Rule of 72 Calculator

COMPOUND GROWTH FINANCE RULE OF 72
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By Checked against sources by , Founder & Architect, RECATOOLS Published Updated Last reviewed

Divide 72 by the rate for a quick doubling time, then compare it with the exact figure for interest added yearly, monthly or continuously. Negative rates give a halving time.

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Rule of 72 Calculator

⚠ Disclaimer: Estimates only. Not investment advice. RECATOOLS is not a licensed or registered investment adviser in any jurisdiction. Past performance does not guarantee future results. Trading and investing carry risk of partial or total loss of capital.

The Rule of 72 is a mental-math shortcut: divide 72 by the annual growth rate to get the number of years it takes for money to double. At 8% growth it says 9 years (72 ÷ 8). The tool also computes the exact doubling time, which depends on how often interest is added to the balance, so you can see how far out the shortcut is and in which direction.

% / year

"Monthly" and "Continuously" read your rate as a nominal yearly rate (an APR) that is split across the year. If your rate is already an effective annual rate (EIR, APY or AER), choose "Once a year": for an effective rate that formula is exact however often interest is credited. A negative rate gives the time for a value to halve.

📅 Research current as of 16 Oct 2026 · Sources: Rule of 72: years ≈ 72 ÷ rate%, against the exact years = ln 2 ÷ (m × ln(1 + r/m)) for m rests a year (ln 2 ÷ r continuously). Reference rates: Damodaran, US historical returns 1928-2025 (January 2026).
Rates, regulations, and lender practices change frequently — verify current figures with your provider or licensed advisor before acting.
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Rule of 72 approximation
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Exact doubling time
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Common rates and their doubling times (exact: Once a year)

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How to Use the Rule of 72 Calculator

Pick the direction

Either: enter a rate to find the doubling time. Or: enter a target time and see what rate is required. Then say how often interest is added to the balance: once a year, monthly or continuously.

Enter the value

At 8% / year growth, money doubles in 72/8 = 9 years. Doubling in 10 years requires 72/10 = 7.2% / year growth.

Compare to exact math

With interest added once a year, the exact formula is years = ln(2) / ln(1 + r). There the rule is within 1% of the exact answer from about 5.7% to 10%. Below that band it gives too many years (1.4% too many at 5%); above it, too few (1.9% too few at 12% and 5.3% too few at 20%). With interest added monthly on an APR, the rule gives 2.6% to 3.9% too many years at every rate from 0.5% to 30%, so it is never within 1% there. The result line says which way it is out.

Apply to inflation, debt and investment scenarios

At 3% annual inflation, prices double, and so the purchasing power of cash halves, in 23.4 years; the rule says 24. At the S&P 500's 10.0% compound annual return from 1928 to 2025 (with dividends, before inflation; Damodaran), money doubled about every 7.3 years. An unpaid debt at a 22% APR doubles in 3.2 years if the interest is added monthly, or 3.5 years if it is added once a year; the rule says 3.3 either way.

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The Rule of 72 — Mental Math for Compound Growth

Where the 72 Comes From

The exact doubling time at a rate r is ln(2) ÷ ln(1 + r). For small r, ln(1 + r) is close to r, so the answer is close to 0.693 ÷ r, or 69.3 ÷ the rate in percent. When interest is added once a year the true figure is a little longer than that, and 72 corrects for it well between about 5% and 10%, while dividing cleanly by 2, 3, 4, 6, 8, 9 and 12. When interest is added monthly or continuously there is almost nothing to correct, and 69.3 or 70 comes closer than 72.

With yearly compounding the exact formula is years to double = ln(2) ÷ ln(1 + r), where r is the rate as a decimal. At r = 0.08, exact = 0.693 ÷ 0.0770 = 9.006 years. Rule of 72 says 72 ÷ 8 = 9 years. At 8% the two differ by less than 0.1%, and the rule stays within 1% from about 5.7% to 10%. Above about 7.85% the rule gives too few years, and the gap widens with the rate: at 20% it says 3.6 years against an exact 3.8. Below 7.85% it gives too many, and the gap widens as the rate falls: at 2% the rule says 36 years against an exact 35.0, a full year (2.8%) too many. Below about 4.9% the rule of 70 is the closer of the two.

