The Mathematics of Money

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

FV
$5,000(2.71828)0.40

FV
$5,000(1.491824296)

FV
$7,459.12

This is the same answer we had when interest compounded every minute. The answers

aren’t really the same; continuous compounding does provide slightly more. However, that

“more” is very slight indeed—less than a penny—and so the miniscule difference is lost in

the rounding.

Frequency Times/ Year I N Formula Future Value

Continuous “Infi nitely many” n/a n/a FV 
 $5,000e0.40 $7,459.12

You may find it strange that the exponent in the above problem was not a whole

number. The most familiar explanation of what an exponent is, and the one we have

used in developing the compound interest formula, defines exponents in terms of

repeated multiplication. Thus e^5 would be understood as five e’s multiplied together, or

(e)(e)(e)(e)(e). But e0.40 makes no sense with this view; how do you multiply together

0.40 e’s? The answer to this puzzle is that our usual understanding of an exponent is

fine when the exponent is a whole number, but there is much more to the story. Expo-

nents have a deeper meaning that allows them to be negative numbers, fractions, deci-

mals, irrational numbers, and worse. A proper explanation of this lies well beyond the

scope of this book. The curious reader can find more information in a college algebra

or precalculus textbook; the less curious reader can be content with the knowledge that

any calculator will be able to handle these exponents, and that will suffice for practical

business purposes.

The astute (and/or paranoid) reader may also notice that in evaluating this formula

we appear to have violated order of operations. Order of operations says that exponents

should be done before multiplication, yet we multiplied (5)(0.08) before we used the

result as an exponent. The reason for this apparent violation of order of operations is that

when we write the “rt” together up in the exponent, it is understood that they make up the

exponent together, and so we must evaluate the result of multiplying them together first.

It is as though there were parentheses around them. In mathematical notation, it is com-

mon practice to not bother to write in parentheses when it is clear from the way things

are written that things are meant to be grouped together in this way. This is known as an

implied grouping.

This does become an issue, though, when working out compound interest with a

calculator. If you are entering the entire formula at once and trusting the calculator to fol-

low order of operations, we have to insert parentheses around the things in the exponent.

When typing an expression into a calculator, there is no “up there” making it clear what is,

or is not, in the exponent. So, to perform the calculation from Example 3.2.6, you would

need to enter:

Correct Operations Result

5000*2.71828^(.08*5)
 7459.12

If you leave out the parentheses, the result is far off the mark

INCORRECT Operations Result

5000*2.71828^.08*5
 27082.18

If you are concerned about forgetting this, you may want to actually write the implied

parentheses into your formula. While this is not the standard way of writing it, it is a correct

alternative form.

3.2 Compounding Frequencies 107