Finance & Economics · Personal Finance
Rule of 72 Calculator
Estimates the number of years required to double an investment given a fixed annual interest rate, using the Rule of 72 approximation.
Calculator
Formula
t is the approximate number of years required to double the investment. r is the annual interest rate expressed as a percentage (e.g., enter 6 for 6%). The constant 72 is a well-established approximation that works best for interest rates between 1% and 20%. For more precision at higher rates, the Rule of 70 or Rule of 69.3 can be used instead.
Source: Pacioli, L. (1494). Summa de arithmetica. Widely attributed as the earliest documented use; reinforced in financial mathematics textbooks including Brealey, Myers & Allen — Principles of Corporate Finance.
How it works
The Rule of 72 derives from the mathematics of compound interest. When money grows at a fixed annual rate r (expressed as a percentage), the exact doubling time is given by the logarithmic formula t = ln(2) / ln(1 + r/100). Since ln(2) ≈ 0.6931, and the denominator approximates to r/100 for small rates, this simplifies to roughly 69.3/r. The number 72 is preferred over 69.3 in practice because it has more integer divisors (1, 2, 3, 4, 6, 8, 9, 12), making mental arithmetic much easier — for instance, 72 divides cleanly by 6, 8, 9, and 12, corresponding to common benchmark interest rates.
The formula is simply t ≈ 72 / r, where t is the approximate doubling time in years and r is the annual interest rate in percent. This approximation is remarkably accurate for rates between roughly 1% and 20%. At 6%, the Rule of 72 gives 12.0 years while the exact answer is 11.90 years — an error of less than 1%. At 20%, the approximation gives 3.6 years versus the exact 3.80 years, still within 5%. The approximation degrades at very low rates (below 1%) and very high rates (above 25%), where the Rule of 69.3 or the exact logarithmic calculation becomes more appropriate.
Beyond investing, the Rule of 72 applies to any quantity growing at a constant percentage rate: GDP growth, inflation eroding purchasing power, population growth, or even the spread of debt. For inflation, dividing 72 by the inflation rate tells you how many years it takes for the cost of goods to double — equivalently, how many years until your money's purchasing power is halved. Financial educators frequently use this rule to illustrate why starting to invest early matters so dramatically: at a 9% average annual return, money doubles roughly every 8 years, meaning a 25-year-old investor has approximately four doubling periods before age 57.
Worked example
Suppose you invest $10,000 at an annual return of 8%. Using the Rule of 72:
Step 1: Identify the annual rate: r = 8%
Step 2: Apply the formula: t ≈ 72 / 8 = 9 years
So your $10,000 is expected to grow to approximately $20,000 in about 9 years.
Verification with exact formula: t = ln(2) / ln(1.08) = 0.6931 / 0.07696 ≈ 9.006 years. The Rule of 72 is essentially perfect at 8% — it overestimates by less than 0.07%.
Extended example — multiple doublings: At 8% annual return, starting with $10,000:
• After 9 years: ~$20,000
• After 18 years: ~$40,000
• After 27 years: ~$80,000
• After 36 years: ~$160,000
This illustrates why long investment horizons are so powerful. An investor who starts 9 years earlier effectively doubles their final outcome.
Inflation example: If inflation runs at 3% per year, then 72 / 3 = 24 years for prices to double — meaning your $50 grocery bill today will cost approximately $100 in 24 years.
Limitations & notes
The Rule of 72 is an approximation and has several important limitations to keep in mind. First, it assumes a constant, fixed annual rate of return — real-world investments fluctuate, and sequence-of-returns risk means the actual doubling time can be significantly longer than estimated. Stock market returns, for example, vary dramatically year to year even if the long-run average is stable. Second, the approximation is less accurate at extreme interest rates: below 1% or above 25%, the error grows meaningfully and the exact logarithmic formula should be used. At 1%, the Rule of 72 gives 72 years while the exact answer is 69.7 years — a difference of over 2 years. At 35%, the Rule of 72 gives 2.06 years while the exact answer is 2.31 years. Third, the rule does not account for taxes, fees, or inflation. If your investment earns 8% but you pay 1.5% in fund fees and 2% in taxes on gains, your effective real rate is closer to 4.5%, giving a doubling time of 16 years rather than 9. Always apply the rule to after-tax, after-fee real returns for meaningful personal finance decisions. Fourth, the rule applies to compound interest only, not simple interest. Finally, it should not be used for irregularly timed cash flows or annuities — use proper NPV or IRR analysis in those cases.
Frequently asked questions
Why is the number 72 used in the Rule of 72, and not 69 or 70?
The mathematically precise constant is 69.3 (derived from 100 × ln(2) ≈ 69.315). However, 72 is preferred in practice because it is highly divisible — it divides evenly by 1, 2, 3, 4, 6, 8, 9, and 12, which correspond to many commonly encountered interest rates. This makes mental arithmetic much easier. For example, 72/6 = 12, 72/8 = 9, and 72/9 = 8 are all clean whole numbers. At typical rates between 6% and 10%, 72 also happens to give a slightly more accurate result than 69.3 due to compounding curvature.
How accurate is the Rule of 72 for common investment return rates?
The Rule of 72 is highly accurate for rates between 4% and 12%, with errors typically under 1%. At 6%, it gives 12.0 years versus the exact 11.9 years (error: 0.8%). At 8%, it gives 9.0 years versus exact 9.006 years (error: 0.07%). At 12%, it gives 6.0 years versus exact 6.116 years (error: 1.9%). Accuracy drops at higher rates: at 20%, the error is about 4.5%, and at 30%, the error exceeds 8%.
Can the Rule of 72 be used in reverse to find the required interest rate?
Yes — this is one of the rule's most practical applications. If you want to double your money in a specific number of years, rearrange the formula: r ≈ 72 / t. For example, to double your money in 6 years, you need an annual return of approximately 72 / 6 = 12%. To double in 10 years, you need roughly 7.2% annually. This reverse application is useful for setting investment return targets or evaluating whether a financial product's promised returns are realistic.
Does the Rule of 72 apply to inflation and debt as well as investments?
Absolutely. The Rule of 72 applies to any quantity growing at a constant exponential rate. For inflation at 3% per year, purchasing power halves in approximately 72 / 3 = 24 years — meaning what costs $100 today will cost $200 in 24 years. For debt growing at a credit card rate of 24% APR, the balance doubles in roughly 72 / 24 = 3 years if no payments are made. This makes the rule a powerful tool for understanding the long-run cost of high-interest debt.
What is the difference between the Rule of 72, Rule of 70, and Rule of 69.3?
All three are approximations for doubling time, differing only in the numerator constant. The Rule of 69.3 is the most mathematically exact (100 × ln(2) = 69.315) and gives the best accuracy for continuous compounding. The Rule of 70 is commonly used in economics and demography because 70 is easy to divide by common percentage rates like 2%, 5%, and 7%. The Rule of 72 is preferred in finance and banking because 72 has more integer factors, simplifying mental calculations for common interest rates like 6%, 8%, and 9%. For most practical purposes, all three give acceptable results within the 1%–20% rate range.
Last updated: 2025-01-15 · Formula verified against primary sources.