The Mathematics of Money

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

FORMULA 3.1.2

The Rule of 72^1

The time required for a sum of money to double at a compound interest rate of x%

is approximately 72/x years.

(The interest rate should not be converted to a decimal.)

Look back at Example 3.1.4, where we had $2,000 invested at 6% compound interest

for 12 years. The rule says that the time required to double your money at 6% compound

interest should be approximately 72/6 
 12 years. And in fact this was pretty much what

happened; the original $2,000 a bit more than doubled in 12 years.

The Rule of 72 is strictly an approximation; it is no substitute for the compound inter-

est formula, but it does have its uses. It is a useful tool for coming up with quick, ballpark

estimates, and can also be useful as a quick way of validating the reasonableness of a com-

pound interest calculation. The following examples will illustrate this.

Example 3.1.6 Jarron deposited $3,200 into a retirement account, which he expects

to earn 7% annually compounded interest. If his expectation about the interest rate is

correct, how much will his deposit grow to between now and when he retires 40 years

from now? Use the Rule of 72 to obtain an approximate answer, then use the compound

interest formula to fi nd the exact value.

Using the Rule of 72, we know that his money should double approximately every 10 years,

since 72/7 
 10.2857, a bit more than 10. So in 40 years, his account should experience

approximately 40/10 
 4 doublings. Thus:

Time Approximate Account Value

Start $3,200

After 10 years $6,400

After 20 years $12,800

After 30 years $25,600

After 40 years $51,200

We could also have arrived that this by doubling his account balance four times using an

exponent:

FV 
 $3,200(2)(2)(2)(2) 
 $3,200(2^4 ) 
 $3,200(16) 
 $51,200

So we expect Jarron’s future value to be “in the neighborhood of $50,000.” Using the com-

pound interest formula, we have

FV 
 $3,200(1.07)^40 
 $47,918.27

The actual result is not particularly close to the $51,200 given by the Rule of 72. The

difference between the two is over $3,000. However, we shouldn’t kid ourselves about

what Rule of 72 is—it is strictly a means of getting a quick and rough estimate. While there

is a gap between the two answers, they are in roughly the same ballpark. If Jarron needs a

precise future value, the Rule of 72 is not all that helpful, but then no one ever claimed that

it would be. However, if he just wants to get a rough idea of how large the account might

grow, the Rule of 72 would suffice. It is also useful as a way of validating the more precise

calculation; $47,918.27 seems like an awfully large amount for $3,200 to grow to, but the

(^1) The Rule of 72 is sometimes instead called the Rule of 70, in which case, as the name suggests we use 70 in
place of 72. This actually gives a slightly better approximation in some cases. However, tradition sides with 72,
which has the advantage of being evenly divisible by more numbers than 70 is. In any case, either rule gives only
a rough approximation, so we could just as well use the “Rule of Whatever Number in the Low 70s That You Like.”
Be careful, though: there is also a Rule of 78, which, despite the similar sounding name, has nothing at all to do
with what we are talking about here.
3.1 Compound Interest: The Basics 95