## Annuities

To solve for the compound annual interest rate (i) implicit in this annuity problem, we make use of future value of an annuity Eq. (3.9) as follows.

Reading across the 8-period row in Table 3.5, we look for the future value interest factor of an annuity (FVIFA) that comes closest to our calculated value of 9.5. In our table, that interest factor is 9.549 and is found inthe 5o/o column. Because 9.549 is slightly larger than 9.5, we conclude that the interest rate implicit in the example situation is actually slightly less than 5 percent. (For a more accurate answer, you would need to rely on trial-and-error testing of different interest rates, interpolation, or a financial calculator.)

Mined, however, is the size of each equal payment or receipt. In a business setting, we most frequently encounter the need to determine periodic annuity pa1'rnents in sinking fund (i.e., building up a fund through equal-dollar payments) and loan amortization (i.e., extinguishing a loan through equal-dollar payments) problems.

Rearrangement of either the basic present or future value annuity equation is necessary to solve for the periodic payment or receipt implicit in an annuity. Because we devote an entire section at the end of this chapter to the important topic of ioan amortization, we will illustrate how to calculate the periodic payment with a sinking fund problem.

How much must one deposit each year end in a savings account earning 5 percent compound annual interest to accumulate \$ 10,000 at the end of 8 years? We compute the pa1'rnent (R) going into the savings account each year with the help of future value of an annuity Eq. (3.9). In addition, we use Table 3.5 to find the value corresponding to FVIF,A',,, and proceed as follows.

Therefore, by making eight year-end deposits of \$1,047.23 each into a savings account earn, compound annual interest, we will build up a sum totaling \$10,000 at the end perpetuity An ordinary eerpetuity. A perpetuity rs an ordinary annuitywhose payrnents or receipts continue forannuity wnose ever. The ability to determine the present value of this special type of annuity will be required payments or receipts when we value perpetual bonds and preferred stock in the next chapter. A look back to PVA, continue forever' in Eq. (3.10) should help us to make short work of this qpe of task. Replacing n in Eq. (3.10) with the value infinity (-) gives.

Thus the present value of a perpetuity is simply the periodic receipt (payment) divided by the interest rate per period. For example, if\$100 is received each year forever and the interest rate is 8 percent, the present value ofthis perpetuity is \$1,250 (that is, \$100/0.08).