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# Chapter 2.

## The Time Value of Money

The "Chapter" worksheet performs the calculations required for Chapter 2, and was used to create many of the chapter exhibits (Tables and Figures). We pasted in a few dialog boxes for specific Excel functions and features and show then off to the right of where they apply, but in general we encourage students who want to know more about Excel to use the Excel Tutorial and refer to it as necessary. We also like to let students know that Excel models can be used to create tables and graphs that can then be copied into Word documents, which is the way we prepared the text manuscript for submission to the publisher. That procedure is used often in business to prepare reports. Although answers to the Self-Test questions within the chapter are generally quite easy and were found with a calculator, we also solved them with Excel as a check and also to provide some information on the solutions in case students have questions. The tabs at the lower part of this screen take you to these solutions. Even if students are not familiar with Excel, they should still be able to see the solution setup and then work out the answer with a calculator. Although we did not create the model specifically for use in lectures, it could be used as such in a classroom where a projector is attached to a computer. The instructor could scroll through the model and lecture on points as they come up. This would be more useful if the students have some familiarity with Excel, but that is not really necessary because everything the model does can also be done with a financial calculator.

## FUTURE VALUES (Section 2.2)

A dollar in hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest it, earn interest, and end up with more than a dollar in the future. The process of going to future values (FVs) from present values (PVs) is called compounding. To illustrate, refer to our 3-year time line and assume that you plan to deposit \$100 in a bank that pays a guaranteed 5% interest each year. How much would you have at the end of Year 3?

## Figure 2-1. Summary of Future Value Calculations

Investment = CF0 = PV = Interest rate = I = No. of periods = N = -\$100.00 5.00% 3 Periods:

Cash Flow Time Line: Step-by-Step Approach: Formula Approach: FVN = PV(1+I)N 3 N

## 0 | -\$100 \$100 FV3 = 5 I/YR

1 |

2 |

3 | FV = ? \$115.76 \$115.76

## \$105.00 \$100(1.05)3 -\$100.00 PV

\$110.25 = \$0 PMT

Calculator Approach:

FV \$115.76

Excel Approach:

## =FV(I,N,0,PV) =FV(0.05,3,0,-100) = =FV(C15,C16,0,C14) =

\$115.76 \$100.00

In the Excel formula, the terms are entered in this sequence: interest, periods, 0 to indicate no intermediate cash flows, and then the PV. The data can be entered as fixed numbers or as cell references.

## The Compounding Process: A Graphic View

Figure 2-2 (shown below) shows how a \$1 investment grows over time at different interest rates. The curves were created by solving for FV at different values for N and I. This allows you to simultaneously see the effects of varying time and the interest rate. The data table used to create this figure is shown to the right. For instruction on data tables, refer to the Excel Tutorial.
6.00 5.00 4.00

Future Value of \$1

Periods

## PRESENT VALUES (Section 2.3)

In many ways the present value is just the opposite of the future value. Instead of compounding a value forward, you discount it back. If you know the PV, you can compound to find the FV, while if you know the FV, you can discount to find the PV.

To illustrate, refer to the time line and assume that \$115.76 is due in 3 years. If a bank pays a guaranteed 5% interest rate each year, how much would you need to deposit now to have \$115.76 in 3 years?

## Figure 2-3. Summary of Present Value Calculations

Future payment = CFN = FV = Interest rate = I = No. of periods = N = \$115.76 5.00% 3 Periods:

Cash Flow Time Line: Step-by-Step Approach: Formula Approach: PV = FVN / (1 + I)N 3 N

0 | PV = ? \$100.00

1 |

2 |

## 3 | \$115.76 \$115.76 \$100.00 \$115.76 FV

\$105.00

\$110.25 = \$0 PMT

PV = \$115.76/(1.05)3 5 I/YR

Calculator Approach:

PV -\$100.00

Excel Approach:

PV = PV = PV =

## =PV(I,N,0,FV) =PV(0.05,3,0,115.76) = =PV(C65,C66,0,C64) =

-\$100.00 -\$100.00

In the Excel function, 0 indicates that there are no intermediate cash flows.

