage of annual sales.
Clear Lake’s forecasted balance sheet ties very closely to both the forecasted income statement and the prior year’s income statement. In , we see in C7 an addition formula using the sum of the current month and three months of prior sales as an estimate of the ending accounts receivable balance. The formula for inventory is similar but forward looking. In C8, inventory is estimated by adding the cost of goods sold for the current month and next three months from the forecasted income statement.
Total current assets in C10 is calculated with a SUM formula that adds together the values in all the selected cells. Amounts such as short-term investments and common stock that are not anticipated to change are simply typed as a number in the cell. Much like in the income statement, subtotals are found in C13 for total assets, C17 for current liabilities, C24 for total equity, and C25 for total liabilities and equity. Retained earnings in C23 pulls the ending retained earnings balance from the end of last year (hidden in column B) and adds the net income for January in the forecasted income statement to get the current month’s ending balance.
Much like the balance sheet, the cash forecast also relies heavily on data from the forecasted income statement as well as the forecasted balance sheet. To begin the year, in , we see that the formula in B4 pulls the cash balance from the forecasted balance sheet. In B6, the formula pulls the sales for the three months prior from the previous year’s income statement. This is because it’s assumed that cash is collected from customers 90 days after the sale. The same approach is used for accounts payable, rent, salaries, and utilities. The formulas pull the expenses from a prior month depending on the assumed timing for payment. Utilities, for example, are assumed to be paid within 30 days, so the cash outflow in February is assumed to be the utilities expense for January from the forecasted income statement. Note that interest payments are assumed to be zero in January and February, but in March, the formula in D14 sums the interest expenses on the forecasted income statement for January, February, and March. This is because interest is paid quarterly.
Finally, note the formula in C4. The beginning cash balance for a given month is the same as the ending cash balance from the prior month; thus, the figure in B18 is linked to C4 to start the new month.
### Using Excel Functions to Balance
Once we get a draft of the forecasts outlined, then the tinkering starts. Additional information can be used to adjust the formulas, as we saw with the 2% reduction in January sales for the forecasted income statement. Because we have linked most (though not all) of our expenses, subtotals, and statements together using formulas, management can also use the forecast workbook to perform scenario and sensitivity analyses, essentially asking “what if?” and looking at the results. When completed, however, before finalizing the forecast, it’s important that the financial statements are in balance (particularly the balance sheet, just as the name implies).
Notice that throughout, we used formulas to calculate subtotals to ensure they are correct and change as needed. We also linked figures, such as the ending and beginning cash balances, to ensure they are in balance. Perhaps the easiest but most important thing to do is to ensure that the balance sheet balances. We can do this with a simple formula that compares total assets to total liabilities and equity. We can see in that subtracting one from the other in cell C27 should result in $0. If there is a difference, the formula will highlight it, forcing us to investigate and correct the sheet so that it balances.
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Excel can be a powerful tool for creating financial forecasts. Formulas that complete mathematical functions and tie accounts and financial statements together are used to create the statements, ensure that they balance, and facilitate scenario and sensitivity analyses.
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# The Importance of Trade Credit and Working Capital in Planning
## Why It Matters
During the COVID-19 pandemic, many families and small businesses realized the importance of financial resiliency. In personal finance, financial resiliency is the ability to overcome financial difficulties such as sudden job loss or significant unexpected expenses—to spring back quickly.
To help promote resiliency, personal financial planners advise clients to maintain liquid assets equal to three to six months of living expenses, keep debt levels low, manage the household budget, keep insurance in force (health, property, and life), establish a solid credit history, and make wise use of credit cards and home equity lines of credit.
In business finance, financial resiliency is not important only during pandemics but is important through the ups and downs of seasonal cycles and economic downturns. Managing cash, accounts receivable, and inventory while making optimal use of trade credit (accounts payable) makes for a business that meets its operating needs and pays its debts when due.
Working capital management is also critical during good times. Even though profits might be rising, a business with growing demand for its products and services still needs to have working capital management tools to pay its bills. Growth in sales and profits do not immediately mean sufficient cash flow, so planning ahead with tools such as a cash budget is key. |
# The Importance of Trade Credit and Working Capital in Planning
## What Is Working Capital?
By the end of this section, you will be able to:
1. Define working capital.
2. Calculate a firm’s operating cycle and cash cycle.
3. Compute inventory days, accounts receivable days, and accounts payable days.
The concept of business capital is often associated with the cash and assets (such as land and equipment) that the owners contributed to the business. Early political economists like Adam Smith and Karl Marx identified this concept of capital, along with labor and entrepreneurship, to be the factors of production.
That general idea of capital is important and critical to a company’s productive capacity. This chapter is about a specific type of capital— working capital—that is just as important as long-term capital. Working capital describes the resources that are needed to meet the daily, weekly, and monthly operating cash flow needs. Employees are paid out of working capital as well as cash from operations, the fulfillment of merchandise orders is possible because of working capital, and the liquidity of a company hinges upon how well management plans and controls working capital.
Understanding working capital begins with the concept of current assets—those resources of a business that are cash, near cash, or expected to be turned into cash within a year through the normal operations of the business. Current assets are necessary for the everyday operation of the firm, and they are synonymous with term gross working capital.
Cash is needed to pay the bills and meet the payroll. Excess cash is invested in cash alternatives such as marketable securities, creating liquidity that can be tapped when operating cash flow needs exceed the amount of cash on hand (checking account balances). Investment in inventory is necessary to meet the demand for products (sales), and if the firm extends credit to its customers so that a sale can be made, the balance sheet will also show accounts receivable—a very common current asset that derives its value from the probability that customers will pay their bills.
Working capital is often spoken about in two versions: gross working capital and net working capital. As was previously stated, gross working capital is equivalent to current assets, particularly those that are cash, cash-like, or will be converted to cash within a short period of time (i.e., in less than one year).
Net working capital (NWC) is a more refined concept of working capital. It is best understood by examining its formula:
### Goal of Working Capital Management
The goal of working capital management is to maintain adequate working capital to
1. meet the operational needs of the company;
2. satisfy obligations (current liabilities) as they come due; and
3. maintain an optimal level of current assets such as cash (provides no return), accounts receivable, and inventory.
Working capital management encompasses all decisions involving a company’s current assets and current liabilities. One very important aspect of working capital management is to provide enough cash to satisfy both maturing short-term obligations and operational expenditures—keeping the company sufficiently liquid.
In summary, working capital management helps a company run smoothly and mitigates the risk of illiquidity. Well-run companies make effective use of current liabilities to finance an optimal level of current assets and maintain sufficient cash balances to meet short-term operating goals and to satisfy short-term obligations. Working capital management is accomplished through
1. cash management;
2. credit and receivables management;
3. inventory management; and
4. accounts payable management.
### Components of Working Capital Management
In contrast to net working capital, gross working capital is synonymous with current assets, particularly those current assets that are either cash or cash equivalents or that will be converted to cash within a short period of time (i.e., in less than one year).
Below is a list of the components of gross working capital.
1. Cash and cash equivalents
2. Marketable securities
3. Accounts receivable
4. Inventory
Here is an example. On December 31, a company has the following balances and gross working capital:
Think of the $1,105,000 of gross working capital as a source of funds for the most pressing obligations (i.e., current liabilities) of the company. Gross working capital is available to pay the bills. However, some of the current assets would need to be converted to cash first. Accounts receivable need to be collected, and inventory would need to be sold before it too can become cash. What if the company had $600,000 of current liabilities? That amount of current obligations could not be paid out of cash until the marketable securities were sold and a significant portion of accounts receivable were collected.
The second, more refined and useful concept of working capital is net working capital:
For example, if a company has $1,000,000 of current assets and $750,000 of current liabilities, its net working capital would be $250,000 ($1,000,000 less $750,000).
NWC provides a better picture because it takes into account the liability “coverage” provided by the current assets. As the above example shows, the current assets would “cover” the current liabilities with an excess of $250,000. Think of it this way: if the current assets could be converted to cash, they could be used to meet the current obligations with another $250,000 of cash leftover.
Current liabilities include
1. accounts payable;
2. dividends payable;
3. notes payable (due within a year);
4. current portion of deferred revenue;
5. current maturities of long-term debt;
6. interest payable;
7. income taxes payable; and
8. accrued expenses such as compensation owed to employees.
Net working capital possibilities can be thought of as a spectrum from negative working capital to positive, as explained in .
Measures of Financial Health provides information on a variety of financial ratios to help users of financial statements understand the strengths and weakness of companies’ financial statements. Three of the financial ratios covered in that chapter are brought back into this chapter’s discussion to demonstrate how financial managers examine working capital and liquidity. Liquidity is the ease with which an asset can be converted into cash. Those ratios are the current ratio, the quick ratio, and the cash ratio. A higher ratio indicates a greater level of liquidity.
The formulas for the three liquidity ratios are:
Notice how the current ratio includes the two elements of net working capital—current assets and current liabilities. It makes for a quick comparison of relative size or proportion.
There are two drawbacks to the current ratio: (1) it is a working capital analytic as of a point in time but is not indicative of future liquidity or future cash flows and (2) as an indicator of liquidity, it can be deceptive if a significant proportion of the current assets are inventory, supplies, or prepaid expenses. Inventory is not very liquid as it can take an extended time period to convert to cash, and assets such as supplies and prepaid expenses never become cash and therefore are not a source of funds to pay bills.
The quick ratio is considered a more conservative indication of liquidity since it does not include a firm’s inventory: .
Working capital ratios, like any financial ratio, are most valuable when examined in light of trends and in comparison to industry/peer averages. For example, a deteriorating current ratio over several quarters (a decline in the company’s current ratio) could indicate a reduced ability to pay bills.
Working capital ratios are also compared to industry averages, which are available in databases produced by such financial publishers as Dun & Bradstreet, Dow Jones Company, and the Risk Management Association (RMA). These information services are available via subscriptions and through many libraries. For example, if a company’s current ratio is 0.9 while the industry average is 2.0, then the company is less liquid than the average company in its industry and strategies, and techniques need to be considered to change things and to better compete with peer groups. Industry averages can be aspirational, motivating management to set liquidity goals and best practices for working capital management.
It is common to think about working capital with a simple assumption: current assets are being “financed” by current liabilities. However, such an assumption may be an oversimplification. Some level of current assets is often necessary to meet longer-term obligations, and in that way, you could think of some amount of current assets as a permanent based of working capital that may need to be financed with longer-term sources of capital.
Think of a company with seasonal business. During busy times, more working capital will be needed than during certain other portions of the year, such as less busy times. But there will always be some level—a permanent base—of working capital needed. Think of it this way: the total working capital of many companies will ebb and flow depending on many variables such as the operating cycle, production needs, and the growth of revenue. Therefore, working capital can be thought of as having a permanent base that is always needed and a total working capital amount that increases when activity levels (i.e., production and sales volume) are higher (see ).
### The Cash Cycle
The cash cycle, also called the cash conversion cycle, is the time period between when a business begins production and acquires resources from its suppliers (for example, acquisition of materials and other forms of inventory) and when it receives cash from its customers. This is offset by the time it takes to pay suppliers (called the payables deferral period).
The cash cycle is measured in days, and it is best understood by examining its formula:
The inventory conversion period is also called the days of inventory. It is the time (days) it takes to convert inventory to sales and is calculated by following these steps:
1. First, calculate the Inventory Turnover Ratio using this formula:
The Average Inventory is arrived at as follows:
2. Then, use the Inventory Turnover Ratio to calculate the Inventory Conversion Period:
The receivables collection period, also called the days sales outstanding (DSO) or the average collection period, is the number of days it typically takes to collect cash from a credit sale. It is calculated by following these steps:
1. First, calculate the Accounts Receivable Turnover using this formula:
The Average Accounts Receivable is arrived at as follows:
2. Then, use the Accounts Receivable Turnover to calculate the Receivables Collection Period:
The payables deferral period, also known as days in payables, is the average number of days its takes for a company to pay its suppliers. It is calculated by following these steps:
1. First, calculate the Accounts Payable Turnover using this formula:
The Average Accounts Payable is arrived at as follows:
2. Then, use the Accounts Payable Turnover to calculate the Payables Deferral Period:
Shortening the inventory conversion period and the receivables collection period or lengthening the payables deferral period shortens the cash conversion cycle. Financial managers monitor and analyze each component of the cash conversion cycle. Ideally, a company’s management should minimize the number of days it takes to convert inventory to cash while maximizing the amount of time it takes to pay suppliers.
Quickly converting inventory to sales speeds up cash inflows and shortens the cash cycle, but it also could help reduce inventory losses as a result of obsolescence. Inventory becomes obsolete because of a variety of factors including time—inventory that has not been sold for a long period of time and is not expected to be sold in the future has to be written down or written off according to accounting rules. Write-offs of inventory can result in significant losses for a company. In the food business, inventory conversion periods take on great importance because of spoilage of perishable goods; in retailing, seasonal items lose value the longer they stay on the shelves.
Various inventory management techniques are used to shorten production time in manufacturing, and in retailing, strategies are used to reduce the amount of time a product sits on the shelf or is stored in the warehouse. Production techniques such as just-in-time inventory systems and marketing and pricing strategies can have an impact on the number of days in the inventory conversion cycle.
For the receivables collection period, a relatively long receivables collection period means that the company is having trouble collecting cash from its customers and so whatever can be done to speed up collections while still offering competitive credit terms should be pursued by financial managers. For example, companies that converted paper invoicing to e-invoicing most likely reduce the average collection period by some number of days, as it makes sense that if a bill is transmitted electronically, lag time is cut (no delays because of “snail mail”) and collections (payments back to the company from customers) may happen sooner. Other credit management techniques, some of which are explained in subsequent sections, can help minimize and control the receivables collection period.
The payables deferral period is the one element that probably cannot be optimized without violating credit terms. Certainly, cash balances can be conserved by delaying payments to vendors for as long as possible; however, payments on trade credit need to be made on time or the company’s relationship with the supplier can suffer. In a worst-case scenario, the company’s credit rating could also deteriorate.
A credit rating, also called a credit score, is a measure produced by an independent agency indicating the likelihood that a company will meet its financial obligations as they come due; it is an indication of the company’s ability to pay its creditors. Three business credit rating services are Equifax Small Business, Experian Business, and Dun & Bradstreet.
### Working Capital Needs by Industry
When comparing working capital needs by industry, you can see some variation. For example, some companies in the grocery business can have very low cash conversion cycles, while construction companies can have very high cash conversion cycles. And some companies, like those in the restaurant business, can have very low numbers and even have negative cash conversion cycles.
Working capital can also differ from one industry to another. An often cited general rule is that a current ratio of 2 is considered optimal. However, general rules of thumb must be treated with caution. A better benchmarking approach is to compare a firm’s ratios—current ratio and quick ratios—to the average of the industry in which the subject company operates.
Take, for example, a home construction company. Such as firm has a long operating cycle because of the production process (building homes), and the “storage of finished goods” can result in very high current ratios—such as 11 or 12 times current liabilities—whereas a retailer like Walmart or Target would have much lower current ratios.
In recent years, Walmart Stores Inc. (NYSE: WMT) has had a current ratio of around 0.9 and has been able to manage its working capital needs by efficient management of its supply chain, quick turnover of inventory, and a very small investment in accounts receivables.Walmart Inc. “2020 Annual Report.” 2020. https://corporate.walmart.com/media-library/document/2020-walmart-annual-report/_proxyDocument?id=00000171-a3ea-dfc0-af71-b3fea8490000 Big retailers like Walmart are effective at negotiating favorable payment terms with their vendors. The ability to generate consistent positive cash flow from operations allows a retailer like Walmart to operate with relatively low amounts of working capital.
The credit policies of a company also affect working capital. A company with a liberal credit policy will require a greater amount of working capital, as collection periods of accounts receivable are longer and therefore tie up more dollars in receivables.
Almost all businesses will have times when additional working capital is needed to pay bills, meet the payroll (salaries and wages), and plan for accrued expenses. The wait for the cash to flow into the company’s treasury from the collection of receivables and cash sales can be longer during tough times.
During the COVID-19 pandemic, the US government made paycheck protection program (PPP) loans available to help alleviate working capital problems for small and large business when the economy slowed because of shutdowns and social distancing. And although 60 percent of the PPP loan proceeds were to go to cover payroll-related costs, 40 percent could be used to bolster working capital to meet rent, utilities costs, and some interest expense while companies were “treading water”—waiting for positive cash flow to pick up under a recovery.US Small Business Administration. “PPP Loan Forgiveness.” n.d. https://www.sba.gov/funding-programs/loans/covid-19-relief-options/paycheck-protection-program/ppp-loan-forgiveness
It isn’t just during downturns that working capital is strained. Growing companies, even if they are extremely profitable, need additional working capital as they ramp up operations by acquiring raw materials, component parts, supplies, or other forms of inventory; hiring temporary or additional employees; and taking on new projects. Whenever additional resources are needed, working capital is also needed.
Some of the current assets and expenditures needed in a growing company may need to be financed from sources that are not spontaneous financing—trade credit (accounts payable). Such forms of external financing such as lines of credit, short-term bank loans, inventory-based loans (also called floor planning), and the factoring of accounts receivables might have to be relied upon.
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Working capital is not only necessary to run a business; it is a resource that will expand and contract with business cycles and must be carefully managed and monitored. The daily, weekly, and monthly needs of business operations are met by cash. Financial managers understand the significance of net working capital (current assets – current liabilities) and various liquidity ratios as they attempt to ensure that bills can be paid. The cash conversion cycle and the cash budget provide additional working capital management tools.
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### How Companies Report Cash Flow
### Trade Credit and Interest Rates on Short-Term Borrowing
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# The Importance of Trade Credit and Working Capital in Planning
## What Is Trade Credit?
By the end of this section, you will be able to:
1. Compute the cost of trade credit.
2. Define cash discount.
3. Define discount period.
4. Define credit period.
Trade credit, also known as accounts payable, is a critical part of a business’s working capital management strategy. Trade credit is granted by vendors to creditworthy companies when those companies purchase materials, inventory, and services.
A company’s purchasing system is usually integrated with other functions such production planning and sales forecasting. Purchasing managers search for and evaluate vendors, negotiate order quantities, and prepare purchase orders. In carrying out the purchasing process, credit terms are granted by the company’s vendors and purchases of inventory and services can be made on trade credit accounts—allowing the purchaser time to pay. The purchaser carries an accounts payable balance until the account is paid.
Trade credit is referred to as spontaneous financing, as it occurs spontaneously with the gearing up of operations and the additional investment in current assets. Think of it this way: If sales are increasing, so too is production. Increased sales mean more current assets (accounts receivable and inventory), and increased sales mean increases in accounts payable (financing happening spontaneously with increased sales and inventory purchases). Compared to other financing arrangements, such as lines of credit and bank loans, trade credit is convenient, simple, and easy to use.
Once a company is approved for trade credit, there is no paperwork or contracts to sign, as is the case with various forms of bank financing. Invoices specify the credit terms, and there is usually no interest expense associated with trade credit. Accounts payable is a type of obligation that is interest-free and is distinguished from debt obligations, such as notes payable, that require the creditor to pay back principal and interest.
### How Trade Credit Works
Trade credit is common in B2B (business to business) transactions and is analogous to consumer spending using a credit card. With a credit card, a consumer opens an account with a credit limit. Most trade credit is offered to a company with an open account that has a credit limit up to which the company can purchase goods or services without having to pay the cash up front. As long as the payments are made in accordance with the terms of the agreement (also called credit terms), no interest or additional fees are charged on the credit balance except possibly for a fee for late payment.
Initially, the vendor’s credit department approves both a trade credit limit and credit payment terms (i.e., number of days after the invoice date that payment is due). Timely payments on accounts payable (trade credit) helps create a credit history for the purchasing firm.
### Trade Credit Terms
Trade credit arrangements often carry credit terms that offer an incentive, called a discount, for a company (the buyer) to pay its bill within a relatively short period of time. Net terms, also referred to as the full credit period, are the number of days that a business (purchaser) has before they must pay their invoice. A common net term is Net 30, with payment due in full within 30 days of the invoice.
Many vendors also offer cash discounts to customers that pay their bill early. A company’s invoice that specifies payment terms of “2/10 n/30” (stated as: “two ten net 30”) would allow a 2 percent discount if the buyer’s account balance is paid within 10 days of the invoice date; otherwise, the net amount owed would be due in 30 days. The “10 days” in the example is the discount period—the number of days the buyer has to take advantage of the cash discount for an early payment, also known as quick payment.
For example, Jackson’s Premium Jams Inc. received a $10,500 invoice for the purchase of jelly jars. The invoice has payment terms of 2/10 n/30. Jackson’s pays the bill within 10 days of the invoice date. Jackson’s payment would be . The effect of taking a discount because of a quick payment is a lowering of the cost of inventory in the case of purchases of materials (for a manufacturer), merchandise (for a retailer or wholesaler), and operating expenses (for any company that “buys” services using trade credit). In Cost of Trade Credit, there is an example that shows the high annualize opportunity cost (36.73 percent) of not taking advantage of cash discounts.
### Cost of Trade Credit
Trade credit is often referred to as a no-cost type of financing. Unlike with other credit arrangements (e.g., bank loans, lines of credit, and commercial paper), there is usually no interest expense associated with trade credit, and as long as your account does not become delinquent, there are no special fees. Some accounts payable arrangements specify an interest penalty or a late fee when the account goes delinquent, but as long as payments are made on time, trade credit is thought of as a low-cost source of working capital.
However, there is one possible cost associated with trade credit for companies that don’t take advantage of cash discounts when offered by sellers. Using accounts payable to purchase goods and services can involve an opportunity cost—a cost of the forgone opportunity of making a quick payment and benefiting from a cash discount. A business that does not take advantage of a cash discount for early payment of trade credit will pay more for goods and services than a business that routinely takes advantage of discounts.
The annual percentage rate of forgoing quick payment discounts can be estimated with the following formula:
Example: Novelty Accessories Inc. (NAI) purchases products from a vendor that offers credit payment terms of 2/10, net 30. The annual cost to NAI of not taking advantage of the discount for quick payment is 36.73 percent.
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Trade credit is very prevalent in the business world, especially in B2B (business-to-business) transactions. Many business exchanges (sales) could not take place without trade credit and the credit terms that are offered. Like any component of working capital, trade credit must be planned and managed. The creditor (the company granting the credit) does so based on an analysis of creditworthiness and must monitor payments and manage slow-paying accounts. The debtor (accounts payable) needs to make payments on time to keep a clean credit history and to take advantage of discounts.
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# The Importance of Trade Credit and Working Capital in Planning
## Cash Management
By the end of this section, you will be able to:
1. Explain why firms hold cash.
2. List instruments available to a financial manager for investing cash balances.
Cash management means efficiently collecting cash from customers and managing cash outflows. To manage cash, the cash budget—a forward-looking document—is an important planning tool. To understand cash management, you must first understand what is meant by cash holdings and the motivations (reasons) for holding cash. A cash budget example is covered in Using Excel to Create the Short-Term Plan.
### Cash Holdings
The cash holdings of a company are more than the currency and coins in the cash registers or the treasury vault. Cash includes currency and coins, but usually those amounts are insignificant compared to the cash holdings of checks to be deposited in the company’s bank account and the balances in the company’s checking accounts.
### Motivations for Holding Cash
The initial answer to the question of why companies hold cash is pretty obvious: because cash is how we pay the bills—it is the medium of exchange. The transactional motive of holding cash means that checks and electronic funds transfers are necessary to meet the payroll (pay the employees), pay the vendors, satisfy creditors (principal and interest payments on loans), and reward stockholders with dividend payments. Cash for transaction is one reason to hold cash, but there is another reason—one that stems from uncertainty and the precautions you might take to be ready for the unexpected.
Just as you keep cash balances in your checking and savings accounts and even a few dollars in your wallet or purse for unexpected expenditures, cash balances are also necessary for a business to provide for unexpected events. Emergencies might require a company to write a check for repairs, for an unexpected breakdown of equipment, or for hiring temporary workers. This motive of holding cash is called the precautionary motive.
Some companies maintain a certain amount of cash instead of investing it in marketable securities or in upgrades or expansion of operations. This is called the speculative motive. Companies that want to quickly take advantage of unexpected opportunities want to be quick to purchase assets or to acquire a business, and a certain amount of cash or quick access to cash is necessary to jump on an opportunity.
Sometimes cash balances may be required by a bank with which a company conducts significant business. These balances are called compensating balances and are typically a minimum amount to be maintained in the company’s checking account.
For example, Jack’s Outback Restaurant Group borrowed $500,000 from First National Bank and Trust. As part of the loan agreement, First National Bank required Jack’s to keep at least $50,000 in its company checking account as a way of compensating the bank for other corporate services it provides to Jack’s Outback Restaurant Group.
### Cash Alternatives
Cash that a company has that is in excess of projected financial needs is often invested in short-term investments, also known as cash equivalents (cash alternatives). The reason for this is that cash does not earn a rate of return; therefore, too much idle cash can affect the profitability of a business.
shows a list of typical investment vehicles used by corporations to earn interest on excess cash. Financial managers search for opportunities that are safe and highly liquid and that will provide a positive rate of return. Cash alternatives, because of their short-term maturities, have low interest rate risk (the risk that an investment’s value will decrease because of changes in market interest rates). In that way, prudent investment of excess cash follows the risk/return trade-off; in order to achieve safe returns, the returns will be lower than the possible returns achieved with risky investments. Cash alternative investments are not committed to the stock market.
shows a note within the 2021 Annual Report (Form 10-K) of Target Corporation. The note discloses the amount of Target’s cash and cash equivalent balances of $8,511,000,000 for January 30, 2021, and $2,577,000,000 for February 1, 2020.
In that note, which is a supplement to the company’s balance sheet, receivables from third-party financial institutions is also considered a cash equivalent. That is because purchases by Target’s customers who use their credit cards (e.g., VISA or MasterCard) create very short-term receivables—amounts that Target is waiting to collect but are very close to a cash sale. So instead of being reported as accounts receivable—a line item on the Target balance sheet that is separate from cash and cash equivalents—these amounts receivable from third-party financial institutions are considered part of the cash and cash equivalents and are a very liquid asst. For example, the amount of $560,000,000 for January 30, 2021, is considered a cash equivalent since the settlement of these accounts will happen in a day or two with cash deposited in Target’s bank accounts. When a retailer sells product and accepts a credit card such as VISA, MasterCard, or American Express, the cash collection happens very soon after the credit card sale—typically within 24 to 72 hours.Creditcardprocessing.com. “How Long Does it Take for a Merchant to Receive Funds?” n.d. https://www.creditcardprocessing.com/resource/article/long-take-merchant-receive-funds/#:~:text=The%20time%20that%20it%20takes,days%20to%20process%20the%20payment
Companies also invest excess funds in marketable securities. These are debt and equity investments such as corporate and government bonds, preferred stock, and common stock of other entities that can be readily sold on a stock or bond exchange. Ford Motor Company has this definition of marketable securities in its 2019 Annual Report (Form 10-K):
“Investments in securities with a maturity date greater than three months at the date of purchase and other securities for which there is more than an insignificant risk of change in value due to interest rate, quoted price, or penalty on withdrawal are classified as Marketable securities.”Ford Motor Company. “2019 Annual Report.” n.d. https://s23.q4cdn.com/725981074/files/doc_downloads/Ford-2019-Printed-Annual-Report.pdf
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Cash management is simply making sure you have enough cash to meet expected obligations and for contingencies (unexpected or emergency cash needs). Excess cash should be invested low-risk and highly liquid marketable securities. The cash budget is a critical tool of cash management.
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# The Importance of Trade Credit and Working Capital in Planning
## Receivables Management
By the end of this section, you will be able to:
1. Discuss how decisions on extending credit are made.
2. Explain how to monitor accounts receivables.
For any business that sells goods or services on credit, effective accounts receivable management is critical for cash flow and profitability planning and for the long-term viability of the company. Receivables management begins before the sale is made when a number of factors must be considered.
1. Can the customer be approved for a credit sale?
2. If the credit is approved, what will be the credit terms (i.e., how long do we give customers to pay their bills)?
3. Will there be a cash discount for quick payment?
4. How much credit should be extended to each customer (credit limit)?
Accounts receivable is not about accepting credit cards. Credit card sales are not technically accounts receivable. When a credit card is accepted, it means that the credit card company (e.g., VISA, MasterCard, or American Express) will guarantee the payment. The cash will be deposited in the merchant’s bank account in a very short period of time.
When a business makes a sale on account, management (e.g., a credit manager or analyst) does its best to distinguish between customers who have a high likelihood of paying and customers who have a low likelihood. Customers with low credit risk are approved; the decision is based on an effective analysis of creditworthiness.
Creditworthiness is judged by looking at a number of factors including an evaluation of the customer’s financial statements, financial ratios, and credit reports (credit scores) based on a customer’s payment history on credits owed to other firms. If a company has a prior relationship with a customer seeking trade credit, the customer’s payment history with the firm is also carefully evaluated before additional credit is granted.
### Determining the Credit Policy
A company’s credit policy encompasses rules of credit granting and procedures for the collections of accounts. It’s how a company will process credit applications, utilize credit scoring and credit bureaus, analyze financial statements, make credit limit decisions, and conduct collection efforts when accounts become delinquent (still outstanding after their due date).
