Rule of 72

Allegedly discovered by Benjamin Franklin, the rule of 72 states:

If you divide the annual percentage rate into 72, you get the number of years it takes for your money to double.

For example, if your money earns 2% interest, it will double in 72 / 2 = 36 years. Of course, it can be used for other time periods. So, if your money grows at 2% a month, it will double in 36 months.

When I first learned this rule I was puzzled. I mean compound interest does not work in a linear way (that's its beauty), so how can there be such a simple rule...

I wrote a computer program just to test it. In it, I used the proper math to calculate how long it takes for money to double with percentage rates from 1% to 72%, and compared it with the result of diving the rate into 72. To my surprise, the rule of 72 did very well. The difference was negligible.

The familiarity with the rule of 72 can come handy whenever financiers do not give you a straight answer.

For example, one day I walked into a bank and asked about the annual percentage rate of some government securities. Instead of giving me the rate, the teller said: "It is designed to double your money in 12 years."

To that I replied, "Hmmm, 6%, that's not too much." She seemed extremely impressed that I knew 12 years to double meant 6%. Then she agreed it was not too much (at that time savings accounts typically offered 5.5% APR).

I wonder if the bank management instructed its tellers to give a potential buyer the number of years for the bonds to double instead of the APR in the hope 12 years would impress people more than 6%, and entice them into buying.

But, as they say, knowledge is power. And in this case, very simple knowledge gives you a lot of power.

The Rule of 72 is based on the truncation of a Taylor Series for ln(1+x). To see how this works, let's consider the nice sum of x dollars on your bank account. Assuming a 5% annual interest, this amount has accumulated to 1.05 × M dollars after one year. After two years, the total amount is 1.05 × 1.05 × M = 1.1025 M. We can formalize this by:

     Mn = M × (1.05)n

where M is the initial amount of money, M_n is the amount after n years, and n is the number of years.

Or, in terms of an arbitrary interest rate i (in %):

     Mn = M × (1+(i/100))n

Now we want to find the number of years over which M doubles, thus M_n = 2 × M:

     2 × M  = M × (1+(i/100))n

     2 = (1+(i/100))n

     ln(2) = n × ln(1+(i/100))

Thus, the exact solution for the doubling problem is:

     n = ln(2) / ln(1+(i/100))

For example: at 2% interest, your money doubles in ln(2)/ln(1.02) = 35.00 years. Apparently the Rule of 72 is pretty close to the exact solution.

However, the exact solution is rather impractical in daily use. To simplify the equation we make use of a Taylor Series:

     ln(1+x) = x - x2/2 + x3/3 - ... 

     ln(1+x) =  x

Combine this with the exact solution for the doubling problem:

     n = ln(2) / (i/100)

     n = 100*ln(2) / i

     n = 69 / i 

Thus, the Rule of 72 has a mathematical basis. The choice of the factor 72 instead of the more correct value 69 is for ease of calculation. For low values of the annual interest, the error is small.