## The Discounting Process: A Graphic View

Figure 2-4 shows how a \$1 payment in the future has a lower and lower present value as the interest rate and time until receipt increase. The data table to the right provides the data used to draw the figure.

## 1.00 0.80 0.60 0.40 0.20 0.00 0 10 20 30 40 50

Present Value of \$1

Periods

## FINDING THE INTEREST RATE (Section 2.4)

Previously, we solved equations to find FV and PV. However, we could just as easily solve for I or N. For example, suppose we know that a given bond has a cost of \$100 and that it will return \$150 after 10 years. Thus, we know PV, FV, and N, and we want to find the rate of return we would earn if we bought the bond. Present value (PV) Future value (FV) No. of years (N) Interest rate (I) Interest rate (I) -\$100.00 \$150.00 10 = RATE(N,0,PV,FV) 4.14%

## FINDING THE NUMBER OF YEARS (Section 2.5)

Sometimes we need to know how long it will take to accumulate a given sum of money, given our beginning funds and the rate we will earn on those funds. For example, suppose we believe that we could retire comfortably if we had \$1 million, and we want to find how long it will take us to reach that goal, assuming that we now have \$500,000 invested at 4.5%.

Present value (PV) Future value (FV) Interest rate (I) No. of years (N) No. of years (N)

## FUTURE VALUE OF AN ORDINARY ANNUITY (Section 2.7)

An ordinary annuity has regular, periodic payments that occur at the end of each period. Methods for solving the future value of an ordinary annuity are shown below.

## Figure 2-5. Summary: Future Value of an Ordinary Annuity

Payment amount = PMT = Interest rate = I = Number of periods = N = Periods: Cash Flow Time Line: Step-By-Step Approach. Multiply each payment by (1+I)N-t and sum these FVs to find FVAN: Formula Approach: FVAN = \$100.00 5.00% 3 0 | 1 | -\$100 2 | -\$100 3 | -\$100 -\$100.00 -\$105.00 -\$110.25 -\$315.25

1I N 1 P M T I I
3 N

\$315.25

Calculator Approach:

5 I/YR

\$0 PV

-\$100.00 PMT

FV \$315.25

\$315.25 \$315.25

## FUTURE VALUE OF AN ANNUITY DUE (Section 2.8)

An annuity due also has regular, periodic payments, but unlike an ordinary annuity, the payments occur at the beginning of each period.

## Summary: Future Value of an Annuity Due (Not in Text)

Payment amount = PMT = Interest rate = I = Number of periods = N = Periods: Cash Flow Time Line: Step-By-Step Approach. Multiply each payment by (1+I)N-t and sum these FVs to find FVAN: Formula Approach: FVAN(due) = \$100.00 5.00% 3 0 | -\$100 1 | -\$100 2 | -\$100 3 | -\$105.00 -\$110.25 -\$115.76 -\$331.01

PM T

1I N 1 1I I I
5 I

\$331.01

3 N

0 PV

-100 PMT

FV 331.01

## =FV(I,N,PMT,PV,Type) =FV(0.05,3,-100,0,1) = =FV(C163,C164,-C162,0,1) =

331.01 331.01

In the Excel formula, the 1 at the end of the formula indicates that cash flows occur at the beginning of each period. A 0 or nothing would indicate end of period payments.

## PRESENT VALUE OF AN ORDINARY ANNUITY (Section 2.9)

The present value of an ordinary annuity is the sum of the PVs of the individual cash flows. Methods for solving the present value of an ordinary annuity are shown below.

## Figure 2-6. Summary: Present Value of an Ordinary Annuity

Payment amount = PMT = Interest rate = I = Number of periods = N = Periods: Cash Flow Time Line: Step-By-Step Approach. Divide each payment by (1+I)t and sum these PVs to find PVAN: Formula Approach: PVAN = \$100.00 5.00% 3 0 | \$95.24 \$90.70 \$86.38 \$272.32 1 | -\$100 2 | -\$100 3 | -\$100

PM T
3 N

1 1 I I 1I N
5 I

\$272.32

Calculator Approach:

PV 272.32

-100 PMT

0 FV

272.32 272.32

## PRESENT VALUE OF AN ANNUITY DUE (not in text)

The difference between the present value of an ordinary annuity and an annuity due is that payments are received earlier in an annuity due.