### Establishing Credit Terms
Trade credit terms were discussed earlier. Recall that part of the terms and conditions of a sale are the credit terms—elements of a sales agreement (contract) that indicate when payment is due, possible discounts (for quick payments), and any late fee charges.
If open credit is for a sales transaction, an agreement is made as to the length of time for which credit is to be granted (payment period) and a discount for early payment. Although companies are free to establish credit terms as they see fit, most companies look to the practice of the particular industry in which they operate. The credit terms offered by the competition are a factor. Net terms usually range between 30 days and 90 days, depending on the industry. Discounts for early payments also differ and are typically from 1 to 3 percent.
Establishing credit terms offered can be thought of as a decision process similar to setting a price for products and services. Just as a price is the result of a market forces, so too are credit terms. If credit terms are not competitive within the industry, sales can suffer. Typically, companies follow standard industry credit terms. If most companies in an industry offer a discount for early payments, then most companies will follow suit and also offer an equal discount.
Once credit terms are established, they can be changed based on both marketing strategies and financial management goals. For example, discounts for early payments can be more generous, or the full credit period can be extended to stimulate additional sales. Both discount periods and full credit periods can be tightened to try to speed up collections. The establishment of and changes to credit terms are usually made in consultation with the sales and financial management departments.
### Monitoring Accounts Receivables
Financial managers monitor accounts receivables using some basic tools. One of those tools is the accounts receivable aging schedule (report). To prepare the aging schedule, a classifying of customer account balances is performed with age as the sorting attribute.
An account receivable begins its life as a credit sale. The age of a receivable is the number of days that have transpired since the credit sale was made (the date of the invoice). For example, if a credit sale was made on June 1 and is still unpaid on July 15, that receivable is 45 days old. Aging of accounts is thought to be a useful tool because of the idea that the longer the time owed, the greater the possibility that individual accounts receivable will prove to be uncollectible.
An aging schedule is a report that organizes the outstanding (unpaid) receivable balances into age categories. The receivables are grouped by the length of time they have been outstanding, and an uncollectible percentage is assigned to each category. The length of uncollectible time increases the percentage assigned. For example, a category might consist of accounts receivable that are 0–30 days past due and is assigned an uncollectible percentage of 6 percent. Another category might be 31–60 days past due and is assigned an uncollectible percentage of 15 percent. All categories of estimated uncollectible amounts are summed to get a total estimated uncollectible balance.
The aging of accounts is useful to the credit and collection managers, both from a global view—estimating how much of the accounts receivable asset might be bad debts—and on a micro basis—being able to drill down to see which specific customers are slow paying or delinquent so as to implement collection tactics.
Accountants and auditors also find the aging of accounts to determine a reasonable amount to be reported as bad debt expense and to establish a sufficient balance in the allowance for doubtful accounts. Bad debt expense is the cost of doing business because some customers will not pay the amounts they owe (accounts receivable), while the allowance for doubtful accounts is a contra-asset (it will be deducted from accounts receivable on the balance sheet) that contains management’s best guess (management’s estimate) as to how much of its accounts receivable will never be collected.
In , Foodinia Inc.’s accounts receivable aging report shows that the total receivables balance is $189,000. The company splits its accounts into four age categories: not due, 30 to 60 days past due, 61 to 90 days past due, and more than 90 days past due. Of the $189,000 owed to Foodinia by its customers, $75,500 ($189,000 less $113,500) of invoices have been outstanding (not paid yet) beyond their due dates.
In addition to preparing aging schedules, financial managers also use financial ratios to monitor receivables. The accounts receivable turnover ratio determines how many times (i.e., how often) accounts receivable are collected during an operating period and converted to cash. A higher number of times indicates that receivables are collected quickly. In contrast, a lower accounts receivable turnover indicates that receivables are collected at a slower rate, taking more days to collect from a customer.
Another receivables ratio is the number of days’ sales in receivables ratio, also called the receivables collection period—the expected days it will take to convert accounts receivable into cash. A comparison of a company’s receivables collection period to the credit terms granted to customers can alert management to collection problems. Both the accounts receivable turnover ratio and receivables collection period are covered, including the formulas for calculating the ratios, in the previous section of this chapter.
### Accounts Receivables and Notes Receivable
An accounts receivable is an informal arrangement between a seller (a company) and customer. Accounts receivable are usually paid within a month or two. Accounts receivable don’t require any complex paperwork, are evidenced by an invoice, and do not involve interest payments. In contrast, a note receivable is a more formal arrangement that is evidence by a legal contract called a promissory note specifying the payment amount and date and interest.
The length of a note receivable can be for any time period including a term longer than the typical account receivable. Some notes receivable have a term greater than a year. The assets of a bank include many notes receivable (a loan made by a bank is an asset for the bank).
A note receivable can be used in exchange for products and services or in exchange for cash (usually in the case of a financial lender). Sometimes a company might request that a slow-paying customer sign a note promissory note to further secure the receivable, charge interest, or add some type of collateral to the arrangement, in which case the receivable would be called a secured promissory note. Several characteristics of notes receivable further define the contract elements and scope of use (see ).
###
Accounts receivables are monitored by management with tools such as the ratios accounts receivable turnover and average collection period and the aging of receivables. The credit managers’ mantra rings true: “The older the receivable, the greater the likelihood that the account will not be collected.”
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# The Importance of Trade Credit and Working Capital in Planning
## Inventory Management
By the end of this section, you will be able to:
1. Outline the costs of holding inventory.
2. Outline the benefits of holding inventory.
Financial managers must consider the impact of inventory management on working capital. Earlier in the chapter, the concept of the inventory conversion cycle was covered. The number of days that goods are held by a business is one of the focal points of inventory management.
Managers look to minimize inventory balances and raise inventory turnover ratios while trying to balance the needs of operations and sales. Purchasing personnel need to order enough inventory to “feed” production or to stock the shelves. The sales force wants to meet or surpass their sales budgets, and the operations people need inventory for the factories, warehouses, and e-commerce sites.
The days in inventory ratio measures the average number of days between acquiring inventory (i.e., purchasing merchandise) and its sale. This ratio is a metric to be watched and monitored by inventory managers and, if possible, minimized. A high days in inventory ratio could mean “aging” inventory. Old inventory could mean obsolesce or, in the case of perishable goods, spoilage. In either case, old inventory means losses.
Imagine a company selling high-tech products such as consumer electronics. A high days in inventory ratio could mean that technologically obsolete products will be sold at a discount. There are similar issues with older inventory in the fashion industry. Last year’s styles are not as appealing to the fashion-conscious consumer and are usually sold at significant discounts. In the accounting world, lower of cost or market value is a test of inventory value to determine if inventory needs to be “written down,” meaning that the company takes an expense for inventory that has lost significant value. Lower of cost or market is required by Generally Accepted Accounting Principles (GAAP) to state inventory valuations at realistic and conservative values.
Inventory is a very significant working capital component for many companies, such as manufacturers, wholesalers, and retailers. For those companies, inventory management involves management of the entire supply chain: sourcing, storing, and selling inventory. At its very basic level, inventory management means having the right amount of stock at the right place and at the right time while also minimizing the cost of inventory. This concept is explained in the next section.
### Inventory Cost
Controlling inventory costs minimizes working capital needs and, ultimately, the cost of goods sold. Inventory management impacts profitability; minimizing cost of goods sold means maximizing gross profit (Gross Profit = Net Sales Less Cost of Goods Sold).
There are four components to inventory cost:
1. Purchasing costs: the invoice amount (after discounts) for inventory; the initial investment in inventory
2. Carrying costs: all costs of having inventory in stock, which includes storage costs (i.e., the cost of the space to store the inventory, such as a warehouse), insurance, inventory obsolescence and spoilage, and even the opportunity cost of the investment in inventory
3. Ordering costs: the costs of placing an order with a vendor; the cost of a purchase and managing the payment process
4. Stockout costs: an opportunity cost incurred when a customer order cannot be filled and the customer goes elsewhere for the product; lost revenue
Minimizing total inventory costs is a combination of many strategies, the scope and complexity of which are beyond the scope of this text. Concepts such as just-in-time (JIT) inventory practices and economic order quantity (EOQ) are tools used by inventory managers, both of which help keep a company lean (minimizing inventory) while making sure the inventory resources are in place in time to complete the sale.
### Benefit of Holding Inventory
Brick-and-mortar stores need goods in stock so that the customer can see and touch the product and be able to acquire it when they need it. Customers are disappointed if they cannot see and touch the item or if they find out upon arrival at the store that it is out of stock.
Customers of all kinds don’t want to wait for the delivery of a purchase. We have become accustomed to Amazon orders being delivered to the door the next day. Product fulfillment and availability is important. Inventory must be in stock, or sales will be lost.
In manufacturing, the inventory of materials and component parts must be in place at the start of the value chain (the conversion process), and finished goods need to be ready to meet scheduled shipments. Holding sufficient inventory meets customer demand, whether it is products on the shelves or in the warehouse that are ready to move through the supply chain and into the hands of the customer.
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Inventory, usually the least liquid of the current assets, presents its own set of management challenges. Finding the optimal level of inventory is probably more of an art than a science. JIT helps to reduce the investment in inventory and lower the costs of storage, but stockout costs can be very damaging to profitability.
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# The Importance of Trade Credit and Working Capital in Planning
## Using Excel to Create the Short-Term Plan
By the end of this section, you will be able to:
1. Create a one-year budget.
2. Create a cash budget.
A cash budget is a tool of cash management and therefore assists financial managers in the planning and control of a critical asset. The cash budget, like any other budget, looks to the future. It projects the cash flows into and out of the company. The budgeting process of a company is a really an integrated process—it links a series of budgets together so that company objectives can be achieved. For example, in a manufacturing company, a series of budgets such as those for sales, production, purchases, materials, overhead, selling and administrative costs, and planned capital expenditures would need to be prepared before cash needs (cash budget) can be predicted.
Just as you might budget your earnings (salary, business income, investment income, etc.) to see if you will be able to cover your expected living expenses and planned savings amounts, to be successful and to increase the odds that sufficient cash will be available in the months ahead, financial managers prepare cash budgets to
1. meet payrolls;
2. allocate dollars for contingencies and emergencies;
3. analyze if planned collections and disbursements policies and procedures result in adequate cash balances; and
4. plan for borrowings on lines of credit and short-term loans that might be needed to balance the cash budget.
A cash budget is a model that often goes through several iterations before managers can approve it as the plan going forward. Changes in any of the “upstream” budgets—budgets that are prepared before the cash budget, such as the sales, purchases, and production budgets—may need to be revised because of changing assumptions. New economic forecasts and even cost-cutting measures will require a revision of the cash budget.
Although a budget might be prepared for each month of a future 12-month period, such as the upcoming fiscal year, a rolling budget is often used. A rolling budget changes often as the planning period (e.g., a fiscal year) plays out. When one month ends, another month is added to the end (the next column) of the budget. For example, if in your budget January is the first month of the planning period, once January is over, next January’s cash budget column would be added—right after December’s column (at the far right of the budget).
### Sample One-Year (Annual) Operating Budget
Preparing an annual operating budget can be a complex task. In essence, a company budget is a series of budgets, many of which are interrelated.
The sales budget is prepared first and has an impact on many other budgets. Take the example of a production budget of a manufacturer. The sales budget impacts what needs to be produced (production budget), and the production budget influences planned purchases of material (purchases budget), overhead resources (overhead budget), and the amount of labor costs for the year ahead (direct labor budget.)
For a merchant (such as a wholesaler or retailer), the annual budget would be less complex than that of a manufacturing firm but would still require an inventory purchases budget and an operating expense budget (such as selling and administrative expenses). For a service firm, a purchase budget for inventory would not be necessary, but an operating budget would be. All businesses need a cash budget, which is the topic of the next section of this chapter.
The example operating budget presented here is of a merchandising company. Budgets are prepared following a process that begins with a sales (or revenue) forecast. The sales forecast is normally based on information obtained from both internal and external sources and predicts the amount of units to be sold in the planning period—usually one year into the future.
A company’s management, in consultation with its marketing and sales executives, would prepare a sales budget by making assumptions about the number of units that are expected to be sold and the prices that will be charged. From the sales budget, projections are made as to cash receipts each month, and therefore assumptions have to be made as to how much of each month’s sales will be cash sales and how much cash will flow into the company from the collection credit sales (including cash flow in from the prior month’s sales). provides an example of a sales budget and projected accounts receivable collections and cash sales for the months of January through December. Keep in mind that projected monthly sales amounts are not equal to cash collected from sales. Because of sales on credit, some cash from sales lags credit sales as collections can extend beyond the month of sale. Credit terms such terms such as net 30 (net amount owed to be paid in 30 days) have to be considered when developing a forecasted cash collection pattern.
Sales budgets “drive” the preparation of other budgets. If sales are expected to increase, purchases of inventory and some operating expenses would also increase. To meet the demand for goods and services (as defined in the sales budget), a purchases (inventory) budget would be prepared. In this example (), the purchases budget shows projected purchases of inventory (merchandise) and the projected payments (also called disbursements) for each month.
Cash outflows as a result of purchases often do not equal the projected purchase amount. That is because payments for purchases are usually on credit (accounts payable), and so purchases for one month typically get spread out over a period of time that encompasses the current month and the month (or months) thereafter. To keep this example simple, the assumption is that the purchases are paid for in the following month (an average days payable outstanding of 30 days). However, in other cases, payment patterns may be based on other payment periods such as 45, 60, or even 90 days, depending on the trade credit terms.
An operating expense budget is prepared next and is basically a prediction of the selling and administrative expenditures of the company. Notice in that in the operating expense budget, cost of goods sold (an expense) is not included, nor are noncash expenses such as depreciation. The cash outlays related to goods sold, at least in a merchandising operation, are accounted for in the purchases budget (payments for purchases of inventory.)
With the sales, purchases, and operating expense budgets prepared, the cash budget can be prepared. Some of the “inputs” to the cash budget are from the sales (collections of cash), purchases (payments), and the operating expense budget (cash expenditures for selling and administrative expenses). A sample cash budget and a discussion of its preparation follows in the next section of this chapter.
### Sample Cash Budget
A cash budget is the last budget to be prepared and is often part of the financial budget (cash budget, budgeted income statement, and budgeted balance sheet). The purpose of the cash budget is to estimate cash flows, to help ensure sufficient cash balances are maintained during the planning period, and to plan for external financing during periods of cash deficits.
When a budget is prepared in Excel, cash budget analysts can play “what if” with different scenarios to see when cash surpluses and deficits are expected. Cash surpluses means that funds can be invested in marketable securities to earn a rate of return, while cash deficits mean that financing, such as a line of credit, will be necessary (assuming forecasts are accurate).
Although the example shown in is a monthly cash budget, a cash budget could be prepared using any useful time elements: weekly, monthly, or quarterly.
One common practice is to use a rolling cash budget. A rolling cash budget is continually updated to add a new budget period, such as a month’s amount of cash flow activity, as the most recent budgeted month expires. For example, assume that a 12-month cash budget is prepared for a period covering January 20X1 to December 20X1. Once the month of January 20X1 has concluded, a 12-month planning period continues by add January 20X2 to the last column of the budget. The rolling cash monthly budget is an extension of the initial cash budget model, adding one month and thereby always extending cash flow projections one year into the future.
Using as an example, shows the formulas that form the skeleton of a monthly cash budget.
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Short-term plans of a business are funded with cash, with cash budgets being a critical tool of planning. The cash budget takes into account a target amount of cash, factoring in all the motives for holding cash. A cash budget looks ahead—predicting cash inflows and outflows, allocating for minimum cash balances to be maintained, and helping management determine short-term financing needed. Although it is the last budget prepared, the preparation of the cash budget is an important financial planning exercise of companies small and large.
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# Risk Management and the Financial Manager
## Why It Matters
Each year, American Airlines consumes approximately four billion gallons of jet fuel.American Airlines. In the spring of 2018, jet fuel prices rose from an average of $2.07 per gallon to a price of $2.19 per gallon.S&P Global. “Platts Jet Fuel.” A $0.12-per-gallon increase in the price of jet fuel may not seem significant, but on an annualized basis, a price increase of this magnitude would increase the company’s jet fuel bill by approximately $500 million.
That added cost cuts into the profits of the company, leaving less money available to provide a return to the company’s investors. Rising costs could even cause the business to become unprofitable and close, causing many employees to lose their jobs. The financial managers of American Airlines are not able to control the price of jet fuel. However, they must be aware of the risk that price volatility poses to the company and consider prudent ways to manage this risk. |
# Risk Management and the Financial Manager
## The Importance of Risk Management
### Learning Outcomes
By the end of this section, you will be able to:
1. Describe risk in the context of financial management.
2. Explain how risk can impact firm value.
3. Distinguish between hedging and speculating.
### What Is Risk?
The job of the financial manager is to maximize the value of the firm for the owners, or shareholders, of the company. The three major areas of focus for the financial manager are the size, the timing, and the riskiness of the cash flows of the company. Broadly, the financial manager should work to
1. increase cash coming into the company and decrease cash going out of the company;
2. speed up cash coming into the company and slow down cash going out of the company; and
3. decrease the riskiness of both money coming in and money going out of the company.
The first item in this list is obvious. The more revenue a company has, the more profitable it will be. Businesspeople talk about “top line” growth when discussing this objective because revenue appears at the top of the company’s income statement. Also, the lower the company’s expenses, the more profitable the company will be. When businesspeople talk about the “bottom line,” they are focused on what will happen to a company’s net income. The net income appears at the bottom of the income statement and reflects the amount of revenue left over after all of the company’s expenses have been paid.
The second item in the list—the speed at which money enters and exits the company—has been addressed throughout this book. One of the basic principles of finance is the time value of money—the idea that a dollar received today is more valuable than a dollar received tomorrow. Many of the topics explored in this book revolve around the issue of the time value of money.
The focus of this chapter is on the third item in the list: risk. In finance, risk is defined as uncertainty. Risk occurs because you cannot predict the future. Compared to other business decisions, financial decisions are generally associated with contracts in which the parties of the contract fulfill their obligations at different points in time. If you choose to purchase a loaf of bread, you pay the baker for the bread as you receive the bread; no future obligation arises for either you or the baker because of this purchase. If you choose to buy a bond, you pay the issuer of the bond money today, and in return, the issuer promises to pay you money in the future. The value of this bond depends on the likelihood that the promise will be fulfilled.
Because financial agreements often represent promises of future payment, they entail risk. Even if the party that is promising to make a payment in the future is ethical and has every intention of honoring the promise, things can happen that can make it impossible for them to do so. Thus, much of financial management hinges on managing this risk.
### Risk and Firm Value
You would expect the managers of Starbucks Corporation to know a lot about coffee. They must also know a lot about risk. It is not surprising that the term coffee appears in the text of the company’s 2020 annual report 179 times, given that the company’s core business is coffee. It might be surprising, however, that the term risk appears in the report 99 times.Starbucks. Given that the text of the annual report is less than 100 pages long, the word risk appears, on average, more than once per page.
Starbucks faces a number of different types of risk. In 2020, corporations experienced an unprecedented risk because of COVID-19. Coffee shops were forced to remain closed as communities experienced government-mandated lockdowns. Locations that were able to service customers through drive-up windows were not immune to declining revenue due to the pandemic. As fewer people gathered in the workplace, Starbucks experienced a declining number of to-go orders from meeting attendees. In addition, Starbucks locations faced the risk of illness spreading as baristas gathered in their buildings to fill to-go orders.
While COVID-19 brought discussions of risk to the forefront of everyday conversations, risk was an important focus of companies such as Starbucks before the pandemic began. (The term risk appeared in the company’s 2019 annual report 82 times.Starbucks. ) Starbucks’s business model revolves around turning coffee beans into a pleasurable drink. Anything that impacts the company’s ability to procure coffee beans, produce a drink, and sell that drink to the customer will impact the company’s profitability.
The investors in the company have allowed Starbucks to use its capital to lease storefronts, purchase espresso machines, and obtain all of the assets necessary for the company to operate. Debt holders expect interest to be paid and their principal to be returned. Stockholders expect a return on their investment. Because investors are risk averse, the riskier they perceive the cash flows they will receive from the business to be, the higher the expected return they will require to let the company use their money. This required return is a cost of doing business. Thus, the riskier the cash flows of a company, the higher the cost of obtaining capital. As any cost of operating a business increases, the value of the firm declines.
In the following sections, you will learn about some of the types of risk that firms commonly face. You will also learn about ways in which firms can reduce their exposure to these risks. When firms take actions to reduce their exposures to risk, they are said to be hedging. Firms hedge to try to protect themselves from losses. Thus, in finance, hedging is a risk management tool.
Certain strategies are commonly used by firms to hedge risk, which is part of corporate financial management. Many of these same strategies can be used by economic players who wish to speculate. Speculating occurs when someone bets on a future outcome. It involves trying to predict the future and profit off of that prediction, knowing that there is some risk that an incorrect prediction will lead to a loss. Speculators bet on the future direction of an asset price. Thus, speculation involves directional bets.
If you are concerned that the price of hand sanitizer is going to rise because people are concerned about a new virus and you purchase a few extra bottles to keep on your shelf “just in case,” you are hedging. If you see this situation as a business opportunity and purchase bottles of hand sanitizer, hoping that you can sell them on eBay in a few weeks at twice what you paid for them, you are speculating.
In the popular press, you will often hear of some of the strategies in this chapter discussed in terms of people using them to speculate. In upper-level finance courses, these strategies are discussed in more depth, including how they might be used to speculate. In this chapter, however, the focus is on the perspective of a financial manager using these strategies to manage risk.
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Risk arises due to uncertainty. The future is unpredictable. One job of the financial manager is to manage the risks of both cash inflows and cash outflows. Investors are risk-averse. The riskier a firm’s cash flows are, the higher the rate of return investors require to provide capital to the company.
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# Risk Management and the Financial Manager
## Commodity Price Risk
### Learning Outcomes
By the end of this section, you will be able to:
1. Describe commodity price risk.
2. Explain the use of long-term contracts as a hedge.
3. Explain the use vertical integration as a hedge.
4. Explain the use of futures contracts as a hedge.
One of the most significant risks that many companies face arises from normal business operations. Companies purchase raw materials to produce the products and provide the services they sell. A change in the market price of these raw materials can significantly impact the profitability of a company.
For example, Starbucks must purchase coffee beans in order to make its coffee drinks. The price of coffee beans is highly volatile. Sample prices of a pound of Arabica coffee beans over the past couple of decades are shown in . Over this period, the price of coffee beans ranged from a low of $0.52 per pound in the summer of 2002 to a high of over $3.00 per pound in the spring of 2011. The costs, and thus the profits, of Starbucks will vary greatly depending on if the company is paying less than $1.00 per pound for coffee or if it is paying three times that much.
### Long-Term Contracts
One method of hedging the risk of volatile input prices is for a firm to enter into long-term contracts with its suppliers. Starbucks, for example, could enter into an agreement with a coffee farmer to purchase a particular quantity of coffee beans at a predetermined price over the next several years.
These long-term contracts can benefit both the buyer and the seller. The buyer is concerned that rising commodity prices will increase its cost of goods sold. The seller, however, is concerned that falling commodity prices will mean lower revenue. By entering into a long-term contract, the buyer is able to lock in a price for its raw materials and the seller is able to lock in its sales price. Thus, both parties are able to reduce uncertainty.
While long-term contracts reduce uncertainty about the commodity price, and thus reduce risk, there are several possible disadvantages to these types of contracts. First, both parties are exposed to the risk that the other party may default and fail to live up to the terms of the contract. Second, these contracts cannot be entered into anonymously; the parties to the contract know each other’s identity. This lack of anonymity may have strategic disadvantages for some firms. Third, the value of this contract cannot be easily determined, making it difficult to track gains and losses. Fourth, canceling the contract may be difficult or even impossible.
### Vertical Integration
A common method of handling the risk associated with volatile input prices is vertical integration, which involves the merger of a company and its supplier. For Starbucks, a vertical integration would involve Starbucks owning a coffee bean farm. If the price of coffee beans rises, the firm’s costs increase and the supplier’s revenues rise. The two companies can offset these risks by merging.
Although vertical integration can reduce commodity price risk, it is not a perfect hedge. Starbucks may decrease its commodity price risk by purchasing a coffee farm, but that action may expose it to other risks, such as land ownership and employment risk.
### Futures Contracts
Another method of hedging commodity price risk is the use of a futures contract. A commodity futures contract is designed to avoid some of the disadvantages of entering into a long-term contract with a supplier. A futures contract is an agreement to trade an asset on some future date at a price locked in today. Futures exist for a range of commodities, including natural resources such as oil, natural gas, coal, silver, and gold and agricultural products such as soybeans, corn, wheat, rice, sugar, and cocoa.
Futures contracts are traded anonymously on an exchange; the market price is publicly observable, and the market is highly liquid. The company can get out of the contract at any time by selling it to a third party at the current market price.
A futures contract does not have the credit risk that a long-term contract has. Futures exchanges require traders to post margin when buying or selling commodities futures contracts. The margin, or collateral, serves as a guarantee that traders will honor their obligations. Additionally, through a procedure known as marking to market, cash flows are exchanged daily rather than only at the end of the contract. Because gains and losses are computed each day based on the change in the price of the futures contract, there is not the same risk as with a long-term contract that the counterparty to the contract will not be able to fulfill their obligation.
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Companies do not know how much they will have to pay for raw materials in future months. The price of raw materials will change as economic conditions change, impacting a company’s cost of goods sold and profits. Some ways that a company can hedge this risk are through vertical integration, long-term contracts, and futures contracts.
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### Hedging at Southwest Airlines
Jet fuel costs represent a major expense for airlines. Southwest Airlines has been known as the most aggressive airline when it comes to hedging the risk of jet fuel cost volatility. In this interview, the CEO of Southwest Airlines, Gary Kelly, discusses crude oil prices in the spring of 2012 and the impact on Southwest Airlines. |
# Risk Management and the Financial Manager
## Exchange Rates and Risk
### Learning Outcomes
By the end of this section, you will be able to:
1. Describe exchange rate risk.
2. Identify transaction, translation, and economic risks.
3. Describe a natural hedge.
4. Explain the use of forward contracts as a hedge.
5. List the characteristics of an option contract.
6. Describe the payoff to the holder and writer of a call option.
7. Describe the payoff to the holder and writer of a put option.
The managers of companies that operate in the global marketplace face additional complications when managing the riskiness of their cash flows compared to domestic companies. Managers must be aware of differing business climates and customs and operate under multiple legal systems. Often, business must be conducted in multiple languages. Geopolitical events can impact business relationships. In addition, the company may receive cash flows and make payments in multiple currencies.
### Exchange Rates
The costs to companies are impacted when the prices of the raw materials they use change. Very little coffee is grown in the United States. This means that all of those coffee beans that Starbucks uses in its espresso machines in Seattle, New York, Miami, and Houston were bought from suppliers outside of the United States. Brazil is the largest coffee-producing country, exporting about one-third of the world’s coffee.Global Agricultural Information Network. When a company purchases raw materials from a supplier in another country, the company needs not just money but the money that is used in that country to make the purchase. Thus, the company is concerned about the exchange rate, or the price of the foreign currency.
The currency used in Brazil is called the Brazilian real. shows how many Brazilian reals could be purchased for $1.00 from 2010 through the first quarter of 2021. In March 2021, 5.4377 Brazilian reals could be purchased for $1.00. This will often be written in the form of
BRL is an abbreviation for Brazilian real, and USD is an abbreviation for the US dollar. This price is known as a currency exchange rate, or the rate at which you can exchange one currency for another currency.
If you know the price of $1.00 is 5.4377 Brazilian reals, you can easily find the price of Brazilian reals in US dollars. Simply divide both sides of the equation by 5.4377, or the price of the US dollar:
If you have US dollars and want to purchase Brazilian reals, it will cost you $0.1839 for each Brazilian real you want to buy.
The foreign exchange rate changes in response to demand for and supply of the currency. In early 2020, the exchange rate was . In other words, $1 purchased fewer reals in early 2020 than in it did a year later. Because you receive more reals for each dollar in 2021 than you would have a year earlier, the dollar is said to have appreciated relative to the Brazilian real. Likewise, because it takes more Brazilian reals to purchase $1.00, the real is said to have depreciated relative to the US dollar.
### Exchange Rate Risks
Starbucks, like other firms that are engaged in international business, faces currency exchange rate risk. Changes in exchange rates can impact a business in several ways. These risks are often classified as transaction, translation, or economic risk.
### Transaction Risk
Transaction risk is the risk that the value of a business’s expected receipts or expenses will change as a result of a change in currency exchange rates. If Starbucks agrees to pay a Brazilian coffee grower seven million Brazilian reals for an order of one million pounds of coffee beans, Starbucks will need to purchase Brazilian reals to pay the bill. How much it will cost Starbucks to purchase these Brazilian reals depends on the exchange rate at the time Starbucks makes the purchase.
In March 2021, with an exchange rate of , it would have cost Starbucks to purchase the reals needed to receive the one million pounds of coffee beans. If, however, Starbucks agreed in March to purchase the coffee beans several months later, in July, Starbucks would not have known then what the exchange rate would be when it came time to complete the transaction. Although Starbucks would have locked in a price of BRL 7,000,000 for one million pounds of coffee beans, it would not have known what the coffee beans would cost the company in terms of US dollars.