The rule is old. It is printed in Luca Pacioli's Summa de arithmetica (Venice, 1494), folio 181 recto, problem 44, which can be read in the Wellcome Collection's digitised copy. Pacioli says to keep 72 in mind and divide it by the yearly interest, with interest added at each year's end, and works two examples: 6% doubles in 12 years and 8% in 9. He gives no derivation.

Three Practical Uses

Investment planning. At 6.5% a year, added once a year, the rule says money doubles every 72 ÷ 6.5 ≈ 11 years. Over 40 years that is about 3.6 doublings, so a single USD 50,000 grows to about USD 620,000 (1.065⁴⁰ ≈ 12.4). The rule makes this intuition immediate: it converts an abstract "compounding" idea into a concrete "money doubles every X years" mental model.

Inflation impact. Inflation compounds the cost of living, so the rule applies to prices just as it does to savings. At 3% annual inflation, prices double in 72 ÷ 3 = 24 years by the rule, or 23.4 years exactly. At 5% inflation, prices double in about 14 years — the same coffee costing twice as much. Halving the purchasing power of cash is the same event as doubling prices, so it takes the same time; it is not a halving calculation.

Debt warning. The arithmetic runs fast on high-rate debt. Take a USD 5,000 balance at a 22% APR with no payments. The rule says it doubles in 72 ÷ 22 = 3.3 years. If interest is added once a year, the exact figure is 3.5 years and the balance after 10 years is about USD 36,500, 7.3 times the start. If interest is added monthly, the exact figure is 3.2 years and the balance after 10 years is about USD 44,200, 8.8 times the start. The rule gives too few years in the first case and too many in the second. Check your agreement for how your lender adds interest. This is why the minimum-payment trap is so destructive: any unpaid interest compounds at the same rate.

Compound growth is a curve, not a line: each doubling adds more than all the growth before it put together.

When the Rule Breaks Down

The Rule of 72 assumes compound growth at a constant annual rate. In real markets, returns vary year-to-year: in Damodaran's S&P 500 series (with dividends), annual returns since 1928 have ranged from −43.8% (1931) to +52.6% (1954), and 2008 was −36.6%. Over long periods (20+ years), the average converges, but in any short window the rule's prediction can be off by years. For variable-return scenarios, use our DCA/SIP Calculator which models actual monthly contributions and variable returns more carefully.

For a value that falls each year, 72 divided by the size of the fall estimates how long it takes to halve, and it is less accurate there than for doubling. Falling 5% a year (with the loss taken once a year), a value halves in 13.5 years; the rule says 14.4, 0.9 years (6.6%) too many. Falling 7% a year, it halves in 9.6 years against the rule's 10.3 (7.7% too many). Enter a negative rate in the calculator to see the exact halving time.

Added yearly, interest keeps the rule within 0.1% at 8%; monthly, it is 3.5% out

01

At 20%, added once a year, the rule says 3.6 years; the exact answer is 3.8.

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Exact formula, interest added once a year: years = ln(2) ÷ ln(1 + r). Monthly: ln(2) ÷ (12 × ln(1 + r/12)). Approximate: years ≈ 72 ÷ rate%.

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Added once a year, the approximation is within 1% from about 5.7% to 10%; at 5% it gives 1.4% too many years and at 12% 1.9% too few.

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The S&P 500 returned 10.0% a year compounded from 1928 to 2025, with dividends (Damodaran): a doubling about every 7.3 years.

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At the Fed's 2% inflation target, prices double in about 35 years.

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At a 22% APR with interest added monthly, an unpaid debt doubles in 3.2 years — the math behind the minimum-payment trap.

