What is the rule of 72?

The rule of 72 is a quick mental shortcut: dividing 72 by the yearly rate of return, in percent, gives roughly the number of years it takes an investment to double. At 6% a year, money doubles in about 12 years; at 9%, in about 8.

The formula

T_{\text{double}} \approx \frac{72}{r}

where $r$ is the yearly return in percent. It also works the other way round: to double money within a given number of years, the required yearly return is about 72 divided by that number.

Where it comes from

The exact doubling time is the $T$ for which $(1 + r)^T = 2$ , with $r$ as a decimal:

T = \frac{\ln 2}{\ln(1 + r)} \approx \frac{0.693}{r}

For small returns, $\ln(1 + r)$ is close to $r$ , which would give a "rule of 69.3". The number 72 is used instead because it corrects the approximation at typical returns of 5–10%, and because it divides evenly by 2, 3, 4, 6, 8, 9 and 12, which makes mental arithmetic easy.

How accurate it is

Yearly return	Rule of 72	Exact doubling time
+2	36.0 years	35.0 years
+4	18.0 years	17.7 years
+6	12.0 years	11.9 years
+8	9.0 years	9.0 years
+10	7.2 years	7.3 years
+15	4.8 years	5.0 years

The approximation tracks the exact answer closely between about 4% and 12% - the range most investors care about.

Between about 4% and 12% it is accurate to within a few months.

A simple illustration

€20,000 invested at 8% a year doubles roughly every nine years:

20{,}000 \rightarrow 40{,}000 \rightarrow 80{,}000 \rightarrow 160{,}000

after about 9, 18 and 27 years. At 4%, the same €20,000 would reach only about €58,000 after 27 years: halving the return far more than halves the long-term result.

Other uses

Inflation: at 3% inflation, prices double — and the purchasing power of money halves — in about 24 years.
Costs: yearly costs of 2% cut an investment's final value roughly in half over 36 years, compared with the same investment without costs.
Debt: a loan at 12% interest doubles in about six years if nothing is repaid.

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