The Mathematics of Money

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

to reason that we would have to apply the exponent before multiplying to avoid including

things to which it should not apply.

Many calculators are programmed to follow order of operations automatically. This

is true of most (but not all) newer models. On those calculators, if you enter the entire

expression 5000*1.08^50 or 5000*(1  .08)^50, the result will be correct. These calculators

understand the order in which the operations should be done. On other models, though, the

calculator performs operations in the order you enter them, and so with those calculators

you must first add 1  .08, then enter 1.08^50, obtain the result, and then multiply by 5000.

Using your calculator, you should at this point try working through Example 3.1.1 (where

you know what the answer should be) to determine how the model you are using works.

If your calculator does not recognize order of operations, it may be worth investing in one

that does, since the work we have ahead will be much easier with calculator we can trust to

observe order of operations correctly.

Before moving on, let’s work one more example to get the hang of the formula:

Example 3.1.2 Suppose you invest $14,075 at 7.5% annually compounded interest.

How much will this grow to over 20 years?

In this case, $14,075, the amount you start with, is the present value, so PV 
 $14,075. The

interest rate is 7.5%, so i 
 0.075. And since the term is 20 years, n 
 20. Plugging those

into the formula gives:

FV 
 PV(1  i)n

FV 
 $14,075(1  .075)^20

FV 
 $14,075(1.075)^20

FV 
 $14,075(4.247851100)

FV 
 $59,788.50

Of course, your calculator may allow you to get this result entering the expression to be cal-

culated all at once. If so, it is not necessary to work it through line by line.

Calculating Compound Interest

The simple interest formula calculates the amount of interest directly. If you want to know

maturity value, you need to take the extra step of adding the interest onto the principal. The

compound interest formula, though, works a bit differently. This formula calculates the

future value directly. No extra adding step is needed. Because it eliminates the need for an

extra step, the compound interest formula is a bit more convenient when the final balance

is our goal.

But what if we instead want to know the total interest earned? Then the situation is

reversed. While the simple interest formula answers that question directly, the compound

interest formula does not. A few moments thought,

though, reveals the solution.

Example 3.1.3 Suppose you invest $14,075 at

7.5% annually compounded interest. How much total

interest will you earn over 20 years?

In Example 3.1.2 we found that the future value for this

account would be $59,788.50. Of that account value,

$14,075 comes from your original principal, so the rest must

be the interest, $59,788.50 – $14,075 
 $45,713.50. So

$45,713.50 is the total interest you will earn.

Be careful when working problems (and also of course

when using these formulas in the real world) to be clear

about whether it is total interest or future value that you

are after. The extra step of subtracting isn’t too great

With a high rate of compound interest, even small investments can grow

to large future values over time. © S Meltzer/PhotoLink/Getty Images/DIL

3.1 Compound Interest: The Basics 93