The Mathematics of Money

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

which leads to:

(0.072)s n (^) |i
(1.072)^5  1
Finally, dividing both sides by 0.072, we get:
s 5 | (^) .072
1.072
(^5)  1
__0.072
Evaluating this gives:
s _ 5 | (^) .072
5.77373311
which is the same annuity factor we calculated earlier. It worked!
Now this may seem like an awfully complicated way of getting at s _n (^) |i , far more
effort than just the direct bucket approach. However, notice that we could repeat this
process for any interest rate and number of payments. There was nothing special about
n
5 and I
0.072. If we have n
40 and i
0.565, the same process would work.
Rather than slog through the details a second time, though, we can just skip to the end
and get:
s 40 | (^) .0565
1.0565
(^40)  1
__0.0565
141.79020544
Since this reasoning would hold up for any n and i, it turns out that what we have found
here is what we were seeking all along; a formula that will enable us to efficiently find
annuity factors, and thus to find annuity future values. We can now generalize this as a
formula:

FORMULA 4.2.2

The Future Value Annuity Factor

s _n|i 

(1  i)n  1

___________i

where

i represents the INTEREST RATE per payment period

and

n represents the NUMBER OF PAYMENTS

The following example will illustrate the use of this formula:

Example 4.2.2 Find the future value annuity factor for a term of 20 years with an

interest rate of 7.9% compounded annually.

Here, i 
 20 and n 
 0.079. Plugging these values into Formula 4.2.2 we get:

s _n (^) |i

(1  i)n  1
___i

(1  0.079)^20  1

____0.079

We need to be careful evaluating this. This formula is far more complicated than others

we have used so far. Order of operations demands that we must fi rst add the 1  0.079

because it is in ( )’s, and next raise the result to the 20th power, to get 4.575398165. Order

of operations next says to do multiplication and division, but clearly it is impossible to divide

4.575398165  1 by 0.079 without fi rst doing the subtraction. In this case, we are allowed

to subtract fi rst. The fraction line is considered an implied grouping symbol, which means

that we treat it as though there were ( )’s around both top and bottom even though we don’t

bother to write them— it is simply assumed that everyone knows that they are there. And so

we next do the subtraction, getting 3.575398165, which we divide by 0.079 to get a fi nal

answer of

s 20 __ (^) | (^) .079
45.258204619
That description was probably a bit hard to follow. It will probably be much easier to follow
if we summarize the calculator steps in a table. If your calculator does not follow order of
4.2 Future Values of Annuities 151