## Summary: Present Value of an Annuity Due (Not in text)

Payment amount = PMT = Interest rate = I = Number of periods = N = Periods: Cash Flow Time Line: Step-By-Step Approach. Divide each payment by (1+I)t and sum these PVs to find PVAN: Formula Approach: PVAN = \$100.00 5.00% 3 0 | -\$100 -\$100.00 -\$95.24 -\$90.70 -\$285.94 1 | -\$100 2 | -\$100 3 |

PM T

1 1 1I = I I 1I N
5 I PV 285.94

\$285.94

3 N

-100 PMT

0 FV

285.94 285.94

## FINDING ANNUITY PAYMENTS, PERIODS, AND INTEREST RATES (Section 2.10)

Fundamentally, this section is no different than previous TVM exercises. When solving for PMT, N, or I, you must be given values for the other variables, and then you solve the problem. FINDING PMT Suppose we need to accumulate \$10,000 and have it available 5 years from now. Suppose further that we can earn a return of 6% on our savings, which are currently zero. No. of years (N) Interest rate (I) Present value (PV) Future value (FV) END MODE Payment (PMT) -\$1,773.96 =PMT(I,N,PV,FV) 5 6% \$0 \$10,000 BEGIN MODE -\$1,673.55 =PMT(I,N,PV,FV,Type)

Payment (PMT)

FINDING N Suppose you decide to make end-of-year deposits, but you can only save \$1,200 per year. Again assuming that you would earn 6%, how long would it take you to reach your \$10,000 goal? Interest rate (I) Present value (PV) Payment (PMT) Future value (FV) No. of years (N) 6% \$0 -\$1,200 \$10,000 6.96 =NPER(I,PMT,PV,FV)

FINDING I Now suppose you can only save \$1,200 annually, but you still want to have the \$10,000 in 5 years. What rate of return would enable you to achieve your goal? No. of years (N) Present value (PV) Payment (PMT) Future value (FV) Interest rate (I) 5 \$0 -\$1,200 \$10,000 25.78% =RATE(N,PMT,PV,FV)

## PERPETUITIES (Section 2.11)

Perpetuities are securities that promise to make payments forever. The present value of a perpetuity can be found with a simple formula: Value = I / r . Note that we cannot calculate the future value of a perpetuity because, since payments go on forever, this value would be infinitely large and thus meaningless.

Consider a British consol that pays a \$25 annual payment. If interest rates are currently 5.2%, what is the value of the consol? Payment (PMT) Interest rate (I) Interest rate (I) \$25 5.2% \$480.77

If an annuity makes constant payments, then adding more payments to the security adds less value for each additional payment. This helps explain why perpetuities' values are finite, while payments are infinite. To see this better, consider the figure below (not in the text). The data used to construct the graph is shown to the right in columns I through L. One hundred payments are analyzed and their present values, the total value of an annuity of N number of years, and the contribution of the Nth payment are all shown in the table.

\$100

Value of 25-Year Annuity: Value of 50-Year Annuity: Value of 100-Year Annuity: Value of Perpetuity:
PV of Additional Payments in an Annuity \$50 Added: Years 1-25: Amt.

## 26-50: 51-100: From 101 to infinity:

\$0 3 9 15 21 27 33 39 45 51 57 63 69 76 82 88 94 100 0 6 12 18 24 30 36 42 48 54 60 66 73 79 85 91 97

Years (N)

## UNEVEN CASH FLOWS (Section 2.12)

An annuity has constant payments. Although many financial decisions do involve annuities, many others involve uneven, or nonconstant, cash flows. With a spreadsheet, the present value of a series of uneven cash flows (called the net present value) can be calculated easily . First, consider a security that pays \$100 for 5 years and a lump sum of \$1,000 at the end of 5 years. We can find the PV using either the PV function or the NPV function.

