The rule of 72 applies to annually compounded
interest, but it's easiest to understand by looking at the case of continuously compounded interest first.
We'll write P for the starting principal and r for the return rate (as a decimal); we're looking for Y to double P:
2P = PeYr
Solve for Y:
Y = ln(2) / r
The log of 2 is about equal to .69, so
Y = .69 / r
You can think of this as The Rule of 69. It's valid for all values of r.
Solving the formula for annually compounded interest is messier:
2P = P(1 + r)Y
Y = ln(2) / ln(1 + r)
We want to approximate this as a neat fraction again,
Y = K / r
where K is some number that will make the approximation pretty good for some ranges of r (and pretty lousy
for others). We'll choose K to make the approximation work for a return rate of ten percent:
ln(2) / ln(1 + r) = K / r
ln(2) / ln(1 + .1) = K / 0.1
K = [ln(2) / ln(1.1)] x 0.1
K = .72
And that's it!
