The Mathematics of Money

96 Chapter 3 Compound Interest

fact that the two calculations are consistent should give him some comfort that he did the

calculation correctly.

Using the Rule of 72 to Find Rates

The Rule of 72 can also be used to estimate the rate needed to achieve a desired future

value. Suppose you have $30,000 in an investment account, and hope that the account

will grow to $1,000,000 by the time you retire 30 years from now. It would be helpful

to know what sort of rate that would require, so that you can figure out whether or not

your hope is realistic.

We can adapt the rule to deal with problems like these. With a little algebraic effort, we

can set up the Rule of 72 to answer this question.

doubling time
____rate^72

Multiplying both sides by the rate gives:

(doubling time)(rate) 
 72

And then dividing by doubling time we get:

rate
_____ doubling time^72

This leads us to:

FORMULA 3.1.3

The Rule of 72 (Alternate Form)

The compound interest rate required for a sum of money to double in x years

is approximately 72/x percent.

Example 3.1.7 What compound interest rate is required to double $50,000 in 5 years?

72/5 
 14.4, and so the interest rate would need to be approximately 14.4%. (The fact that

the amount was $50,000 is irrelevant.)

Returning to the question we posed a bit earlier, consider this example:

Example 3.1.8 What compound interest rate is required for $30,000 to grow to

$1,000,000 in 30 years?

This does not ask for just a single doubling, and so we fi rst must determine how many dou-

blings are required. One way to do this would be to take the initial amount and repeatedly

double it until it reaches the target FV. Doing so gives:

Number of Doublings Account Value

Start

1 2 3 4 5 6

$30,000

$60,000

$120,000

$240,000

$480,000

$960,000

$1,920,000

From this table we can see that we reach $1,000,000 just a bit after 5 doublings.

Alternatively, we could note that $1,000,000/$30,000 
 33.33. Now, no number of dou-

blings will give you exactly 33.33 times your original money, but we will try to get as close as

possible. One doubling gives you twice your original money, two doublings gives you (2)(2) 


22 
 4 times your money, and so on. Keeping this up, we soon see that fi ve doublings would

give 2^5 
 (2)(2)(2)(2)(2) 
 32 times your original money, and so we see that we need a bit

more than 5 doublings.