The Rule of 72 is the most famous shortcut in personal finance. Divide 72 by an annual growth rate in percent, and you get roughly how many years it takes money to double. At 8%, 72 ÷ 8 = 9 years. It is fast and memorable, but how close is it really? This guide compares the rule against the exact answer at rates from 1% to 30%. It explains why the rule works, where it drifts, and why you will sometimes see 69.3 or 70 used instead. The exact figures come from the same function that powers our Rule of 72 calculator.
The exact doubling-time formula
With annual compounding at rate r (as a decimal), money doubles when (1 + r)^t = 2. Taking natural logs of both sides and solving for t:
This is the exact answer for annual compounding at a constant rate. The Rule of 72 is an approximation of it:
Why 72 works at all
For small r, ln(1 + r) is very close to r itself. Substituting that into the exact formula gives t ≈ ln(2) ÷ r ≈ 0.693 ÷ r. Writing r in percent, that is 69.3 ÷ rate. So the “natural” constant is 69.3, not 72.
But ln(1 + r) is always a little smaller than r, and the gap grows as r grows. A smaller denominator means a longer doubling time, so 69.3 ÷ rate always underestimates. Bumping the numerator up to 72 compensates for that at typical investment rates. 72 also has a practical advantage: it divides evenly by 1, 2, 3, 4, 6, 8, 9 and 12, which makes mental math easy.
Accuracy table: 72 vs 70 vs 69.3
Each row compares the three shortcuts with the exact annual-compounding doubling time. Error is the absolute difference divided by the exact value.
| Rate | Exact (yrs) | 72 ÷ r | Error | 70 ÷ r | Error | 69.3 ÷ r | Error |
|---|---|---|---|---|---|---|---|
| 1% | 69.66 | 72.00 | 3.36% | 70.00 | 0.49% | 69.30 | 0.52% |
| 2% | 35.00 | 36.00 | 2.85% | 35.00 | 0.01% | 34.65 | 1.01% |
| 3% | 23.45 | 24.00 | 2.35% | 23.33 | 0.50% | 23.10 | 1.49% |
| 4% | 17.67 | 18.00 | 1.85% | 17.50 | 0.98% | 17.32 | 1.97% |
| 5% | 14.21 | 14.40 | 1.36% | 14.00 | 1.45% | 13.86 | 2.44% |
| 6% | 11.90 | 12.00 | 0.88% | 11.67 | 1.93% | 11.55 | 2.91% |
| 7% | 10.24 | 10.29 | 0.40% | 10.00 | 2.39% | 9.90 | 3.37% |
| 8% | 9.01 | 9.00 | 0.07% | 8.75 | 2.85% | 8.66 | 3.82% |
| 9% | 8.04 | 8.00 | 0.54% | 7.78 | 3.30% | 7.70 | 4.27% |
| 10% | 7.27 | 7.20 | 1.00% | 7.00 | 3.75% | 6.93 | 4.71% |
| 12% | 6.12 | 6.00 | 1.90% | 5.83 | 4.63% | 5.77 | 5.58% |
| 15% | 4.96 | 4.80 | 3.22% | 4.67 | 5.90% | 4.62 | 6.85% |
| 20% | 3.80 | 3.60 | 5.31% | 3.50 | 7.94% | 3.46 | 8.86% |
| 25% | 3.11 | 2.88 | 7.28% | 2.80 | 9.86% | 2.77 | 10.76% |
| 30% | 2.64 | 2.40 | 9.16% | 2.33 | 11.68% | 2.31 | 12.56% |
What the table shows
- The Rule of 72 is most accurate near 8%. At 8% the error is just 0.07% (9.00 vs 9.01 years). Between about 5% and 10%, it stays within roughly 1.4%, which covers most stock-market and long-run planning assumptions.
- At low rates it overestimates. At 2% the rule says 36.00 years while the exact answer is 35.00. That is off by 2.85%. In that range 70 does better: at 2%, 70 ÷ 2 = 35.00, almost exactly right (0.01% error). This matters for savings accounts, CDs and inflation.
