The Rule of 72 is a rule of thumb that states that the number of time periods (such as years) that it takes for a sum to double at compound interest is very nearly 72 divided by the percentage interest rate per period. Thus, for example, it takes (almost exactly) 9 years for a sum to double at 8% interest compounded yearly.

1. Using the rule, find the annual rate of increase for an investment that has quadrupled in 24 years. Compare to the exact value.

2. At what interest rate is the rule exact?

3. Justify the rule using mathematical analysis and a few numerical calculations.

The formula for continuous compounding is Pe^(r*t)

For the principal to double we need e^(r*t)=2 so r*t = ln(2) or about .6931 (69.31 if r is a percent) 
This is close to 72 but a little less.  The reason it can be less is that continuous compounding increases faster than annual compounding.

This will be exact doubling for any values (not just whole numbers of years) whose product is ln(2).

-Jer

Posted by Jer on 2005-01-25 18:08:11