Principles of Finance/Section 1/Chapter 3/Applications of Time Value of Money/Annuities

The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in academic discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money concepts.

An annuity can be valued with the following formula, where C is the amount of the payment, r is the discount rate, and t is the number of periods for which the payment will be received.

PVannuity = $$C\left[\frac{1}{r}-\frac{1}{r\left(1+r\right)^t}\right]$$

Thus, we can see that a 10 year, $100 per year annuity would be worth $614.46 with a 10% discount rate.

We can also find the present value of an annuity whose payments grow by a fixed rate g.

PVgrowing annuity = $${C \over (r-g)}\left[ 1- \left({1+g \over 1+r}\right)^t \right] $$

Solving a similar problem, with a payment of $100 that grows at a rate of 5% for 10 years, with a 10% discount rate, we get a present value of $743.98.

Note that the preceding two formulas assume that the payment will be made at the end of each period. If the payment is to be received at the beginning of each period, it is called an Annuity Due. The preceding two formulas can still be used in the case of an annuity due, with some modification. Simply subtract 1 from the number of periods, and then add the amount of the payment to the resulting answer. For example, if the first problem we did were an annuity due, it would look like this:

PVannuity due = $$100\left[\frac{1}{0.1}-\frac{1}{0.1\left(1.1\right)^9}\right] + 100$$

The answer, of course, is $675.90. Note that an annuity due will always have a higher present value than an equivalent standard annuity, due to the time value of money. That is, a dollar received at an earlier date is more valuable than a dollar received at a later date.