If the US dollar appreciated so that it cost less to purchase each Brazilian real in July, Starbucks would find that it was paying less than $1,287,300 for the coffee beans. For example, suppose the dollar appreciated so that the exchange rate was in July 2021. Then the coffee beans would only cost Starbucks .
On the other hand, if the US dollar depreciated and it cost more to purchase each Brazilian real, then Starbucks would find that its dollar cost for the coffee beans was higher than it expected. If the US dollar depreciated (and the Brazilian real appreciated) so that the exchange rate was in July 2021, then the coffee beans would cost Starbucks . This uncertainty regarding the dollar cost of the coffee beans Starbucks would purchase to make its lattes is an example of transaction risk.
A global company such as Starbucks has transaction risk not only because it is purchasing raw materials in foreign countries but also because it is selling its product—and thus collecting revenue—in foreign countries. Customers in Japan, for example, spend Japanese yen when they purchase a Starbucks cappuccino, coffee mug, or bag of coffee beans. Starbucks must then convert these Japanese yen to US dollars to pay the expenses that it incurs in the United States to produce and distribute these products.
The Japanese yen–US dollar foreign exchange rates from 2011 through the first quarter of 2021 are shown in . In 2012, $1.00 could be purchased with fewer than 80 Japanese yen. In 2015, it took over 120 yen to purchase $1.00.
If a company is receiving yen from customers and paying expenses in dollars, the company is harmed when the yen depreciates relative to the dollar, meaning that the yen the company receives from its customers can be exchanged for fewer dollars. Conversely, when the yen appreciates, it takes fewer yen to purchase each dollar; this appreciation of the yen benefits companies with revenues in yen and expenses in dollars.
### Translation Risk
In addition to the transaction risk, if Starbucks holds assets in a foreign country, it faces translation risk. Translation risk is an accounting risk. Starbucks might purchase a coffee plantation in Costa Rica for 120 million Costa Rican colones. This land is an asset for Starbucks, and as such, the value of it should appear on the company’s balance sheet.
The balance sheet for Starbucks is created using US dollar values. Thus, the value of the coffee plantation has to be translated to dollars. Because exchange rates are volatile, the dollar value of the asset will vary depending on the day on which the translation takes place. If the exchange rate is 500 colones to the dollar, then this coffee plantation is an asset with a value of $240,000. If the Costa Rican colón depreciates to 600 colones to the dollar, then the asset has a value of only $200,000 when translated using this exchange rate.
Although it is the same piece of land with the same productive capacity, the value of the asset, as reported on the balance sheet, falls as the Costa Rican colón depreciates. This decrease in the value of the company’s assets must be offset by a decrease in the stockholders’ equity for the balance sheet to balance. The loss is due simply to changes in exchange rates and not the underlying profitability of the company.
### Economic Risk
Economic risk is the risk that a change in exchange rates will impact a business’s number of customers or its sales. Even a company that is not involved in international transactions can face this type of risk. Consider a company located in Mississippi that makes shirts using 100% US-grown cotton. All of the shirts are made in the United States and sold to retail outlets in the United States. Thus, all of the company’s expenses and revenues are in US dollars, and the company holds no assets outside of the United States.
Although this firm has no financial transactions involving international currency, it can be impacted by changes in exchange rates. Suppose the US dollar strengthens relative to the Vietnamese dong. This will allow US retail outlets to purchase more Vietnamese dong, and thus more shirts from Vietnamese suppliers, for the same amount of US dollars. Because of this, the retail outlets experience a drop in the cost of procuring the Vietnamese shirts relative to the shirts produced by the firm in Mississippi. The Mississippi company will lose some of its customers to these Vietnamese producers simply because of a change in the exchange rate.
### Hedging
Just as companies may practice hedging techniques to reduce their commodity risk exposure, they may choose to hedge to reduce their currency risk exposure. The types of futures contracts that we discussed earlier in this chapter exist for currencies as well as for commodities. A company that knows that it will need Korean won later this year to purchase raw materials from a South Korean supplier, for example, can purchase a futures contract for Korean won.
While futures contracts allow companies to lock in prices today for a future commitment, these contracts are not flexible enough to meet the risk management needs of all companies. Futures contracts are standardized contracts. This means that the contracts have set sizes and maturity dates. Futures contracts for Korean won, for example, have a contract size of 125 million won. A company that needs 200 million won later this year would need to either purchase one futures contract, hedging only a portion of its needs, or purchase two futures contracts, hedging more than it needs. Either way, the company has remaining currency risk.
In this next section, we will explore some additional hedging techniques.
### Forward Contracts
Suppose a company needs access to 200 million Korean won on March 1. In addition to a specified contract size, currency futures contracts have specified days on which the contracts are settled. For most currency futures contracts, this occurs on the third Wednesday of the month. If the company needed 125 million Korean won (the basic contract size) on the third Wednesday of March (the standard settlement date), the futures contract could be useful. Because the company needs a different number of Korean won on a different date from those specified in the standard contract, the futures contract is not going to meet the specific risk management needs of the company.
Another type of contract, the forward contract, can be used by this company to meet its specific needs. A forward contract is simply a contractual agreement between two parties to exchange a specified amount of currencies at a future date. A company can approach its bank, for example, saying that it will need to purchase 200 million Korean won on March 1. The bank will quote a forward rate, which is a rate specified today for the sale of currency on a future date, and the company and the bank can enter into a forward contract to exchange dollars for 200 million Korean won at the quoted rate on March 1.
Because a forward contract is a contract between two parties, those two parties can specify the amount that will be traded and the date the trade will occur. This contract is similar to your agreeing with a hotel that you will arrive on March 1 and rent a room for three nights at $200 per night. You are agreeing today to show up at the hotel on a future (specified) date and pay the quoted price when you arrive. The hotel agrees to provide you the room on March 1 and cannot change the price of the room when you arrive. With a forward contract, you are also agreeing that you will indeed make the purchase and you cannot change your mind; so, using the hotel room analogy, this would mean that the hotel will definitely charge your credit card for the agreed-upon $200 per night on March 1.
The forward contract is an individualized contract between the buyer and the seller; they are both under a contractual obligation to honor the contract. Because this contract is not standardized like the futures contract (so that it can be traded on an exchange), it can be tailored to the needs of the two parties. While the forward contract has the advantage of being fine-tuned to meet the company’s needs, it has a risk, known as counterparty risk, that the futures contract does not have. The forward contract is only as good as the promise of the counterparty. If the company enters into a forward contract to purchase 200 million Korean won on March 1 from its bank and the bank goes out of business before March 1, the company will not be able to make the exchange with a nonexistent bank. The exchanges on which futures contracts are traded guard the purchaser of a futures contract from this type of risk by guaranteeing the contract.
### Natural Hedges
A hedge simply refers to a reduction in the risk or exposure that a company has to volatility and uncertainty. We have been focusing on how a company might use financial market instruments to hedge, but sometimes a company can use a natural hedge to mitigate risk. A natural hedge occurs when a business can offset its risk simply through its own operations. With a natural hedge, when a risk occurs that would decrease the value of a company, an offsetting event occurs within the firm that increases the value of the company.
As an example, consider a British-based travel agency. One of the major tours the company offers is a tour of Italy. The company arranges for transportation, lodging, meals, and sightseeing for Brits to visit the highlights of Rome, Florence, and Venice. Because the company charges customers in British pounds but must pay the bus companies, hotels, and other service providers in Italy in euros, the travel agency faces significant transaction exposure. If the value of the British pound depreciates after the company sets the price it will charge for the tour but before it pays the Italian suppliers, the company will be harmed. In fact, if the British pound depreciates by a great deal, the company could end up in a situation in which the British pounds it collects are not enough to purchase the euros it needs to pay its suppliers.
The company could create a natural hedge by offering tours of London to individuals living in the European Union. The travel agency could charge people who live in Germany, Italy, Spain, or any other country that has the euro as its currency for a travel package to London. Then the agency would pay British restaurants, tour guides, hotels, and bus companies in British pounds. This segment of the business also has currency risk. If the British pound depreciates, the company gains because the euros it collects from its EU customers will purchase more British pounds than before.
Thus, the company has created a situation in which if the British pound depreciates, the decrease in value of its tours of Italy is exactly offset by the increase in value of its tours of London. If the British pound appreciates, the opposite occurs: the company experiences a gain in its division that charges British pounds for tourists traveling to Italy and an offsetting loss in its division that charges euros for tourists traveling to London.
### Options
A financial option gives the owner the right, but not the obligation, to purchase or sell an asset for a specified price at some future date. Options are considered derivative securities because the value of a derivative is derived from, or comes from, the value of another asset.
### Options Terminology
Specific terminology is used in the finance industry to describe the details of an options contract. If the owner of an option decides to purchase or sell the asset according to the terms of the options contract, the owner is said to be exercising the option. The price the option holder pays if purchasing the asset or receives if selling the asset is known as the strike price or exercise price. The price the owner of the option paid for the option is known as the premium.
An option contract will have an expiration date. The most common kinds of options are American options, which allow the holder to exercise the option at any time up to and including the expiration date. Holders of European options may exercise their options only on the expiration date. The labels American option and European option can be confusing as they have nothing to do with the location where the options are traded. Both American and European options are traded worldwide.
Option contracts are written for a variety of assets. The most common option contracts are options on shares of stock. Options are traded for US Treasury securities, currencies, gold, and oil. There are also options on agricultural products such as wheat, soybeans, cotton, and orange juice. Thus, options can be used by financial managers to hedge many types of risk, including currency risk, interest rate risk, and the risk that arises from fluctuations in the prices of raw materials.
Options are divided into two main categories, call options and put options. A call option gives the owner of the option the right, but not the obligation, to buy the underlying asset. A put option gives the owner the right, but not the obligation, to sell the underlying asset.
### Call Options
If a Korean company knows that it will need pay a $100,000 bill to a US supplier in six months, it knows how many US dollars it will need to pay the bill. As a Korean company, however, its bank account is denominated in Korean won. In six months, it will need to use its Korean won to purchase 100,000 US dollars.
The company can determine how many Korean won it would take to purchase $100,000 today. If the current exchange rate is , then it will need KWN 110,000,000 to pay the bill. The current exchange rate is known as the spot rate.
The company, however, does not need the US dollars for another six months. The company can purchase a call option, which is a contract that will allow it to purchase the needed US dollars in six months at a price stated in the contract. This allows the company to guarantee a price for dollars in six months, but it does not obligate the company to purchase the dollars at that price if it can find a better price when it needs the dollars in six months.
The price that is in the contract is called the strike price (exercise price). Suppose the company purchases a call contract for US dollars with a strike price of KWN 1,200/USD. While this contract would be for a set size, or a certain number of US dollars, we will talk about this transaction as if it were per one US dollar to highlight how options contracts work.
The company must pay a price, known as the premium, to purchase this call option contract. For our example, let’s assume the premium for the call option contract is KWN 50. In other words, the company has paid KWN 50 for the right to buy US dollars in six months for a price of KWN 1,200/USD.
In six months, the company makes a choice to either (1) pay the strike price of KWN 1,200/USD or (2) let the option expire. If the company chooses to pay the strike price and purchase the US dollars, it is exercising the option. How does the company choose which to do? It simply compares the strike price of KWN 1,200/USD to the market, or spot, exchange rate at the time the option is expiring.
If, six months from now, the spot exchange rate is , it will be cheaper for the company to buy the US dollars it needs at the spot price than it would be to buy the dollars with the option. In fact, if the spot rate is anything below , the company will not choose to exercise the option. If, however, the spot exchange rate in six months is , the company will exercise the option and purchase each US dollar for only KWN 1,200.
The profitability, or the payoff, to the owner of a call option is represented by the chart in below. Possible spot prices are measured from left to right, and the financial gain or loss to the company of the option contract is measured vertically. If the spot price is anything less than KWN 1,200/USD, the option expires without being exercised. The company paid KWN 50 for something that ended up being worthless.
If, in six months, the spot exchange rate is , then the company will choose to exercise the option. The company will be saving KWN 25 for each dollar purchased, but the company originally paid 50 KWN for the contract. So, the company will be 25 KWN worse off than if it had never purchased the call option.
If the spot exchange rate is , the company will be in exactly the same position having purchased and exercised the call option as it would have been if it had not purchased the option. At any spot price higher than KWN 1,250/USD, the firm will be in a better financial position, or will have a positive payoff, because it purchased the call option. The more the Korean won depreciates over the next six months, the higher the payoff to the firm of owning the call contract. Purchasing the call contract is a way that the company can protect itself from the currency exposure it faces.
For any transaction, there must be two parties—a buyer and a seller. For the company to have purchased the call option, another party must have sold the call option. The seller of a call option is called the option writer. Let’s consider the potential benefits and risks to the writer of the call option.
When the company purchases the call option, it pays the premium to the writer. The writer of the option does not have a choice regarding whether the option will be exercised. The purchaser of the option has the right to make the choice; in essence, the writer of the option sold the right to make that decision to the purchasers of the call option.
shows the payoff to the writer of the call option. Recall that the buyer of the call option will let the option expire if the spot rate is less than when the call option matures in six months. If this occurs, the writer of the option collected the KWN 50 option premium when the contract was sold and then never hears from the purchaser again. This is what the writer of the option is hoping for; the writer of the call option profits when the options contract is not exercised
If the spot rate is above , then the holder of the option will choose to exercise the right to purchase the won at the option strike price. Then the writer of the option will be obligated to sell the Korean won at a price of KWN 1,200/USD. If the spot rate is , the option writer will be obligated to sell the dollars for KWN 50 less than what they are worth; because the option writer was initially paid a KWN 50 premium for taking on that obligation, the option writer will just break even. For any exchange rate higher than , the writer of the call option will have a loss.
The option contract is a zero-sum game. Any payoff the owner of the option receives is exactly equal to the loss the writer of the option has. Any loss the owner of the option has is exactly equal to the payoff the writer of the option receives.
### Put Options
While the call option you just considered gives the owner the right to buy an underlying asset, the put option gives the owner to right to sell an underlying asset. Take, for example, an Indian company that has a contract to provide graphic artwork for a US company. The US company will pay the Indian company 200,000 US dollars in three months.
While the Indian company receives US dollars, it must pay its workers in Indian rupees. Because the company does not know what the spot exchange rate will be in three months, it faces transaction risk and may be interested in hedging this exposure using a put option.
The company knows that the current spot rate is , meaning that the company would be able to use $200,000 to purchase if it possessed the $200,000 today. If the Indian rupee appreciates relative to the US dollar over the next three months, however, the company will receive fewer rupees when it makes the exchange; perhaps the company will not be able to purchase enough rupees to cover the wages of its employees.
Assume the company can purchase a put option that gives it the right to sell US dollars in three months at a strike price of INR 75/USD; the premium for this put option is INR 5. By purchasing this put option, the company is spending INR 5 to guarantee that it can sell its US dollars for rupees in three months at a price of INR 75/USD.
If, in three months, when the company receives payment in US dollars, the spot exchange rate is higher than , the company will simply exchange the US dollars for rupees at that exchange rate, allowing the put option to expire without exercising it. The payoff to the company for the option is INR -5, the premium that was paid for the option that was never used (see ).
If, however, in three months, the spot exchange rate is anything less than , then the company will choose to exercise the option. If the spot rate is between and , the payoff for the option is negative. For example, if the spot exchange rate is , the company will exercise the option and receive three more Indian rupees per dollar than it would in the spot market. However, the company had to spend INR 5 for the option, so the payoff is INR -2. At a spot exchange rate of , the company has a zero payoff; the benefit of exercising the option, INR 5, is exactly equal to the price of purchasing the option, the premium of INR 5.
If, in three months, the spot exchange rate is anything below , the payoff of the put option is positive. At the theoretical extreme, if the USD became worthless and would purchase no rupees in the spot market when the company received the dollars, the company could exercise its option and receive INR 75/USD, and its payoff would be INR 70.
Now that we have considered the payoff to a purchaser of a put contract, let’s consider the opposite side of the contract: the seller, or writer, of the put option. The writer of a put option is selling the right to sell dollars to the purchaser of the put option. The writer of the put option collects a premium for this. The writer of the put has no choice as to whether the put option will be exercised; the writer only has an obligation to honor the contract if the owner of the put option chooses to exercise it.
The owner of the option will choose to let the option expire if the spot exchange rate is anything above . If that is the case, the writer of the put option collects the INR 5 premium for writing the put, as shown by the horizontal line in . This is what the writer of the put is hoping will occur.
The owner of the option will choose to exercise the option if the exchange rate is less than . If the spot exchange rate is between and , the writer of the put option has a positive payoff. Although the writer must now purchase US dollars for a price higher than what the dollars are worth, the INR 5 premium that the writer received when entering into the position is more than enough to offset that loss.
If the spot exchange rate drops below , however, the writer of the put option is losing more than INR 5 when the option is exercised, leaving the writer with a negative payoff. In the extreme, the writer of the put will have to purchase worthless US dollars for INR 75/USD, resulting in a loss of INR 70.
Notice that the payoff to the writer of the put is the negative of the payoff to the holder of the put at every spot price. The highest payoff occurs to the writer of the put when the option is never exercised. In that instance, the payoff to the writer is the premium that the holder of the put paid when purchasing the option (see ).
provides a summary of the positions that the parties who enter into options contract are in. Remember that the buyer of an option is always the one purchasing the right to do something. The seller or writer of an option is selling the right to make a decision; the seller has the obligation to fulfill the contract should the buyer of the option choose to exercise the option. The most the seller of an option can ever profit is by the premium that was paid for the option; this occurs when the option is not exercised.
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Exchange rates are unpredictable. This leads to transaction risk, translation risk, and economic risk as currency values change. A forward contract is an agreement between two parties to make an exchange at a particular rate on a given date in the future. Companies can use options to mitigate the risks. A call option gives the holder the right, but not the obligation, to purchase an underlying asset. A put option give the holder the right, but not the obligation, to sell an underlying asset.
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### BMW in the United States
While the name BMW may sound German, a significant amount of BMW’s production occurs outside of Germany. Watch this video to learn about this international activity of BMW. |
# Risk Management and the Financial Manager
## Interest Rate Risk
### Learning Outcomes
By the end of this section, you will be able to:
1. Describe interest rate risk.
2. Explain how a change in interest rates changes the value of cash flows.
3. Describe the use of an interest rate swap.
An interest rate is simply the price of borrowing money. Just as other prices are volatile, interest rates are also volatile. Just as volatility in other prices leads to uncertain cash flows for a company, volatility in interest rates can also lead to uncertain cash flows.
### Measuring Interest Rate Risk
Suppose that a company is supposed to pay a bill of $1,000 in 10 years. The present value of this bill depends on the level of interest rates. If the interest rate is 5%, the present value of the bill is . If the interest rate rises to 6%, the present value of the bill is . The increase in the interest rate by 1% causes the present value of the expected cash flow to fall by .
Interest rate risk can be highlighted by looking at bonds. Consider two $1,000 face value bonds with a 5% coupon rate, paid semiannually. One of the bonds matures in five years, and the other bond matures in 30 years. If the market interest rate is 5%, each of these bonds will sell for face value, or $1,000. If, instead, the market interest rate is 6%, the five-year bond will sell for $957.35 and the 30-year bond will sell for $861.62.
Notice that as the interest rate rises, the price of both of these bonds will fall. However, the price of the longer-term bond will fall by more than the price of the shorter-term bond. The longer-term bond price will fall by 1.38%; the shorter-term bond price will fall by only 0.43%.
Consider two additional $1,000 face value bonds. The difference is that these bonds have a 6% coupon rate, paid semiannually. If a bond has a 6% coupon rate and matures in five years, it will sell for $1,043.76 when the market interest rate is 5%. A 30-year bond that matures in 30 years and has a 6% coupon rate will sell for $1,154.54 when the market interest rate is 5%. However, if the interest rate in the economy is 6%, both of these bonds will sell for a price of $1,000. The price of the five-year bond will drop by 4.19%; the price of the 30-year bond will drop by 13.39%.
The sensitivity of bond prices to changes in the interest rate is known as interest rate risk. Duration is an important measure of interest rate risk that incorporates the maturity and coupon rate of a bond as well as the level of current market interest rates. Calculating duration is a complex topic that is beyond the scope of this introductory textbook, but it is useful to note that
1. the higher the duration of a bond, the more sensitive the price of the bond will be to interest rate changes;
2. the duration of a bond will be higher when market yields are lower, all else being equal;
3. the duration of a bond will be higher the longer the maturity of the bond, all else being equal; and
4. the duration of a bond will be higher the lower the coupon rate on the bond, all else being equal.
### Swap-Based Hedging
As the name suggests, a swap involves two parties agreeing to swap, or exchange, something. Generally, the two parties, known as counterparties, are swapping obligations to make specified payment streams.
To illustrate the basics of how an interest rate swap works, let’s consider two hypothetical companies, Alpha and Beta. Alpha is a strong, well-established company with a AAA (triple-A) bond rating. This means that Alpha has the highest rating a company can have. With this high rating, Alpha can borrow at relatively low interest rates. Often, companies in this situation will borrow at a floating rate. This means that their interest rate goes up and down as interest rates in the overall economy vary. The floating rate will be tied to a benchmark rate that is widely quoted in the financial press. Historically, companies have often used the London Interbank Offered Rate (LIBOR) as the benchmark rate. Because published quotes for LIBOR will be phased out by 2023, firms are beginning to use alternative rates. As of yet, no single alternative has emerged as the most commonly used rate; therefore, LIBOR will be used in our example. Suppose that Alpha finds that it can borrow money at rate equal to ; thus, if LIBOR is 2.75%, the company will pay 3.0% to borrow. If the company wants to borrow at a long-term fixed rate, its cost of borrowing will be 5.0%.
Beta has a BBB bond rating. Although this is considered a good, investment-grade rating, it is lower than the rating of Alpha. Because Beta is less creditworthy and a bit riskier than Alpha, it will have to pay a higher interest rate to borrow money. If Beta wants to borrow money at a floating rate, it will need to pay If LIBOR is 2.75%, Beta must pay 3.5% on its floating rate debt. In order for Beta to borrow at a long-term fixed rate, its cost of borrowing will be 6.75%.
Let’s consider how these two companies can enter into a swap in which both parties benefit. summarizes the situation and the rates at which Alpha and Beta can borrow. It also illustrates a way in which an interest rate swap can benefit both Alpha and Beta.
Alpha borrows in the capital markets at a fixed rate of 5%. Beta chooses to borrow at a floating rate that equals Beta also agrees to pay Alpha a fixed rate of 5.5%. In essence, Beta is paying 5.5% to Alpha, 0.75% to its lender, and LIBOR to its lender.
In return, Alpha promises to pay Beta LIBOR. The exact amount that Alpha will pay to Beta fluctuates as LIBOR fluctuates. However, from Beta’s perspective, the payment of LIBOR it receives from Alpha exactly offsets the payment of LIBOR it makes to its lender. When LIBOR increases, the rate of that Beta is paying to its lender increases, but the LIBOR rate it receives from Alpha also increases. When LIBOR decreases, Beta receives less from Alpha, but it also pays less to its lender. Because the LIBOR it receives from Alpha is exactly equal to the LIBOR it pays to its lender, Beta’s net amount of interest paid is 6.25%—the 5.5% it pays to Alpha plus the 0.75% it pays to its lender.
Alpha is in the position of paying 5.0% to its lender and LIBOR to Beta while receiving 5.5% from Beta. This means that Alpha’s net interest paid is Alpha is said to have swapped its fixed interest rate for a floating rate. Because it is paying , it will experience fluctuating interest rates; however, as a company with a AAA bond rating, it is a strong, creditworthy company that can withstand that interest rate exposure. It would have cost Alpha to borrow the money from its lenders at a variable rate. By participating in this swap arrangement, Alpha has been able to lower its interest rate by 0.75%.
Through this swap arrangement, Beta has been able to fix its interest rate at 6.25% rather than having a variable rate. This predictability is a benefit for a company, especially one that is in a bit more precarious position as far as its creditworthiness and stability. The 6.25% Beta pays as a result of this arrangement is 0.5% below the 6.75% it would have paid if it simply borrowed from its lenders at a fixed rate.
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When interest rates increase, the present value of future cash flows decreases. Duration is a measure of interest rate risk. A swap involves two parties agreeing to exchange something, often specified payment streams.
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### CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute. |
# The Americas, Europe, and Africa Before 1492
## Introduction
Globalization, the ever-increasing interconnectedness of the world, is not a new phenomenon, but it accelerated when western Europeans discovered the riches of the East. During the Crusades (1095–1291), Europeans developed an appetite for spices, silk, porcelain, sugar, and other luxury items from the East, for which they traded fur, timber, and Slavic people they captured and sold (hence the word slave). But when the Silk Road, the long overland trading route from China to the Mediterranean, became costlier and more dangerous to travel, Europeans searched for a more efficient and inexpensive trade route over water, initiating the development of what we now call the Atlantic World.
In pursuit of commerce in Asia, fifteenth-century traders unexpectedly encountered a “New World” populated by millions and home to sophisticated and numerous peoples. Mistakenly believing they had reached the East Indies, these early explorers called its inhabitants “Indians.” West Africa, a diverse and culturally rich area, soon entered the stage as other nations exploited its slave trade and brought its peoples to the New World in chains. Although Europeans would come to dominate the New World, they could not have done so without Africans and Native peoples (). |
# The Americas, Europe, and Africa Before 1492
## The Americas
Most Native American origin stories assert that Native nations have always called the Americas home; however, some scholars believe that between nine and fifteen thousand years ago, a land bridge existed between Asia and North America that we now call Beringia. The first inhabitants of what would be named the Americas migrated across this bridge in search of food. When the glaciers melted, water engulfed Beringia, and the Bering Strait was formed. Later settlers came by boat across the narrow strait. (The fact that Asians and Native Americans share genetic markers on a Y chromosome lends credibility to this migration theory.) Continually moving southward, the settlers eventually populated both North and South America, creating unique cultures that ranged from the highly complex and urban Aztec civilization in what is now Mexico City to the woodland tribes of eastern North America. Recent research along the west coast of South America suggests that migrant populations may have traveled down this coast by water as well as by land.
Researchers believe that about ten thousand years ago, humans also began the domestication of plants and animals, adding agriculture as a means of sustenance to hunting and gathering techniques. With this agricultural revolution, and the more abundant and reliable food supplies it brought, populations grew and people were able to develop a more settled way of life, building permanent settlements. Nowhere in the Americas was this more obvious than in Mesoamerica ().
### THE FIRST AMERICANS: THE OLMEC
Mesoamerica is the geographic area stretching from north of Panama up to the desert of central Mexico. Although marked by great topographic, linguistic, and cultural diversity, this region cradled a number of civilizations with similar characteristics. Mesoamericans were polytheistic; their gods possessed both male and female traits and demanded blood sacrifices of enemies taken in battle or ritual bloodletting. Corn, or maize, domesticated by 5000 BCE, formed the basis of their diet. They developed a mathematical system, built huge edifices, and devised a calendar that accurately predicted eclipses and solstices and that priest-astronomers used to direct the planting and harvesting of crops. Most important for our knowledge of these peoples, they created the only known written language in the Western Hemisphere; researchers have made much progress in interpreting the inscriptions on their temples and pyramids. Though the area had no overarching political structure, trade over long distances helped diffuse culture. Weapons made of obsidian, jewelry crafted from jade, feathers woven into clothing and ornaments, and cacao beans that were whipped into a chocolate drink formed the basis of commerce. The mother of Mesoamerican cultures was the Olmec civilization.
Flourishing along the hot Gulf Coast of Mexico from about 1200 to about 400 BCE, the Olmec produced a number of major works of art, architecture, pottery, and sculpture. Most recognizable are their giant head sculptures () and the pyramid in La Venta. The Olmec built aqueducts to transport water into their cities and irrigate their fields. They grew maize, squash, beans, and tomatoes. They also bred small domesticated dogs which, along with fish, provided their protein. Although no one knows what happened to the Olmec after about 400 BCE, in part because the jungle reclaimed many of their cities, their culture was the base upon which the Maya and the Aztec built. It was the Olmec who worshipped a rain god, a maize god, and the feathered serpent so important in the future pantheons of the Aztecs (who called him Quetzalcoatl) and the Maya (to whom he was Kukulkan). The Olmec also developed a system of trade throughout Mesoamerica, giving rise to an elite class.
### THE MAYA
After the decline of the Olmec, a city rose in the fertile central highlands of Mesoamerica. One of the largest population centers in pre-Columbian America and home to more than 100,000 people at its height in about 500 CE, Teotihuacan was located about thirty miles northeast of modern Mexico City. The ethnicity of this settlement’s inhabitants is debated; some scholars believe it was a multiethnic city. Large-scale agriculture and the resultant abundance of food allowed time for people to develop special trades and skills other than farming. Builders constructed over twenty-two hundred apartment compounds for multiple families, as well as more than a hundred temples. Among these were the Pyramid of the Sun (which is two hundred feet high) and the Pyramid of the Moon (one hundred and fifty feet high). Near the Temple of the Feathered Serpent, graves have been uncovered that suggest humans were sacrificed for religious purposes. The city was also the center for trade, which extended to settlements on Mesoamerica’s Gulf Coast.
