The Mathematics of Money

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

same as the one we made with compound interest when we moved from annual compounding

to other compounding frequencies.

Example 4.2.4 Find the future value of quarterly payments of $750 for 5 years,

assuming an 8% interest rate.

We fi rst determine the values of i and n. This is done exactly the same we did it with com-

pound interest in Chapter 3:

i
0.08_____ 4
0.02

and

n 
 5(4) 
 20

Before we can proceed, we need to fi nd the annuity factor. To obtain the factor by using the

formula, we plug in these values of n and i to get:

sn _ (^) | (^) i

( 1  i)n  1
___i

( 1  0.02)^20  1

___0.02

Evaluating this on the calculator step by step we get:

Operation Result

1 .02
 1.02

^20
 1.48594740

 1 
 0.48594740

/.079
 24.2973698

Alternatively:

Operation Result

(1.02)^20
 1.48594740

 1 
 0.48594740

/.079
 24.2973698

(You should fi nd the same value for the annuity factor if you are using a table, or if you are

using a preprogrammed calculator feature as well.)

Now that we have the annuity factor, we can complete the calculation. Plugging into our

annuity formula, we get:

FV
PMT s _n (^) | (^) i
FV
($750) s 20
0.02
FV
($750)(24.2973698)
FV
$18,223.03
While lengthy, this last problem was not much different from a problem with annual pay-
ments. Calculating annuity factors for nonannual annuities can become a bit stickier, though,
since nonannual compounding means that we will usually have to divide to get i. If the result
does not come out evenly, this can require an awful lot of tedious typing with rich potential for
typos. The following example will illustrate ways to accurately complete this calculation.
Example 4.2.5 Find the future value annuity factor for a monthly annuity, assuming
the term is 15 years and the interest rate is 7.1% compounded monthly.
Here,
i
____0.071 12
0.005916667
and
n
(15)(12)
180
4.2 Future Values of Annuities 153