# Today I Learned: The Rule of 72

I came over a .pdf on linkedin, which I’m unfortunately unable to find again, containing some lessons of investing from Benjamin Graham’s mentor1. One of them was the rule of 72. The rule of 72 is used to find out how long it takes to double your investment given an annual return rate:

$\frac{72}{\text{Rate of Return}} \approx \text{Years to Double}$

I never heard about it before, but it works pretty well for a reasonable range of returns. This plot shows how close the curves are:

This made me wonder where this rule came from and how one could come up with such a useful heuristic. It was referenced already in Summa de Arithmetica2 by Luca Pacioli in 1494, also applied to investments. The rule is not explained so it is assumed that the rule predates the book. Fascinating!

The number is somehow derived from $$log(2) = 0.693147 \dots$$, which is a transcendental number. Wikipedia also refers to the rule of 69.3, which is more exact. However, it is impractical. 72 is preferable over 69 or 69.3 as it is a mental arithmetic heuristic for the simple reason that 72 is divisible by 2, 3, 4, 6, 8, 9, 12 (and 18, 24, 36, but the heuristic comes from a list of tips for the investor; not the speculator…). However, logarithms were not really around in Europe in 1494. They only started being used from the 17th century onwards. So how would one compute $$log(2)$$ in those days?

1. Benjamin Graham wrote The Intelligent Investor and is also well-known as Warren Buffett’s mentor. ↩︎