## Annuities

It is very helpful to begin solving time value of money problems by frst drawitg a time line on which you position the relevant cash flows. The time line helps you focus on the probIem and reduces the chance for error. When we get to mi-xed cash flows, this will become even more apparent.

An abbreviated listing of FVIFAs appears in Table 3.5. Almore complete listing appears in Table III in the Appendix at the end of this book.

This answer is identical to that shown in Figure 3.4. (Note: Use of a table rather than a formula subjects us to some slight rounding error. Had we used Eq. (3.8), our answer would have been 40 cents more. Therefore, when extreme accuracy is called for, use formulas rather than tables.)

Return for the moment to Figure 3.3. Only now let's assume the cash flows of \$1,000 a year for three years represent withdrawals from a savings account earning B percent compound annual interest. How much money would you have to place in the account right now (time period 0) such that you would end up with a zero balance after the last \$1,000 withdrawal?

As can be seen from Figure 3.5, solving for the present vaiue of an annuity boils down to determining the sum of a series of individual present values. Therefore, we can write the general formula for the present value of an (ordinary) annuity for n periods (PVA,).

Notice that our formula reduces lo PVA,, being equal to the periodic receipt (R) times the "sum of the present value interest factors at I percent for time periods l to n." Mathematically, this is equivalent to, and can be expressed even more simply.

We can make use of Table 3.6 to solve for the present value of the \$1,000 annuity for three years at 8 percent shown in Figure 3.5. The PVIFAB,*3 is found from the table to be2.577. (Notice this figure is nothing more than the sum of the first three numbers under the 8o/o column in Table 3.4, which gives PVIFs.) Employing Eq. (3.11), we get PVA3= \$1,000(PVlFABwl

A rearrangement of the basic future value (present vaiue) of an annuity equation can be used to solve for the compound interest (discount) rate implicit in an annuityif we know: (1) the annuity's future (present) value, (2) the periodic pa)..rnent or receipt, and (3) the number of periods involved. Suppose that you need to have at least \$9,500 at the end of8 years in order to send your parents on a luxury cruise. To accumulate this sum, you have decided io deposit \$1,000 at the end of each of the next 8 years in a bank savings account. If the bank compounds interest annually, what minimum compound annual interest rate must the bank offer for your savings plan to work?