# Numeracy #12: Rule of 72

Imagine that we just opened a new savings account that pays 10% interest compounded annually. In reality, we would get something closer to 0.1% interest compounded continuously, but let’s keep the numbers large and whole for now.

If we opened the account with \$100, after a year we would \$110, and after two years we would have \$121. How long will it take us to double our money? You may have heard of the Rule of 72 which is a trick for answering this question. Before we get into how it works, let’s review how compound interest works.

### Compound Interest

In our example, we received \$10 of interest in year one and \$11 in year two. Where did that extra dollar come from? With compound interest, we earn interest on top of interest, so the extra dollar just comes from \$10 * 10% = \$1. How much money will we have after three years?

YR1: 100(1.10) = \$110
YR2: 100(1.10)(1.10) = \$121
YR3: 100(1.10)(1.10)(1.10) = \$133.1

You should start to see a pattern. After n years, how much money will we have?

YRn: 100(1.10)^n

Or more generally:

FV = PV(1+r)^n

FV is future value
PV is present value
r is the interest rate
n is the number of years

We can make this more complicated by adding different compounding periods. For example, let’s look at our balance after one year with:

Monthly compounding: 100(1+(.10/12))^12 = \$110.47
Continuous compounding: 100*e^(.10) = \$110.52

Using what we know so far, we could figure out the doubling time by just plugging in numbers for n until we hit \$200. But there’s an easier way.

### The Rule of 72

Now that we have a general equation for compound interest (FV = PV(1+r)^n), we can derive the equation for doubling time. In our example, \$200 is the FV we want, so we can plug-in numbers and solve for n.

FV = PV(1+i)^n

200 = 100(1 + .10)^n
2 = 1.10^n
n = log1.102
n = ln(2)/ln(1.10)
n = 7.27 years

Or more generally:

DT = ln(2) / ln(1 + r)

DT is doubling time
r is the interest rate

We have our answer, but working with logarithms and exponents is rarely fun. This is where the Rule of 72 comes in. Instead of doing all that math, the Rule of 72 states that we can approximate the doubling time by dividing our interest rate into 72. In our example, the interest rate was 10%, so we get:

DT = 72/10 = 7.2 years

Pretty close approximation for a lot less work, but where did 72 come from? Well, if we look back at our general formula for doubling times, the numerator is ln(2) which equals .693. The Rule of 69.3 doesn’t sound as good, but it does give exact results in continuous compounding situations. We use 72 because of its convenience (it’s easily divisible by 1, 2, 3, 4, 6, 8, 9, 12) and since it better approximates annual compounding.

### Accuracy

The following graph shows the actual doubling time at a given interest rate vs the doubling time for the Rule of 72:

For the most part, the Rule of 72 is indistinguishable from the actual doubling time. However, for smaller interest rates it overestimates the doubling time and for larger interest rates it underestimates the doubling time. If you ever want an accurate doubling time for 40%, you might be better off doing the math.

### Conclusion

The Rule of 72 is a very easy way to find out how long it takes to double your money at a given interest rate. For example, if you find an investment that pays 9% compounded annually, it would take about 72/9 = 8 years for you to double your money. Since all of investing is a game of compounded interest, the Rule of 72 remains a useful tool.