The shortcut against the exact doubling formula, rate by rate, and what it shows about fees and inflation over a working life.
Very accurate in the range that matters for long-term investing. At 7 per cent the rule of 72 gives a doubling time of 72 ÷ 7 = 10.29 years against an exact 10.24, an error of about half a month. The rule is exact at 7.85 per cent, drifts to 2.8 per cent too slow at 2 per cent and to 5.3 per cent too fast at 20 per cent.
Worked in full in Stock Market Investing for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The rule answers one question: how many years does it take for money compounding at a fixed annual rate to double? Everything below uses illustrative rates with annual compounding and no withdrawals. Market returns are never fixed, so treat the doubling time as a property of the rate, not a promise about any particular decade.
| Input | Value |
|---|---|
| Annual return, illustrative | 7% |
| Compounding | annual |
| Starting amount | 10,000 |
| Horizon, years | 40 |
| Inflation, illustrative | 3% |
Money doubles when (1 + r)t = 2. Take logarithms of both sides and solve for t.
Exact: t = ln 2 ÷ ln(1 + r)
At 7 per cent: 0.6931 ÷ 0.0677 = 10.24 years
Rule of 72: t ≈ 72 ÷ (rate in per cent) = 72 ÷ 7 = 10.29 years
Check: 1.0710.24 = 2.0000, while 1.0710.29 = 2.0055
Excel: =LN(2)/LN(1+7%) for the exact figure, or =NPER(7%,0,-1,2)
Why 72? For small rates ln(1 + r) is close to r, so the exact answer is close to 0.6931 ÷ r, which is the rule of 69.3. That version is exact for continuous compounding: 69.3 ÷ 7 = 9.90 years, the same as ln 2 ÷ 0.07. With annual compounding ln(1 + r) is a little smaller than r, so the true numerator drifts upwards with the rate. Around the returns people actually use for equities, it sits near 72. And 72 has a practical advantage: it divides cleanly by 2, 3, 4, 6, 8, 9 and 12.
| Rate | Exact | Rule of 72 | Error, years | Error | Rule of 70 |
|---|---|---|---|---|---|
| 2% | 35.00 | 36.00 | +1.00 | +2.8% | 35.00 |
| 4% | 17.67 | 18.00 | +0.33 | +1.9% | 17.50 |
| 6% | 11.90 | 12.00 | +0.10 | +0.9% | 11.67 |
| 7% | 10.24 | 10.29 | +0.04 | +0.4% | 10.00 |
| 8% | 9.01 | 9.00 | −0.01 | −0.1% | 8.75 |
| 10% | 7.27 | 7.20 | −0.07 | −1.0% | 7.00 |
| 12% | 6.12 | 6.00 | −0.12 | −1.9% | 5.83 |
| 15% | 4.96 | 4.80 | −0.16 | −3.2% | 4.67 |
| 20% | 3.80 | 3.60 | −0.20 | −5.3% | 3.50 |
Between 4 and 12 per cent the rule of 72 is within 2 per cent of the truth, and between 6 and 10 per cent within about 1 per cent. It crosses zero error at 7.85 per cent. The rule of 70 is better for low rates such as inflation or cash yields, exact at 2 per cent, and steadily worse above it.
If you want one shortcut that holds across the whole table, adjust the numerator by a third of the distance from 8 per cent: 72 + (r − 8) ÷ 3. At 2 per cent that gives 70.00 and 35.00 years; at 12 per cent 73.33 and 6.11 years against 6.12 exact; at 20 per cent 76.00 and 3.80 years, exact to two decimals.
The real use of the rule is counting doublings in your head. At 7 per cent, money doubles roughly every ten years, so 10,000 left alone for 40 years doubles about four times. Four doublings would be 16 times, or 160,000. The precise counts are 3.89 doublings by the rule and 3.90 exactly, so the true figure is 1.0740 × 10,000 = 149,745. The rule's own estimate, 148,140, is within about 1 per cent of it. Rounding the doublings to four is where the larger error comes from, not the rule.
The same arithmetic works on costs. Inflation at 3 per cent halves purchasing power in 72 ÷ 3 = 24 years (23.45 exactly): after 24 years, 100 buys what 49.19 buys today. And it makes fees vivid. A 7 per cent return doubles money every 10.24 years; after one percentage point of fees the 6 per cent net return takes 11.90. Over 36 years that is 3.51 doublings against 3.03: 11.42 times the money against 8.15 times, 28.7 per cent less at the end. Half a doubling sounds small and is not. The full fee calculation is in what one percentage point of annual fees actually costs.
The rule also runs in reverse: the return needed to double money in n years is about 72 ÷ n.
| Years to double | Rule of 72 | Exact |
|---|---|---|
| 5 | 14.40% | 14.87% |
| 8 | 9.00% | 9.05% |
| 10 | 7.20% | 7.18% |
This is the quickest test of a sales pitch. Anyone promising to double your money in five years is promising nearly 15 per cent a year, compounded, every year, which no honest diversified investment can promise.
The error is not in the rule but in what goes into it. People plug in a return before fees and before inflation, then read the answer as if it were real wealth. At 7 per cent nominal and 3 per cent inflation, money doubles in about ten years on the statement but in roughly eighteen in purchasing power. The second mistake is trusting the rule at high rates: at 20 per cent it is a fifth of a year fast, and at 30 per cent or more, use the exact formula.
The rule describes a constant rate. Real portfolios compound unevenly, so the doubling time of a fund over any particular ten years can be far from 72 divided by its long-run average.
Put your own rate, horizon and fee into the blank calculator in the free workbook for this case to see doublings turn into money.
Both estimate doubling time by dividing a constant by the rate in per cent. The rule of 70 is closer for low rates: at 2 per cent it gives 35.00 years, exactly right, where 72 gives 36.00. Above about 5 per cent the rule of 72 is better: at 7 per cent it gives 10.29 years against an exact 10.24, while 70 gives 10.00.
With annual compounding, ln 2 / ln 1.10 = 7.27 years. The rule of 72 says 7.20, about a month short, and the rule of 70 says 7.00. At 10 per cent the rule of 72 understates the true doubling time by about 1.0 per cent.
Yes, in reverse: it tells you how long prices take to double, or money to halve in value. At 3 per cent inflation, 72 / 3 gives 24 years against an exact 23.45, and after 24 years 100 buys what 49.19 buys today. For low rates like this the rule of 70 is slightly closer.
Compounding and costs are the subject of the workbook for chapters 4 and 8 of Stock Market Investing for Beginners. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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