0.8 Mathematics of finance  (Page 5/9)

Annuities and sinking funds

Section overview

In this section, you will learn to:

1. Find the future value of an annuity.
2. Find the amount of payments to a sinking fund.

In [link] and [link] , we did problems where an amount of money was deposited lump sum in an account and was left there for the entire time period. Now we will do problems where timely payments are made in an account. When a sequence of payments of some fixed amount are made in an account at equal intervals of time, we call that an annuity . And this is the subject of this section.

To develop a formula to find the value of an annuity, we will need to recall the formula for the sum of a geometric series.

A geometric series is of the form: $a+\text{ar}+{\text{ar}}^2+{\text{ar}}^3+\text{.}\text{.}\text{.}+{\text{ar}}^n$ .

The following are some examples of geometric series.

In a geometric series, each subsequent term is obtained by multiplying the preceding term by a number, called the common ratio. And a geometric series is completely determined by knowing its first term, the common ratio, and the number of terms.

In the example, $a+\text{ar}+{\text{ar}}^2+{\text{ar}}^3+\text{.}\text{.}\text{.}+{\text{ar}}^{n-1}$ the first term of the series is $a$ , the common ratio is $r$ , and the number of terms are $n$ .

In your algebra class, you developed a formula for finding the sum of a geometric series. The formula states that the sum of a geometric series is

We will use this formula to find the value of an annuity.

Consider the following example.

When the payments are made at the end of each period rather than at the beginning, we call it an ordinary annuity .