The rule of 72

An investor often needs a quick estimate: in how many years does a capital double at a given rate? The rule of 72 provides an answer in one’s head, without a calculator.

Observation — Rule of 72

To estimate how many years are needed to double a capital at the rate $i\%$ under the compound regime: $t_{2\times}\approx 72/i$. At $6\%$: $\approx 12$ years; at $4\%$: $\approx 18$ years; at $8\%$: $\approx 9$ years. It derives from $\ln 2/\ln(1+i)\approx 0{,}693/i$ for small $i$, and $72$ is chosen because it has many divisors and gives “round” results for typical rates. We shall revisit it in the chapter on logarithms.

The approximation arises from $\ln 2 \approx 0{,}693$ and from $\ln(1+i)\approx i$ for small rates; the number $72$ (rather than $69{,}3$) is preferred because it is divisible by many common rates. The link with logarithms will be clarified later on.

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Topics: Percentages
Concepts: Compound interest · Rule of 72
Skills: Estimating