The Mathematics of Money

92 Chapter 3 Compound Interest

would simply take the place of the $5,000. We can sum up all of these observations in a

formula:

FORMULA 3.1.1

The Compound Interest Formula

FV 
 PV(1  i)n

where

FV represents the FUTURE VALUE (the ending amount)

PV represents the PRESENT VALUE (the starting amount)

i represents the INTEREST RATE (per time period)

and

n represents the NUMBER OF TIME PERIODS

This formula introduces two new terms: present value and future value. As defined

above, present value refers to the amount at the beginning of the term. In this context,

the present value is essentially the same idea as the principal. Likewise, since the future

value is the amount at the end of the time period in question, it can be thought of as

maturity value. (However, future value is not always the same as maturity value. We

might use the formula to figure out an account balance at some point prior to a note’s

actual maturity.)

To see how the formula works, let’s revisit a problem we’ve already worked out.

Example 3.1.1 Use the compound interest formula to fi nd how much $5,000 will

grow to in 50 years at 8% annual compound interest.

FV 
 PV(1  i)n

FV 
 $5,000(1  0.08)^50

FV 
 $5,000(1.08)^50

FV 
 $5,000(46.9016125132)

FV 
 $234,508.06

Order of Operations

Notice that in the above calculation, we used the exponent before multiplying, even though

reading from left to right it might appear as though the multiplication should have come

first. For that matter, we added what was inside the parentheses before multiplying, also

defying left to right order. While in English we read from left to right, there is a different

set of “rules of the road” in mathematics known as order of operations, which determines

the order in which calculations are performed. These rules are summarized below:

Order Operation

1 st

2 nd

3 rd

4 th

Parentheses

Exponents

Multiplication/division

Addition/subtraction

Following order of operations, we must first do whatever is inside parentheses, and so we

first added 1  0.08 to get 1.08. Next come exponents, ahead of multiplication, so this

requires that we evaluate 1.08^50 before multiplying by $5,000.

Aside from needing to play by the rules, there is another very good reason for doing

things in this order as well. If we had worked from left to right and multiplied $5,000(1.08)

and then raised the result to the 50th power, the number the exponent applies to would have

included the $5,000 in it. So instead of just multiplying fifty 1.08s (which we know we

want), we also would have been throwing in fifty $5,000s (which we don’t want). It stands