The Maya were one Mesoamerican culture that had strong ties to Teotihuacan. The Maya’s architectural and mathematical contributions were significant. Flourishing from roughly 2000 BCE to 900 CE in what is now Mexico, Belize, Honduras, and Guatemala, the Maya perfected the calendar and written language the Olmec had begun. They devised a written mathematical system to record crop yields and the size of the population, and to assist in trade. Surrounded by farms relying on primitive agriculture, they built the city-states of Copan, Tikal, and Chichen Itza along their major trade routes, as well as temples, statues of gods, pyramids, and astronomical observatories (). However, because of poor soil and a drought that lasted nearly two centuries, their civilization declined by about 900 CE and they abandoned their large population centers.
The Spanish found little organized resistance among the weakened Maya upon their arrival in the 1520s. However, they did find Mayan history, in the form of glyphs, or pictures representing words, recorded in folding books called codices (the singular is codex). In 1562, Bishop Diego de Landa, who feared the converted Native people had reverted to their traditional religious practices, collected and burned every codex he could find. Today only a few survive.
### THE AZTEC
When the Spaniard Hernán Cortés arrived on the coast of Mexico in the sixteenth century, at the site of present-day Veracruz, he soon heard of a great city ruled by an emperor named Moctezuma. This city was tremendously wealthy—filled with gold—and took in tribute from surrounding tribes. The riches and complexity Cortés found when he arrived at that city, known as Tenochtitlán, were far beyond anything he or his men had ever seen.
According to legend, a warlike people called the Aztec (also known as the Mexica) had left a city called Aztlán and traveled south to the site of present-day Mexico City. In 1325, they began construction of Tenochtitlán on an island in Lake Texcoco. By 1519, when Cortés arrived, this settlement contained upwards of 200,000 inhabitants and was certainly the largest city in the Western Hemisphere at that time and probably larger than any European city (). One of Cortés’s soldiers, Bernal Díaz del Castillo, recorded his impressions upon first seeing it: “When we saw so many cities and villages built in the water and other great towns on dry land we were amazed and said it was like the enchantments . . . on account of the great towers and cues and buildings rising from the water, and all built of masonry. And some of our soldiers even asked whether the things that we saw were not a dream? . . . I do not know how to describe it, seeing things as we did that had never been heard of or seen before, not even dreamed about.”
Unlike the dirty, fetid cities of Europe at the time, Tenochtitlán was well planned, clean, and orderly. The city had neighborhoods for specific occupations, a trash collection system, markets, two aqueducts bringing in fresh water, and public buildings and temples. Unlike the Spanish, Aztecs bathed daily, and wealthy homes might even contain a steam bath. A labor force of enslaved people from subjugated neighboring tribes had built the fabulous city and the three causeways that connected it to the mainland. To farm, the Aztec constructed barges made of reeds and filled them with fertile soil. Lake water constantly irrigated these , or “floating gardens,” which are still in use and can be seen today in Xochimilco, a district of Mexico City.
Each god in the Aztec pantheon represented and ruled an aspect of the natural world, such as the heavens, farming, rain, fertility, sacrifice, and combat. A ruling class of warrior nobles and priests performed ritual human sacrifice daily to sustain the sun on its long journey across the sky, to appease or feed the gods, and to stimulate agricultural production. The sacrificial ceremony included cutting open the chest of a criminal or captured warrior with an obsidian knife and removing the still-beating heart ().
### THE INCA
In South America, the most highly developed and complex society was that of the Inca, whose name means “lord” or “ruler” in the Andean language called Quechua. At its height in the fifteenth and sixteenth centuries, the Inca Empire, located on the Pacific coast and straddling the Andes Mountains, extended some twenty-five hundred miles. It stretched from modern-day Colombia in the north to Chile in the south and included cities built at an altitude of 14,000 feet above sea level. Its road system, kept free of debris and repaired by workers stationed at varying intervals, rivaled that of the Romans and efficiently connected the sprawling empire. The Inca, like all other pre-Columbian societies, did not use axle-mounted wheels for transportation. They built stepped roads to ascend and descend the steep slopes of the Andes; these would have been impractical for wheeled vehicles but worked well for pedestrians. These roads enabled the rapid movement of the highly trained Incan army. Also like the Romans, the Inca were effective administrators. Runners called traversed the roads in a continuous relay system, ensuring quick communication over long distances. The Inca had no system of writing, however. They communicated and kept records using a system of colored strings and knots called the ().
The Inca people worshipped their lord who, as a member of an elite ruling class, had absolute authority over every aspect of life. Much like feudal lords in Europe at the time, the ruling class lived off the labor of the peasants, collecting vast wealth that accompanied them as they went, mummified, into the next life. The Inca farmed corn, beans, squash, quinoa (a grain cultivated for its seeds), and the indigenous potato on terraced land they hacked from the steep mountains. Peasants received only one-third of their crops for themselves. The Inca ruler required a third, and a third was set aside in a kind of welfare system for those unable to work. Huge storehouses were filled with food for times of need. Each peasant also worked for the Inca ruler a number of days per month on public works projects, a requirement known as the . For example, peasants constructed rope bridges made of grass to span the mountains above fast-flowing icy rivers. In return, the lord provided laws, protection, and relief in times of famine.
The Inca worshipped the sun god Inti and called gold the “sweat” of the sun. Unlike the Maya and the Aztecs, they rarely practiced human sacrifice and usually offered the gods food, clothing, and coca leaves. In times of dire emergency, however, such as in the aftermath of earthquakes, volcanoes, or crop failure, they resorted to sacrificing prisoners. The ultimate sacrifice was children, who were specially selected and well fed. The Inca believed these children would immediately go to a much better afterlife.
In 1911, the American historian Hiram Bingham uncovered the lost Incan city of Machu Picchu (). Located about fifty miles northwest of Cusco, Peru, at an altitude of about 8,000 feet, the city had been built in 1450 and inexplicably abandoned roughly a hundred years later. Scholars believe the city was used for religious ceremonial purposes and housed the priesthood. The architectural beauty of this city is unrivaled. Using only the strength of human labor and no machines, the Inca constructed walls and buildings of polished stones, some weighing over fifty tons, that were fitted together perfectly without the use of mortar. In 1983, UNESCO designated the ruined city a World Heritage Site.
### NATIVE AMERICANS
With few exceptions, the North American Native cultures were much more widely dispersed than the Mayan, Aztec, and Incan societies, and did not have their population size or organized social structures. Although the cultivation of corn had made its way north, many Native people still practiced hunting and gathering. Horses, first introduced by the Spanish, allowed the Plains Natives to more easily follow and hunt the huge herds of bison. A few societies had evolved into relatively complex forms, but they were already in decline at the time of Christopher Columbus’s arrival.
In the southwestern part of today’s United States dwelled several groups we collectively call the Pueblo. The Spanish first gave them this name, which means “town” or “village,” because they lived in towns or villages of permanent stone-and-mud buildings with thatched roofs. Like present-day apartment houses, these buildings had multiple stories, each with multiple rooms. The three main groups of the Pueblo people were the Mogollon, Hohokam, and Anasazi.
The Mogollon thrived in the Mimbres Valley (New Mexico) from about 150 BCE to 1450 CE. They developed a distinctive artistic style for painting bowls with finely drawn geometric figures and wildlife, especially birds, in black on a white background. Beginning about 600 CE, the Hohokam built an extensive irrigation system of canals to irrigate the desert and grow fields of corn, beans, and squash. By 1300, their crop yields were supporting the most highly populated settlements in the southwest. The Hohokam decorated pottery with a red-on-buff design and made jewelry of turquoise. In the high desert of New Mexico, the Anasazi, whose name means “ancient enemy” or “ancient ones,” carved homes from steep cliffs accessed by ladders or ropes that could be pulled in at night or in case of enemy attack ().
Roads extending some 180 miles connected the Pueblos’ smaller urban centers to each other and to Chaco Canyon, which by 1050 CE had become the administrative, religious, and cultural center of their civilization. A century later, however, probably because of drought, the Pueblo peoples abandoned their cities. Their present-day descendants include the Hopi and Zuni tribes.
The Indigenous groups who lived in the present-day Ohio River Valley and achieved their cultural apex from the first century CE to 400 CE are collectively known as the Hopewell culture. Their settlements, unlike those of the southwest, were small hamlets. They lived in wattle-and-daub houses (made from woven lattice branches “daubed” with wet mud, clay, or sand and straw) and practiced agriculture, which they supplemented by hunting and fishing. Utilizing waterways, they developed trade routes stretching from Canada to Louisiana, where they exchanged goods with other tribes and negotiated in many different languages. From the coast they received shells; from Canada, copper; and from the Rocky Mountains, obsidian. With these materials they created necklaces, woven mats, and exquisite carvings. What remains of their culture today are huge burial mounds and earthworks. Many of the mounds that were opened by archaeologists contained artworks and other goods that indicate their society was socially stratified.
Perhaps the largest indigenous cultural and population center in North America was located along the Mississippi River near present-day St. Louis. At its height in about 1100 CE, this five-square-mile city, now called Cahokia, was home to more than ten thousand residents; tens of thousands more lived on farms surrounding the urban center. The city also contained one hundred and twenty earthen mounds or pyramids, each dominating a particular neighborhood and on each of which lived a leader who exercised authority over the surrounding area. The largest mound covered fifteen acres. Cahokia was the hub of political and trading activities along the Mississippi River. After 1300 CE, however, this civilization declined—possibly because the area became unable to support the large population.
### NATIVE PEOPLES OF THE EASTERN WOODLAND
Encouraged by the wealth found by the Spanish in the settled civilizations to the south, sixteenth- and seventeenth-century English, Dutch, and French explorers expected to discover the same in North America. What they found instead were small, disparate communities, many already ravaged by European diseases brought by the Spanish and transmitted among the Native peoples. Rather than gold and silver, there was an abundance of land, and the timber and fur that land could produce.
The Native peoples living east of the Mississippi did not construct the large and complex societies of those to the west. Because they lived in small autonomous clans or tribal units, each group adapted to the specific environment in which it lived (). These groups were by no means unified, and warfare among tribes was common as they sought to increase their hunting and fishing areas. Still, these tribes shared some common traits. A leader or group of tribal elders made decisions, and although the leader was a man, usually the women selected and counseled him. Gender roles were not as fixed as they were in the patriarchal societies of Europe, Mesoamerica, and South America.
Women typically cultivated corn, beans, and squash and harvested nuts and berries, while men hunted, fished, and provided protection. But both took responsibility for raising children, and most major Native societies in the east were matriarchal. In tribes such as the Iroquois, Lenape, Muscogee, and Cherokee, women had both power and influence. They counseled and passed on the traditions of the tribe. These complementary gender roles changed dramatically with the coming of the Europeans, who introduced, sometimes forcibly, their own customs and traditions to the natives.
Clashing beliefs about land ownership and use of the environment would be the greatest area of conflict with Europeans. Although tribes often claimed the right to certain hunting grounds—usually identified by some geographical landmark—Native peoples did not practice, or in general even have the concept of, private ownership of land. The European Christian worldview, on the other hand, viewed land as the source of wealth. According to the Christian Bible, God created humanity in his own image with the command to use and subdue the rest of creation, which included not only land, but also all animal life. Land, and the game that populated it, they believed, were there for the taking.
### Section Summary
Great civilizations had risen and fallen in the Americas before the arrival of the Europeans. In North America, the complex Pueblo societies including the Mogollon, Hohokam, and Anasazi as well as the city at Cahokia had peaked and were largely memories. The Eastern Woodland peoples were thriving, but they were soon overwhelmed as the number of English, French, and Dutch settlers increased.
Mesoamerica and South America had also witnessed the rise and fall of cultures. The once-mighty Mayan population centers were largely empty. In 1492, however, the Aztecs in Mexico City were at their peak. Subjugating surrounding tribes and requiring tribute of both humans for sacrifice and goods for consumption, the island city of Tenochtitlán was the hub of an ever-widening commercial center and the equal of any large European city until Cortés destroyed it. Further south in Peru, the Inca linked one of the largest empires in history through the use of roads and disciplined armies. Without the use of the wheel, they cut and fashioned stone to build Machu Picchu high in the Andes before abandoning the city for unknown reasons. Thus, depending on what part of the New World they explored, the Europeans encountered peoples that diverged widely in their cultures, traditions, and numbers.
### Review Questions
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# The Americas, Europe, and Africa Before 1492
## Europe on the Brink of Change
The fall of the Roman Empire (476 CE) and the beginning of the European Renaissance in the late fourteenth century roughly bookend the period we call the Middle Ages. Without a dominant centralized power or overarching cultural hub, Europe experienced political and military discord during this time. Its inhabitants retreated into walled cities, fearing marauding pillagers including Vikings, Mongols, Arabs, and Magyars. In return for protection, they submitted to powerful lords and their armies of knights. In their brief, hard lives, few people traveled more than ten miles from the place they were born.
The Christian Church remained intact, however, and emerged from the period as a unified and powerful institution. Priests, tucked away in monasteries, kept knowledge alive by collecting and copying religious and secular manuscripts, often adding beautiful drawings or artwork. Social and economic devastation arrived in 1340s, however, when Genoese merchants returning from the Black Sea unwittingly brought with them a rat-borne and highly contagious disease, known as the bubonic plague. In a few short years, it had killed many millions, about one-third of Europe’s population. A different strain, spread by airborne germs, also killed many. Together these two are collectively called the Black Death (). Entire villages disappeared. A high birth rate, however, coupled with bountiful harvests, meant that the population grew during the next century. By 1450, a newly rejuvenated European society was on the brink of tremendous change.
### LIFE IN FEUDAL EUROPE
During the Middle Ages, most Europeans lived in small villages that consisted of a manorial house or castle for the lord, a church, and simple homes for the peasants or serfs, who made up about 60 percent of western Europe’s population. Hundreds of these castles and walled cities remain all over Europe ().
Europe’s feudal society was a mutually supportive system. The lords owned the land; knights gave military service to a lord and carried out his justice; serfs worked the land in return for the protection offered by the lord’s castle or the walls of his city, into which they fled in times of danger from invaders. Much land was communally farmed at first, but as lords became more powerful they extended their ownership and rented land to their subjects. Thus, although they were technically free, serfs were effectively bound to the land they worked, which supported them and their families as well as the lord and all who depended on him. The Catholic Church, the only church in Europe at the time, also owned vast tracts of land and became very wealthy by collecting not only tithes (taxes consisting of 10 percent of annual earnings) but also rents on its lands.
A serf’s life was difficult. Women often died in childbirth, and perhaps one-third of children died before the age of five. Without sanitation or medicine, many people perished from diseases we consider inconsequential today; few lived to be older than forty-five. Entire families, usually including grandparents, lived in one- or two-room hovels that were cold, dark, and dirty. A fire was kept lit and was always a danger to the thatched roofs, while its constant smoke affected the inhabitants’ health and eyesight. Most individuals owned no more than two sets of clothing, consisting of a woolen jacket or tunic and linen undergarments, and bathed only when the waters melted in spring.
In an agrarian society, the seasons dictate the rhythm of life. Everyone in Europe’s feudal society had a job to do and worked hard. The father was the unquestioned head of the family. Idleness meant hunger. When the land began to thaw in early spring, peasants started tilling the soil with primitive wooden plows and crude rakes and hoes. Then they planted crops of wheat, rye, barley, and oats, reaping small yields that barely sustained the population. Bad weather, crop disease, or insect infestation could cause an entire village to starve or force the survivors to move to another location.
Early summer saw the first harvesting of hay, which was stored until needed to feed the animals in winter. Men and boys sheared the sheep, now heavy with wool from the cold weather, while women and children washed the wool and spun it into yarn. The coming of fall meant crops needed to be harvested and prepared for winter. Livestock was butchered and the meat smoked or salted to preserve it. With the harvest in and the provisions stored, fall was also the time for celebrating and giving thanks to God. Winter brought the people indoors to weave yarn into fabric, sew clothing, thresh grain, and keep the fires going. Everyone celebrated the birth of Christ in conjunction with the winter solstice.
### THE CHURCH AND SOCIETY
After the fall of Rome, the Christian Church—united in dogma but unofficially divided into western and eastern branches—was the only organized institution in medieval Europe. In 1054, the eastern branch of Christianity, led by the Patriarch of Constantinople (a title that became roughly equivalent to the western Church’s pope), established its center in Constantinople and adopted the Greek language for its services. The western branch, under the pope, remained in Rome, becoming known as the Roman Catholic Church and continuing to use Latin. Following this split, known as the Great Schism, each branch of Christianity maintained a strict organizational hierarchy. The pope in Rome, for example, oversaw a huge bureaucracy led by cardinals, known as “princes of the church,” who were followed by archbishops, bishops, and then priests. During this period, the Roman Church became the most powerful international organization in western Europe.
Just as agrarian life depended on the seasons, village and family life revolved around the Church. The sacraments, or special ceremonies of the Church, marked every stage of life, from birth to maturation, marriage, and burial, and brought people into the church on a regular basis. As Christianity spread throughout Europe, it replaced pagan and animistic views, explaining supernatural events and forces of nature in its own terms. A benevolent God in heaven, creator of the universe and beyond the realm of nature and the known, controlled all events, warring against the force of darkness, known as the Devil or Satan, here on earth. Although ultimately defeated, Satan still had the power to trick humans and cause them to commit evil or sin.
All events had a spiritual connotation. Sickness, for example, might be a sign that a person had sinned, while crop failure could result from the villagers’ not saying their prayers. Penitents confessed their sins to the priest, who absolved them and assigned them penance to atone for their acts and save themselves from eternal damnation. Thus the parish priest held enormous power over the lives of his parishioners.
Ultimately, the pope decided all matters of theology, interpreting the will of God to the people, but he also had authority over temporal matters. Because the Church had the ability to excommunicate people, or send a soul to hell forever, even monarchs feared to challenge its power. It was also the seat of all knowledge. Latin, the language of the Church, served as a unifying factor for a continent of isolated regions, each with its own dialect; in the early Middle Ages, nations as we know them today did not yet exist. The mostly illiterate serfs were thus dependent on those literate priests to read and interpret the Bible, the word of God, for them.
### CHRISTIANITY ENCOUNTERS ISLAM
The year 622 brought a new challenge to Christendom. Near Mecca, Saudi Arabia, a prophet named Muhammad received a revelation that became a cornerstone of the Islamic faith. The Koran contained his message, affirming monotheism but identifying Christ not as God but as a prophet like Moses, Abraham, David, and Muhammad. Following Muhammad’s death in 632, Islam spread by both conversion and military conquest across the Middle East and Asia Minor to India and northern Africa, crossing the Straits of Gibraltar into Spain in the year 711 ().
The Islamic conquest of Europe continued until 732. Then, at the Battle of Tours (in modern France), Charles Martel, nicknamed the Hammer, led a Christian force in defeating the army of Abdul Rahman al-Ghafiqi. Muslims, however, retained control of much of Spain, where Córdoba, known for leather and wool production, became a major center of learning and trade. By the eleventh century, a major Christian holy war called the Reconquista, or reconquest, had begun to slowly push Muslims from Spain. With the start of the Crusades, the wars between Christians and Muslims for domination of the Holy Land (the Biblical region of Palestine), Christians in Spain and around Europe began to see the Reconquista as part of a larger religious struggle with Islam.
### JERUSALEM AND THE CRUSADES
The city of Jerusalem is a holy site for Jews, Christians, and Muslims. It was here King Solomon built the Temple in the tenth century BCE. It was here the Romans crucified Jesus in 33 CE, and from here, Christians maintain, he ascended into heaven, promising to return. From here, Muslims believe, Muhammad traveled to heaven in 621 to receive instructions about prayer. Thus claims on the area go deep, and emotions about it run high, among followers of all three faiths. Evidence exists that the three religions lived in harmony for centuries. In 1095, however, European Christians decided not only to retake the holy city from the Muslim rulers but also to conquer what they called the Holy Lands, an area that extended from modern-day Turkey in the north along the Mediterranean coast to the Sinai Peninsula and that was also held by Muslims. The Crusades had begun.
Religious zeal motivated the knights who participated in the four Crusades. Adventure, the chance to win land and a title, and the Church’s promise of wholesale forgiveness of sins also motivated many. The Crusaders, mostly French knights, retook Jerusalem in June 1099 amid horrific slaughter. A French writer who accompanied them recorded this eyewitness account: “On the top of Solomon’s Temple, to which they had climbed in fleeing, many were shot to death with arrows and cast down headlong from the roof. Within this Temple, about ten thousand were beheaded. If you had been there, your feet would have been stained up to the ankles with the blood of the slain. What more shall I tell? Not one of them was allowed to live. They did not spare the women and children.” A Muslim eyewitness also described how the conquerors stripped the temple of its wealth and looted private homes.
In 1187, under the legendary leader Saladin, Muslim forces took back the city. Reaction from Europe was swift as King Richard I of England, the Lionheart, joined others to mount yet another action. The battle for the Holy Lands did not conclude until the Crusaders lost their Mediterranean stronghold at Acre (in present-day Israel) in 1291 and the last of the Christians left the area a few years later.
The Crusades had lasting effects, both positive and negative. On the negative side, the wide-scale persecution of Jews began. Christians classed them with the infidel Muslims and labeled them “the killers of Christ.” In the coming centuries, kings either expelled Jews from their kingdoms or forced them to pay heavy tributes for the privilege of remaining. Muslim-Christian hatred also festered, and intolerance grew.
On the positive side, maritime trade between East and West expanded. As Crusaders experienced the feel of silk, the taste of spices, and the utility of porcelain, desire for these products created new markets for merchants. In particular, the Adriatic port city of Venice prospered enormously from trade with Islamic merchants. Merchants’ ships brought Europeans valuable goods, traveling between the port cities of western Europe and the East from the tenth century on, along routes collectively labeled the Silk Road. From the days of the early adventurer Marco Polo, Venetian sailors had traveled to ports on the Black Sea and established their own colonies along the Mediterranean Coast. However, transporting goods along the old Silk Road was costly, slow, and unprofitable. Muslim middlemen collected taxes as the goods changed hands. Robbers waited to ambush the treasure-laden caravans. A direct water route to the East, cutting out the land portion of the trip, had to be found. As well as seeking a water passage to the wealthy cities in the East, sailors wanted to find a route to the exotic and wealthy Spice Islands in modern-day Indonesia, whose location was kept secret by Muslim rulers. Longtime rivals of Venice, the merchants of Genoa and Florence also looked west.
### THE IBERIAN PENINSULA
Although Norse explorers such as Leif Ericson, the son of Eric the Red who first settled Greenland, had reached Canada roughly five hundred years prior to Christopher Columbus’s voyage, it was explorers sailing for Portugal and Spain who traversed the Atlantic throughout the fifteenth century and ushered in an unprecedented age of exploration and permanent contact with North America.
Located on the extreme western edge of Europe, Portugal, with its port city of Lisbon, soon became the center for merchants desiring to undercut the Venetians’ hold on trade. With a population of about one million and supported by its ruler Prince Henry, whom historians call “the Navigator,” this independent kingdom fostered exploration of and trade with western Africa. Skilled shipbuilders and navigators who took advantage of maps from all over Europe, Portuguese sailors used triangular sails and built lighter vessels called caravels that could sail down the African coast.
Just to the east of Portugal, King Ferdinand of Aragon married Queen Isabella of Castile in 1469, uniting two of the most powerful independent kingdoms on the Iberian peninsula and laying the foundation for the modern nation of Spain. Isabella, motivated by strong religious zeal, was instrumental in beginning the Inquisition in 1480, a brutal campaign to root out Jews and Muslims who had seemingly converted to Christianity but secretly continued to practice their faith, as well as other heretics. This powerful couple ruled for the next twenty-five years, centralizing authority and funding exploration and trade with the East. One of their daughters, Catherine of Aragon, became the first wife of King Henry VIII of England.
The year 1492 witnessed some of the most significant events of Ferdinand and Isabella’s reign. The couple oversaw the final expulsion of North African Muslims (Moors) from the Kingdom of Granada, bringing the nearly eight-hundred-year Reconquista to an end. In this same year, they also ordered all unconverted Jews to leave Spain.
Also in 1492, after six years of lobbying, a Genoese sailor named Christopher Columbus persuaded the monarchs to fund his expedition to the Far East. Columbus had already pitched his plan to the rulers of Genoa and Venice without success, so the Spanish monarchy was his last hope. Christian zeal was the prime motivating factor for Isabella, as she imagined her faith spreading to the East. Ferdinand, the more practical of the two, hoped to acquire wealth from trade.
Most educated individuals at the time knew the earth was round, so Columbus’s plan to reach the East by sailing west was plausible. Though the calculations of Earth’s circumference made by the Greek geographer Eratosthenes in the second century BCE were known (and, as we now know, nearly accurate), most scholars did not believe they were dependable. Thus Columbus would have no way of knowing when he had traveled far enough around the Earth to reach his goal—and in fact, Columbus greatly underestimated the Earth’s circumference.
In August 1492, Columbus set sail with his three small caravels (). After a voyage of about three thousand miles lasting six weeks, he landed on an island in the Bahamas named Guanahani by the native Lucayans. He promptly christened it San Salvador, the name it bears today.
### Section Summary
One effect of the Crusades was that a larger portion of western Europe became familiar with the goods of the East. A lively trade subsequently developed along a variety of routes known collectively as the Silk Road to supply the demand for these products. Brigands and greedy middlemen made the trip along this route expensive and dangerous. By 1492, Europe—recovered from the Black Death and in search of new products and new wealth—was anxious to improve trade and communications with the rest of the world. Venice and Genoa led the way in trading with the East. The lure of profit pushed explorers to seek new trade routes to the Spice Islands and eliminate Muslim middlemen.
Portugal, under the leadership of Prince Henry the Navigator, attempted to send ships around the continent of Africa. Ferdinand of Aragon and Isabella of Castile hired Columbus to find a route to the East by going west. As strong supporters of the Catholic Church, they sought to bring Christianity to the East and any newly found lands, as well as hoping to find sources of wealth.
### Review Questions
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# The Americas, Europe, and Africa Before 1492
## West Africa and the Role of Slavery
It is difficult to generalize about West Africa, which was linked to the rise and diffusion of Islam. This geographical unit, central to the rise of the Atlantic World, stretches from modern-day Mauritania to the Democratic Republic of the Congo and encompasses lush rainforests along the equator, savannas on either side of the forest, and much drier land to the north. Until about 600 CE, most Africans were hunter-gatherers. Where water was too scarce for farming, herders maintained sheep, goats, cattle, or camels. In the more heavily wooded area near the equator, farmers raised yams, palm products, or plantains. The savanna areas yielded rice, millet, and sorghum. Sub-Saharan Africans had little experience in maritime matters. Most of the population lived away from the coast, which is connected to the interior by five main rivers—the Senegal, Gambia, Niger, Volta, and Congo.
Although there were large trading centers along these rivers, most West Africans lived in small villages and identified with their extended family or their clan. Wives, children, and dependents (including enslaved people) were a sign of wealth among men, and polygyny, the practice of having more than one wife at a time, was widespread. In time of need, relatives, however far away, were counted upon to assist in supplying food or security. Because of the clannish nature of African society, “we” was associated with the village and family members, while “they” included everyone else. Hundreds of separate dialects emerged; in modern Nigeria, nearly five hundred are still spoken.
### THE MAJOR AFRICAN EMPIRES
Following the death of the prophet Muhammad in 632 CE, Islam continued to spread quickly across North Africa, bringing not only a unifying faith but a political and legal structure as well. As lands fell under the control of Muslim armies, they instituted Islamic rule and legal structures as local chieftains converted, usually under penalty of death. Only those who had converted to Islam could rule or be engaged in trade. The first major empire to emerge in West Africa was the Ghana Empire (). By 750, the Soninke farmers of the sub-Sahara had become wealthy by taxing the trade that passed through their area. For instance, the Niger River basin supplied gold to the Berber and Arab traders from west of the Nile Valley, who brought cloth, weapons, and manufactured goods into the interior. Huge Saharan salt mines supplied the life-sustaining mineral to the Mediterranean coast of Africa and inland areas. By 900, the monotheistic Muslims controlled most of this trade and had converted many of the African ruling elite. The majority of the population, however, maintained their tribal animistic practices, which gave living attributes to nonliving objects such as mountains, rivers, and wind. Because Ghana’s king controlled the gold supply, he was able to maintain price controls and afford a strong military. Soon, however, a new kingdom emerged.
By 1200 CE, under the leadership of Sundiata Keita, Mali had replaced Ghana as the leading state in West Africa. After Sundiata’s rule, the court converted to Islam, and Muslim scribes played a large part in administration and government. Miners then discovered huge new deposits of gold east of the Niger River. By the fourteenth century, the empire was so wealthy that while on a hajj, or pilgrimage to the holy city of Mecca, Mali’s ruler Mansu Musa gave away enough gold to create serious price inflation in the cities along his route. Timbuktu, the capital city, became a leading Islamic center for education, commerce and the slave trade. Meanwhile, in the east, the city of Gao became increasingly strong under the leadership of Sonni Ali and soon eclipsed Mali’s power. Timbuktu sought Ali’s assistance in repelling the Tuaregs from the north. By 1500, however, the Tuareg empire of Songhay had eclipsed Mali, where weak and ineffective leadership prevailed.