Summary of Uneven Cash Flow Present Value Calculations (Annuity plus Lump Sum)
Interest rate Periods: Annuity CFs:
Lump sum CFs:

I 0 | \$0

12% 1 | \$100 \$100 2 | \$100 \$100 3 | \$100 \$100 4 | \$100 \$100 5 | \$100 \$1,000 \$1,100

Total CFs:

## = PV of cash flow stream = value of the asset

PV = =PV(0.12,5,-100,-1000) Fixed inputs: PV = =PV(B335,G337,-C339,-G340) Cell references NPV = =NPV(0.12,100,100,100,100,1100) Fixed inputs: =NPV(B335,C341:G341) Cell references: NPV = Now consider an irregular cash flow stream (where CFs can take on any value). Excel Function Approach:

## Figure 2-7. PV of an Uneven Cash Flow Stream

Interest rate Periods: = I = 12% 1 | \$100 2 | \$300 3 | \$300 4 | \$300 5 | \$500

## Fixed inputs: Cell references:

NPV = NPV =

=NPV(0.12,100,300,300,300,500) =NPV(B358,C362:G362)

1,016.35 1,016.35

Our Excel formula ignores the initial cash flow (in Year 0). When entering a cash flow range, Excel assumes that the first value entered occurs at the end of the first year. If there is an initial cash flow, as we will see later, that cash flow must be separately added to the NPV formula result. Notice too that you can enter cash flows one-by-one, or if the cash flows appear in consecutive cells, you can enter the cell range.

## FUTURE VALUE OF AN UNEVEN CASH FLOW STREAM (Section 2.13)

We find the future value of uneven cash flow streams by compounding rather than discounting. The stepby-step approach works the same, but unfortunately, Excel does not have a net future value (NFV) function. One way around this is to solve for the NPV and find the FV of this amount to the end of the cash flow stream.

## Figure 2-8. FV of an Uneven Cash Flow Stream

Interest rate Periods: CF Time Line: = I 0 | \$0 = 12% 1 | \$100 2 | \$300 3 | \$300 4 | \$300 5 | \$500 \$500.00 336.00 376.32 421.48

## 157.35 0.00 \$1,791.15

Excel Function Approach: First find NPV: Then compound NPV for 5 years:

NPV = NFV =

=NPV(B383,C387:G387) =FV(B383,G385,0,-G397)

1,016.35 1,791.15

The NFV result using the Excel formulas is a negative number. This is because we used Excel's FV function and entered the NPV as a positive value as the PV.

## SOLVING FOR I WITH UNEVEN CASH FLOWS (Section 2.14)

Assume that an investment with the following positive cash flows has a cost of \$927.90. Find the rate of return on this investment.

## Finding the Interest Rate, Annuity Plus Lump Sum

Annuity pmts
Future lump sum

\$100 \$1,000 0 | -\$927.90 1 | \$100 Cell references: Cell references: 2 | \$100 RATE = IRR = 3 | \$100 4 | \$100 5 | \$1,100 12.00% 12.00%

## Excel Function Approach: Excel Function Approach:

=RATE(G411,B408,B413,B409)

=IRR(B413:G413)

## Figure 2-9. IRR of an Uneven Cash Flow Stream

Periods: CF Time Line: 0 | -\$1,000 1 | \$100 Cell references: 2 | \$300 IRR = 3 | \$300 =IRR(B423:G423) 4 | \$300 5 | \$500 12.55%

## SEMIANNUAL AND OTHER COMPOUNDING PERIODS (Section 2.15)

If \$100 is invested in an account at an annual nominal interest rate of 10% for 5 years, what are the future values and effective interest rates for annual, semiannual, quarterly, monthly and daily compounding?