- At high rates it underestimates. At 20% the rule says 3.60 years versus an exact 3.80 (5.31% error). At 30% the error reaches 9.16%. For very high rates, a common adjustment is to add 1 to the numerator for every 3 points above 8%. At 20%, that gives 76 ÷ 20 = 3.8 years, close to the exact 3.80.
- 69.3 is for continuous compounding. It underestimates at every rate in the table under annual compounding. With continuous compounding, though, the doubling time is exactly ln(2) ÷ r, so 69.3 is the right constant there.
Worked example: doubling at 8%
Example inputs: $10,000 growing at 8% a year for 36 years. The rule says 72 ÷ 8 = 9 years per doubling, so 36 years is four doublings: $10,000 → $20,000 → $40,000 → $80,000 → $160,000. The exact calculation, 10,000 × 1.08^36, gives $159,682. The exact doubling time at 8% is 9.01 years, so 36 years is 4.00 doublings. For an estimate you can do in your head, being within a few hundred dollars over 36 years is remarkable.
Worked example: inflation halving your money
The same rule runs in reverse for inflation. At a constant 3% inflation rate, 72 ÷ 3 = 24 years for prices to double, which means a dollar’s purchasing power halves. Checking with our inflation calculator math, $1,000 of goods today would cost $2,033 after 24 years at 3%. A dollar would buy $0.492 worth of today’s goods. That is slightly less than half, because the precise halving time at 3% is 23.45 years, a little shorter than 24. The rule slightly overestimates at low rates, exactly as the table predicts.
Worked example: comparing two rates
The rule is most useful for comparisons. Suppose you are weighing an account that earns 4% against an investment you assume will earn 8%. Example inputs only:
- At 4%: 72 ÷ 4 = 18 years to double (exact 17.67).
- At 8%: 72 ÷ 8 = 9 years to double (exact 9.01).
Doubling the rate roughly halves the doubling time. Over 36 years, that is two doublings (4×) versus four doublings (16×). This is the clearest way to see why a few percentage points of return, or of fees, matter so much over decades. A 1% annual fee taken out of an 8% return leaves 7%, which stretches the doubling time from about 9 to about 10.3 years.
Where the high-rate adjustment comes from
The “add 1 for every 3 points above 8%” tweak isn’t magic. The exact doubling time is ln(2) ÷ ln(1 + r). A slightly better approximation of ln(1 + r) than plain r is r − r²/2. Using it, the ideal numerator works out to roughly 69.3 + (rate in percent) ÷ 3. At 8% that is 69.3 + 2.67 ≈ 72, which is exactly why 72 is so accurate there. At 20% it is 69.3 + 6.67 ≈ 76, matching the adjusted rule above. At 2% it is about 70, which is why 70 works better for low rates. One formula explains all three constants.
Tripling and quadrupling: the rules of 114 and 144
The same logic works for other multiples. Money triples when (1 + r)^t = 3, so t = ln(3) ÷ ln(1 + r). Since ln(3) ≈ 1.0986, the matching shortcut is about 110 ÷ rate. People often round it to 114 so it matches mid-range rates better. At 8%, 114 ÷ 8 = 14.25 years, and the exact tripling time is 14.27 years. Quadrupling is just two doublings, so the rule becomes 144 ÷ rate: 144 ÷ 8 = 18 years versus an exact 18.01.
The rule works for debt too
Compounding doesn’t care whether the balance is an asset or a debt. Example input: an unpaid balance growing at 24% a year with interest added and no payments. The Rule of 72 says 72 ÷ 24 = 3 years to double. The exact figure is 3.22 years. Here the rule underestimates by about 7%, as the table predicts for high rates. Either way the message is the same: at credit-card-level rates, an untouched balance doubles in roughly three years. Real card interest compounds daily or monthly and depends on payments, so treat this only as an illustration of scale.
When not to rely on it
- Variable returns. Markets don’t deliver a smooth 8%. The rule describes a constant rate, so real outcomes will vary widely around it.