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Rule of 70 is a variant used in some economics textbooks; 69.3 is near-exact for continuous compounding, because 100 × ln(2) = 69.31.

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Added once a year, above about 7.85% the rule undershoots: at 15% it says 4.8 years against an exact 4.96.

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At 22% the rule says 3.3 years. Added once a year the exact answer is 3.5, so the rule is 6.1% short; added monthly it is 3.2, so the rule is 2.9% long.

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At 7% a year, money doubles about 3.9 times in 40 years (1.07⁴⁰ ≈ 15).

Frequently Asked Questions

  • Math: the exact doubling-time formula is ln(2) ÷ ln(1 + r). Multiply by 100 to convert rate-as-decimal to percentage: 100 × ln(2) ≈ 69.3. So "Rule of 69.3" would be mathematically perfect — but 72 is easier to divide mentally (it has many factors: 2, 3, 4, 6, 8, 9, 12...). When interest is added once a year, the 72 vs 69.3 difference creates a small bias that cancels the second-order error in the approximation exactly at about 7.85%, and nearly so between about 5.7% and 10%. So with yearly compounding, Rule of 72 is more accurate than Rule of 69.3 from about 3.9% upwards. With monthly or continuous compounding there is no such error to cancel, and 69.3 is the more accurate of the two at every rate from 0.5% to 20%.
  • It depends on how often interest is added. Added once a year, the rule is within 1% from about 5.7% to 10%. At 8%, rule says 9 years, exact says 9.006 — 0.07% too few. At 6%, rule says 12, exact says 11.90 — 0.9% too many. At 15%, rule says 4.8, exact says 4.96 — 3.2% too few (the rule undershoots above about 7.85%). Around the S&P 500's long-run 10.0% (Damodaran, 1928-2025) the rule is about 1% short. Added monthly on an APR, the rule gives 2.6% to 3.9% too many years at every rate from 0.5% to 30%: at 8% it says 9 years against an exact 8.69. Use this tool's "exact" figure when you need precision, the rule when you're doing mental math.
  • Yes — it's the same math working in the other direction. At 3% inflation, prices double in 72 ÷ 3 = 24 years. At 7% inflation, prices double in just 10 years. For purchasing-power planning, this is the math that matters: a retirement target of USD 1M today requires roughly USD 2.4M in 30 years at 3% inflation, and about USD 4.3M at 5%. The rule makes the impact of inflation feelable in a way that abstract "compounding inflation" doesn't.
  • No one can tell you the future return, so test a few. For the past, Damodaran's data for 1928 to 2025 gives compound annual returns of 10.0% for the S&P 500 with dividends, 4.5% for 10-year Treasuries and 3.4% for 3-month T-bills, all before inflation. Subtract the inflation you expect to get a real rate: at 5% real, the Rule of 72 says money doubles every 14.4 years.
  • No. The rule operates on whatever rate you give it. If you want post-tax doubling time, use the after-tax return; if you want post-fee, subtract management fees first. For example, if 2.5 points of a 7% return go on fees and tax, the 4.5% that remains doubles in 16 years instead of about 10. This is why expense ratios and tax efficiency matter so much over multi-decade horizons.
  • Rule of 70 is closer to the mathematical exact (which is 69.3, since 100 × ln(2) ≈ 69.3). Rule of 72 has more integer factors and is easier to divide mentally (72 = 8×9 = 6×12 = 4×18). Some economics textbooks prefer Rule of 70. With interest added once a year, 70 is the closer of the two below about 4.9% and 72 above it: at 2.5%, 70 gives 28.0 years against an exact 28.07, while 72 gives 28.8. Neither is negligible everywhere: at 15%, 72 is 3.2% too short and 70 is 5.9% too short. With interest added monthly or continuously, 70 is the closer of the two at every rate from 0.5% to 30%.
  • For a value that falls each year, 72 divided by the size of the fall estimates how long it takes to halve, and it always gives too many years. Falling 5% a year (loss taken once a year), a value halves in 13.5 years; the rule says 14.4 (6.6% too many). At −10% it halves in 6.6 years against the rule's 7.2 (9.4% too many). The error is larger than for doubling at the same rate, so use the exact figure: enter a negative rate above. Deflation is different: falling prices raise the purchasing power of cash rather than halving it.
  • The quote is often attributed to Albert Einstein, though the attribution is disputed. The intuition behind the metaphor: linear thinking expects money that doubles in 9 years to triple in 18 and quadruple in 27 — compounding gives 4× in 18 years, 8× in 27 and 16× in 36. Compound growth is exponential, not linear, and the Rule of 72 makes this immediate. USD 10,000 invested at 8% real return for 36 years (4 doublings) becomes USD 160,000 — far more than the linear intuition suggests. The "wonder" is that compounding produces outcomes that feel disproportionate to the inputs, until you do the math.
  • The math is universal; only the rate and how often interest is added change, and bank and bond rates move often, so use the rate your bank or the bond's prospectus quotes today. The low rates common on savings are where the rule of 72 is weakest. Singapore's CPF Board, for example, quotes 2.5% a year on the Ordinary Account and 4% on the Special, MediSave and Retirement Accounts for October to December 2026 (the legislated minimum and the floor rate), and says interest is computed monthly and compounded yearly. Leaving out extra interest, 2.5% compounded yearly doubles in 28.07 years: the rule of 72 says 28.8 (2.6% too many) and the rule of 70 says 28.0. At 4% the exact figure is 17.67 years; 72 says 18.0 and 70 says 17.5. This is arithmetic on quoted rates, not a forecast. Comparing savings in different currencies also means weighing the exchange-rate risk, which the Rule of 72 does not capture.