## Nominal annual rate = Amount invested = Number of years =

10% \$100 5

Table 2-1. The Impact of Frequent Compounding Number of periods per year (M) 1 2 4 12

## Nominal Annual Rate 10% 10% 10% 10%

Periodic Effective Interest Rate Annual Rate Future Value 10.00% 10.000% \$161.05 5.00% 10.250% \$162.89 2.50% 10.381% \$163.86 0.83% 10.471% \$164.53

Daily

10%

365

0.03%

10.516%

\$164.86

## AMORTIZED LOANS (Section 2.17)

If a loan is to be repaid in equal amounts on a monthly, quarterly, or annual basis it is said to be an amortized loan. Table 2-4 (replicated below) illustrates the amortization process. A homeowner borrows \$100,000 on a mortgage loan, and the loan is to be repaid in 5 equal payments at the end of each of the next 5 years. The lender charges 6% on the balance at the beginning of each year.

First, we solve for the required payment, then we construct an amortization table. N I PV FV PMT 5 6% \$100,000 \$0 -\$23,739.64

Table 2-2. Loan Amortization Schedule, \$100,000 at 6% for 5 Years Amount borrowed: \$100,000 Years: 5 Rate: 6% PMT: \$23,739.64

Year 1 2 3 4 5
a

## Interesta (3) \$6,000.00 \$4,935.62 \$3,807.38 \$2,611.44 \$1,343.75

Repayment of Principalb (2) - (3) = (4) \$17,739.64 \$18,804.02 \$19,932.26 \$21,128.20 \$22,395.89

Ending Balance (1) - (4) = (5) \$82,260.36 \$63,456.34 \$43,524.08 \$22,395.89 \$0.00

Interest in each period is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefore, interest in Year 1 is \$100,000(0.06) = \$6,000; in Year 2 it is \$82,260.36(0.06) = \$4,935.62; and so on. Repayment of principal is equal to the payment of \$23,739.64 minus the interest charge for the year.
b

## Growing Annuities (Section 2.18)

SECTION 2.2
SOLUTIONS TO SELF-TEST 2a What would the future value of \$100 be after 5 years at 10% compound interest? N I PV PMT 5 10% \$100 \$0

FV

\$161.05

3a Suppose you currently have \$2,000 and plan to purchase a 3-year certificate of deposit (CD) that pays 4 percent interest compounded annually. How much will you have when the CD matures? N I PV PMT 3 4% \$2,000 \$0

FV

\$2,249.73

3b How would your answer change if the interest rate were 5%, or 6%, or 20%? Interest rate 5% 6% 20% \$2,249.73 \$2,315.25 \$2,382.03 \$3,456.00

4 A companys sales in 2006 were \$100 million. If sales grow at 8 percent, what will they be 10 years later, in 2016? N I PV (\$M) PMT 10 8% \$100 \$0

FV (\$M)

\$215.89

5a How much would \$1, growing at 5% per year, be worth after 100 years? N I PV PMT 100 5% \$1 \$0

FV

\$131.50

5b What would FV be if the growth rate were 10%? N I PV PMT 100 10% \$1 \$0

FV

\$13,780.61

SECTION 2.3
SOLUTIONS TO SELF-TEST 3a Suppose a U.S. government bond promises to pay \$2,249.73 three years from now. If the going interest rate on 3-year government bonds is 4%, how much is the bond worth today? N I PMT FV 3 4% \$0 \$2,250

PV

\$2,000.00

3b How would your answer change if the bond matured in 5 rather than 3 years? N I PMT FV 5 4% \$0 \$2,250

PV

\$1,849.11

3c What if the interest rate on the 5-year bond were 6% rather than 4%? N I PMT FV 5 6% \$0 \$2,250

PV

\$1,681.13

4a How much would \$1,000,000 due in 100 years be worth today if the discount rate were 5%? N I PMT FV 100 5% \$0 \$1,000,000

PV

\$7,604.49

## 4b If the discount rate were 20%? N I PMT FV 100 20% \$0 \$1,000,000

PV

\$0.0121

SECTION 2.4
SOLUTIONS TO SELF-TEST 1a The U.S. Treasury offers to sell you a bond for \$585.43. No payments will be made until the bond matures 10 years from now, at which time it will be redeemed for \$1,000. What interest rate would you earn if you bought this bond for \$585.43? N PMT PV FV 10 \$0 \$585.43 \$1,000

