Rule of 72

The rule of 72 is a mental shortcut for compound growth: divide 72 by a yearly growth rate in per cent, and the answer is roughly how many years an amount takes to double. It works on savings, inflation, debt and whole economies alike — one of the few pieces of finance arithmetic that can be done in your head in a second.

The rule in one line

At 6% a year, money doubles in about 72 ÷ 6 = 12 years. At 9%, about eight years. At 3%, about 24. The rule also runs backwards: to double money in six years, you need a return of about 72 ÷ 6 = 12% a year.

The answer is always a number of periods, in the same unit as the rate. Feed the rule a monthly rate and it returns months; feed it a yearly rate — the usual case — and it returns years.

Doubling is a useful unit because doublings stack. Money that doubles every 12 years is four times larger after 24 years and eight times larger after 36. Long horizons that are hard to picture as percentages become easy to count as doublings, which is why the rule turns up wherever compound interest does.

Why 72, and how accurate it is

The exact doubling time for a rate r, compounded once a year, is ln 2 ÷ ln(1 + r), where ln is the natural logarithm. Few people carry logarithms around, so the rule replaces the whole expression with one division.

The number comes from ln 2, which is about 0.693. If growth were continuous — compounding at every instant — the exact doubling time would be 69.3 ÷ the rate. Compounding once a year grows a little more slowly than that, so the honest numerator creeps upward as rates rise, reaching 72 at around 8%. And 72 has one more thing going for it: it divides cleanly by 1, 2, 3, 4, 6, 8, 9 and 12. That convenience is a large part of why 72, not 69 or 70, is the number people remember.

Between roughly 4% and 12% — the range that covers most savings rates, bond yields and long-run stock returns — the rule is right to within a few months. Outside that range it drifts: slightly too long at very low rates, increasingly too short at high ones. A simple correction keeps it sharp: add 1 to 72 for every three percentage points above 8%, and subtract 1 for every three below. At 11%, 73 ÷ 11 gives 6.64 years, against an exact 6.64; at 5%, 71 ÷ 5 gives 14.2, against an exact 14.2.

Where it came from

The rule is older than modern finance. Its earliest known appearance is in Luca Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita, printed in Venice in 1494 — the same book that set out double-entry bookkeeping for a wide readership. Pacioli gives the rule as a way to find how many years money earning a given yearly rate of interest takes to double, and offers no derivation. That silence suggests merchants were already using it; who first worked it out is unknown.

Nobody in Pacioli’s Venice could have explained why it works. Logarithms, which supply the reason, were not published until 1614. For more than a century, the rule of 72 was simply a trick that gave the right answer.

Four places the rule earns its keep

Saving and investing. A fund returning 7% a year doubles in roughly ten years; one returning 6% takes twelve. Over 36 years that single percentage point — about the size of a high expense ratio — is the difference between ending with about 11.4 times the starting sum and about 8.1 times. A one-point fee costs close to 30% of the final pot, which is the drag fund costs put on returns expressed in a single number.

Inflation. Run the rule on inflation and it says how fast prices double — or, put the other way, how fast cash loses half its buying power. At 3% inflation that takes about 24 years, which is inside many retirements; at 6%, about 12. Seeing the halving date is the plainest cure for money illusion, the habit of judging money by its face value rather than what it buys.

Debt. The rule works on what you owe as well as on what you own. A credit-card balance charged 24% APR — about 2% a month — doubles in roughly three years if nothing is repaid: 72 ÷ 2 = 36 months. Card issuers usually work interest out daily or monthly; how credit-card interest is calculated covers the detail, but the doubling date barely moves.

Economies. Economists apply the same arithmetic to GDP growth, usually as the “rule of 70”, because their growth rates are small and closer to continuous. An economy growing 7% a year doubles its output in about a decade; one growing 2% a year takes about 35 years.

The gap between 2% and 7% growth is the gap between a generation and a decade.

Where the rule breaks down

The rule is only as good as the rate it is given, and only for the kind of growth it describes.

• Simple interest. The rule assumes interest earns interest. Under simple interest, money doubles when rate × years = 100, so 5% simple interest takes 20 years to double, not 14.
• Very high rates. Above about 20% the rule increasingly understates the time. At 36% it says two years; the truth is two and a quarter. For hyperinflation, where prices can rise tens of per cent a month, use the exact formula.
• Volatile returns. The rate must be the compound (annualised) return, not the simple average of yearly returns. A portfolio that gains 50% and then loses 50% has an average return of zero but is down 25%. Feed an average into the rule and it overstates how fast a volatile investment grows. Once withdrawals begin, the order of returns matters too — see sequence-of-returns risk.
• Nominal versus real. The rule doubles whatever number it is given. A nominal rate doubles nominal money; to see when purchasing power doubles, use the real interest rate, roughly the nominal rate minus inflation. A 7% return with 3% inflation doubles real wealth in about 18 years, not ten.
• Taxes, fees and contributions. Taxes and fees lower the rate, so subtract them first. Regular additions, such as monthly savings into an index fund, are outside the rule entirely: it describes a single lump sum left alone.

The close cousins: 70, 69.3 and 144

The rule of 72 has a small family, each member tuned to a different job.

• Rule of 69.3 — exact for continuous compounding, since 69.3 is 100 × ln 2. Used where interest compounds daily or continuously.
• Rule of 70 — the economists’ version. It is more accurate than 72 at low rates and divides neatly by 2, 5, 7 and 10, which suits growth rates for output and population.
• Rule of 144 — the time to quadruple: two doublings, so 2 × 72. At 8%, money quadruples in about 18 years.

All three answer the same question as the original. The rule of 72 survives as the default because it is accurate across the rates most people meet and divides by the numbers they most often use.

See also