### THE ROLE OF SLAVERY
The institution of slavery is not a recent phenomenon. Most civilizations have practiced some form of human bondage and servitude, and African empires were no different (). Famine or fear of stronger enemies might force one tribe to ask another for help and give themselves in a type of bondage in exchange. Similar to the European serf system, those seeking protection, or relief from starvation, would become the servants of those who provided relief. Debt might also be worked off through a form of servitude. Typically, these servants became a part of the extended tribal family. There is some evidence of chattel slavery, in which people are treated as personal property to be bought and sold, in the Nile Valley. It appears there was a slave-trade route through the Sahara that brought sub-Saharan Africans to Rome, which had enslaved people from all over the world.
Arab slave trading, which exchanged enslaved people for goods from the Mediterranean, existed long before Islam’s spread across North Africa. Muslims later expanded this trade and enslaved not only Africans but also Europeans, especially from Spain, Sicily, and Italy. Male captives were forced to build coastal fortifications and serve as enslaved galley people. Women were added to the harem.
The major European slave trade began with Portugal’s exploration of the west coast of Africa in search of a trade route to the East. By 1444, enslaved people were being brought from Africa to work on the sugar plantations of the Madeira Islands, off the coast of modern Morocco. The slave trade then expanded greatly as European colonies in the New World demanded an ever-increasing number of workers for the extensive plantations growing tobacco, sugar, and eventually rice and cotton ().
In the New World, the institution of slavery assumed a new aspect when the mercantilist system demanded a permanent, identifiable, and plentiful labor supply. Enslaved Africans were both easily identified (by their skin color) and plentiful, because of the thriving slave trade. This led to a race-based slavery system in the New World unlike any bondage system that had come before. Initially, the Spanish and Portuguese tried to force Native people to farm their crops. However, enslaved Native people often became sick or died from disease or from the overwork and cruel treatment they were subjected to. Although he later repented of his ideas, the great defender of the Native peoples, Bartolomé de Las Casas, seeing the near extinction of the native population, suggested the Spanish send Black (and White) laborers to the Indies. The Portuguese did the same in their portions of the New World, particularly for the sugar plantations in Brazil. These enslaved people were less susceptible to disease, and within fifty years, a change took place: the profitability of the African slave trade, coupled with the seemingly limitless number of potential enslaved people and the Catholic Church’s denunciation of the enslavement of Christians, led race to become a dominant factor in the institution of slavery.
In the English colonies along the Atlantic coast, indentured servants initially filled the need for labor in the North, where family farms were the norm. In the South, however, labor-intensive crops such as tobacco, rice, and indigo prevailed, and eventually the supply of indentured servants was insufficient to meet the demand. These workers served only for periods of three to seven years before being freed; a more permanent labor supply was needed. Thus, whereas in Africa permanent, inherited slavery was unknown, and children of those bound in slavery to the tribe usually were free and intermarried with their captors, this changed in the Americas; slavery became permanent, and children born to enslaved people became enslaved. This development, along with slavery’s identification with race, forever changed the institution and shaped its unique character in the New World.
### Section Summary
Before 1492, Africa, like the Americas, had experienced the rise and fall of many cultures, but the continent did not develop a centralized authority structure. African peoples practiced various forms of slavery, all of which differed significantly from the racial slavery that ultimately developed in the New World. After the arrival of Islam and before the Portuguese came to the coast of West Africa in 1444, Arabs and Berbers controlled the slave trade out of Africa, which expanded as European powers began to colonize the New World. Driven by a demand for labor, slavery in the Americas developed a new form: It was based on race, and the status of slave was both permanent and inherited.
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# Early Globalization: The Atlantic World, 1492–1650
## Introduction
The story of the Atlantic World is the story of global migration, a migration driven in large part by the actions and aspirations of the ruling heads of Europe. Columbus is hardly visible in this illustration of his ships making landfall on the Caribbean island of Hispaniola (). Instead, Ferdinand II of Spain (in the foreground) sits on his throne and points toward Columbus’s landing. As the ships arrive, the Arawak people tower over the Spanish, suggesting the native population density of the islands.
This historic moment in 1492 sparked new rivalries among European powers as they scrambled to create New World colonies, fueled by the quest for wealth and power as well as by religious passions. Almost continuous war resulted. Spain achieved early preeminence, creating a far-flung empire and growing rich with treasures from the Americas. Native Americans who confronted the newcomers from Europe suffered unprecedented losses of life, however, as previously unknown diseases sliced through their populations. They also were victims of the arrogance of the Europeans, who viewed themselves as uncontested masters of the New World, sent by God to bring Christianity to the “Indians.” |
# Early Globalization: The Atlantic World, 1492–1650
## Portuguese Exploration and Spanish Conquest
Portuguese colonization of Atlantic islands in the 1400s inaugurated an era of aggressive European expansion across the Atlantic. In the 1500s, Spain surpassed Portugal as the dominant European power. This age of exploration and the subsequent creation of an Atlantic World marked the earliest phase of globalization, in which previously isolated groups—Africans, Native Americans, and Europeans—first came into contact with each other, sometimes with disastrous results.
### PORTUGUESE EXPLORATION
Portugal’s Prince Henry the Navigator spearheaded his country’s exploration of Africa and the Atlantic in the 1400s. With his support, Portuguese mariners successfully navigated an eastward route to Africa, establishing a foothold there that became a foundation of their nation’s trade empire in the fifteenth and sixteenth centuries.
Portuguese mariners built an Atlantic empire by colonizing the Canary, Cape Verde, and Azores Islands, as well as the island of Madeira. Merchants then used these Atlantic outposts as debarkation points for subsequent journeys. From these strategic points, Portugal spread its empire down the western coast of Africa to the Congo, along the western coast of India, and eventually to Brazil on the eastern coast of South America. It also established trading posts in China and Japan. While the Portuguese didn’t rule over an immense landmass, their strategic holdings of islands and coastal ports gave them almost unrivaled control of nautical trade routes and a global empire of trading posts during the 1400s.
The travels of Portuguese traders to western Africa introduced them to the African slave trade, already brisk among African states. Seeing the value of this source of labor in growing the profitable crop of sugar on their Atlantic islands, the Portuguese soon began exporting enslaved Africans along with African ivory and gold. Sugar fueled the Atlantic slave trade, and the Portuguese islands quickly became home to sugar plantations. The Portuguese also traded these enslaved people, introducing much-needed human capital to other European nations. In the following years, as European exploration spread, slavery spread as well. In time, much of the Atlantic World would become a gargantuan sugar-plantation complex in which Africans labored to produce the highly profitable commodity for European consumers.
### SPANISH EXPLORATION AND CONQUEST
The Spanish established the first European settlements in the Americas, beginning in the Caribbean and, by 1600, extending throughout Central and South America. Thousands of Spaniards flocked to the Americas seeking wealth and status. The most famous of these Spanish adventurers are Christopher Columbus (who, though Italian himself, explored on behalf of the Spanish monarchs), Hernán Cortés, and Francisco Pizarro.
The history of Spanish exploration begins with the history of Spain itself. During the fifteenth century, Spain hoped to gain advantage over its rival, Portugal. The marriage of Ferdinand of Aragon and Isabella of Castile in 1469 unified Catholic Spain and began the process of building a nation that could compete for worldwide power. Since the 700s, much of Spain had been under Islamic rule, and King Ferdinand II and Queen Isabella I, arch-defenders of the Catholic Church against Islam, were determined to defeat the Muslims in Granada, the last Islamic stronghold in Spain. In 1492, they completed the Reconquista: the centuries-long Christian conquest of the Iberian Peninsula. The Reconquista marked another step forward in the process of making Spain an imperial power, and Ferdinand and Isabella were now ready to look further afield.
Their goals were to expand Catholicism and to gain a commercial advantage over Portugal. To those ends, Ferdinand and Isabella sponsored extensive Atlantic exploration. Spain’s most famous explorer, Christopher Columbus, was actually from Genoa, Italy. He believed that, using calculations based on other mariners’ journeys, he could chart a westward route to India, which could be used to expand European trade and spread Christianity. Starting in 1485, he approached Genoese, Venetian, Portuguese, English, and Spanish monarchs, asking for ships and funding to explore this westward route. All those he petitioned—including Ferdinand and Isabella at first—rebuffed him; their nautical experts all concurred that Columbus’s estimates of the width of the Atlantic Ocean were far too low. However, after three years of entreaties, and, more important, the completion of the Reconquista, Ferdinand and Isabella agreed to finance Columbus’s expedition in 1492, supplying him with three ships: the Nina, the Pinta, and the Santa Maria. The Spanish monarchs knew that Portuguese mariners had reached the southern tip of Africa and sailed the Indian Ocean. They understood that the Portuguese would soon reach Asia and, in this competitive race to reach the Far East, the Spanish rulers decided to act.
Columbus held erroneous views that shaped his thinking about what he would encounter as he sailed west. He believed the earth to be much smaller than its actual size and, since he did not know of the existence of the Americas, he fully expected to land in Asia. On October 12, 1492, however, he made landfall on an island in the Bahamas. He then sailed to an island he named Hispaniola (present-day Dominican Republic and Haiti) (). Believing he had landed in the East Indies, Columbus called the native Taínos he found there “Indios,” giving rise to the term “Indian” for any native people of the New World. Upon Columbus’s return to Spain, the Spanish crown bestowed on him the title of Admiral of the Ocean Sea and named him governor and viceroy of the lands he had discovered. As a devoted Catholic, Columbus had agreed with Ferdinand and Isabella prior to sailing west that part of the expected wealth from his voyage would be used to continue the fight against Islam.
Columbus’s 1493 letter—or (proof of merit)—describing his “discovery” of a New World did much to inspire excitement in Europe. Probanzas de méritos were reports and letters written by Spaniards in the New World to the Spanish crown, designed to win royal patronage. Today they highlight the difficult task of historical work; while the letters are primary sources, historians need to understand the context and the culture in which the conquistadors, as the Spanish adventurers came to be called, wrote them and distinguish their bias and subjective nature. While they are filled with distortions and fabrications, probanzas de méritos are still useful in illustrating the expectation of wealth among the explorers as well as their view that native peoples would not pose a serious obstacle to colonization.
In 1493, Columbus sent two copies of a probanza de mérito to the Spanish king and queen and their minister of finance, Luis de Santángel. Santángel had supported Columbus’s voyage, helping him to obtain funding from Ferdinand and Isabella. Copies of the letter were soon circulating all over Europe, spreading news of the wondrous new land that Columbus had “discovered.” Columbus would make three more voyages over the next decade, establishing Spain’s first settlement in the New World on the island of Hispaniola. Many other Europeans followed in Columbus’s footsteps, drawn by dreams of winning wealth by sailing west. Another Italian, Amerigo Vespucci, sailing for the Portuguese crown, explored the South American coastline between 1499 and 1502. Unlike Columbus, he realized that the Americas were not part of Asia but lands unknown to Europeans. Vespucci’s widely published accounts of his voyages fueled speculation and intense interest in the New World among Europeans. Among those who read Vespucci’s reports was the German mapmaker Martin Waldseemuller. Using the explorer’s first name as a label for the new landmass, Waldseemuller attached “America” to his map of the New World in 1507, and the name stuck.
The 1492 Columbus landfall accelerated the rivalry between Spain and Portugal, and the two powers vied for domination through the acquisition of new lands. In the 1480s, Pope Sixtus IV had granted Portugal the right to all land south of the Cape Verde islands, leading the Portuguese king to claim that the lands discovered by Columbus belonged to Portugal, not Spain. Seeking to ensure that Columbus’s finds would remain Spanish, Spain’s monarchs turned to the Spanish-born Pope Alexander VI, who issued two papal decrees in 1493 that gave legitimacy to Spain’s Atlantic claims at the expense of Portugal. Hoping to salvage Portugal’s Atlantic holdings, King João II began negotiations with Spain. The resulting Treaty of Tordesillas in 1494 drew a north-to-south line through South America (); Spain gained territory west of the line, while Portugal retained the lands east of the line, including the east coast of Brazil.
Columbus’s discovery opened a floodgate of Spanish exploration. Inspired by tales of rivers of gold and timid, malleable natives, later Spanish explorers were relentless in their quest for land and gold. Hernán Cortés hoped to gain hereditary privilege for his family, tribute payments and labor from natives, and an annual pension for his service to the crown. Cortés arrived on Hispaniola in 1504 and took part in the conquest of that island. In anticipation of winning his own honor and riches, Cortés later explored the Yucatán Peninsula. In 1519, he entered Tenochtitlán, the capital of the Aztec (Mexica) Empire. He and his men were astonished by the incredibly sophisticated causeways, gardens, and temples in the city, but they were horrified by the practice of human sacrifice that was part of the Aztec religion. Above all else, the Aztec wealth in gold fascinated the Spanish adventurers.
Hoping to gain power over the city, Cortés took Moctezuma, the Aztec ruler, hostage. The Spanish then murdered hundreds of high-ranking Mexica during a festival to celebrate Huitzilopochtli, the god of war. This angered the people of Tenochtitlán, who rose up against the interlopers in their city. Cortés and his people fled for their lives, running down one of Tenochtitlán’s causeways to safety on the shore. Smarting from their defeat at the hands of the Aztec, Cortés slowly created alliances with native peoples who resented Aztec rule. It took nearly a year for the Spanish and the tens of thousands of native allies who joined them to defeat the Mexica in Tenochtitlán, which they did by laying siege to the city. Only by playing upon the disunity among the diverse groups in the Aztec Empire were the Spanish able to capture the grand city of Tenochtitlán. In August 1521, having successfully fomented civil war as well as fended off rival Spanish explorers, Cortés claimed Tenochtitlán for Spain and renamed it Mexico City.
The traditional European narrative of exploration presents the victory of the Spanish over the Aztec as an example of the superiority of the Europeans over the "savage Indians." However, the reality is far more complex. When Cortés explored central Mexico, he encountered a region simmering with conflict. Far from being unified and content under Aztec rule, many peoples in Mexico resented it and were ready to rebel. One group in particular, the Tlaxcalan, threw their lot in with the Spanish, providing as many as 200,000 fighters in the siege of Tenochtitlán. The Spanish also brought smallpox into the valley of Mexico. The disease took a heavy toll on the people in Tenochtitlán, playing a much greater role in the city’s demise than did Spanish force of arms.
Cortés was also aided by a Nahua woman called Malintzin (also known as La Malinche or Doña Marina, her Spanish name), whom the natives of Tabasco gave him as tribute. Malintzin translated for Cortés in his dealings with Moctezuma and, whether willingly or under pressure, entered into a physical relationship with him. Their son, Martín, may have been the first mestizo (person of mixed indigenous American and European descent). Malintzin remains a controversial figure in the history of the Atlantic World; some people view her as a traitor because she helped Cortés conquer the Aztecs, while others see her as a victim of European expansion. In either case, she demonstrates one way in which native peoples responded to the arrival of the Spanish. Without her, Cortés would not have been able to communicate, and without the language bridge, he surely would have been less successful in destabilizing the Aztec Empire. By this and other means, native people helped shape the conquest of the Americas.
Spain’s acquisitiveness seemingly knew no bounds as groups of its explorers searched for the next trove of instant riches. One such explorer, Francisco Pizarro, made his way to the Spanish Caribbean in 1509, drawn by the promise of wealth and titles. He participated in successful expeditions in Panama before following rumors of Inca wealth to the south. Although his first efforts against the Inca Empire in the 1520s failed, Pizarro captured the Inca emperor Atahualpa in 1532 and executed him one year later. In 1533, Pizarro founded Lima, Peru. Like Cortés, Pizarro had to combat not only the natives of the new worlds he was conquering, but also competitors from his own country; a Spanish rival assassinated him in 1541.
Spain’s drive to enlarge its empire led other hopeful conquistadors to push further into the Americas, hoping to replicate the success of Cortés and Pizarro. Hernando de Soto had participated in Pizarro’s conquest of the Inca, and from 1539 to 1542 he led expeditions to what is today the southeastern United States, looking for gold. He and his followers explored what is now Florida, Georgia, the Carolinas, Tennessee, Alabama, Mississippi, Arkansas, Oklahoma, Louisiana, and Texas. Everywhere they traveled, they brought European diseases, which claimed thousands of native lives as well as the lives of the explorers. In 1542, de Soto himself died during the expedition. The surviving Spaniards, numbering a little over three hundred, returned to Mexico City without finding the much-anticipated mountains of gold and silver.
Francisco Vásquez de Coronado was born into a noble family and went to Mexico, then called New Spain, in 1535. He presided as governor over the province of Nueva Galicia, where he heard rumors of wealth to the north: a golden city called Quivira. Between 1540 and 1542, Coronado led a large expedition of Spaniards and native allies to the lands north of Mexico City, and for the next several years, they explored the area that is now the southwestern United States (). During the winter of 1540–41, the explorers waged war against the Tiwa in present-day New Mexico. Rather than leading to the discovery of gold and silver, however, the expedition simply left Coronado bankrupt.
### THE SPANISH GOLDEN AGE
The exploits of European explorers had a profound impact both in the Americas and back in Europe. An exchange of ideas, fueled and financed in part by New World commodities, began to connect European nations and, in turn, to touch the parts of the world that Europeans conquered. In Spain, gold and silver from the Americas helped to fuel a golden age, the Siglo de Oro, when Spanish art and literature flourished. Riches poured in from the colonies, and new ideas poured in from other countries and new lands. The Habsburg dynasty, which ruled a collection of territories including Austria, the Netherlands, Naples, Sicily, and Spain, encouraged and financed the work of painters, sculptors, musicians, architects, and writers, resulting in a blooming of Spanish Renaissance culture. One of this period’s most famous works is the novel The Ingenious Gentleman Don Quixote of La Mancha, by Miguel de Cervantes. This two-volume book (1605 and 1618) told a colorful tale of an hidalgo (gentleman) who reads so many tales of chivalry and knighthood that he becomes unable to tell reality from fiction. With his faithful sidekick Sancho Panza, Don Quixote leaves reality behind and sets out to revive chivalry by doing battle with what he perceives as the enemies of Spain.
Spain attracted innovative foreign painters such as El Greco, a Greek who had studied with Italian Renaissance masters like Titian and Michelangelo before moving to Toledo. Native Spaniards created equally enduring works. Las Meninas (The Maids of Honor), painted by Diego Velázquez in 1656, is one of the best-known paintings in history. Velázquez painted himself into this imposingly large royal portrait (he’s shown holding his brush and easel on the left) and boldly placed the viewer where the king and queen would stand in the scene ().
### Section Summary
Although Portugal opened the door to exploration of the Atlantic World, Spanish explorers quickly made inroads into the Americas. Spurred by Christopher Columbus’s glowing reports of the riches to be found in the New World, throngs of Spanish conquistadors set off to find and conquer new lands. They accomplished this through a combination of military strength and strategic alliances with native peoples. Spanish rulers Ferdinand and Isabella promoted the acquisition of these new lands in order to strengthen and glorify their own empire. As Spain’s empire expanded and riches flowed in from the Americas, the Spanish experienced a golden age of art and literature.
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# Early Globalization: The Atlantic World, 1492–1650
## Religious Upheavals in the Developing Atlantic World
Until the 1500s, the Catholic Church provided a unifying religious structure for Christian Europe. The Vatican in Rome exercised great power over the lives of Europeans; it controlled not only learning and scholarship but also finances, because it levied taxes on the faithful. Spain, with its New World wealth, was the bastion of the Catholic faith. Beginning with the reform efforts of Martin Luther in 1517 and John Calvin in the 1530s, however, Catholic dominance came under attack as the Protestant Reformation, a split or schism among European Christians, began.
During the sixteenth century, Protestantism spread through northern Europe, and Catholic countries responded by attempting to extinguish what was seen as the Protestant menace. Religious turmoil between Catholics and Protestants influenced the history of the Atlantic World as well, since different nation-states competed not only for control of new territories but also for the preeminence of their religious beliefs there. Just as the history of Spain’s rise to power is linked to the Reconquista, so too is the history of early globalization connected to the history of competing Christian groups in the Atlantic World.
### MARTIN LUTHER
Martin Luther () was a German Catholic monk who took issue with the Catholic Church’s practice of selling indulgences, documents that absolved sinners of their errant behavior. He also objected to the Catholic Church’s taxation of ordinary Germans and the delivery of Mass in Latin, arguing that it failed to instruct German Catholics, who did not understand the language.
Many Europeans had called for reforms of the Catholic Church before Martin Luther did, but his protest had the unintended consequence of splitting European Christianity. Luther compiled a list of what he viewed as needed Church reforms, a document that came to be known as The Ninety-Five Theses, and nailed it to the door of a church in Wittenberg, Germany, in 1517. He called for the publication of the Bible in everyday language, took issue with the Church’s policy of imposing tithes (a required payment to the Church that appeared to enrich the clergy), and denounced the buying and selling of indulgences. Although he had hoped to reform the Catholic Church while remaining a part of it, Luther’s action instead triggered a movement called the Protestant Reformation that divided the Church in two. The Catholic Church condemned him as a heretic, but a doctrine based on his reforms, called Lutheranism, spread through northern Germany and Scandinavia.
### JOHN CALVIN
Like Luther, the French lawyer John Calvin advocated making the Bible accessible to ordinary people; only by reading scripture and reflecting daily about their spiritual condition, he argued, could believers begin to understand the power of God. In 1535, Calvin fled Catholic France and led the Reformation movement from Geneva, Switzerland.
Calvinism emphasized human powerlessness before an omniscient God and stressed the idea of predestination, the belief that God selected a few chosen people for salvation while everyone else was predestined to damnation. Calvinists believed that reading scripture prepared sinners, if they were among the elect, to receive God’s grace. In Geneva, Calvin established a Bible commonwealth, a community of believers whose sole source of authority was their interpretation of the Bible, not the authority of any prince or monarch. Soon Calvin’s ideas spread to the Netherlands and Scotland.
### PROTESTANTISM IN ENGLAND
Protestantism spread beyond the German states and Geneva to England, which had been a Catholic nation for centuries. Luther’s idea that scripture should be available in the everyday language of worshippers inspired English scholar William Tyndale to translate the Bible into English in 1526. The seismic break with the Catholic Church in England occurred in the 1530s, when Henry VIII established a new, Protestant state religion.
A devout Catholic, Henry had initially stood in opposition to the Reformation. Pope Leo X even awarded him the title “Defender of the Faith.” The tides turned, however, when Henry desired a male heir to the Tudor monarchy. When his Spanish Catholic wife, Catherine (the daughter of Ferdinand and Isabella), did not give birth to a boy, the king sought an annulment to their marriage. When the Pope refused his request, Henry created a new national Protestant church, the Church of England, with himself at its head. This left him free to annul his own marriage and marry Anne Boleyn.
Anne Boleyn also failed to produce a male heir, and when she was accused of adultery, Henry had her executed. His third wife, Jane Seymour, at long last delivered a son, Edward, who ruled for only a short time before dying in 1553 at the age of fifteen. Mary, the daughter of Henry VIII and his discarded first wife Catherine, then came to the throne, committed to restoring Catholicism. She earned the nickname “Bloody Mary” for the many executions of Protestants, often by burning alive, that she ordered during her reign.
Religious turbulence in England was finally quieted when Elizabeth, the Protestant daughter of Henry VIII and Anne Boleyn, ascended the throne in 1558. Under Elizabeth, the Church of England again became the state church, retaining the hierarchical structure and many of the rituals of the Catholic Church. However, by the late 1500s, some English members of the Church began to agitate for more reform. Known as Puritans, they worked to erase all vestiges of Catholicism from the Church of England. At the time, the term “puritan” was a pejorative one; many people saw Puritans as holier-than-thou frauds who used religion to swindle their neighbors. Worse still, many in power saw Puritans as a security threat because of their opposition to the national church.
Under Elizabeth, whose long reign lasted from 1558 to 1603, Puritans grew steadily in number. After James I died in 1625 and his son Charles I ascended the throne, Puritans became the target of increasing state pressure to conform. Many crossed the Atlantic in the 1620s and 1630s instead to create a New England, a haven for reformed Protestantism where Puritan was no longer a term of abuse. Thus, the religious upheavals that affected England so much had equally momentous consequences for the Americas.
### RELIGIOUS WAR
By the early 1500s, the Protestant Reformation threatened the massive Spanish Catholic empire. As the preeminent Catholic power, Spain would not tolerate any challenge to the Holy Catholic Church. Over the course of the 1500s, it devoted vast amounts of treasure and labor to leading an unsuccessful effort to eradicate Protestantism in Europe.
Spain’s main enemies at this time were the runaway Spanish provinces of the North Netherlands. By 1581, these seven northern provinces had declared their independence from Spain and created the Dutch Republic, also called Holland, where Protestantism was tolerated. Determined to deal a death blow to Protestantism in England and Holland, King Philip of Spain assembled a massive force of over thirty thousand men and 130 ships, and in 1588 he sent this navy, the Spanish Armada, north. But English sea power combined with a maritime storm destroyed the fleet.
The defeat of the Spanish Armada in 1588 was but one part of a larger but undeclared war between Protestant England and Catholic Spain. Between 1585 and 1604, the two rivals sparred repeatedly. England launched its own armada in 1589 in an effort to disable the Spanish fleet and capture Spanish treasure. However, the foray ended in disaster for the English, with storms, disease, and the strength of the Spanish Armada combining to bring about defeat.
The conflict between Spain and England dragged on into the early seventeenth century, and the newly Protestant nations, especially England and the Dutch Republic, posed a significant challenge to Spain (and also to Catholic France) as imperial rivalries played out in the Atlantic World. Spain retained its mighty American empire, but by the early 1600s, the nation could no longer keep England and other European rivals—the French and Dutch—from colonizing smaller islands in the Caribbean ().
Religious intolerance characterized the sixteenth and seventeenth centuries, an age of powerful state religions with the authority to impose and enforce belief systems on the population. In this climate, religious violence was common. One of the most striking examples is the St. Bartholomew’s Day Massacre of 1572, in which French Catholic troops began to kill unarmed French Protestants (). The murders touched off mob violence that ultimately claimed nine thousand lives, a bloody episode that highlights the degree of religious turmoil that gripped Europe in the aftermath of the Protestant Reformation.
### Section Summary
The sixteenth century witnessed a new challenge to the powerful Catholic Church. The reformist doctrines of Martin Luther and John Calvin attracted many people dissatisfied with Catholicism, and Protestantism spread across northern Europe, spawning many subgroups with conflicting beliefs. Spain led the charge against Protestantism, leading to decades of undeclared religious wars between Spain and England, and religious intolerance and violence characterized much of the sixteenth and seventeenth centuries. Despite the efforts of the Catholic Church and Catholic nations, however, Protestantism had taken hold by 1600.
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# Early Globalization: The Atlantic World, 1492–1650
## Challenges to Spain’s Supremacy
For Europeans, the discovery of an Atlantic World meant newfound wealth in the form of gold and silver as well as valuable furs. The Americas also provided a new arena for intense imperial rivalry as different European nations jockeyed for preeminence in the New World. The religious motives for colonization spurred European expansion as well, and as the Protestant Reformation gained ground beginning in the 1520s, rivalries between Catholic and Protestant Christians spilled over into the Americas.
### ENGLISH EXPLORATION
Disruptions during the Tudor monarchy—especially the creation of the Protestant Church of England by Henry VIII in the 1530s, the return of the nation to Catholicism under Queen Mary in the 1550s, and the restoration of Protestantism under Queen Elizabeth—left England with little energy for overseas projects. More important, England lacked the financial resources for such endeavors. Nonetheless, English monarchs carefully monitored developments in the new Atlantic World and took steps to assert England’s claim to the Americas. As early as 1497, Henry VII of England had commissioned John Cabot, an Italian mariner, to explore new lands. Cabot sailed from England that year and made landfall somewhere along the North American coastline. For the next century, English fishermen routinely crossed the Atlantic to fish the rich waters off the North American coast. However, English colonization efforts in the 1500s were closer to home, as England devoted its energy to the colonization of Ireland.
Queen Elizabeth favored England’s advance into the Atlantic World, though her main concern was blocking Spain’s effort to eliminate Protestantism. Indeed, England could not commit to large-scale colonization in the Americas as long as Spain appeared ready to invade Ireland or Scotland. Nonetheless, Elizabeth approved of English privateers, sea captains to whom the home government had given permission to raid the enemy at will. These skilled mariners cruised the Caribbean, plundering Spanish ships whenever they could. Each year the English took more than £100,000 from Spain in this way; English privateer Francis Drake first made a name for himself when, in 1573, he looted silver, gold, and pearls worth £40,000.
Elizabeth did sanction an early attempt at colonization in 1584, when Sir Walter Raleigh, a favorite of the queen’s, attempted to establish a colony at Roanoke, an island off the coast of present-day North Carolina. The colony was small, consisting of only 117 people, who suffered a poor relationship with the local Croatans, and struggled to survive in their new land (). Their governor, John White, returned to England in late 1587 to secure more people and supplies, but events conspired to keep him away from Roanoke for three years. By the time he returned in 1590, the entire colony had vanished. The only trace the colonists left behind was the word Croatoan carved into a fence surrounding the village. Governor White never knew whether the colonists had decamped for nearby Croatoan Island (now Hatteras) or whether some disaster had befallen them all. Roanoke is still called “the lost colony.”
