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 equation for compound interest, as Jer said, is P(1+r)^t.  If this equals 2P, we can solve for T.

P(1+r)^t = 2P
(1+r)^t = 2
t ln(1+r) = ln(2)
t = ln(2)/ln(1+r)

While t isn't exactly inversely proportional to r, we must remember that r is very small; r is .08 if the interest rate is 8%.  For this reason, ln(1+r)/ln(2) can almost be considered linear,   and ln(2)/ln(1+r) almost a hyperbole.

As for the number 72, I suppose you would have to decide what range of interest rates are most important, and choose a slope that matches most accurately.  71 would match a lower range, while 73 would match a higher range.  According to my graphing calculator, the rule intersects at about 7.8468715%, so that might give you an idea of what the range is.  Also, 72 is a much more round number, easier to use than the primes 71 and 73.

For the sake of completeness, the answer to part 1 is 6%, though the exact value is around 5.9463094%.

Posted by Tristan on 2005-01-25 19:45:13