The Mathematics of Money

152 Chapter 4 Annuities

operations, or if you do not want to try to enter a formula this large all at once, the steps

would be:

Operation Result

1 .079
 1.079

^20
 4.575398165

 1 
 3.575398165

/.079
 45.25820462

If in previous work you have been entering the entire formulas into the calculator in one

step, you can do that here also. You must, though, be careful about the implied grouping.

Since you cannot enter things on top of one another, the calculator does not know about the

implied grouping. You have to make the implied grouping explicit to the calculator, and so

you must make sure to put in parentheses around the top yourself.

Operation Result

((1.079)^201)/(.079)
 45.25820462

Make sure not to overlook these implied parentheses! The result of leaving them out will

usually be an answer that is miles away from the correct one. Because of the length of the

formula and the many opportunities it presents to make a mistake typing it in, the all-at-once

approach is not recommended.

A third alternative would be to enter by combining some, but not all of the steps. For example,

we could calculate this annuity factor as:

Operation Result

(1.079)^20
 4.575398165

 1 
 3.575398165

/.079
 45.25820462

You may want to try calculating annuity factors each way for yourself to see which approach

you like best.

Now that we’ve figured out how to calculate the annuity factors, we can put them to work

to find actual future values.

Example 4.2.3 Suppose that $750 is deposited each year into an account paying

7.9% interest compounded annually. What will the future value of the account be?

Since the payments are equal and made at regular intervals, this is an annuity; since the

timing of the payments is unspecifi ed, we assume it to be an ordinary annuity and use

Formula 4.2.1. The annuity factor we need is the one calculated in Example 4.2.2.

FV
PMTs _n (^) | (^) i
FV
$750 s __ 20 | (^) .079
FV
($750)(45.258204619)
FV
$33,943.65
Getting comfortable with these calculations will no doubt take additional practice. There
will be numerous opportunities to get that practice from the remaining examples and exer-
cises in this section.
Nonannual Annuities
Even though the annuities we’ve worked with so far in this section have all had annual
payments, there is no reason why we can’t use the same techniques with an annuity whose
payments are quarterly, monthly, weekly, or any other frequency. The transition is basically the
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