The Mathematics of Money

94 Chapter 3 Compound Interest

an inconvenience, but it is easy to overlook. Read questions carefully to make sure that,

whether you are finding future value or total interest, you actually are answering the

question asked.

Example 3.1.4 Suppose that $2,000 is deposited at a compound interest rate of

6% annually. Find (a) the total account value after 12 years and (b) the total interest

earned in those 12 years.

(a) Finding the future account value is just a matter of using the formula. To wit:

FV 
 PV(1  i)n

FV 
 $2,000(1  0.06)^12

FV 
 $4,024.39

(b) The difference between the future value and present value is the total interest. Thus:

To tal interest 
 $4,024.39  $2,000 
 $2,024.39

In all of our examples in this section, we are assuming that the interest is credited each

year, just as it did in the example we used to start this chapter. Because the interest com-

pounds each year, we say that it is compounded annually. It is possible to have interest

that compounds more often than once a year, though we will not see any examples of this

until Section 3.2.

Finding Present Value

The algebraic tools we have developed in prior sections are still useful with the compound

interest formula as well.

Example 3.1.5 How much money should I deposit today into an account earning 7^3 ⁄ 8 %

annually compounded interest in order to have $2,000 in the account 5 years from now?

Before beginning with the formula, it is important to notice that in this case the $2,000 is

the FV, not the PV.

FV 
 PV(1  i)n

$2,000 
 PV(1  0.07375)^5

$2,000 
 PV(1.42730203237)

It is the PV that we are after. Using the same reasoning that we used in Chapters 1 and 2,

we can divide both sides by the 1.42730203237 to get PV by itself. (Remember that it may

be helpful to use your calculator’s memory to avoid having to type in the long decimal when

you divide.)

Doing this gives us a fi nal answer of:

PV 
 $1,401.25

I should therefore deposit $1,401.25.

In Chapter 1 we used algebra on the I 
 PRT formula to obtain each of the quantities

included in that formula. We solved for P, then solved for R, then solved for T. In Chapter 2,

we did the same thing with the simple discount formula. So it is natural to expect that our

next move would be to look at problems where we need to find the interest rate i or the

time n in the compound interest formula. Unfortunately, solving for these values requires a

much more significant algebraic investment, and so we will dodge those questions for now.

We can, though, find approximate answers using a handy tool known as the Rule of 72.

The Rule of 72

The Rule of 72 is a useful rule of thumb for estimating how quickly money will grow at

a given compound interest rate.