English promoters of colonization pushed its commercial advantages and the religious justification that English colonies would allow the establishment of Protestantism in the Americas. Both arguments struck a chord. In the early 1600s, wealthy English merchants and the landed elite began to pool their resources to form joint stock companies. In this novel business arrangement, which was in many ways the precursor to the modern corporation, investors provided the capital for and assumed the risk of a venture in order to reap significant returns. The companies gained the approval of the English crown to establish colonies, and their investors dreamed of reaping great profits from the money they put into overseas colonization.
The first permanent English settlement was established by a joint stock company, the Virginia Company. Named for Elizabeth, the “virgin queen,” the company gained royal approval to establish a colony on the east coast of North America, and in 1606, it sent 144 men and boys to the New World. In early 1607, this group sailed up Chesapeake Bay. Finding a river they called the James in honor of their new king, James I, they established a ramshackle settlement and named it Jamestown. Despite serious struggles, the colony survived.
Many of Jamestown’s settlers were desperate men; although they came from elite families, they were younger sons who would not inherit their father’s estates. The Jamestown adventurers believed they would find instant wealth in the New World and did not actually expect to have to perform work. George Percy, born in England to the eighth Earl of Northumberland, was among them. His account, excerpted below, illustrates the hardships the English confronted in Virginia in 1607.
By any measure, England came late to the race to colonize. As Jamestown limped along in the 1610s, the Spanish Empire extended around the globe and grew rich from its global colonial project. Yet the English persisted, and for this reason the Jamestown settlement has a special place in history as the first permanent English colony in what later became the United States.
After Jamestown’s founding, English colonization of the New World accelerated. In 1609, a ship bound for Jamestown foundered in a storm and landed on Bermuda. (Some believe this incident helped inspire Shakespeare’s 1611 play The Tempest.) The admiral of the ship, George Somers, claimed the island for the English crown. The English also began to colonize small islands in the Caribbean, an incursion into the Spanish American empire. They established themselves on small islands such as St. Christopher (1624), Barbados (1627), Nevis (1628), Montserrat (1632), and Antigua (1632).
From the start, the English West Indies had a commercial orientation, for these islands produced cash crops: first tobacco and then sugar. Very quickly, by the mid-1600s, Barbados had become one of the most important English colonies because of the sugar produced there. Barbados was the first English colony dependent on enslaved people, and it became a model for other English slave societies on the American mainland. These differed radically from England itself, where slavery was not practiced.
English Puritans also began to colonize the Americas in the 1620s and 1630s. These intensely religious migrants dreamed of creating communities of reformed Protestantism where the corruption of England would be eliminated. One of the first groups of Puritans to move to North America, known as Pilgrims and led by William Bradford, had originally left England to live in the Netherlands. Fearing their children were losing their English identity among the Dutch, however, they sailed for North America in 1620 to settle at Plymouth, the first English settlement in New England. The Pilgrims differed from other Puritans in their insistence on separating from what they saw as the corrupt Church of England. For this reason, Pilgrims are known as Separatists.
Like Jamestown, Plymouth occupies an iconic place in American national memory. The tale of the 102 migrants who crossed the Atlantic aboard the and their struggle for survival is a well-known narrative of the founding of the country. Their story includes the signing of the Mayflower Compact, a written agreement whereby the English voluntarily agreed to help each other. Some interpret this 1620 document as an expression of democratic spirit because of the cooperative and inclusive nature of the agreement to live and work together. In 1630, a much larger contingent of Puritans left England to escape conformity to the Church of England and founded the Massachusetts Bay Colony. In the following years, thousands more arrived to create a new life in the rocky soils and cold climates of New England.
In comparison to Catholic Spain, however, Protestant England remained a very weak imperial player in the early seventeenth century, with only a few infant colonies in the Americas in the early 1600s. The English never found treasure equal to that of the Aztec city of Tenochtitlán, and England did not quickly grow rich from its small American outposts. The English colonies also differed from each other; Barbados and Virginia had a decidedly commercial orientation from the start, while the Puritan colonies of New England were intensely religious at their inception. All English settlements in America, however, marked the increasingly important role of England in the Atlantic World.
### FRENCH EXPLORATION
Spanish exploits in the New World whetted the appetite of other would-be imperial powers, including France. Like Spain, France was a Catholic nation and committed to expanding Catholicism around the globe. In the early sixteenth century, it joined the race to explore the New World and exploit the resources of the Western Hemisphere. Navigator Jacques Cartier claimed northern North America for France, naming the area New France. From 1534 to 1541, he made three voyages of discovery on the Gulf of St. Lawrence and the St. Lawrence River. Like other explorers, Cartier made exaggerated claims of mineral wealth in America, but he was unable to send great riches back to France. Due to resistance from the native peoples as well as his own lack of planning, he could not establish a permanent settlement in North America.
Explorer Samuel de Champlain occupies a special place in the history of the Atlantic World for his role in establishing the French presence in the New World. Champlain explored the Caribbean in 1601 and then the coast of New England in 1603 before traveling farther north. In 1608 he founded Quebec, and he made numerous Atlantic crossings as he worked tirelessly to promote New France. Unlike other imperial powers, France—through Champlain’s efforts—fostered especially good relationships with native peoples, paving the way for French exploration further into the continent: around the Great Lakes, around Hudson Bay, and eventually to the Mississippi. Champlain made an alliance with the Huron confederacy and the Algonquins and agreed to fight with them against their enemy, the Iroquois ().
The French were primarily interested in establishing commercially viable colonial outposts, and to that end, they created extensive trading networks in New France. These networks relied on native hunters to harvest furs, especially beaver pelts, and to exchange these items for French glass beads and other trade goods. (French fashion at the time favored broad-brimmed hats trimmed in beaver fur, so French traders had a ready market for their North American goods.) The French also dreamed of replicating the wealth of Spain by colonizing the tropical zones. After Spanish control of the Caribbean began to weaken, the French turned their attention to small islands in the West Indies, and by 1635 they had colonized two, Guadeloupe and Martinique. Though it lagged far behind Spain, France now boasted its own West Indian colonies. Both islands became lucrative sugar plantation sites that turned a profit for French planters by relying on African slave labor.
### DUTCH COLONIZATION
Dutch entrance into the Atlantic World is part of the larger story of religious and imperial conflict in the early modern era. In the 1500s, Calvinism, one of the major Protestant reform movements, had found adherents in the northern provinces of the Spanish Netherlands. During the sixteenth century, these provinces began a long struggle to achieve independence from Catholic Spain. Established in 1581 but not recognized as independent by Spain until 1648, the Dutch Republic, or Holland, quickly made itself a powerful force in the race for Atlantic colonies and wealth. The Dutch distinguished themselves as commercial leaders in the seventeenth century (), and their mode of colonization relied on powerful corporations: the Dutch East India Company, chartered in 1602 to trade in Asia, and the Dutch West India Company, established in 1621 to colonize and trade in the Americas.
While employed by the Dutch East India Company in 1609, the English sea captain Henry Hudson explored New York Harbor and the river that now bears his name. Like many explorers of the time, Hudson was actually seeking a northwest passage to Asia and its wealth, but the ample furs harvested from the region he explored, especially the coveted beaver pelts, provided a reason to claim it for the Netherlands. The Dutch named their colony New Netherlands, and it served as a fur-trading outpost for the expanding and powerful Dutch West India Company. With headquarters in New Amsterdam on the island of Manhattan, the Dutch set up several regional trading posts, including one at Fort Orange—named for the royal Dutch House of Orange-Nassau—in present-day Albany. (The color orange remains significant to the Dutch, having become particularly associated with William of Orange, Protestantism, and the Glorious Revolution of 1688.) A brisk trade in furs with local Algonquian and Iroquois peoples brought the Dutch and native peoples together in a commercial network that extended throughout the Hudson River Valley and beyond.
The Dutch West India Company in turn established colonies on Aruba, Bonaire, and Curaçao, St. Martin, St. Eustatius, and Saba. With their outposts in New Netherlands and the Caribbean, the Dutch had established themselves in the seventeenth century as a commercially powerful rival to Spain. Amsterdam became a trade hub for all the Atlantic World.
### Section Summary
By the beginning of the seventeenth century, Spain’s rivals—England, France, and the Dutch Republic—had each established an Atlantic presence, with greater or lesser success, in the race for imperial power. None of the new colonies, all in the eastern part of North America, could match the Spanish possessions for gold and silver resources. Nonetheless, their presence in the New World helped these nations establish claims that they hoped could halt the runaway growth of Spain’s Catholic empire. English colonists in Virginia suffered greatly, expecting riches to fall into their hands and finding reality a harsh blow. However, the colony at Jamestown survived, and the output of England’s islands in the West Indies soon grew to be an important source of income for the country. New France and New Netherlands were modest colonial holdings in the northeast of the continent, but these colonies’ thriving fur trade with native peoples, and their alliances with those peoples, helped to create the foundation for later shifts in the global balance of power.
### Review Questions
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# Early Globalization: The Atlantic World, 1492–1650
## New Worlds in the Americas: Labor, Commerce, and the Columbian Exchange
European promoters of colonization claimed the Americas overflowed with a wealth of treasures. Burnishing national glory and honor became entwined with carving out colonies, and no nation wanted to be left behind. However, the realities of life in the Americas—violence, exploitation, and particularly the need for workers—were soon driving the practice of slavery and forced labor. Everywhere in America a stark contrast existed between freedom and slavery. The Columbian Exchange, in which Europeans transported plants, animals, and diseases across the Atlantic in both directions, also left a lasting impression on the Americas.
### LABOR SYSTEMS
Physical power—to work the fields, build villages, process raw materials—is a necessity for maintaining a society. During the sixteenth and seventeenth centuries, humans could derive power only from the wind, water, animals, or other humans. Everywhere in the Americas, a crushing demand for labor bedeviled Europeans because there were not enough colonists to perform the work necessary to keep the colonies going. Spain granted —legal rights to native labor—to conquistadors who could prove their service to the crown. This system reflected the Spanish view of colonization: the king rewarded successful conquistadors who expanded the empire. Some native peoples who had sided with the conquistadors, like the Tlaxcalan, also gained encomiendas; Malintzin, the Nahua woman who helped Cortés defeat the Mexica, was granted one.
The Spanish believed native peoples would work for them by right of conquest, and, in return, the Spanish would bring them Catholicism. In theory the relationship consisted of reciprocal obligations, but in practice the Spaniards ruthlessly exploited it, seeing native people as little more than beasts of burden. Convinced of their right to the land and its peoples, they sought both to control native labor and to impose what they viewed as correct religious beliefs upon the land’s inhabitants. Native peoples everywhere resisted both the labor obligations and the effort to change their ancient belief systems. Indeed, many retained their religion or incorporated only the parts of Catholicism that made sense to them.
The system of encomiendas was accompanied by a great deal of violence (). One Spaniard, Bartolomé de Las Casas, denounced the brutality of Spanish rule. A Dominican friar, Las Casas had been one of the earliest Spanish settlers in the Spanish West Indies. In his early life in the Americas, he enslaved Native people and was the recipient of an encomienda. However, after witnessing the savagery with which encomenderos (recipients of encomiendas) treated the native people, he reversed his views. In 1515, Las Casas released his enslaved natives, gave up his encomienda, and began to advocate for humane treatment of native peoples. He lobbied for new legislation, eventually known as the New Laws, which would eliminate slavery and the encomienda system.
Las Casas’s writing about the Spaniards’ horrific treatment of Native people helped inspire the so-called Black Legend, the idea that the Spanish were bloodthirsty conquerors with no regard for human life. Perhaps not surprisingly, those who held this view of the Spanish were Spain’s imperial rivals. English writers and others seized on the idea of Spain’s ruthlessness to support their own colonization projects. By demonizing the Spanish, they justified their own efforts as more humane. All European colonizers, however, shared a disregard for Native peoples.
Native peoples were not the only source of cheap labor in the Americas; by the middle of the sixteenth century, Africans formed an important element of the labor landscape, producing the cash crops of sugar and tobacco for European markets. Europeans viewed Africans as non-Christians, which they used as a justification for enslavement. Denied control over their lives, enslaved people endured horrendous conditions. At every opportunity, they resisted enslavement, and their resistance was met with violence. Indeed, physical, mental, and sexual violence formed a key strategy among European slaveholders in their effort to assert mastery and impose their will. The Portuguese led the way in the evolving transport of captive enslaved people across the Atlantic; slave “factories” on the west coast of Africa, like Elmina Castle in Ghana, served as holding pens for enslaved people brought from Africa’s interior. In time, other European imperial powers would follow in the footsteps of the Portuguese by constructing similar outposts on the coast of West Africa.
The Portuguese traded or sold enslaved people to Spanish, Dutch, and English colonists in the Americas, particularly in South America and the Caribbean, where sugar was a primary export. Thousands of enslaved Africans found themselves growing, harvesting, and processing sugarcane in an arduous routine of physical labor. Enslaved people had to cut the long cane stalks by hand and then bring them to a mill, where the cane juice was extracted. They boiled the extracted cane juice down to a brown, crystalline sugar, which then had to be cured in special curing houses to have the molasses drained from it. The result was refined sugar, while the leftover molasses could be distilled into rum. Every step was labor-intensive and often dangerous.
Las Casas estimated that by 1550, there were fifty thousand enslaved people on Hispaniola. However, it is a mistake to assume that during the very early years of European exploration all Africans came to America as captives; some were free men who took part in expeditions, for example, serving as conquistadors alongside Cortés in his assault on Tenochtitlán. Nonetheless, African slavery was one of the most tragic outcomes in the emerging Atlantic World.
### COMMERCE IN THE NEW WORLD
The economic philosophy of mercantilism shaped European perceptions of wealth from the 1500s to the late 1700s. Mercantilism held that only a limited amount of wealth, as measured in gold and silver bullion, existed in the world. In order to gain power, nations had to amass wealth by mining these precious raw materials from their colonial possessions. During the age of European exploration, nations employed conquest, colonization, and trade as ways to increase their share of the bounty of the New World. Mercantilists did not believe in free trade, arguing instead that the nation should control trade to create wealth. In this view, colonies existed to strengthen the colonizing nation. Mercantilists argued against allowing their nations to trade freely with other nations.
Spain’s mercantilist ideas guided its economic policy. Every year, enslaved laborers or native workers loaded shipments of gold and silver aboard Spanish treasure fleets that sailed from Cuba for Spain. These ships groaned under the sheer weight of bullion, for the Spanish had found huge caches of silver and gold in the New World. In South America, for example, Spaniards discovered rich veins of silver ore in the mountain called Potosí and founded a settlement of the same name there. Throughout the sixteenth century, Potosí was a boom town, attracting settlers from many nations as well as native people from many different cultures.
Colonial mercantilism, which was basically a set of protectionist policies designed to benefit the nation, relied on several factors: colonies rich in raw materials, cheap labor, colonial loyalty to the home government, and control of the shipping trade. Under this system, the colonies sent their raw materials, harvested by enslaved laborers or native workers, back to their mother country. The mother country sent back finished materials of all sorts: textiles, tools, clothing. The colonists could purchase these goods only from their mother country; trade with other countries was forbidden.
The 1500s and early 1600s also introduced the process of commodification to the New World. American silver, tobacco, and other items, which were used by native peoples for ritual purposes, became European commodities with a monetary value that could be bought and sold. Before the arrival of the Spanish, for example, the Inca people of the Andes consumed chicha, a corn beer, for ritual purposes only. When the Spanish discovered chicha, they bought and traded for it, turning it into a commodity instead of a ritual substance. Commodification thus recast native economies and spurred the process of early commercial capitalism. New World resources, from plants to animal pelts, held the promise of wealth for European imperial powers.
### THE COLUMBIAN EXCHANGE
As Europeans traversed the Atlantic, they brought with them plants, animals, and diseases that changed lives and landscapes on both sides of the ocean. These two-way exchanges between the Americas and Europe/Africa are known collectively as the Columbian Exchange ().
Of all the commodities in the Atlantic World, sugar proved to be the most important. Indeed, sugar carried the same economic importance as oil does today. European rivals raced to create sugar plantations in the Americas and fought wars for control of some of the best sugar production areas. Although refined sugar was available in the Old World, Europe’s harsher climate made sugarcane difficult to grow, and it was not plentiful. Columbus brought sugar to Hispaniola in 1493, and the new crop was growing there by the end of the 1490s. By the first decades of the 1500s, the Spanish were building sugar mills on the island. Over the next century of colonization, Caribbean islands and most other tropical areas became centers of sugar production.
Though of secondary importance to sugar, tobacco achieved great value for Europeans as a cash crop as well. Native peoples had been growing it for medicinal and ritual purposes for centuries before European contact, smoking it in pipes or powdering it to use as snuff. They believed tobacco could improve concentration and enhance wisdom. To some, its use meant achieving an entranced, altered, or divine state; entering a spiritual place.
Tobacco was unknown in Europe before 1492, and it carried a negative stigma at first. The early Spanish explorers considered natives’ use of tobacco to be proof of their savagery and, because of the fire and smoke produced in the consumption of tobacco, evidence of the Devil’s sway in the New World. Gradually, however, European colonists became accustomed to and even took up the habit of smoking, and they brought it across the Atlantic. As did the Native Americans, Europeans ascribed medicinal properties to tobacco, claiming that it could cure headaches and skin irritations. Even so, Europeans did not import tobacco in great quantities until the 1590s. At that time, it became the first truly global commodity; English, French, Dutch, Spanish, and Portuguese colonists all grew it for the world market.
Native peoples also introduced Europeans to chocolate, made from cacao seeds and used by the Aztec in Mesoamerica as currency. Mesoamerican Natives consumed unsweetened chocolate in a drink with chili peppers, vanilla, and a spice called achiote. This chocolate drink—xocolatl—was part of ritual ceremonies like marriage and an everyday item for those who could afford it. Chocolate contains theobromine, a stimulant, which may be why native people believed it brought them closer to the sacred world.
Spaniards in the New World considered drinking chocolate a vile practice; one called chocolate “the Devil’s vomit.” In time, however, they introduced the beverage to Spain. At first, chocolate was available only in the Spanish court, where the elite mixed it with sugar and other spices. Later, as its availability spread, chocolate gained a reputation as a love potion.
The crossing of the Atlantic by plants like cacao and tobacco illustrates the ways in which the discovery of the New World changed the habits and behaviors of Europeans. Europeans changed the New World in turn, not least by bringing Old World animals to the Americas. On his second voyage, Christopher Columbus brought pigs, horses, cows, and chickens to the islands of the Caribbean. Later explorers followed suit, introducing new animals or reintroducing ones that had died out (like horses). With less vulnerability to disease, these animals often fared better than humans in their new home, thriving both in the wild and in domestication.
Europeans encountered New World animals as well. Because European Christians understood the world as a place of warfare between God and Satan, many believed the Americas, which lacked Christianity, were home to the Devil and his minions. The exotic, sometimes bizarre, appearances and habits of animals in the Americas that were previously unknown to Europeans, such as manatees, sloths, and poisonous snakes, confirmed this association. Over time, however, they began to rely more on observation of the natural world than solely on scripture. This shift—from seeing the Bible as the source of all received wisdom to trusting observation or empiricism—is one of the major outcomes of the era of early globalization.
Travelers between the Americas, Africa, and Europe also included microbes: silent, invisible life forms that had profound and devastating consequences. Native peoples had no immunity to diseases from across the Atlantic, to which they had never been exposed. European explorers unwittingly brought with them chickenpox, measles, mumps, and smallpox, which ravaged native peoples despite their attempts to treat the diseases, decimating some populations and wholly destroying others ().
In eastern North America, some native peoples interpreted death from disease as a hostile act. Some groups, including the Iroquois, engaged in raids or “mourning wars,” taking enemy prisoners in order to assuage their grief and replace the departed. In a special ritual, the prisoners were “requickened”—assigned the identity of a dead person—and adopted by the bereaved family to take the place of their dead. As the toll from disease rose, mourning wars intensified and expanded.
### Section Summary
In the minds of European rulers, colonies existed to create wealth for imperial powers. Guided by mercantilist ideas, European rulers and investors hoped to enrich their own nations and themselves, in order to gain the greatest share of what was believed to be a limited amount of wealth. In their own individual quest for riches and preeminence, European colonizers who traveled to the Americas blazed new and disturbing paths, such as the encomienda system of forced labor and the enslavement of tens of thousands of Africans.
All Native inhabitants of the Americas who came into contact with Europeans found their worlds turned upside down as the new arrivals introduced their religions and ideas about property and goods. Europeans gained new foods, plants, and animals in the Columbian Exchange, turning whatever they could into a commodity to be bought and sold, and Native peoples were introduced to diseases that nearly destroyed them. At every turn, however, Native Americans placed limits on European colonization and resisted the newcomers’ ways.
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# Creating New Social Orders: Colonial Societies, 1500–1700
## Introduction
By the mid-seventeenth century, the geopolitical map of North America had become a patchwork of imperial designs and ambitions as the Spanish, Dutch, French, and English reinforced their claims to parts of the land. Uneasiness, punctuated by violent clashes, prevailed in the border zones between the Europeans’ territorial claims. Meanwhile, still-powerful native peoples waged war to drive the invaders from the continent. In the Chesapeake Bay and New England colonies, conflicts erupted as the English pushed against their native neighbors ().
The rise of colonial societies in the Americas brought Native Americans, Africans, and Europeans together for the first time, highlighting the radical social, cultural, and religious differences that hampered their ability to understand each other. European settlement affected every aspect of the land and its people, bringing goods, ideas, and diseases that transformed the Americas. Reciprocally, Native American practices, such as the use of tobacco, profoundly altered European habits and tastes. |
# Creating New Social Orders: Colonial Societies, 1500–1700
## Spanish Exploration and Colonial Society
During the 1500s, Spain expanded its colonial empire to the Philippines in the Far East and to areas in the Americas that later became the United States. The Spanish dreamed of mountains of gold and silver and imagined converting thousands of eager Native Americans to Catholicism. In their vision of colonial society, everyone would know his or her place. Patriarchy (the rule of men over family, society, and government) shaped the Spanish colonial world. Women occupied a lower status. In all matters, the Spanish held themselves to be atop the social pyramid, with Native peoples and Africans beneath them. Both Africans and native peoples, however, contested Spanish claims to dominance. Everywhere the Spanish settled, they brought devastating diseases, such as smallpox, that led to a horrific loss of life among native peoples. European diseases killed far more native inhabitants than did Spanish swords.
The world Native peoples had known before the coming of the Spanish was further upset by Spanish colonial practices. The Spanish imposed the encomienda system in the areas they controlled. Under this system, authorities assigned Native workers to mine and plantation owners with the understanding that the recipients would defend the colony and teach the workers the tenets of Christianity. In reality, the encomienda system exploited native workers. It was eventually replaced by another colonial labor system, the , which required Native towns to supply a pool of labor for Spanish overlords.
### ST. AUGUSTINE, FLORIDA
Spain gained a foothold in present-day Florida, viewing that area and the lands to the north as a logical extension of their Caribbean empire. In 1513, Juan Ponce de León had claimed the area around today’s St. Augustine for the Spanish crown, naming the land Pascua Florida (Feast of Flowers, or Easter) for the nearest feast day. Ponce de León was unable to establish a permanent settlement there, but by 1565, Spain was in need of an outpost to confront the French and English privateers using Florida as a base from which to attack treasure-laden Spanish ships heading from Cuba to Spain. The threat to Spanish interests took a new turn in 1562 when a group of French Protestants (Huguenots) established a small settlement they called Fort Caroline, north of St. Augustine. With the authorization of King Philip II, Spanish nobleman Pedro Menéndez led an attack on Fort Caroline, killing most of the colonists and destroying the fort. Eliminating Fort Caroline served dual purposes for the Spanish—it helped reduce the danger from French privateers and eradicated the French threat to Spain’s claim to the area. The contest over Florida illustrates how European rivalries spilled over into the Americas, especially religious conflict between Catholics and Protestants.
In 1565, the victorious Menéndez founded St. Augustine, now the oldest European settlement in the Americas. In the process, the Spanish displaced the local Timucua Natives from their ancient town of Seloy, which had stood for thousands of years (). The Timucua suffered greatly from diseases introduced by the Spanish, shrinking from a population of around 200,000 pre-contact to fifty thousand in 1590. By 1700, only one thousand Timucua remained. As in other areas of Spanish conquest, Catholic priests worked to bring about a spiritual conquest by forcing the surviving Timucua, demoralized and reeling from catastrophic losses of family and community, to convert to Catholicism.
Spanish Florida made an inviting target for Spain’s imperial rivals, especially the English, who wanted to gain access to the Caribbean. In 1586, Spanish settlers in St. Augustine discovered their vulnerability to attack when the English pirate Sir Francis Drake destroyed the town with a fleet of twenty ships and one hundred men. Over the next several decades, the Spanish built more wooden forts, all of which were burnt by raiding European rivals. Between 1672 and 1695, the Spanish constructed a stone fort, Castillo de San Marcos (), to better defend St. Augustine against challengers.
### SANTA FE, NEW MEXICO
Farther west, the Spanish in Mexico, intent on expanding their empire, looked north to the land of the Pueblo Natives. Under orders from King Philip II, Juan de Oñate explored the American southwest for Spain in the late 1590s. The Spanish hoped that what we know as New Mexico would yield gold and silver, but the land produced little of value to them. In 1610, Spanish settlers established themselves at Santa Fe—originally named La Villa Real de la Santa Fe de San Francisco de Asís, or “Royal City of the Holy Faith of St. Francis of Assisi”—where many Pueblo villages were located. Santa Fe became the capital of the Kingdom of New Mexico, an outpost of the larger Spanish Viceroyalty of New Spain, which had its headquarters in Mexico City.
As they had in other Spanish colonies, Franciscan missionaries labored to bring about a spiritual conquest by converting the Pueblo to Catholicism. At first, the Pueblo adopted the parts of Catholicism that dovetailed with their own long-standing view of the world. However, Spanish priests insisted that natives discard their old ways entirely and angered the Pueblo by focusing on the young, drawing them away from their parents. This deep insult, combined with an extended period of drought and increased attacks by local Apache and Navajo in the 1670s—troubles that the Pueblo came to believe were linked to the Spanish presence—moved the Pueblo to push the Spanish and their religion from the area. Pueblo leader Popé demanded a return to native ways so the hardships his people faced would end. To him and to thousands of others, it seemed obvious that “when Jesus came, the Corn Mothers went away.” The expulsion of the Spanish would bring a return to prosperity and a pure, native way of life.
In 1680, the Pueblo launched a coordinated rebellion against the Spanish. The Pueblo Revolt killed over four hundred Spaniards and drove the rest of the settlers, perhaps as many as two thousand, south toward Mexico. However, as droughts and attacks by rival tribes continued, the Spanish sensed an opportunity to regain their foothold. In 1692, they returned and reasserted their control of the area. Some of the Spanish explained the Pueblo success in 1680 as the work of the Devil. Satan, they believed, had stirred up the Pueblo to take arms against God’s chosen people—the Spanish—but the Spanish, and their God, had prevailed in the end.
### Section Summary
In their outposts at St. Augustine and Santa Fe, the Spanish never found the fabled mountains of gold they sought. They did find many native people to convert to Catholicism, but their zeal nearly cost them the colony of Santa Fe, which they lost for twelve years after the Pueblo Revolt. In truth, the grand dreams of wealth, conversion, and a social order based on Spanish control never came to pass as Spain envisioned them.
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# Creating New Social Orders: Colonial Societies, 1500–1700
## Colonial Rivalries: Dutch and French Colonial Ambitions
Seventeenth-century French and Dutch colonies in North America were modest in comparison to Spain’s colossal global empire. New France and New Netherland remained small commercial operations focused on the fur trade and did not attract an influx of migrants. The Dutch in New Netherland confined their operations to Manhattan Island, Long Island, the Hudson River Valley, and what later became New Jersey. Dutch trade goods circulated widely among the native peoples in these areas and also traveled well into the interior of the continent along preexisting native trade routes. French habitants, or farmer-settlers, eked out an existence along the St. Lawrence River. French fur traders and missionaries, however, ranged far into the interior of North America, exploring the Great Lakes region and the Mississippi River. These pioneers gave France somewhat inflated imperial claims to lands that nonetheless remained firmly under the dominion of native peoples.
### FUR TRADING IN NEW NETHERLAND
The Dutch Republic emerged as a major commercial center in the 1600s. Its fleets plied the waters of the Atlantic, while other Dutch ships sailed to the Far East, returning with prized spices like pepper to be sold in the bustling ports at home, especially Amsterdam. In North America, Dutch traders established themselves first on Manhattan Island.
One of the Dutch directors-general of the North American settlement, Peter Stuyvesant, served from 1647 to 1664. He expanded the fledgling outpost of New Netherland east to present-day Long Island, and for many miles north along the Hudson River. The resulting elongated colony served primarily as a fur-trading post, with the powerful Dutch West India Company controlling all commerce. Fort Amsterdam, on the southern tip of Manhattan Island, defended the growing city of New Amsterdam. In 1655, Stuyvesant took over the small outpost of New Sweden along the banks of the Delaware River in present-day New Jersey, Pennsylvania, and Delaware. He also defended New Amsterdam from Native American attacks by ordering enslaved Africans to build a protective wall on the city’s northeastern border, giving present-day Wall Street its name ().
