What is the Rule of 72?
The Rule of 72 is a mental-math shortcut for estimating how many years it takes an investment to double at a fixed annual return. Just divide 72 by your expected return rate. At 8%, your money doubles in roughly 72 ÷ 8 = 9 years. At 6%, it takes about 12 years. At 12%, just 6.
The rule works because of how exponential growth behaves: doubling time is closely tied to the natural logarithm of 2 (≈ 0.693). The number 72 is convenient because it divides cleanly by 1, 2, 3, 4, 6, 8, 9, and 12 — most realistic return rates.
Doubling time at common rates
| Annual return | Years to double (Rule of 72) | Exact years |
|---|---|---|
| 2% | 36.0 | 35.0 |
| 4% | 18.0 | 17.7 |
| 6% | 12.0 | 11.9 |
| 8% | 9.0 | 9.0 |
| 10% | 7.2 | 7.3 |
| 12% | 6.0 | 6.1 |
Frequently asked questions
- How accurate is the Rule of 72?
- Very accurate for return rates between 4% and 12%, where it's typically off by less than 1%. For very low or very high rates, the exact formula (years = ln(2) / ln(1 + r)) is more precise.
- Does the Rule of 72 work for inflation?
- Yes — it works for any compounding rate. At 3% inflation, your money loses half its purchasing power in 72 ÷ 3 = 24 years.
- Where does the rule come from?
- It dates back to at least 1494, when Italian mathematician Luca Pacioli mentioned it in his treatise Summa de Arithmetica. The math has been useful ever since.
The 500-year-old shortcut, derived
The full formula for doubling time is t = ln(2) / ln(1 + r). For small rates, ln(1 + r) ≈ r, so t ≈ ln(2) / r ≈ 0.693 / r. Multiplying numerator and denominator by 100 gives 69.3 / r%. Pacioli rounded to 72 in 1494 because 72 divides evenly by 1, 2, 3, 4, 6, 8, 9, and 12 — a much friendlier number for mental math five centuries before pocket calculators.
Modern variants exist for slightly higher precision: the Rule of 70 is used by demographers because population growth rates are usually in the 1-3% range. TheRule of 69.3 is the mathematically exact version for continuous compounding.
Real-world applications beyond investing
Anywhere something compounds, the Rule of 72 applies. A few practical examples:
| Scenario | Rate | Time to double |
|---|---|---|
| U.S. credit-card APR (avg) | 24% | 3 years |
| Inflation cuts purchasing power in half | 3% | 24 years |
| GDP growth (China, 1980-2010) | 10% | 7 years |
| Population growth (global, 2024) | 0.9% | 80 years |
| High-yield savings account | 4.5% | 16 years |
The same arithmetic explains why carrying credit-card debt is so dangerous: at 24% APR, an unpaid balance doubles every three years — faster than almost any investment can grow it back.
Common mistakes
Using the rule with very high rates
At 25%+ returns the approximation breaks down. Use the exact ln(2)/ln(1+r) formula or a calculator.
Mixing nominal and real rates
A 10% nominal return at 3% inflation only doubles your purchasing power in ~10 years, not 7.2. Decide which one you care about, then apply the rule.
Forgetting that contributions change the math
The Rule of 72 assumes a lump sum sitting and growing. If you're also adding money each month, your portfolio doubles faster than the rule suggests.
Sources & further reading
- Luca Pacioli — Summa de Arithmetica (1494)— The first known printed reference to the doubling-time shortcut.
- Federal Reserve — Consumer Credit (G.19)— Official source for U.S. credit card APR averages cited above.
- World Bank — GDP Growth Data— Country-level growth rates for the doubling-time examples.