The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

4.1 What Is an Annuity?

All of the situations we have considered so far, whether simple interest, simple discount, or

compound interest, have had something in common. In every case, a sum of money is lent

and then the loan is repaid in full all at once at the maturity date. While the terminology and

formulas have varied, timelines for any of these loans have looked like:

Money

borrowed

Loan

repaid

This is all well and good, but there are many financial situations where this sort of time-

line is far too simple. Take a car loan for example. Suppose you borrow $14,500 at 8%

compounded monthly for 5 years to buy a car. The deal is unlikely to be that you get the

$14,500 up front, do nothing for 5 years, and then repay the entire loan at the end, all in a

single lump sum. Such an arrangement would not make sense either to you or to the lender.

Instead, the usual arrangement requires you pay off the loan by making payments every

month. A timeline for your car loan would instead look more like this:

Money

borrowed

Payments made monthly

This loan’s consistent and regular payments are an example of an annuity. Clearly, we will

need to go beyond the mathematics we have so far to deal with such situations. Before we

do that, though, we’ll need to first define some new terminology.

Definition 4.1.1

An annuity is any collection of equal payments made at regular time intervals.

Annuities are very common in business and personal finance. A number of examples are

given below; with a few moments’ thought you can probably come up with many others.

Some Examples of Annuities

Monthly payments on a car loan, student loan, or mortgage

Your paycheck (if salaried or if hourly with hours that are always the same)

The payments on a car loan

are an example of an annuity.

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4.1 What Is an Annuity? 141