## Saturday, August 20, 2011

### Compound Interest

You may have heard of the famous rule of 72 (about compound interest). If an investment returns X% annually, then the number of years required to double the principal is 72/X. So if you get 7% on something, it will take you approximately 72/7 = 10 years to double your initial investment.

A friend asked me yesterday what the origin of this was, and I figured it wouldn't be hard to discover what someone had already invented.

Consider the standard compound interest formula with compounding rate represented x=X/100 (in decimal rather than %), final amount = initial amount * (1+x)^n, where ^ denotes a power operation. Since we are looking for a double (final amount = 2 * initial amount), we can take the logarithm of both sides of the equation as:

log 2 = n * log (1+x)

For small values of x (compared to 1), log(1+x) is approximately x (from Taylor series expansion). Hence,

n = log(2)/x = 0.693/x = 69.3/X

We probably use 72 instead of 69.3 because it has a large number of factors.