what is the present value of an annuity 
the present value of an annuity is the current value of future payments from an annuity given a specified rate of return or discount rate the higher the discount rate the lower the present value of the annuity present value pv is an important calculation that relies on the concept of the time value of money whereby a dollar today is relatively more valuable in terms of its purchasing power than a dollar in the future understanding the present value of an annuityan annuity is a financial product that provides a stream of payments to an individual over a period of time typically in the form of regular installments annuities can be either immediate or deferred depending on when the payments begin immediate annuities start paying out right away while deferred annuities have a delay before payments begin because of the time value of money money received today is worth more than the same amount of money in the future because it can be invested in the meantime by the same logic 5 000 received today is worth more than the same amount spread over five annual installments of 1 000 each present value is an important concept for annuities because it allows individuals to compare the value of receiving a series of payments in the future to the value of receiving a lump sum payment today by calculating the present value of an annuity individuals can determine whether it is more beneficial for them to receive a lump sum payment or to receive an annuity spread out over a number of years this can be particularly important when making financial decisions such as whether to take a lump sum payment from a pension plan or to receive a series of payments from an annuity the pension provider will determine the commuted value of the payment due to the beneficiary 1 they do this to ensure they are able to meet future payment obligations present value calculations can also be used to compare the relative value of different annuity options such as annuities with different payment amounts or different payment schedules the discount rate is a key factor in calculating the present value of an annuity the discount rate is an assumed rate of return or interest rate that is used to determine the present value of future payments the discount rate reflects the time value of money which means that a dollar today is worth more than a dollar in the future because it can be invested and potentially earn a return 2 the higher the discount rate the lower the present value of the annuity because the future payments are discounted more heavily conversely a lower discount rate results in a higher present value for the annuity because the future payments are discounted less heavily in general the discount rate used to calculate the present value of an annuity should reflect the individual s opportunity cost of capital or the return they could expect to earn by investing in other financial instruments for example if an individual could earn a 5 return by investing in a high quality corporate bond they might use a 5 discount rate when calculating the present value of an annuity the smallest discount rate used in these calculations is the risk free rate of return u s treasury bonds are generally considered to be the closest thing to a risk free investment so their return is often used for this purpose 3it s important to note that the discount rate used in the present value calculation is not the same as the interest rate that may be applied to the payments in the annuity the discount rate reflects the time value of money while the interest rate applied to the annuity payments reflects the cost of borrowing or the return earned on the investment the opposite of present value is future value fv the fv of money is also calculated using a discount rate but extends into the future formula and calculation of the present value of an annuitythe formula for the present value of an ordinary annuity is below an ordinary annuity pays interest at the end of a particular period rather than at the beginning 4p pmt 1 1 1 r n r where p present value of an annuity stream pmt dollar amount of each annuity payment r interest rate also known as discount rate n number of periods in which payments will be made begin aligned text p text pmt times frac 1 big frac 1 1 r n big r textbf where text p text present value of an annuity stream text pmt text dollar amount of each annuity payment r text interest rate also known as discount rate n text number of periods in which payments will be made end aligned p pmt r1 1 r n1 where p present value of an annuity streampmt dollar amount of each annuity paymentr interest rate also known as discount rate n number of periods in which payments will be made example of the present value of an annuityassume a person has the opportunity to receive an ordinary annuity that pays 50 000 per year for the next 25 years with a 6 discount rate or take a 650 000 lump sum payment which is the better option using the above formula the present value of the annuity is present value 50 000 1 1 1 0 06 25 0 06 639 168 begin aligned text present value 50 000 times frac 1 big frac 1 1 0 06 25 big 0 06 639 168 end aligned present value 50 000 0 061 1 0 06 251 639 168 given this information the annuity is worth 10 832 less on a time adjusted basis so the person would come out ahead by choosing the lump sum payment over the annuity annuity vs annuity duean ordinary annuity makes payments at the end of each time period while an annuity due makes them at the beginning all else being equal the annuity due will be worth more in the present 5 in the case of an annuity due since payments are made at the beginning of each period the formula is slightly different to find the value of an annuity due simply multiply the above formula by a factor of 1 r 6p pmt 1 1 1 r n r 1 r begin aligned text p text pmt times frac 1 big frac 1 1 r n big r times 1 r end aligned p pmt r1 1 r n1 1 r so if the example above referred to an annuity due rather than an ordinary annuity its value would be as follows present value 50 000 1 1 1 0 06 25 0 06 1 06 677 518 begin aligned text present value 50 000 times frac 1 big frac 1 1 0 06 25 big 0 06 times 1 06 677 518 end aligned present value 50 000 0 061 1 0 06 251 1 06 677 518 in this case the person should choose the annuity due option because it is worth 27 518 more than the 650 000 lump sum 