## Valuing an Annuity

The ordinary annuity cash flow analysis assumes that cash flows occur at the end of each period. However, there is another fairly common cash flow pattern in which level cash flows occur at regular intervals, but the first cash flow occurs immediately. This pattern of cash flows is called an annuity due. For example, if you win the Florida Lottery Lotto grand prize, you will receive your winnings in 20 installments (after taxes, of course). The 20 installments are paid out annually, beginning immediately. The lottery winnings are therefore an annuity due.

Like the cash flows we have considered thus far, the future value of an annuity due can be determined by calculating the future value of each cash flow and summing them. And, the present value of an annuity due is determined in the same way as a present value of any stream of cash flows.

Let's consider first an example of the future value of an annuity due, comparing the values of an ordinary annuity and an annuity due, each comprising three cash flows of \$500, compounded at the interest rate of 4% per period. The calculation of the future value of both the ordinary annuity and the annuity due at the end of three periods is:

Ordinary annuity Annuity due

FV = \$500 1 + 0.04)3 -1 FVdue = \$500 1 + 0.04)3 -1 +1

The future value of each of the \$500 payments in the annuity due calculation is compounded for one more period than for the ordinary annuity. For example, the first deposit of \$500 earns interest for two periods in the ordinary annuity situation [\$500 (1 + 0.04)2], whereas the first \$500 in the annuity due case earns interest for three periods [\$500 (1 + 0.04)3]. In general terms,

which is equal to the future value of an ordinary annuity multiplied by a factor of 1 + i:

FVdue = CF[Future value annuity factor (ordinary) for N and i](1 + i)

The present value of the annuity due is calculated in a similar manner, adjusting the ordinary annuity formula for the different number of discount periods:

Since the cash flows in the annuity due situation are each discounted one less period than the corresponding cash flows in the ordinary annuity, the present value of the annuity due is greater than the present value of the ordinary annuity for an equivalent amount and number of cash flows. Like the future value an annuity due, we can specify the present value in terms of the ordinary annuity factor:

PVdue = CF[Present value annuity factor (ordinary) for N and i](1 + i)

Financial calculators make your calculations easier by automatically adjusting the present or future value annuity factor if you specify the "begin" or "due" mode. For example, if you are using the HP12C calculator and want to calculate the future value of \$500 to be received at the beginning of each of three periods, you first put the calculator in the annuity due mode [ g BEG ], then specify the cash flow (the \$500), the number of payments (3), and the interest rate (4%). 