Rule of 72

Question

An investment compounds at 8% per year. Roughly how long until it doubles? State the rule, use it, and explain why it works.

Accepted Answer

Rule of 72: doubling time approx frac72textrate in %. At 8%: t approx frac728 = mathbf9 text years. (Exact: fracln 2ln 1.08 = frac0.69310.0770 = 9.006, the rule is almost eerily good here.) Why it works: solving (1+r)^t = 2 gives t = fracln 2ln(1+r) approx frac0.693r for small r, which suggests a "rule of 69.3". Using 72 instead does two jobs: it partially corrects the ln(1+r) < r error for typical rates (5–12%), and 72 is divisible by 2, 3, 4, 6, 8, 9, 12, built for mental division. (For continuous compounding, 69.3 is exact; for very high rates use ~76.) Companion reflexes: money at r% for t years approx 1 + rt for small rt (simple interest approximation), and e^x approx 1 + x + fracx^22 for quick compounding corrections. Trading interviews use these to test whether you can move between exact and approximate appropriately, knowing the error direction matters more than decimal places.

Rule of 72

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