5.50%

1b What rate would you earn if you could buy the bond for \$550? N PMT PV FV 1c For \$600? N PMT PV FV 10 \$0 \$600.00 \$1,000 10 \$0 \$550.00 \$1,000

6.16%

5.24%

2a Microsoft earned \$0.12 per share in 1994. Ten years later, in 2004, it earned \$1.04. What was the growth rate in Microsofts earnings per share (EPS) over the 10-year period? N PMT PV FV 10 \$0 \$0.12 \$1.04

24.10%

2b If EPS in 2004 had been \$0.65 rather than \$1.04, what would the growth rate have been? N PMT PV FV 10 \$0 \$0.12 \$1

18.41%

SECTION 2.5
SOLUTIONS TO SELF-TEST

1a How long would it take \$1,000 to double if it were invested in a bank that pays 6% per year? I PMT PV FV 6% \$0 \$1,000 \$2,000

11.90

1b How long would it take if the rate were 10%? I PMT PV FV 10% \$0 \$1,000 \$2,000

7.27

2a Microsofts 2004 earnings per share were \$1.04, and its growth rate during the prior 10 years was 24.1% per year. If that growth rate were maintained, how long would it take for Microsofts EPS to double? I PMT PV FV 24.1% \$0 \$1.04 \$2.08

3.21

SECTION 2.7
SOLUTIONS TO SELF-TEST

1a For an ordinary annuity with 5 annual payments of \$100 and a 10% interest rate, how many years will the 1st payment earn interest, and what will this payments value be at the end? N I PMT PV 5 10% -\$100 \$0

## Years of int Payments FV

4 \$146.41

1b Answer this same question for the 5th payment. N I PMT PV 5 10% -\$100 \$0

## Years of int Payments FV

0 \$100.00

2a Assume that you plan to buy a condo 5 years from now, and you estimate that you can save \$2,500 per year. You plan to deposit the money in a bank that pays 4% interest, and you will make the first deposit at the end of the year. How much will you have after 5 years? N I PMT PV 5 4% -\$2,500 \$0

FV

\$13,540.81

2b How would your answer change if the interest rate were increased to 6%, or lowered to 3%? N I PMT PV N I PMT PV 5 6% -\$2,500 \$0 5 3% -\$2,500 \$0

FV

\$14,092.73

FV

\$13,272.84

SECTION 2.8
SOLUTIONS TO SELF-TEST 3a Assume that you plan to buy a condo 5 years from now, and you need to save for a down payment. You plan to save \$2,500 per year, with the first payment made immediately, and you will deposit the funds in a bank account that pays 4%. How much will you have after 5 years? N I PV PMT 5 4% \$0 -\$2,500

FV

\$14,082.44

2b How much would you have if you made the deposits at the end of each year? N I PV PMT 5 4% \$0 -\$2,500

FV

\$13,540.81

SECTION 2.9
SOLUTIONS TO SELF-TEST 3a What is the PVA of an ordinary annuity with 10 payments of \$100 if the appropriate interest rate is 10%? N I PMT FV 10 10% -\$100 \$0

PV

\$614.46

3b What would PVA be if the interest rate were 4%? N I PMT FV 10 4% -\$100 \$0

PV

\$811.09

## 3c What if the interest rate were 0%? N I PMT FV 10 0% -\$100 \$0

PV

\$1,000.00

3d How would the PVA values differ if we were dealing with annuities due? Part a N I PMT FV PV 10 10% -\$100 \$0 \$675.90 N I PMT FV PV Part b 10 4% -\$100 \$0 \$843.53 N I PMT FV PV Part c 10 0% -\$100 \$0 \$1,000.00

4a Assume that you are offered an annuity that pays \$100 at the end of each year for 10 years. You could earn 8% on your money in other investments with equal risk. What is the most you should pay for the annuity? N I PMT FV 10 8% -\$100 \$0