- Contributions. The rule covers a single lump sum. If you add money regularly, use a full projection such as the SCHD calculator or dividend snowball calculator.
- Very high rates. Above about 15%, use the exact formula or the adjusted numerator. Credit-card debt is an example where this matters.
- APR vs APY. If a rate compounds monthly or daily, the effective annual yield is a bit higher than the stated APR. Use the APY in the rule. Our CD calculator shows both figures.
Counting doublings over a long horizon
The real use of the rule is to count doublings. Over a fixed horizon, the number of doublings is the horizon divided by the doubling time, and every extra doubling multiplies the ending balance by two again. Here is a hypothetical $10,000 left alone for 30 years at constant example rates, using the exact doubling time and the calculator’s ending balance:
| Rate (example) | Exact doubling time | Doublings in 30 years | Ending balance |
|---|---|---|---|
| 4% | 17.67 yrs | 1.70 | $32,434 |
| 6% | 11.90 yrs | 2.52 | $57,435 |
| 8% | 9.01 yrs | 3.33 | $100,627 |
| 10% | 7.27 yrs | 4.13 | $174,494 |
Moving from 4% to 8% does not double the ending balance. It roughly doubles the number of doublings, which is a much bigger effect: the 8% balance is $100,627 against $32,434 at 4%. This is why small differences in rate matter so much over long periods, and why the rule is a useful way to build intuition about compounding. For the full mechanics, including contributions and compounding frequency, see our guide to compound interest explained.
Using the rule for costs, not just returns
The rule works on any steady rate, including costs that eat into growth. A one-percentage-point difference in yearly return, whether from fees, cash drag or a lower-return investment, changes the doubling time noticeably. In the hypothetical example above, the gap between 8% and 7% over 30 years is $24,504 on a $10,000 starting balance, all from one point of yearly return. Over a working lifetime, a gap like that can be the difference between reaching a goal on time and needing several extra years of saving, which is why it is worth checking the costs of any fund you hold and comparing them with similar options. The rule makes this easy to see: 72 ÷ 8 = 9 years per doubling, while 72 ÷ 7 ≈ 10.3 years.
The same thinking applies to inflation. Dividing 72 by an inflation rate estimates how long it takes prices to double, or a dollar’s purchasing power to halve. It also applies in retirement planning, where withdrawals work against growth; our safe withdrawal rate guide covers how those two forces interact. In each case the rule gives a quick mental estimate, and the exact formula or a calculator gives the precise answer when it matters.
One caution: the rule assumes a constant rate. Investment returns vary from year to year, and the average of yearly returns is higher than the compound growth rate you actually experience when returns are volatile. If you plug an average annual return into the rule, the estimate of the doubling time will usually be too optimistic. Use a compound annual growth rate where you have one. For example, a portfolio that gains 50% one year and loses 50% the next has an average yearly return of 0%, yet it ends with only 75% of its starting value, because each percentage applies to a different balance. The rule cannot see that difference unless you give it the compound rate. Testing a few rates, such as a cautious, a middle and an optimistic case, gives a range of doubling times that is more honest than any single estimate, and the calculator makes that comparison quick.
How the calculator reports accuracy
Our Rule of 72 calculator shows both numbers side by side for whatever rate you enter. It shows the 72 ÷ rate estimate, the exact ln(2) ÷ ln(1 + r) answer, and the percentage error between them. It also lists each doubling milestone at multiples of the exact doubling time, plus the ending balance after your chosen horizon using principal × (1 + r)^years. Try entering 2%, 8% and 20% one after another. You will see the error shrink to almost nothing and then grow again, which is the pattern in the table above.
Quick reference
For everyday use: use 72 for rates between about 6% and 10%, use 70 for rates under about 5%, and use the exact ln(2) ÷ ln(1 + r) formula, or our calculator, for anything higher or whenever the answer really matters. The rule is a sanity check, not a plan, but it is a very good sanity check. If you use it to think about reinvested dividends, the DRIP compounding guide shows the rule lining up with a full simulation.