New Netherland failed to attract many Dutch colonists; by 1664, only nine thousand people were living there. Conflict with Native peoples, as well as dissatisfaction with the Dutch West India Company’s trading practices, made the Dutch outpost an undesirable place for many migrants. The small size of the population meant a severe labor shortage, and to complete the arduous tasks of early settlement, the Dutch West India Company imported some 450 enslaved Africans between 1626 and 1664. (The company had involved itself heavily in the slave trade and in 1637 captured Elmina, the slave-trading post on the west coast of Africa, from the Portuguese.) The shortage of labor also meant that New Netherland welcomed non-Dutch immigrants, including Protestants from Germany, Sweden, Denmark, and England, and embraced a degree of religious tolerance, allowing Jewish immigrants to become residents beginning in the 1650s. Thus, a wide variety of people lived in New Netherland from the start. Indeed, one observer claimed eighteen different languages could be heard on the streets of New Amsterdam. As new settlers arrived, the colony of New Netherland stretched farther to the north and the west ().
The Dutch West India Company found the business of colonization in New Netherland to be expensive. To share some of the costs, it granted Dutch merchants who invested heavily in it patroonships, or large tracts of land and the right to govern the tenants there. In return, the shareholder who gained the patroonship promised to pay for the passage of at least thirty Dutch farmers to populate the colony. One of the largest patroonships was granted to Kiliaen van Rensselaer, one of the directors of the Dutch West India Company; it covered most of present-day Albany and Rensselaer Counties. This pattern of settlement created a yawning gap in wealth and status between the tenants, who paid rent, and the wealthy patroons.
During the summer trading season, Native Americans gathered at trading posts such as the Dutch site at Beverwijck (present-day Albany), where they exchanged furs for guns, blankets, and alcohol. The furs, especially beaver pelts destined for the lucrative European millinery market, would be sent down the Hudson River to New Amsterdam. There, enslaved laborers or workers would load them aboard ships bound for Amsterdam.
### COMMERCE AND CONVERSION IN NEW FRANCE
After Jacques Cartier’s voyages of discovery in the 1530s, France showed little interest in creating permanent colonies in North America until the early 1600s, when Samuel de Champlain established Quebec as a French fur-trading outpost. Although the fur trade was lucrative, the French saw Canada as an inhospitable frozen wasteland, and by 1640, fewer than four hundred settlers had made their home there. The sparse French presence meant that colonists depended on the local native Algonquian people; without them, the French would have perished. French fishermen, explorers, and fur traders made extensive contact with the Algonquian. The Algonquian, in turn, tolerated the French because the colonists supplied them with firearms for their ongoing war with the Iroquois. Thus, the French found themselves escalating native wars and supporting the Algonquian against the Iroquois, who received weapons from their Dutch trading partners. These seventeenth-century conflicts centered on the lucrative trade in beaver pelts, earning them the name of the Beaver Wars. In these wars, fighting between rival native peoples spread throughout the Great Lakes region.
A handful of French Jesuit priests also made their way to Canada, intent on converting the native inhabitants to Catholicism. The Jesuits were members of the Society of Jesus, an elite religious order founded in the 1540s to spread Catholicism and combat the spread of Protestantism. The first Jesuits arrived in Quebec in the 1620s, and for the next century, their numbers did not exceed forty priests. Like the Spanish Franciscan missionaries, the Jesuits in the colony called New France labored to convert the native peoples to Catholicism. They wrote detailed annual reports about their progress in bringing the faith to the Algonquian and, beginning in the 1660s, to the Iroquois. These documents are known as the Jesuit Relations (), and they provide a rich source for understanding both the Jesuit view of the Native Americans and the Native response to the colonizers.
One Native convert to Catholicism, a Mohawk woman named Kateri Tekakwitha, so impressed the priests with her piety that a Jesuit named Claude Chauchetière attempted to make her a saint in the Church. However, the effort to canonize Tekakwitha faltered when leaders of the Church balked at elevating a “savage” to such a high status; she was eventually canonized in 2012. French colonizers pressured the native inhabitants of New France to convert, but they virtually never saw Native peoples as their equals.
### Section Summary
The French and Dutch established colonies in the northeastern part of North America: the Dutch in present-day New York, and the French in present-day Canada. Both colonies were primarily trading posts for furs. While they failed to attract many colonists from their respective home countries, these outposts nonetheless intensified imperial rivalries in North America. Both the Dutch and the French relied on native peoples to harvest the pelts that proved profitable in Europe.
### Review Questions
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# Creating New Social Orders: Colonial Societies, 1500–1700
## English Settlements in America
At the start of the seventeenth century, the English had not established a permanent settlement in the Americas. Over the next century, however, they outpaced their rivals. The English encouraged emigration far more than the Spanish, French, or Dutch. They established nearly a dozen colonies, sending swarms of immigrants to populate the land. England had experienced a dramatic rise in population in the sixteenth century, and the colonies appeared a welcoming place for those who faced overcrowding and grinding poverty at home. Thousands of English migrants arrived in the Chesapeake Bay colonies of Virginia and Maryland to work in the tobacco fields. Another stream, this one of pious Puritan families, sought to live as they believed scripture demanded and established the Plymouth, Massachusetts Bay, New Haven, Connecticut, and Rhode Island colonies of New England ().
### THE DIVERGING CULTURES OF THE NEW ENGLAND AND CHESAPEAKE COLONIES
Promoters of English colonization in North America, many of whom never ventured across the Atlantic, wrote about the bounty the English would find there. These boosters of colonization hoped to turn a profit—whether by importing raw resources or providing new markets for English goods—and spread Protestantism. The English migrants who actually made the journey, however, had different goals. In Chesapeake Bay, English migrants established Virginia and Maryland with a decidedly commercial orientation. Though the early Virginians at Jamestown hoped to find gold, they and the settlers in Maryland quickly discovered that growing tobacco was the only sure means of making money. Thousands of unmarried, unemployed, and impatient young Englishmen, along with a few Englishwomen, pinned their hopes for a better life on the tobacco fields of these two colonies.
A very different group of English men and women flocked to the cold climate and rocky soil of New England, spurred by religious motives. Many of the Puritans crossing the Atlantic were people who brought families and children. Often they were following their ministers in a migration “beyond the seas,” envisioning a new English Israel where reformed Protestantism would grow and thrive, providing a model for the rest of the Christian world and a counter to what they saw as the Catholic menace. While the English in Virginia and Maryland worked on expanding their profitable tobacco fields, the English in New England built towns focused on the church, where each congregation decided what was best for itself. The Congregational Church is the result of the Puritan enterprise in America. Many historians believe the fault lines separating what later became the North and South in the United States originated in the profound differences between the Chesapeake and New England colonies.
The source of those differences lay in England’s domestic problems. Increasingly in the early 1600s, the English state church—the Church of England, established in the 1530s—demanded conformity, or compliance with its practices, but Puritans pushed for greater reforms. By the 1620s, the Church of England began to see leading Puritan ministers and their followers as outlaws, a national security threat because of their opposition to its power. As the noose of conformity tightened around them, many Puritans decided to remove to New England. By 1640, New England had a population of twenty-five thousand. Meanwhile, many loyal members of the Church of England, who ridiculed and mocked Puritans both at home and in New England, flocked to Virginia for economic opportunity.
The troubles in England escalated in the 1640s when civil war broke out, pitting Royalist supporters of King Charles I and the Church of England against Parliamentarians, the Puritan reformers and their supporters in Parliament. In 1649, the Parliamentarians gained the upper hand and, in an unprecedented move, executed Charles I. In the 1650s, therefore, England became a republic, a state without a king. English colonists in America closely followed these events. Indeed, many Puritans left New England and returned home to take part in the struggle against the king and the national church. Other English men and women in the Chesapeake colonies and elsewhere in the English Atlantic World looked on in horror at the mayhem the Parliamentarians, led by the Puritan insurgents, appeared to unleash in England. The turmoil in England made the administration and imperial oversight of the Chesapeake and New England colonies difficult, and the two regions developed divergent cultures.
### THE CHESAPEAKE COLONIES: VIRGINIA AND MARYLAND
The Chesapeake colonies of Virginia and Maryland served a vital purpose in the developing seventeenth-century English empire by providing tobacco, a cash crop. However, the early history of Jamestown did not suggest the English outpost would survive. From the outset, its settlers struggled both with each other and with the Native inhabitants, the powerful Powhatan, who controlled the area. Jealousies and infighting among the English destabilized the colony. One member, John Smith, whose famous map begins this chapter, took control and exercised near-dictatorial powers, which furthered aggravated the squabbling. The settlers’ inability to grow their own food compounded this unstable situation. They were essentially employees of the Virginia Company of London, an English joint-stock company, in which investors provided the capital and assumed the risk in order to reap the profit, and they had to make a profit for their shareholders as well as for themselves. Most initially devoted themselves to finding gold and silver instead of finding ways to grow their own food.
### Early Struggles and the Development of the Tobacco Economy
Poor health, lack of food, and fighting with Native peoples took the lives of many of the original Jamestown settlers. The winter of 1609–1610, which became known as “the starving time,” came close to annihilating the colony. By June 1610, the few remaining settlers had decided to abandon the area; only the last-minute arrival of a supply ship from England prevented another failed colonization effort. The supply ship brought new settlers, but only twelve hundred of the seventy-five hundred who came to Virginia between 1607 and 1624 survived.
By the 1620s, Virginia had weathered the worst and gained a degree of permanence. Political stability came slowly, but by 1619, the fledgling colony was operating under the leadership of a governor, a council, and a House of Burgesses. Economic stability came from the lucrative cultivation of tobacco. Smoking tobacco was a long-standing practice among native peoples, and English and other European consumers soon adopted it. In 1614, the Virginia colony began exporting tobacco back to England, which earned it a sizable profit and saved the colony from ruin. A second tobacco colony, Maryland, was formed in 1634, when King Charles I granted its charter to the Calvert family for their loyal service to England. Cecilius Calvert, the second Lord Baltimore, conceived of Maryland as a refuge for English Catholics.
Growing tobacco proved very labor-intensive (), and the Chesapeake colonists needed a steady workforce to do the hard work of clearing the land and caring for the tender young plants. The mature leaf of the plant then had to be cured (dried), which necessitated the construction of drying barns. Once cured, the tobacco had to be packaged in hogsheads (large wooden barrels) and loaded aboard ship, which also required considerable labor.
To meet these labor demands, early Virginians relied on indentured servants. An indenture is a labor contract that young, impoverished, and often illiterate Englishmen and occasionally Englishwomen signed in England, pledging to work for a number of years (usually between five and seven) growing tobacco in the Chesapeake colonies. In return, indentured servants received paid passage to America and food, clothing, and lodging. At the end of their indenture, servants received “freedom dues,” usually food and other provisions, including, in some cases, land provided by the colony. The promise of a new life in America was a strong attraction for members of England’s underclass, who had few if any options at home. In the 1600s, some 100,000 indentured servants traveled to the Chesapeake Bay. Most were poor young men in their early twenties.
Life in the colonies proved harsh, however. Indentured servants could not marry, and they were subject to the will of the tobacco planters who bought their labor contracts. Treated much like property, the contracted servants could be essentially sold or traded among those with means to purchase them. Some contract holders did not feed or house their servants well. If an indentured servant committed a crime or disobeyed those who held their contracts, they found their terms of service lengthened, often by several years. Female indentured servants faced special dangers in what was essentially a bachelor colony. Many were exploited by unscrupulous tobacco planters who seduced them with promises of marriage. If the women became pregnant, the planters would then sell them to other tobacco planters to avoid the costs of raising a child.
Nonetheless, those indentured servants who completed their term of service often began new lives as tobacco planters. To entice even more migrants to the New World, the Virginia Company also implemented the headright system, in which those who paid their own passage to Virginia received fifty acres plus an additional fifty for each servant or family member they brought with them. The headright system and the promise of a new life for servants acted as powerful incentives for English migrants to hazard the journey to the New World.
### The Anglo-Powhatan Wars
By choosing to settle along the rivers on the banks of the Chesapeake, the English unknowingly placed themselves at the center of the Powhatan Empire, a powerful Algonquian confederacy of thirty native groups with perhaps as many as twenty-two thousand people. The territory of the equally impressive Susquehannock people also bordered English settlements at the north end of the Chesapeake Bay.
Tensions ran high between the English and the Powhatan, and near-constant war prevailed. The First Anglo-Powhatan War (1609–1614) resulted not only from the English colonists’ intrusion onto Powhatan land, but also from their refusal to follow cultural protocol by giving gifts. English actions infuriated and insulted the Powhatan. In 1613, the settlers captured Pocahontas (also called Matoaka), the daughter of a Powhatan headman named Wahunsonacook, and gave her in marriage to Englishman John Rolfe. Their union, and her choice to remain with the English, helped quell the war in 1614. Pocahontas converted to Christianity, changing her name to Rebecca, and sailed with her husband and several other Powhatan to England where she was introduced to King James I (). Promoters of colonization publicized Pocahontas as an example of the good work of converting the Powhatan to Christianity.
Peace in Virginia did not last long. The Second Anglo-Powhatan War (1620s) broke out because of the expansion of the English settlement nearly one hundred miles into the interior, and because of the continued insults and friction caused by English activities. The Powhatan attacked in 1622 and succeeded in killing almost 350 English, about a third of the settlers. The English responded by annihilating every Powhatan village around Jamestown and from then on became even more intolerant. The Third Anglo-Powhatan War (1644–1646) began with a surprise attack in which the Powhatan killed around five hundred English colonists. However, their ultimate defeat in this conflict forced the Powhatan to acknowledge King Charles I as their sovereign. The Anglo-Powhatan Wars, spanning nearly forty years, illustrate the degree of native resistance that resulted from English intrusion into the Powhatan confederacy.
### The Rise of Slavery in the Chesapeake Bay Colonies
The transition from indentured servitude to slavery as the main labor source for some English colonies happened first in the West Indies. On the small island of Barbados, colonized in the 1620s, English planters first grew tobacco as their main export crop, but in the 1640s, they converted to sugarcane and began increasingly to rely on African enslaved people. In 1655, England wrestled control of Jamaica from the Spanish and quickly turned it into a lucrative sugar island, run on forced labor, for its expanding empire. While slavery was slower to take hold in the Chesapeake colonies, by the end of the seventeenth century, both Virginia and Maryland had also adopted chattel slavery—which legally defined Africans as property and not people—as the dominant form of labor to grow tobacco. Chesapeake colonists also enslaved Native people.
When the first Africans arrived in Virginia in 1619, slavery—which did not exist in England—had not yet become an institution in colonial America. Many Africans worked as servants and, like their White counterparts, could acquire land of their own. Some Africans who converted to Christianity became free landowners with White servants. The change in the status of Africans in the Chesapeake to that of enslaved people occurred in the last decades of the seventeenth century.
Bacon’s Rebellion, an uprising of both White people and Black people who believed that the Virginia government was impeding their access to land and wealth and seemed to do little to clear the land of Native Americans, hastened the transition to African slavery in the Chesapeake colonies. The rebellion takes its name from Nathaniel Bacon, a wealthy young Englishman who arrived in Virginia in 1674. Despite an early friendship with Virginia’s royal governor, William Berkeley, Bacon found himself excluded from the governor’s circle of influential friends and councilors. He wanted land on the Virginia frontier, but the governor, fearing war with neighboring tribes, forbade further expansion. Bacon marshaled others, especially former indentured servants who believed the governor was limiting their economic opportunities and denying them the right to own tobacco farms. Bacon’s followers believed Berkeley’s frontier policy didn’t protect English settlers enough. Worse still in their eyes, Governor Berkeley tried to keep peace in Virginia by signing treaties with various local Native peoples. Bacon and his followers, who saw all Native peoples as an obstacle to their access to land, pursued a policy of extermination.
Tensions between the English and the Native peoples in the Chesapeake colonies led to open conflict. In 1675, war broke out when Susquehannock warriors attacked settlements on Virginia’s frontier, killing English planters and destroying English plantations, including one owned by Bacon. In 1676, Bacon and other Virginians attacked the Susquehannock without the governor’s approval. When Berkeley ordered Bacon’s arrest, Bacon led his followers to Jamestown, forced the governor to flee to the safety of Virginia’s eastern shore, and then burned the city. The civil war known as Bacon’s Rebellion, a vicious struggle between supporters of the governor and those who supported Bacon, ensued. Reports of the rebellion traveled back to England, leading Charles II to dispatch both royal troops and English commissioners to restore order in the tobacco colonies. By the end of 1676, Virginians loyal to the governor gained the upper hand, executing several leaders of the rebellion. Bacon escaped the hangman’s noose, instead dying of dysentery. The rebellion fizzled in 1676, but Virginians remained divided as supporters of Bacon continued to harbor grievances over access to Native land.
Bacon’s Rebellion helped to catalyze the creation of a system of racial slavery in the Chesapeake colonies. At the time of the rebellion, indentured servants made up the majority of laborers in the region. Wealthy White people worried over the presence of this large class of laborers and the relative freedom they enjoyed, as well as the alliance that Black and White servants had forged in the course of the rebellion. Replacing indentured servitude with Black slavery diminished these risks, alleviating the reliance on White indentured servants, who were often dissatisfied and troublesome, and creating a caste of racially defined laborers whose movements were strictly controlled. It also lessened the possibility of further alliances between Black and White workers. Racial slavery even served to heal some of the divisions between wealthy and poor White people, who could now unite as members of a “superior” racial group.
While colonial laws in the tobacco colonies had made slavery a legal institution before Bacon’s Rebellion, new laws passed in the wake of the rebellion severely curtailed Black freedom and laid the foundation for racial slavery. Virginia passed a law in 1680 prohibiting free Black people and enslaved people from bearing arms, banning Black people from congregating in large numbers, and establishing harsh punishments for enslaved people who assaulted Christians or sought freedom. Two years later, another Virginia law stipulated that all Africans brought to the colony would be enslaved for life. Thus, the increasing reliance on enslaved people in the tobacco colonies—and the draconian laws instituted to control them—not only helped planters meet labor demands, but also served to assuage English fears of further uprisings and alleviate class tensions between rich and poor White people.
### PURITAN NEW ENGLAND
The second major area to be colonized by the English in the first half of the seventeenth century, New England, differed markedly in its founding principles from the commercially oriented Chesapeake tobacco colonies. Settled largely by waves of Puritan families in the 1630s, New England had a religious orientation from the start. In England, reform-minded men and women had been calling for greater changes to the English national church since the 1580s. These reformers, who followed the teachings of John Calvin and other Protestant reformers, were called Puritans because of their insistence on “purifying” the Church of England of what they believed to be un-scriptural, especially Catholic elements that lingered in its institutions and practices.
Many who provided leadership in early New England were learned ministers who had studied at Cambridge or Oxford but who, because they had questioned the practices of the Church of England, had been deprived of careers by the king and his officials in an effort to silence all dissenting voices. Other Puritan leaders, such as the first governor of the Massachusetts Bay Colony, John Winthrop, came from the privileged class of English gentry. These well-to-do Puritans and many thousands more left their English homes not to establish a land of religious freedom, but to practice their own religion without persecution. Puritan New England offered them the opportunity to live as they believed the Bible demanded. In their “New” England, they set out to create a model of reformed Protestantism, a new English Israel.
The conflict generated by Puritanism had divided English society, because the Puritans demanded reforms that undermined the traditional festive culture. For example, they denounced popular pastimes like bear-baiting—letting dogs attack a chained bear—which were often conducted on Sundays when people had a few leisure hours. In the culture where William Shakespeare had produced his masterpieces, Puritans called for an end to the theater, censuring playhouses as places of decadence. Indeed, the Bible itself became part of the struggle between Puritans and James I, who headed the Church of England. Soon after ascending the throne, James commissioned a new version of the Bible in an effort to stifle Puritan reliance on the Geneva Bible, which followed the teachings of John Calvin and placed God’s authority above the monarch’s. The King James Version, published in 1611, instead emphasized the majesty of kings.
During the 1620s and 1630s, the conflict escalated to the point where the state church prohibited Puritan ministers from preaching. In the Church’s view, Puritans represented a national security threat, because their demands for cultural, social, and religious reforms undermined the king’s authority. Unwilling to conform to the Church of England, many Puritans found refuge in the New World. Yet those who emigrated to the Americas were not united. Some called for a complete break with the Church of England, while others remained committed to reforming the national church.
### Plymouth: The First Puritan Colony
The first group of Puritans to make their way across the Atlantic was a small contingent known as the Pilgrims. Unlike other Puritans, they insisted on a complete separation from the Church of England and had first migrated to the Dutch Republic in Europe seeking religious freedom. Although they found they could worship without hindrance there, they grew concerned that they were losing their Englishness as they saw their children begin to learn the Dutch language and adopt Dutch ways. In addition, the English Pilgrims (and others in Europe) feared another attack on the Dutch Republic by Spain. Therefore, in 1620, they moved on to found the Plymouth Colony in present-day Massachusetts. The governor of Plymouth, William Bradford, was a Separatist, a proponent of complete separation from the English state church. Bradford and the other Pilgrim Separatists represented a major challenge to the prevailing vision of a unified English national church and empire. On board the Mayflower, which was bound for Virginia but landed on the tip of Cape Cod, Bradford and forty other adult men signed the Mayflower Compact (), which presented a religious (rather than an economic) rationale for colonization. The compact expressed a community ideal of working together. When a larger exodus of Puritans established the Massachusetts Bay Colony in the 1630s, the Pilgrims at Plymouth welcomed them and the two colonies cooperated with each other.
Different labor systems also distinguished early Puritan New England from the Chesapeake colonies. Puritans expected young people to work diligently at their calling, and all members of their large families, including children, did the bulk of the work necessary to run homes, farms, and businesses. Very few migrants came to New England as laborers; in fact, New England towns protected their disciplined homegrown workforce by refusing to allow outsiders in, assuring their sons and daughters of steady employment. New England’s labor system produced remarkable results, notably a powerful maritime-based economy with scores of oceangoing ships and the crews necessary to sail them. New England mariners sailing New England–made ships transported Virginian tobacco and West Indian sugar throughout the Atlantic World.
### “A City upon a Hill”
A much larger group of English Puritans left England in the 1630s, establishing the Massachusetts Bay Colony, the New Haven Colony, the Connecticut Colony, and Rhode Island. Unlike the exodus of young males to the Chesapeake colonies, these migrants were families with young children and their university-trained ministers. Their aim, according to John Winthrop (), the first governor of Massachusetts Bay, was to create a model of reformed Protestantism—a “city upon a hill,” a new English Israel. The idea of a “city upon a hill” made clear the religious orientation of the New England settlement, and the charter of the Massachusetts Bay Colony stated as a goal that the colony’s people “may be soe religiously, peaceablie, and civilly governed, as their good Life and orderlie Conversacon, maie wynn and incite the Natives of Country, to the Knowledg and Obedience of the onlie true God and Saulor of Mankinde, and the Christian Fayth.” To illustrate this, the seal of the Massachusetts Bay Company () shows a Native American who entreats more of the English to “come over and help us.”
Puritan New England differed in many ways from both England and the rest of Europe. Protestants emphasized literacy so that everyone could read the Bible. This attitude was in stark contrast to that of Catholics, who refused to tolerate private ownership of Bibles in the vernacular. The Puritans, for their part, placed a special emphasis on reading scripture, and their commitment to literacy led to the establishment of the first printing press in English America in 1636. Four years later, in 1640, they published the first book in North America, the Bay Psalm Book. As Calvinists, Puritans adhered to the doctrine of predestination, whereby a few “elect” would be saved and all others damned. No one could be sure whether they were predestined for salvation, but through introspection, guided by scripture, Puritans hoped to find a glimmer of redemptive grace. Church membership was restricted to those Puritans who were willing to provide a conversion narrative telling how they came to understand their spiritual estate by hearing sermons and studying the Bible.
Although many people assume Puritans escaped England to establish religious freedom, they proved to be just as intolerant as the English state church. When dissenters, including Puritan minister Roger Williams and Anne Hutchinson, challenged Governor Winthrop in Massachusetts Bay in the 1630s, they were banished. Roger Williams questioned the Puritans’ taking of Native land. Williams also argued for a complete separation from the Church of England, a position other Puritans in Massachusetts rejected, as well as the idea that the state could not punish individuals for their beliefs. Although he did accept that nonbelievers were destined for eternal damnation, Williams did not think the state could compel true orthodoxy. Puritan authorities found him guilty of spreading dangerous ideas, but he went on to found Rhode Island as a colony that sheltered dissenting Puritans from their brethren in Massachusetts. In Rhode Island, Williams wrote favorably about native peoples, contrasting their virtues with Puritan New England’s intolerance.
Anne Hutchinson also ran afoul of Puritan authorities for her criticism of the evolving religious practices in the Massachusetts Bay Colony. In particular, she held that Puritan ministers in New England taught a shallow version of Protestantism emphasizing hierarchy and actions—a “covenant of works” rather than a “covenant of grace.” Literate Puritan women like Hutchinson presented a challenge to the male ministers’ authority. Indeed, her major offense was her claim of direct religious revelation, a type of spiritual experience that negated the role of ministers. Because of Hutchinson’s beliefs and her defiance of authority in the colony, especially that of Governor Winthrop, Puritan authorities tried and convicted her of holding false beliefs. In 1638, she was excommunicated and banished from the colony. She went to Rhode Island and later, in 1642, sought safety among the Dutch in New Netherland. The following year, Algonquian warriors killed Hutchinson and her family. In Massachusetts, Governor Winthrop noted her death as the righteous judgment of God against a heretic.
Like many other Europeans, the Puritans believed in the supernatural. Every event appeared to be a sign of God’s mercy or judgment, and people believed that witches allied themselves with the Devil to carry out evil deeds and deliberate harm such as the sickness or death of children, the loss of cattle, and other catastrophes. Hundreds were accused of witchcraft in Puritan New England, including townspeople whose habits or appearance bothered their neighbors or who appeared threatening for any reason. Women, seen as more susceptible to the Devil because of their supposedly weaker constitutions, made up the vast majority of suspects and those who were executed. The most notorious cases occurred in Salem Village in 1692. Many of the accusers who prosecuted the suspected witches had been traumatized by the Native wars on the frontier and by unprecedented political and cultural changes in New England. Relying on their belief in witchcraft to help make sense of their changing world, Puritan authorities executed nineteen people and caused the deaths of several others.
### Puritan Relationships with Native Peoples
Like their Spanish and French Catholic rivals, English Puritans in America took steps to convert native peoples to their version of Christianity. John Eliot, the leading Puritan missionary in New England, urged natives in Massachusetts to live in “praying towns” established by English authorities for converted Native Americans, and to adopt the Puritan emphasis on the centrality of the Bible. In keeping with the Protestant emphasis on reading scripture, he translated the Bible into the local Algonquian language and published his work in 1663. Eliot hoped that as a result of his efforts, some of New England’s native inhabitants would become preachers.
Tensions had existed from the beginning between the Puritans and the native people who controlled southern New England (). Relationships deteriorated as the Puritans continued to expand their settlements aggressively and as European ways increasingly disrupted native life. These strains led to King Philip’s War (1675–1676), a massive regional conflict that was nearly successful in pushing the English out of New England.
When the Puritans began to arrive in the 1620s and 1630s, local Algonquian peoples had viewed them as potential allies in the conflicts already simmering between rival Native groups. In 1621, the Wampanoag, led by Massasoit, concluded a peace treaty with the Pilgrims at Plymouth. In the 1630s, the Puritans in Massachusetts and Plymouth allied themselves with the Narragansett and Mohegan people against the Pequot, who had recently expanded their claims into southern New England. In May 1637, the Puritans attacked a large group of several hundred Pequot along the Mystic River in Connecticut. To the horror of their Native allies, the Puritans massacred all but a handful of the men, women, and children they found.
By the mid-seventeenth century, the Puritans had pushed their way further into the interior of New England, establishing outposts along the Connecticut River Valley. There seemed no end to their expansion. Wampanoag leader Metacom or Metacomet, also known as King Philip among the English, was determined to stop the encroachment. The Wampanoag, along with the Nipmuck, Pocumtuck, and Narragansett, took up arms to drive the English from the land. In the ensuing conflict, called King Philip’s War, Native forces succeeded in destroying half of the frontier Puritan towns; however, in the end, the English (aided by Mohegans and Christian Native Americans) prevailed and sold many captives into slavery in the West Indies. (The severed head of King Philip was publicly displayed in Plymouth.) The war also forever changed the English perception of Native peoples; from then on, Puritan writers took great pains to vilify the Native people as bloodthirsty savages. A new type of racial hatred became a defining feature of Native-English relationships in the Northeast.
### Section Summary
The English came late to colonization of the Americas, establishing stable settlements in the 1600s after several unsuccessful attempts in the 1500s. After Roanoke Colony failed in 1587, the English found more success with the founding of Jamestown in 1607 and Plymouth in 1620. The two colonies were very different in origin. The Virginia Company of London founded Jamestown with the express purpose of making money for its investors, while Puritans founded Plymouth to practice their own brand of Protestantism without interference.
Both colonies battled difficult circumstances, including poor relationships with neighboring Native American tribes. Conflicts flared repeatedly in the Chesapeake Bay tobacco colonies and in New England, where a massive uprising against the English in 1675 to 1676—King Philip’s War—nearly succeeded in driving the intruders back to the sea.
