The Rule of 72 is most accurate for annual rates between about 5% and 12%, where its estimate lands within roughly 1% of the exact answer. Below 5% it slightly overestimates the doubling time, and above 12% the error grows, reaching about 5% too high near a 20% rate. For the precise benchmark, the true doubling time is ln(2) / ln(1 + r), which the shortcut is designed to approximate.
The exact benchmark to measure against
Doubling time has an exact formula. If r is the rate written as a decimal (8% = 0.08), the number of periods to double is:
t = ln(2) / ln(1 + r)
Here ln is the natural logarithm and ln(2) is about 0.6931. The Rule of 72 is a clever rearrangement of this: because ln(1 + r) is close to r for small rates, the exact formula simplifies to roughly 0.693 / r. Multiplying by 100 gives about 69.3 / (rate as a percent), and 72 is chosen because it is close to 69.3 yet divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is why mental math is easy. Run any rate through the Rule of 72 calculator to see the estimate next to that benchmark.
The accuracy table
Here is the Rule of 72 estimate against the exact ln(2)/ln(1+r) doubling time, with the error at each rate. Every figure below is recomputed:
| Annual rate | Rule of 72 | Exact (ln 2 / ln(1+r)) | Error |
|---|---|---|---|
| 2% | 36.0 yrs | 35.0 yrs | +2.8% |
| 4% | 18.0 yrs | 17.7 yrs | +1.9% |
| 6% | 12.0 yrs | 11.9 yrs | +0.9% |
| 8% | 9.0 yrs | 9.01 yrs | -0.1% |
| 10% | 7.2 yrs | 7.27 yrs | -1.0% |
| 12% | 6.0 yrs | 6.12 yrs | -1.9% |
| 15% | 4.8 yrs | 4.96 yrs | -3.2% |
| 20% | 3.6 yrs | 3.80 yrs | -5.3% |
The sweet spot is unmistakable: around 8% the error is almost zero (within 0.1%), and across the everyday 5%-to-12% range the error never exceeds about 2%. That is the band where most stock-market and balanced-portfolio assumptions live, which is exactly why the rule earned its reputation.
Which way does the error point?
At low rates the Rule of 72 reads high, and at high rates it reads low relative to a higher base like 72 itself -- but the practical takeaway is simpler. For rates under 8%, the shortcut gives a doubling time that is slightly too long (conservative). For rates above 8%, it gives a doubling time that is slightly too short, so it is mildly optimistic. The further you move from 8% in either direction, the bigger the gap.
- 2% rate: the rule says 36 years, reality is 35 -- about a year too pessimistic.
- 20% rate: the rule says 3.6 years, reality is 3.8 -- about two and a half months too optimistic.
For a savings account or bond yield near 2%-4%, that small bias barely matters. For a speculative 20%-plus assumption, the rule is only a rough sketch, and you should compute the exact balance with the Compound Interest Calculator instead.
A quick fix for high rates
To sharpen the estimate at higher rates, nudge the numerator up: add about 1 to 72 for every 3 percentage points the rate sits above 8%. At a 16% rate that means using roughly 74 instead of 72. Check it: 74 / 16 = 4.625 years, versus the exact 4.67 years -- far closer than 72 / 16 = 4.5 years. This adjustment is rarely worth the trouble for everyday rates, but it explains why some people quote the "Rule of 70" or even "Rule of 69.3" for very low, continuously compounding rates: those numbers track the ln(2) benchmark more tightly at the bottom of the range.
Why it is an estimate, not a calculator
The Rule of 72 answers exactly one question: roughly how long to double? It deliberately trades precision for speed. It does not tell you your ending balance, it does not handle contributions, and it does not account for taxes or fees. Those are jobs for a real projector. When you need the actual dollar figure, the Future Value Calculator and Compound Interest Calculator apply the full formula with no approximation. Use the Rule of 72 calculator for the instinct, then those tools for the decision.
When the small error is fine vs. when it is not
| Situation | Rule of 72 good enough? |
|---|---|
| Mental check of a 7% portfolio doubling time | Yes -- error under 1% |
| Comparing two investments at 5% vs 9% | Yes -- the gap is what matters |
| Estimating doubling at 20%+ returns | No -- use exact compounding |
| Reporting a precise retirement balance | No -- use a full calculator |
| Planning around 2%-4% inflation | Yes -- within a fraction of a year |
For the formal definitions behind compound growth and why the natural log appears, the U.S. SEC's investor.gov compound interest reference is a trustworthy, ad-free source.
The bottom line
The Rule of 72 is genuinely accurate where it counts -- within about 1% from 5% to 12%, and nearly perfect at 8%. Treat it as a fast, reliable instinct for doubling time, lean on the ln(2)/ln(1+r) benchmark when you want to know the exact answer, and switch to a full compound interest or future value calculation whenever real dollars are on the line.
Try it yourself
Run your own numbers in the free Rule of 72 Calculator — instant, private, no sign-up.
Open the Rule of 72 Calculator →Frequently asked questions
- How accurate is the Rule of 72?
- It is most accurate for rates between about 5% and 12%, where the estimate is within roughly 1% of the exact answer. At 8% the error is almost zero, while at 20% the shortcut is about 5% too optimistic.
- What is the exact formula for doubling time?
- The exact doubling time is t = ln(2) / ln(1 + r), where r is the rate as a decimal and ln is the natural logarithm. The Rule of 72 is a simplified approximation of this formula.
- Why is 72 used instead of 69.3?
- The exact shortcut works out to about 69.3 divided by the rate, but 72 is used because it divides evenly by 2, 3, 4, 6, 8, 9 and 12, making mental math far easier. The small trade-off in precision is usually worth the speed.
- At what rate is the Rule of 72 most accurate?
- It is essentially exact around 8%, where 72 / 8 = 9.0 years matches the precise 9.01 years within 0.1%. Accuracy stays high across the surrounding 5%-to-12% range.
- Does the Rule of 72 overestimate or underestimate?
- For rates below 8% it slightly overestimates the doubling time, giving a conservative answer; above 8% it slightly underestimates, giving an optimistic one. The error grows the further the rate is from 8%.
- How can I make the Rule of 72 more accurate at high rates?
- Add about 1 to the numerator for every 3 percentage points the rate is above 8%. At a 16% rate, using 74 instead of 72 gives 4.625 years, much closer to the exact 4.67 years than 72 / 16 = 4.5.
- Should I use the Rule of 72 or a full calculator?
- Use the Rule of 72 for a fast mental estimate of doubling time, and a full compound interest or future value calculator when you need exact dollar amounts. The rule estimates time to double but cannot compute a precise ending balance.
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