the equal payment is made at the beginning of the year.
Ordinary Annuity (OA)
The present value of an ordinary annuity (Pn) can be found by using the following equation:
where PVF-OA(i,n) = T4(i,n) represents the present value of an annuity of $1 discounted at i percent for n years and is found in Table 4.
Example 13
Assume that the cash inflows in Example 11 form an annuity of $10,000 for 3 years. Then the present value is
An excerpt from Table 4 is given opposite.
Table 4: The Present Value of an ordinary annuity of $1.00 (Discounted Amount of an ordinary annuity of $1.00) = PVF-OA (i.n) = T4(i,n)
Example 14
Judy has been offered a 5-year annuity of $2,000 a year or a lump sum payment today. Since Judy wants to invest the money in a security paying 10 percent interest, she decides to take the lump-sum payment today. How large should the lump-sum payment be to equal the 5-year, $2,000 annual annuity at 10 percent interest?
Using the present-value interest factor for an ordinary annuity (PVIFA) of 5 years paying 10 percent interest provides an easy solution:
The lump-sum payment today for Judy should be $7,598 to equal the 5-year, $2000 annual annuity at 10 percent interest.
Annuity Due (AD)
The formula for the computation of the present value of an annuity due must take into consideration one additional year of compounding since the payment occurs at the beginning of the year. Therefore, the future formula must be modified as follows:
using the present-value factor for an ordinary annuity T4 or present-value factor for an annuity due = PVF-AD(i,n) = T5(i,n).
Example 15
Using the same information as in Example 14 and the present-value actor for an annuity T4 of 5 years paying 10 percent interest modified for an annuity due:
Or alternatively,
An excerpt from Table 5 is given below.
Table 5: The Present Value of an annuity due of $1.00 (Discounted Amount of an annuity due of $1.00) = PVF-AD (i,n) = T5(i,n)
Note: The present value of an annuity due of $1can also be found by adding 1 (one extra payment) to the future value of an ordinary annuity of $1 for one less period. That is: Pn = A × [T4(i,n-1) + 1]. In this example, then, Pn = $2,000 × [T4(10%, 4 years) + 1] = $2,000(3.170 + 1) = $2,000(4.170) = $8,340.
Present-Value Difference between Ordinary Annuity and Annuity Due
The present-value difference between an ordinary annuity and an annuity due is substantial. In the above examples, the difference is $8,340 - $7,582 = $756. If an annuity due and an ordinary annuity have the same number of equal payments and the same interest rates. Then the present value of the annuity due is greater than the present value of the ordinary annuity, since an additional year of compounding is essentially received.
Can a Computer Help?
Computer software can be extremely helpful in making present-value calculations. For example, PV(rate,nper,pmt,fv,type) of Excel determines the present value of an investment, based on a series of equal payments, discounted at a periodic interest rate over the number of periods, To calculate the present value of an annuity due, use the following formula: PV x (1 + interest). Financial calculators can do this too.
Perpetuities
Some annuities go on forever, called perpetuities. An example of a perpetuity is preferred stock which yields a constant dollar dividend indefinitely. The present value of a perpetuity is found as follows:
Example 16
Assume that a perpetual bond has an $80-per-year interest payment and that the discount rate is 10 percent. The present value of this perpetuity is:
Deferred Annuities
A deferred annuity is an annuity in which the payments begin after a specified number of periods. A deferred annuity does not begin to produce payments until two or more periods have expired. For example, “an ordinary annuity of six annual payments deferred
4 years” means that no payments will occur during the first 4 years, and that the first of the six payments will occur at the end of the fifth year. “An annuity due of six annual payments deferred 4 years” means that no payments will occur during the first 4 years, and that the first of six payments will occur at the beginning of the fifth year.
Future Value of a Deferred Annuity
Computing the future value of a deferred annuity is relatively straightforward. Because there is no accumulation or investment on which interest may accrue, the future value of a deferred annuity is the same as the future value of an annuity not deferred. That is, computing the future value simply ignores the deferred period.
Example 17
Assume that Allison Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters. Because of cash flow problems, Allison budgets deposits of $80,000, on which it expects to earn 6% annually, only at the end of the fourth, fifth, and sixth periods. What future value will Allison have accumulated at the end of the sixth year? Allison determines the value accumulated by using the standard formula for the future value of an ordinary annuity:
Present Value of a Deferred Annuity
Computing the present value of a deferred annuity must recognize the interest that accrues on the original investment during the deferral period. Two options are available to compute the present value of a deferred annuity, which are illustrated below.
Example 18
Joy has developed and copyrighted an online CPE course for CPAs. He agrees to sell the copyright to a CPE provider for