## Note on Annuities

So far, we have only discussed ordinary annuities. These are the most important, but there is a variation that is fairly common. Remember that with an ordinary annuity, the cash flows occur at the end of each period. When you take out a loan with monthly payments, for example, the first loan payment normally occurs one month after you get the loan. However, when you lease an apartment, the first lease payment is usually due immediately. The second payment is due at the beginning of the second month, and so on. A lease is an example of an annuity due. An annuity due is an annuity for which the cash flows occur at the beginning of each period. Almost any type of arrangement in which we have to prepay the same amount each period is an annuity due.

There are several different ways to calculate the value of an annuity due. With a financial calculator, you simply switch it into "due" or "beginning" mode. It is very important to remember to switch it back when you are done! Another way to calculate the present value of an annuity due can be illustrated with a time line. Suppose an annuity due has five payments of \$400 each, and the relevant discount rate is 10 percent. The time line looks like this:

annuity due

An annuity for which the cash flows occur at the beginning of the period.

174 PART THREE Valuation of Future Cash Flows

Notice how the cash flows here are the same as those for a four-year ordinary annuity, except that there is an extra \$400 at Time 0. For practice, check to see that the value of a four-year ordinary annuity at 10 percent is \$1,267.95. If we add on the extra \$400, we get \$1,667.95, which is the present value of this annuity due.

There is an even easier way to calculate the present or future value of an annuity due. If we assume cash flows occur at the end of each period when they really occur at the beginning, then we discount each one by one period too many. We could fix this by simply multiplying our answer by (1 + r), where r is the discount rate. In fact, the relationship between the value of an annuity due and an ordinary annuity is just:

Annuity due value = Ordinary annuity value X (1 + r) [6.3]

This works for both present and future values, so calculating the value of an annuity due involves two steps: (1) calculate the present or future value as though it were an ordinary annuity, and (2) multiply your answer by (1 + r). 