### Review Questions
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# Creating New Social Orders: Colonial Societies, 1500–1700
## The Impact of Colonization
As Europeans moved beyond exploration and into colonization of the Americas, they brought changes to virtually every aspect of the land and its people, from trade and hunting to warfare and personal property. European goods, ideas, and diseases shaped the changing continent.
As Europeans established their colonies, their societies also became segmented and divided along religious and racial lines. Most people in these societies were not free; they labored as servants or enslaved people, doing the work required to produce wealth for others. By 1700, the American continent had become a place of stark contrasts between slavery and freedom, between the haves and the have-nots.
### THE INSTITUTION OF SLAVERY
Everywhere in the American colonies, a crushing demand for labor existed to grow New World cash crops, especially sugar and tobacco. This need led Europeans to rely increasingly on Africans, and after 1600, the movement of Africans across the Atlantic accelerated. The English crown chartered the Royal African Company in 1672, giving the company a monopoly over the transport of enslaved African people to the English colonies. Over the next four decades, the company transported around 350,000 Africans from their homelands. By 1700, the tiny English sugar island of Barbados had a population of fifty thousand enslaved people, and the English had encoded the institution of chattel slavery into colonial law.
This new system of African slavery came slowly to the English colonists, who did not have slavery at home and preferred to use servant labor. Nevertheless, by the end of the seventeenth century, the English everywhere in America—and particularly in the Chesapeake Bay colonies—had come to rely on enslaved Africans. While Africans had long practiced slavery among their own people, it had not been based on race. Africans enslaved other Africans as war captives, for crimes, and to settle debts; they generally used enslaved people for domestic and small-scale agricultural work, not for growing cash crops on large plantations. Additionally, African slavery was often a temporary condition rather than a lifelong sentence, and, unlike New World slavery, it was typically not heritable (passed from an enslaved mother to her children).
The growing slave trade with Europeans had a profound impact on the people of West Africa, giving prominence to local chieftains and merchants who traded enslaved people for European textiles, alcohol, guns, tobacco, and food. Africans also charged Europeans for the right to trade in enslaved people and imposed taxes on enslaved people purchases. Different African groups and kingdoms even staged large-scale raids on each other to meet the demand for enslaved people.
Once sold to traders, all captured people sent to America endured the hellish Middle Passage, the transatlantic crossing, which took one to two months. By 1625, more than 325,800 Africans had been shipped to the New World, though many thousands perished during the voyage. An astonishing number, some four million, were transported to the Caribbean between 1501 and 1830. When they reached their destination in America, Africans found themselves trapped in shockingly brutal slave societies. In the Chesapeake colonies, they faced a lifetime of harvesting and processing tobacco.
Everywhere, Africans resisted slavery, and running away was common. In Jamaica and elsewhere, escaped enslaved people created maroon communities, groups that resisted recapture and eked out a living from the land, rebuilding their communities as best they could. When possible, they adhered to traditional ways, following spiritual leaders such as Vodun priests.
### CHANGES TO NATIVE LIFE
While the Americas remained firmly under the control of native peoples in the first decades of European settlement, conflict increased as colonization spread and Europeans placed greater demands upon the native populations, including expecting them to convert to Christianity (either Catholicism or Protestantism). Throughout the seventeenth century, the still-powerful native peoples and confederacies that retained control of the land waged war against the invading Europeans, achieving a degree of success in their effort to drive the newcomers from the continent.
At the same time, European goods had begun to change Native life radically. In the 1500s, some of the earliest objects Europeans introduced to Native Americans were glass beads, copper kettles, and metal utensils. Native people often adapted these items for their own use. For example, some cut up copper kettles and refashioned the metal for other uses, including jewelry that conferred status on the wearer, who was seen as connected to the new European source of raw materials.
As European settlements grew throughout the 1600s, European goods flooded Native communities. Soon Native people were using these items for the same purposes as the Europeans. For example, many Native inhabitants abandoned their animal-skin clothing in favor of European textiles. Similarly, clay cookware gave way to metal cooking implements, and Native Americans found that European flint and steel made starting fires much easier ().
The abundance of European goods gave rise to new artistic objects. For example, iron awls made the creation of shell beads among the native people of the Eastern Woodlands much easier, and the result was an astonishing increase in the production of wampum, shell beads used in ceremonies and as jewelry and currency. Native peoples had always placed goods in the graves of their departed, and this practice escalated with the arrival of European goods. Archaeologists have found enormous caches of European trade goods in the graves of Native Americans on the East Coast.
Native weapons changed dramatically as well, creating an arms race among the peoples living in European colonization zones. Native Americans refashioned European brassware into arrow points and turned axes used for chopping wood into weapons. The most prized piece of European weaponry to obtain was a musket, or light, long-barreled European gun. In order to trade with Europeans for these, Native peoples intensified their harvesting of beaver, commercializing their traditional practice.
The influx of European materials made warfare more lethal and changed traditional patterns of authority among tribes. Formerly weaker groups, if they had access to European metal and weapons, suddenly gained the upper hand against once-dominant groups. The Algonquian, for instance, traded with the French for muskets and gained power against their enemies, the Iroquois. Eventually, native peoples also used their new weapons against the European colonizers who had provided them.
### ENVIRONMENTAL CHANGES
The European presence in America spurred countless changes in the environment, setting into motion chains of events that affected native animals as well as people. The popularity of beaver-trimmed hats in Europe, coupled with Native American’s desire for European weapons, led to the overhunting of beaver in the Northeast. Soon, beavers were extinct in New England, New York, and other areas. With their loss came the loss of beaver ponds, which had served as habitats for fish as well as water sources for deer, moose, and other animals. Furthermore, Europeans introduced pigs, which they allowed to forage in forests and other wildlands. Pigs consumed the foods on which deer and other indigenous species depended, resulting in scarcity of the game native peoples had traditionally hunted.
European ideas about owning land as private property clashed with natives’ understanding of land use. Native peoples did not believe in private ownership of land; instead, they viewed land as a resource to be held in common for the benefit of the group. The European idea of usufruct—the right to common land use and enjoyment—comes close to the native understanding, but colonists did not practice usufruct widely in America. Colonizers established fields, fences, and other means of demarcating private property. Native peoples who moved seasonally to take advantage of natural resources now found areas off limits, claimed by colonizers because of their insistence on private-property rights.
### The Introduction of Disease
Perhaps European colonization’s single greatest impact on the North American environment was the introduction of disease. Microbes to which native inhabitants had no immunity led to death everywhere Europeans settled. Along the New England coast between 1616 and 1618, epidemics claimed the lives of 75 percent of the native people. In the 1630s, half the Huron and Iroquois around the Great Lakes died of smallpox. As is often the case with disease, the very young and the very old were the most vulnerable and had the highest mortality rates. The loss of the older generation meant the loss of knowledge and tradition, while the death of children only compounded the trauma, creating devastating implications for future generations.
Some native peoples perceived disease as a weapon used by hostile spiritual forces, and they went to war to exorcise the disease from their midst. These “mourning wars” in eastern North America were designed to gain captives who would either be adopted (“requickened” as a replacement for a deceased loved one) or ritually tortured and executed to assuage the anger and grief caused by loss.
### The Cultivation of Plants
European expansion in the Americas led to an unprecedented movement of plants across the Atlantic. A prime example is tobacco, which became a valuable export as the habit of smoking, previously unknown in Europe, took hold (). Another example is sugar. Columbus brought sugarcane to the Caribbean on his second voyage in 1494, and thereafter a wide variety of other herbs, flowers, seeds, and roots made the transatlantic voyage.
Just as pharmaceutical companies today scour the natural world for new drugs, Europeans traveled to America to discover new medicines. The task of cataloging the new plants found there helped give birth to the science of botany. Early botanists included the English naturalist Sir Hans Sloane, who traveled to Jamaica in 1687 and there recorded hundreds of new plants (). Sloane also helped popularize the drinking of chocolate, made from the cacao bean, in England.
Native Americans, who possessed a vast understanding of local New World plants and their properties, would have been a rich source of information for those European botanists seeking to find and catalog potentially useful plants. Enslaved Africans, who had a tradition of the use of medicinal plants in their native land, adapted to their new surroundings by learning the use of New World plants through experimentation or from the native inhabitants. Native peoples and Africans employed their knowledge effectively within their own communities. One notable example was the use of the peacock flower to induce abortions: Native American and enslaved African women living in oppressive colonial regimes are said to have used this herb to prevent the birth of children into slavery. Europeans distrusted medical knowledge that came from African or native sources, however, and thus lost the benefit of this source of information.
### Section Summary
The development of the Atlantic slave trade forever changed the course of European settlement in the Americas. Other transatlantic travelers, including diseases, goods, plants, animals, and even ideas like the concept of private land ownership, further influenced life in America during the sixteenth and seventeenth centuries. The exchange of pelts for European goods including copper kettles, knives, and guns played a significant role in changing the material cultures of native peoples. During the seventeenth century, native peoples grew increasingly dependent on European trade items. At the same time, many native inhabitants died of European diseases, while survivors adopted new ways of living with their new neighbors.
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# Rule Britannia! The English Empire, 1660–1763
## Introduction
The eighteenth century witnessed the birth of Great Britain (after the union of England and Scotland in 1707) and the expansion of the British Empire. By the mid-1700s, Great Britain had developed into a commercial and military powerhouse; its economic sway ranged from India, where the British East India Company had gained control over both trade and territory, to the West African coast, where British slave traders predominated, and to the British West Indies, whose lucrative sugar plantations, especially in Barbados and Jamaica, provided windfall profits for British planters. Meanwhile, the population rose dramatically in Britain’s North American colonies. In the early 1700s the population in the colonies had reached 250,000. By 1750, however, over a million British migrants and enslaved Africans had established a near-continuous zone of settlement on the Atlantic coast from Maine to Georgia.
During this period, the ties between Great Britain and the American colonies only grew stronger. Anglo-American colonists considered themselves part of the British Empire in all ways: politically, militarily, religiously (as Protestants), intellectually, and racially. The portrait of the Royall family () exemplifies the colonial American gentry of the eighteenth century. Successful and well-to-do, they display fashions, hairstyles, and furnishings that all speak to their identity as proud and loyal British subjects. |
# Rule Britannia! The English Empire, 1660–1763
## Charles II and the Restoration Colonies
When Charles II ascended the throne in 1660, English subjects on both sides of the Atlantic celebrated the restoration of the English monarchy after a decade of living without a king as a result of the English Civil Wars. Charles II lost little time in strengthening England’s global power. From the 1660s to the 1680s, Charles II added more possessions to England’s North American holdings by establishing the Restoration colonies of New York and New Jersey (taking these areas from the Dutch) as well as Pennsylvania and the Carolinas. In order to reap the greatest economic benefit from England’s overseas possessions, Charles II enacted the mercantilist Navigation Acts, although many colonial merchants ignored them because enforcement remained lax.
### CHARLES II
The chronicle of Charles II begins with his father, Charles I. Charles I ascended the English throne in 1625 and soon married a French Catholic princess, Henrietta Maria, who was not well liked by English Protestants because she openly practiced Catholicism during her husband’s reign. The most outspoken Protestants, the Puritans, had a strong voice in Parliament in the 1620s, and they strongly opposed the king’s marriage and his ties to Catholicism. When Parliament tried to contest his edicts, including the king’s efforts to impose taxes without Parliament’s consent, Charles I suspended Parliament in 1629 and ruled without one for the next eleven years.
The ensuing struggle between the king and Parliament led to the outbreak of war. The English Civil War lasted from 1642 to 1649 and pitted the king and his Royalist supporters against Oliver Cromwell and his Parliamentary forces. After years of fighting, the Parliamentary forces gained the upper hand, and in 1649, they charged Charles I with treason and beheaded him. The monarchy was dissolved, and England became a republic: a state without a king. Oliver Cromwell headed the new English Commonwealth, and the period known as the English interregnum, or the time between kings, began.
Though Cromwell enjoyed widespread popularity at first, over time he appeared to many in England to be taking on the powers of a military dictator. Dissatisfaction with Cromwell grew. When he died in 1658 and control passed to his son Richard, who lacked the political skills of his father, a majority of the English people feared an alternate hereditary monarchy in the making. They had had enough and asked Charles II to be king. In 1660, they welcomed the son of the executed king Charles I back to the throne to resume the English monarchy and bring the interregnum to an end (). The return of Charles II is known as the Restoration.
Charles II was committed to expanding England’s overseas possessions. His policies in the 1660s through the 1680s established and supported the Restoration colonies: the Carolinas, New Jersey, New York, and Pennsylvania. All the Restoration colonies started as proprietary colonies, that is, the king gave each colony to a trusted individual, family, or group.
### THE CAROLINAS
Charles II hoped to establish English control of the area between Virginia and Spanish Florida. To that end, he issued a royal charter in 1663 to eight trusted and loyal supporters, each of whom was to be a feudal-style proprietor of a region of the province of Carolina.
These proprietors did not relocate to the colonies, however. Instead, English plantation owners from the tiny Caribbean island of Barbados, already a well-established English sugar colony fueled by slave labor, migrated to the southern part of Carolina to settle there. In 1670, they established Charles Town (later Charleston), named in honor of Charles II, at the junction of the Ashley and Cooper Rivers (). As the settlement around Charles Town grew, it began to produce livestock for export to the West Indies. In the northern part of Carolina, settlers used sap from pine trees to create tar and pitch used to waterproof wooden ships. Political disagreements between settlers in the northern and southern parts of Carolina escalated in the 1710s through the 1720s and led to the creation, in 1729, of two colonies, North and South Carolina. The southern part of Carolina had been producing rice and indigo (a plant that yields a dark blue dye used by English royalty) since the 1700s, and South Carolina continued to depend on these main crops. North Carolina continued to produce items for ships, especially turpentine and tar, and its population increased as Virginians moved there to expand their tobacco holdings. Tobacco was the primary export of both Virginia and North Carolina, which also traded in deerskins and captured people from Africa.
Slavery developed quickly in the Carolinas, largely because so many of the early migrants came from Barbados, where slavery was well established. By the end of the 1600s, a very wealthy class of rice planters, who relied on enslaved people for labor, had attained dominance in the southern part of the Carolinas, especially around Charles Town. By 1715, South Carolina had a Black majority because of the number of enslaved people in the colony. The legal basis for slavery was established in the early 1700s as the Carolinas began to pass slave laws based on the Barbados slave codes of the late 1600s. These laws reduced Africans to the status of property to be bought and sold as other commodities.
As in other areas of English settlement, native peoples in the Carolinas suffered tremendously from the introduction of European diseases. Despite the effects of disease, Native Americans in the area endured and, following the pattern elsewhere in the colonies, grew dependent on European goods. Local Yamasee and Creek tribes built up a trade deficit with the English, trading deerskins and captive people for European guns. English settlers exacerbated tensions with local Native American tribes, especially the Yamasee, by expanding their rice and tobacco fields into Native American lands. Worse still, English traders took Native women captive as payment for debts.
The outrages committed by traders, combined with the seemingly unstoppable expansion of English settlement onto native land, led to the outbreak of the Yamasee War (1715–1718), an effort by a coalition of local tribes to drive away the European invaders. This native effort to force the newcomers back across the Atlantic nearly succeeded in annihilating the Carolina colonies. Only when the Cherokee allied themselves with the English did the coalition’s goal of eliminating the English from the region falter. The Yamasee War demonstrates the key role native peoples played in shaping the outcome of colonial struggles and, perhaps most important, the disunity that existed between different native groups.
### NEW YORK AND NEW JERSEY
Charles II also set his sights on the Dutch colony of New Netherland. The English takeover of New Netherland originated in the imperial rivalry between the Dutch and the English. During the Anglo-Dutch wars of the 1650s and 1660s, the two powers attempted to gain commercial advantages in the Atlantic World. During the Second Anglo-Dutch War (1664–1667), English forces gained control of the Dutch fur trading colony of New Netherland, and in 1664, Charles II gave this colony (including present-day New Jersey) to his brother James, Duke of York (later James II). The colony and city were renamed New York in his honor. The Dutch in New York chafed under English rule. In 1673, during the Third Anglo-Dutch War (1672–1674), the Dutch recaptured the colony. However, at the end of the conflict, the English had regained control ().
The Duke of York had no desire to govern locally or listen to the wishes of local colonists. It wasn’t until 1683, therefore, almost 20 years after the English took control of the colony, that colonists were able to convene a local representative legislature. The assembly’s 1683 Charter of Liberties and Privileges set out the traditional rights of Englishmen, like the right to trial by jury and the right to representative government.
The English continued the Dutch patroonship system, granting large estates to a favored few families. The largest of these estates, at 160,000 acres, was given to Robert Livingston in 1686. The Livingstons and the other manorial families who controlled the Hudson River Valley formed a formidable political and economic force. Eighteenth-century New York City, meanwhile, contained a variety of people and religions—as well as Dutch and English people, it held French Protestants (Huguenots), Jews, Puritans, Quakers, Anglicans, and a large population of enslaved people. As they did in other zones of colonization, native peoples played a key role in shaping the history of colonial New York. After decades of war in the 1600s, the powerful Five Nations of the Iroquois, composed of the Mohawk, Oneida, Onondaga, Cayuga, and Seneca, successfully pursued a policy of neutrality with both the English and, to the north, the French in Canada during the first half of the 1700s. This native policy meant that the Iroquois continued to live in their own villages under their own government while enjoying the benefits of trade with both the French and the English.
### PENNSYLVANIA
The Restoration colonies also included Pennsylvania, which became the geographic center of British colonial America. Pennsylvania (which means “Penn’s Woods” in Latin) was created in 1681, when Charles II bestowed the largest proprietary colony in the Americas on William Penn () to settle the large debt he owed the Penn family. William Penn’s father, Admiral William Penn, had served the English crown by helping take Jamaica from the Spanish in 1655. The king personally owed the Admiral money as well.
Like early settlers of the New England colonies, Pennsylvania’s first colonists migrated mostly for religious reasons. William Penn himself was a Quaker, a member of a new Protestant denomination called the Society of Friends. George Fox had founded the Society of Friends in England in the late 1640s, having grown dissatisfied with Puritanism and the idea of predestination. Rather, Fox and his followers stressed that everyone had an “inner light” inside him or her, a spark of divinity. They gained the name Quakers because they were said to quake when the inner light moved them. Quakers rejected the idea of worldly rank, believing instead in a new and radical form of social equality. Their speech reflected this belief in that they addressed all others as equals, using “thee” and “thou” rather than terms like “your lordship” or “my lady” that were customary for privileged individuals of the hereditary elite.
The English crown persecuted Quakers in England, and some colonial governments were equally harsh; Massachusetts even executed several early Quakers who had gone to proselytize there. To avoid such persecution, Quakers and their families created a community on the sugar island of Barbados or moved to the religiously tolerant colony of Rhode Island. Soon after its founding, however, Pennsylvania became the destination of choice. Quakers flocked to Pennsylvania as well as New Jersey, where they could preach and practice their religion in peace. Unlike much of New England, whose official religion was Puritanism, Pennsylvania did not establish an official church. Indeed, with the notable exception of Rhode Island, the Pennsylvania colony allowed a degree of religious tolerance found nowhere else in English America. To help encourage immigration to his colony, Penn promised fifty acres of land to people who agreed to come to Pennsylvania and completed their term of service. Not surprisingly, those seeking a better life came in large numbers, so much so that Pennsylvania relied on indentured servants more than any other colony.
One of the primary tenets of Quakerism is pacifism, which led William Penn to establish friendly
relationships with local native peoples. He formed a covenant of friendship with the Lenni Lenape (Delaware) tribe, buying their land for a fair price instead of taking it by force. In 1701, he also signed a treaty with the Susquehannocks to avoid war. Unlike other colonies, Pennsylvania did not experience war on the frontier with native peoples during its early history.
As an important port city, Philadelphia grew rapidly. Quaker merchants there established contacts throughout the Atlantic world and participated in the thriving African slave trade. Some Quakers, who were deeply troubled by the contradiction between their belief in the “inner light” and the practice of slavery, rejected the practice and engaged in efforts to abolish it altogether. Philadelphia also acted as a magnet for immigrants, who came not only from England, but from all over Europe by the hundreds of thousands. The city, and indeed all of Pennsylvania, appeared to be the best country for poor men and women, many of whom arrived as servants and dreamed of owning land. A very few, like the fortunate Benjamin Franklin, a runaway from Puritan Boston, did extraordinarily well. Other immigrant groups in the colony, most notably Germans and Scots-Irish (families from Scotland and England who had first lived in Ireland before moving to British America), greatly improved their lot in Pennsylvania. Of course, Africans brought into the colony to labor for White enslavers fared far worse.
### THE NAVIGATION ACTS
Creating wealth for the Empire remained a primary goal, and in the second half of the seventeenth century, especially during the Restoration, England attempted to gain better control of trade with the American colonies. The mercantilist policies by which it tried to achieve this control are known as the Navigation Acts.
The 1651 Navigation Ordinance, a product of Cromwell’s England, required that only English ships carry goods between England and the colonies, and that the captain and three-fourths of the crew had to be English. The ordinance further listed “enumerated articles” that could be transported only to England or to English colonies, including the most lucrative commodities like sugar and tobacco as well as indigo, rice, molasses, and naval stores such as turpentine. All were valuable goods not produced in England and in demand by the British navy. After ascending the throne, Charles II approved the 1660 Navigation Act, which restated the 1651 act to ensure a monopoly on imports from the colonies.
Other Navigation Acts included the 1663 Staple Act and the 1673 Plantation Duties Act. The Staple Act barred colonists from importing goods that had not been made in England, creating a profitable monopoly for English exporters and manufacturers. The Plantation Duties Act taxed enumerated articles exported from one colony to another, a measure aimed principally at New Englanders, who transported great quantities of molasses from the West Indies, including smuggled molasses from French-held islands, to make into rum.
In 1675, Charles II organized the Lords of Trade and Plantation, commonly known as the Lords of Trade, an administrative body intended to create stronger ties between the colonial governments and the crown. However, the 1696 Navigation Act created the Board of Trade, replacing the Lords of Trade. This act, meant to strengthen enforcement of customs laws, also established vice-admiralty courts where the crown could prosecute customs violators without a jury. Under this act, customs officials were empowered with warrants known as “writs of assistance” to board and search vessels suspected of containing smuggled goods.
Despite the Navigation Acts, however, Great Britain exercised lax control over the English colonies during most of the eighteenth century because of the policies of Prime Minister Robert Walpole. During his long term (1721–1742), Walpole governed according to his belief that commerce flourished best when it was not encumbered with restrictions. Historians have described this lack of strict enforcement of the Navigation Acts as salutary neglect. In addition, nothing prevented colonists from building their own fleet of ships to engage in trade. New England especially benefited from both salutary neglect and a vibrant maritime culture made possible by the scores of trading vessels built in the northern colonies. The case of the 1733 Molasses Act illustrates the weaknesses of British mercantilist policy. The 1733 act placed a sixpence-per-gallon duty on raw sugar, rum, and molasses from Britain’s competitors, the French and the Dutch, in order to give an advantage to British West Indian producers. Because the British did not enforce the 1733 law, however, New England mariners routinely smuggled these items from the French and Dutch West Indies more cheaply than they could buy them on English islands.
### Section Summary
After the English Civil War and interregnum, England began to fashion a stronger and larger empire in North America. In addition to wresting control of New York and New Jersey from the Dutch, Charles II established the Carolinas and Pennsylvania as proprietary colonies. Each of these colonies added immensely to the Empire, supplying goods not produced in England, such as rice and indigo. The Restoration colonies also contributed to the rise in population in English America as many thousands of Europeans made their way to the colonies. Their numbers were further augmented by the forced migration of enslaved Africans. Starting in 1651, England pursued mercantilist policies through a series of Navigation Acts designed to make the most of England’s overseas possessions. Nonetheless, without proper enforcement of Parliament’s acts and with nothing to prevent colonial traders from commanding their own fleets of ships, the Navigation Acts did not control trade as intended.
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# Rule Britannia! The English Empire, 1660–1763
## The Glorious Revolution and the English Empire
During the brief rule of King James II, many in England feared the imposition of a Catholic absolute monarchy by the man who modeled his rule on that of his French Catholic cousin, Louis XIV. Opposition to James II, spearheaded by the English Whig party, overthrew the king in the Glorious Revolution of 1688–1689. This paved the way for the Protestant reign of William of Orange and his wife Mary (James’s Protestant daughter).
### JAMES II AND THE GLORIOUS REVOLUTION
King James II (), the second son of Charles I, ascended the English throne in 1685 on the death of his brother, Charles II. James then worked to model his rule on the reign of the French Catholic King Louis XIV, his cousin. This meant centralizing English political strength around the throne, giving the monarchy absolute power. Also like Louis XIV, James II practiced a strict and intolerant form of Roman Catholicism after he converted from Protestantism in the late 1660s. He had a Catholic wife, and when they had a son, the potential for a Catholic heir to the English throne became a threat to English Protestants. James also worked to modernize the English army and navy. The fact that the king kept a standing army in times of peace greatly alarmed the English, who believed that such a force would be used to crush their liberty. As James’s strength grew, his opponents feared their king would turn England into a Catholic monarchy with absolute power over her people.
In 1686, James II applied his concept of a centralized state to the colonies by creating an enormous colony called the Dominion of New England. The Dominion included all the New England colonies (Massachusetts, New Hampshire, Plymouth, Connecticut, New Haven, and Rhode Island) and in 1688 was enlarged by the addition of New York and New Jersey. James placed in charge Sir Edmund Andros, a former colonial governor of New York. Loyal to James II and his family, Andros had little sympathy for New Englanders. His regime caused great uneasiness among New England Puritans when it called into question the many land titles that did not acknowledge the king and imposed fees for their reconfirmation. Andros also committed himself to enforcing the Navigation Acts, a move that threatened to disrupt the region’s trade, which was based largely on smuggling.
In England, opponents of James II’s efforts to create a centralized Catholic state were known as Whigs. The Whigs worked to depose James, and in late 1688 they succeeded, an event they celebrated as the Glorious Revolution while James fled to the court of Louis XIV in France. William III (William of Orange) and his wife Mary II ascended the throne in 1689.
The Glorious Revolution spilled over into the colonies. In 1689, Bostonians overthrew the government of the Dominion of New England and jailed Sir Edmund Andros as well as other leaders of the regime (). The removal of Andros from power illustrates New England’s animosity toward the English overlord who had, during his tenure, established Church of England worship in Puritan Boston and vigorously enforced the Navigation Acts, to the chagrin of those in port towns. In New York, the same year that Andros fell from power, Jacob Leisler led a group of Protestant New Yorkers against the dominion government. Acting on his own authority, Leisler assumed the role of King William’s governor and organized intercolonial military action independent of British authority. Leisler’s actions usurped the crown’s prerogative and, as a result, he was tried for treason and executed. In 1691, England restored control over the Province of New York.
The Glorious Revolution provided a shared experience for those who lived through the tumult of 1688 and 1689. Subsequent generations kept the memory of the Glorious Revolution alive as a heroic defense of English liberty against a would-be tyrant.
### ENGLISH LIBERTY
The Glorious Revolution led to the establishment of an English nation that limited the power of the king and provided protections for English subjects. In October 1689, the same year that William and Mary took the throne, the 1689 Bill of Rights established a constitutional monarchy. It stipulated Parliament’s independence from the monarchy and protected certain of Parliament’s rights, such as the right to freedom of speech, the right to regular elections, and the right to petition the king. The 1689 Bill of Rights also guaranteed certain rights to all English subjects, including trial by jury and habeas corpus (the requirement that authorities bring an imprisoned person before a court to demonstrate the cause of the imprisonment).
John Locke (1632–1704), a doctor and educator who had lived in exile in Holland during the reign of James II and returned to England after the Glorious Revolution, published his Two Treatises of Government in 1690. In it, he argued that government was a form of contract between the leaders and the people, and that representative government existed to protect “life, liberty and property.” Locke rejected the divine right of kings and instead advocated for the central role of Parliament with a limited monarchy. Locke’s political philosophy had an enormous impact on future generations of colonists and established the paramount importance of representation in government.
The Glorious Revolution also led to the English Toleration Act of 1689, a law passed by Parliament that allowed for greater religious diversity in the Empire. This act granted religious tolerance to nonconformist Trinitarian Protestants (those who believed in the Holy Trinity of God the Father, Son, and Holy Ghost), such as Baptists (those who advocated adult baptism) and Congregationalists (those who followed the Puritans’ lead in creating independent churches). While the Church of England remained the official state religious establishment, the Toleration Act gave much greater religious freedom to nonconformists. However, this tolerance did not extend to Catholics, who were routinely excluded from political power. The 1689 Toleration Act extended to the British colonies, where several colonies—Pennsylvania, Rhode Island, Delaware, and New Jersey—refused to allow the creation of an established colonial church, a major step toward greater religious diversity.
### Section Summary
The threat of a Catholic absolute monarchy prompted not only the overthrow of James II but also the adoption of laws and policies that changed English government. The Glorious Revolution restored a Protestant monarchy and at the same time limited its power by means of the 1689 Bill of Rights. Those who lived through the events preserved the memory of the Glorious Revolution and the defense of liberty that it represented. Meanwhile, thinkers such as John Locke provided new models and inspirations for the evolving concept of government.
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