The Rule of 72

Nearly anyone who has learned the time value of money has heard of the Rule of 72. The "rule" is a way to estimate the number of periods required to double an investment at a given interest rate. Conveniently, one can also use the rule to estimate the interest rate need to double principal invested at a given interest rate.

For example, at an interest rate of 6%, it takes approximately 12 years (72/6) to double your investment. Or, if one wanted to double their money in 9 years, you would need an interest rate of 8% (72/9) to achieve your goal. The rule just gives an approximate answer; however, the results are very good for a wide range of interest rates (or periods of time). The error in time compared to using compound interest formulas is under 4% for interest rates between 1 and 16% and the time error is under 2% for interest rates between 4 and 12%. Especially since 72 is easily dividable by many numbers, the rule is a good one for those willing to accept the small error.

While I've heard many people talk about the rule, I've never seen an explanation of why the rule works. Certainly, as given by examples, one can see how it works but I spent some time to determine why the rule works.

Here is my explanation:

Without knowing, the "rule  of 72," suppose someone stated that there exists a number, x, for which the equation: NI = x holds true for the number of periods, N, at a given interest rate, I (in %), that an investment doubles. Could we now solve for the "unknown" number, x?

For our further calculations, we need to express interest in decimal format. Therefore, let i = I/100.

Given the double rule, and using the formula for compound interest:

Since N=x/I, and i=I/100, N=x/(100i). Substituting for N in the above equation:

Take the log of both sides of the equation:

 

Now, solving for x:

Now, let's substitute some interest rates into this equation:

Rate , i	x
.04	70.69
.05	71.03
.06	71.37
.07	71.71
.08	72.05
.09	72.39
.10	72.76

While the value, x, varies with each interest rate, for a wide range of rates, x is close to 72, and that is why the "rule" of 72 works. For very low interest rates, one could say it is better to use the rule of 70; however, 72 remains popular because it is easily divisible by many numbers and this makes the approximations easier.