Method & sources

How it computes

Rule of 72: years = 72 / |rate%|, rate = 72 / years. Exact, m rests a year (1 yearly, 12 monthly on an APR): years = ln 2 / (m ln(1 + r/m)), ln 2 / r continuously; rate = m(2^(1/(m years)) - 1). A negative rate gives the halving time. The error is signed: too many or too few years. Reference rows: Damodaran 1928-2025 (S&P 500 10.02%, 10-year Treasuries 4.54%, 3-month T-bills 3.37%). DoublingKernel, tested.

What this tool implements

  • Corrected arithmetic: the rule is within 1% only from about 6% to 10% (the page said 5-12%; at 5% it is 1.4% out, at 12% 1.9%); at higher rates it undershoots, not overshoots; 7% for 40 years is 3.9 doublings, not 3.6; USD 50,000 at 6.5% for 40 years is about USD 620,000, not USD 700,000+.
  • Corrected against sources: the Fed's inflation target is 2%, not 3%; S&P 500 annual returns since 1928 range from -43.8% (1931) to +52.6% (1954), not -38% to +37% (Damodaran). Removed as unverified: the Pacioli 1494 origin (MacTutor confirms the Summa and its bookkeeping, not the rule), the Einstein quote, 'post-COVID peak' inflation figures, a Vanguard forecast, typical fee and tax drags, and ASEAN savings rates.
  • Corrected 2026-10-16: the exact figure now states its compounding - once a year, monthly on an APR, or continuous; an effective annual rate (EIR/APY) uses the yearly formula. With monthly compounding the rule gives 2.6-3.9% too many years at every rate from 0.5% to 30%. The error is shown with its direction; a negative rate gives an exact halving time (the page had said the rule "predicts negative doubling times"); yearly, the rule undershoots above about 7.85%, not 10%; 10-year Treasuries 4.54%, not 4.53%.
  • Restored 2026-10-16, now checked on the printed page: the rule appears in Pacioli's Summa de arithmetica (Venice, 1494), fol. 181r, problem 44, with interest added once a year and examples at 6% (12 years) and 8% (9 years), and no derivation. CPF Board's quoted 2.5% and 4% (Oct-Dec 2026, compounded yearly) are used as arithmetic test points only.

Sources

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