PV

\$671.01

4b If the payments began immediately, how much would the annuity be worth? N I PMT FV 10 8% -\$100 \$0

PV

\$724.69

SECTION 2.10
SOLUTIONS TO SELF-TEST 1a Suppose you inherited \$100,000 and invested it at 7% per year. How much could you withdraw at the end of each of the next 10 years? N I PV FV 10 7% \$100,000 \$0

PMT

-\$14,237.75

1b How would your answer change if you made withdrawals at the beginning of each year? N I PV FV 10 7% \$100,000 \$0

PMT

-\$13,306.31

2a If you had \$100,000 that was invested at 7% and you wanted to withdraw \$10,000 at the end of each year, how long would your funds last? I PV PMT FV 7.0% \$100,000 -\$10,000 \$0

17.8

2b How long would they last if you earned 0%? I PV PMT FV 0.0% \$100,000 -\$10,000 \$0

10.0

2c How long would they last if you earned the 7% but limited your withdrawal to \$7,000 per year? I PV PMT FV 7.0% \$100,000 -\$7,000 \$0

* This result means that with \$7,000 withdrawals, you would never #VALUE! exhaust the funds.

3 Your rich uncle named you as the beneficiary of his life insurance policy. The insurance company gives you a choice of \$100,000 today or a 12-year annuity of \$12,000 at the end of each year. What rate of return is the insurance company offering? N PMT PV FV 12 \$12,000 \$100,000 \$0

6.11%

4a Assume that you just inherited an annuity that will pay you \$10,000 per year for 10 years, with the first payment being made today. A friend of your mother offers to give you \$60,000 for the annuity. If you sell it, what rate of return would your mothers friend earn on his investment? N PMT PV FV 10 -\$10,000 \$60,000 \$0

13.70%

4b If you think a fair return would be 6%, how much should you ask for the annuity? N I PMT FV 10 6% -\$10,000 \$0

PV

\$78,016.92

SECTION 2.11
SOLUTIONS TO SELF-TEST 1a Whats the present value of a perpetuity that pays \$1,000 per year, beginning one year from now, if the appropriate interest rate is 5%? PMT I \$1,000 5%

PV

\$20,000

1b What would the value be if the annuity began its payments immediately? PMT I \$1,000 5% **The perpetuity value formula values payments 1 through infinity. If a payment is received immediately, it must be added to the formula result.

PV

\$21,000

SECTION 2.12
SOLUTIONS TO SELF-TEST 2a Whats the present value of a 5-year ordinary annuity of \$100 plus an additional \$500 at the end of Year 5 if the interest rate is 6%? Interest rate Year Ann Pmt Lump Sum Total CFs NPV 6% 0 \$0 \$0 \$794.87 1 \$100 \$100 2 \$100 \$100 3 \$100 \$100 4 \$100 \$100 5 \$100 \$500 \$600

2b How would the PV change if the \$100 payments occurred in Years 1 through 10 and the \$500 came at the end of Year 10? Interest rate Year Ann Pmt Lump Sum Total CFs NPV 6% 0 \$0 \$0 \$1,015.21 1 \$100 \$100 2 \$100 \$100 3 \$100 \$100 4 \$100 \$100 5 \$100 \$100 6 \$100 \$100 7 \$100 \$100 8 \$100 \$100 9 \$100 \$100 10 \$100 \$500 \$600

3a Whats the present value of the following uneven cash flow stream: \$0 at Time 0, \$100 in Year 1 (or at Time 1), \$200 in Year 2, \$0 in Year 3, and \$400 in Year 4 if the interest rate is 8%? Interest rate Year CFs 8% 0 \$0 1 \$100 2 \$200 3 \$0 4 \$400

NPV

\$558.07

SECTION 2.13
SOLUTIONS TO SELF-TEST 3a What is the future value of this cash flow stream: \$100 at the end of 1 year, \$150 due after 2 years, and \$300 due after 3 years if the appropriate interest rate is 15%? Interest rate Year CFs FV of CFs NFV 15% 0 \$0 \$0.00 \$604.75 1 \$100 \$132.25 2 \$150 \$172.50 3 \$300 \$300.00

