## Annuities

Annuity Due. In contrast to an ordinary annuity, where payrnents or receipts occur at the end of each period, an annuity due calls for a series of equal papnents occurring at the beginning of each period. Luckily, only a slight modification to the procedures already outlined for the treatment of ordinary annuities will allow us to solve annuity due problems.

Figure 3.6 compares and contrasts the calculation for the future value of a \$1,000 ordinary annuity for three years at 8 percent (FyA3) with that of the future value of a \$1,000 annuity due for three years at 8 percent (FVA4). Notice that the cash flows for the ordinary annuity are perceived to occur at the end of periods 1,2, and 3, and those for the annuity due are perceived to occur at the beginning of periods 2, 3, and 4.

Notice that the future value of the three-year annuity due is simply equal to the future value of a comparable three-year ordinary annuity compounded for one more period. Thus the future value of an annuity due at I percent for n periods (FVAD") is determined.

Whether a cash flow appears to occur at the beginning or end of a period often depends on your perspective, however. (In a similar vein, is midnight the end of one day or the beginning of the next?) Therefore, the real key to distinguishing betrveen the future value of an ordinary annuity and an annuity due is the point at which the future value is calculated. For an ordinary annuity, future value is calculated as of the last cash flow. For an annuity due, future value is calculated as of one period after the last cash flow.

The determination of the present value of an annuity due at i percent for n periods (PVAD,) is best understood by example. Figure 3.7 illustrates the calculations necessary to determine both the present value of a \$1,000 ordinary annuity at 8 percent for three years (PV&) and the present value of a \$1,000 annuity due at 8 percent for three years (PVADT).

As can be seen in Figure 3.7, the present value of a three-year annuity due is equal to the present value of a two-year ordinary annuity plus one nondiscounted periodic receipt or payment. This can be generalized as follows.

Alternatively, we could view the present value of an annuity due as the present value of an ordinary annuity that had been brought back one period too far. That is, we want the present value one period later than the ordinary annuity approach provides. Therefore, we could calculate the present value of an n-period annuity and then compound it one period forward.

Proves by example that both approaches to determinin g PVAD,work equally well. However, the use of Eq. seems to be the more obvious approach. The time-line approach taken in Figure 3.7 also helps us recognize the major differences between the present value ofan ordinary annuity and an annuity due.

In solving for the present value of an ordinary annuity, we consider the cash flows as occurring atthe end of periods (in our Figure 3.7 example, the end of periods 1,2, and 3) and calculate the present value as of one period before the first cash flow. Determination of the present value of an annuity due calls for us to consider the cash flows as occurring at the beginning of periods (in our example, the beginning of periods 1,2, and 3) and to calculate the present vaiue as of the 6rst cash flow.