SECTION 2.14
SOLUTIONS TO SELF-TEST 1 An investment costs \$465 and is expected to produce cash flows of \$100 at the end of each of the next 4 years, then an extra lump sum payment of \$200 at the end of the 4th year. What is the expected rate of return on this investment? Interest rate Year Ann Pmt Lump Sum Total CFs IRR 6% 0 -\$465 -\$465 9.05% 1 \$100 \$100 2 \$100 \$100 3 \$100 \$100 4 \$100 \$200 \$300

2 An investment costs \$465 and is expected to produce cash flows of \$100 at the end Year 1, \$200 at the end or Year 2, and \$300 at the end of Year 3. What is the expected rate of return on this investment? Year CFs IRR 0 -\$465 11.71% 1 \$100 2 \$200 3 \$300

SECTION 2.15
SOLUTIONS TO SELF-TEST 2a Whats the future value of \$100 after 3 years if the appropriate interest rate is 8%, compounded annually? N I PV PMT 3 8% -\$100 \$0

FV

\$125.97

## 2b Compounded monthly? N I PV PMT 36 0.67% -\$100 \$0

FV

\$127.02

3a Whats the present value of \$100 due in 3 years if the appropriate interest rate is 8%, compounded annually? N I PMT FV 3 8% \$0 \$100

PV

\$79.38

## 3b Compounded monthly? N I PMT FV 36 1% \$0 \$100

PV

\$78.73

6 Credit card issuers must by law print their annual percentage rate (APR) on their monthly statements. A common APR is 18%, with interest paid monthly. What is the EFF % on such a loan? Nominal rate Comp/year Effective rate 18% 12 19.56%

SECTION 2.16
SOLUTIONS TO SELF-TEST 1a Suppose a company borrowed \$1 million at a rate of 9%, simple interest, with interest paid at the end of each month. The bank uses a 360-day year. How much interest would the firm have to pay in a 30-day month? Loan \$1,000,000 Interest rate 9% Days/year 360 Interest pd (days) 30 Interest paid \$7,500

1b What would the interest be if the bank used a 365-day year? Loan \$1,000,000 Interest rate 9% Days/year 365 Interest pd (days) 30 Interest paid \$7,397.26

2a Suppose you deposited \$1,000 in a credit union that pays 7% with daily compounding and a 365-day year. What is the EFF%, and how much could you withdraw after 7/12 of a year? Loan Interest rate Comp/year Effective rate \$1,000 7% 365 7.250098%

## Time period (months) Account value

7 \$1,041.67

SECTION 2.17
SOLUTIONS TO SELF-TEST 1 Suppose you borrowed \$30,000 on a student loan at a rate of 8% and now must repay it in 3 equal installments at the end of each of the next 3 years. How large would your payments be, how much of the first payment would represent interest, how much would be principal, and what would your ending balance be after the first year? N I PV FV PMT 3 8% \$30,000 \$0 -\$11,641.01

Loan Amortization Schedule, \$30,000 at 8% for 3 Years Amount borrowed: \$30,000 Years: 3 Rate: 8% PMT: -\$11,641.01
Repayment of Principal (4)

Year 1 2 3

Beginning Payment Interest Amount (1) (2) (3) \$30,000.00 \$11,641.01 \$2,400.00 \$20,758.99 \$11,641.01 \$1,660.72 \$10,778.71 \$11,641.01 \$862.30

## Ending Balance (5) \$9,241.01 \$20,758.99 \$9,980.29 \$10,778.71 \$10,778.71 \$0.00

Rather than focus on Year 1 data, it was easier to just construct a full amortization schedule.

SECTION 2.18
SOLUTIONS TO SELF-TEST

3 If the nominal interest rate is 10% and the expected inflation rate is 5%, what is the expected real rate of return? rNOM Inflation rr 10% 5% 4.7619%
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