The Rule of 70 is a simple mental‑math tool that estimates how many years it will take for a quantity to double given a constant annual growth rate. You calculate it by dividing 70 by the annual growth rate (expressed as a percent)
Years to double ≈ 70 / (annual growth rate in percent)
The Rule of 70 is widely used because it gives a quick, intuitive answer without logarithms. It is a rough estimate, most accurate when growth rates are moderate and approximately constant.
Key takeaways
– Purpose: Provides a quick estimate of the doubling time for investments, populations, GDP, etc.
– Formula: Years to double ≈ 70 ÷ growth rate(%).
– Origin: Numeric approximation of the exact formula based on natural logs (ln2 ≈ 0.6931).
– Alternatives: Rules of 69 and 72 are similar approximations; 69 is tied to continuous compounding, 72 is a convenient mental‑math number for many integer rates.
– Limitations: Assumes a constant growth rate and ignores volatility, taxes, fees, inflation (unless you use a real rate), and changing compounding frequency.
Formula and calculation of the Rule of 70
Exact doubling time (discrete compounding)
t = ln(2) / ln(1 + r)
• r is the growth rate as a decimal (for 5% use r = 0.05).
– ln(2) ≈ 0.693147.
Approximation used by the Rule of 70
Because ln(1 + r) ≈ r for small r, the exact expression becomes:
t ≈ 0.693147 / r (r as decimal) → multiply numerator and denominator by 100 to use r as a percent:
t ≈ 69.3147 / r%
Rounding 69.3147 to a convenient whole number yields the Rule of 69, 70 or 72 (70 is common for ease of use).
What the Rule of 70 can tell you
– How long it takes for an investment (or any quantity growing at a steady rate) to double.
– A quick comparison between different growth rates (which will double faster).
– An intuitive sense of the long‑term impact of small differences in growth rates.
Examples of how to use the Rule of 70
1) Investment example
– Rate: 5% per year.
– Rule of 70: Years to double ≈ 70 / 5 = 14 years.
– Exact discrete compounding: t = ln(2)/ln(1.05) ≈ 14.21 years.
Interpretation: The rule gives a close and usable estimate.
2) Population example (from source)
– U.S. growth rate example: 0.62% per year.
– Rule of 70: 70 / 0.62 ≈ 112.9 → about 113 years for the population to double at that growth rate.
3) GDP example
– Country growing at 10% per year.
– Rule of 70: 70 / 10 = 7 years for GDP to double (assuming real growth and a constant rate).
Rule of 70 vs. real growth
If you want doubling time in real (inflation‑adjusted) terms, use the real growth rate rather than the nominal rate. Convert nominal rate and inflation to a real rate:
real rate ≈ (1 + nominal) / (1 + inflation) − 1
Then use the Rule of 70 with the real rate (in percent). This matters because nominal returns that look large may be eroded by inflation.
Compound interest and the Rule of 70
– Compound interest accelerates doubling relative to simple interest. Rule of 70 implicitly assumes reinvestment (compound growth).
– If interest is not reinvested (simple interest), doubling time = 100 / r% (because at simple interest you need 100% total return).
– For continuous compounding, exact doubling time = ln(2) / r (with r as decimal), which equals about 69.3147 / r% — this is why the Rule of 69 is a better approximation under continuous compounding assumptions.
Practical steps to use the Rule of 70
1) Clarify what you’re measuring
• Investment balance, real GDP, population, a business metric, etc.
2) Determine the appropriate growth rate
• Use the expected annual growth rate in percent.
• For investments, choose an expected annual return after taxes and fees if you want an after‑cost estimate.
• For “real” doubling (purchasing power or GDP), use the real growth rate adjusted for inflation.
3) Pick a rule or compute exact doubling time
• Quick mental estimate: use Rule of 70 (70 ÷ r%).
• Continuous compounding: use 69.31 or Rule of 69 (69.31 ÷ r%).
• If you want exact, compute t = ln(2) / ln(1 + r) with r as decimal.
4) Run the calculation and interpret
• Example: expected after‑fees return = 7%. Years to double ≈ 70 / 7 = 10 years.
• Compare outcomes for different expected rates (sensitivity testing).
5) Stress‑test and update assumptions
• Check doubling time under lower/higher rates (e.g., −1% and +1%).
• Revisit periodically as real returns and macro conditions change.
6) Apply results to decisions
• Use doubling time to set time horizons for goals (retirement, education).
• If the doubling time is longer than your horizon, consider changing allocations or expectations.
Limitations of the Rule of 70
– Assumes a constant growth rate indefinitely — not realistic for most investments or economies.
– Ignores volatility: average returns with high volatility do not behave the same as steady returns (sequence‑of‑returns risk).
– Does not account for taxes, fees, or cash flows (contributions/withdrawals).
– Less accurate at very high growth rates (approximation breaks down).
– Uses nominal rates unless you explicitly convert to real rates — inflation can materially change outcomes.
– For precise planning, use the exact logarithmic formula and include realistic assumptions about compounding frequency, taxes, fees, and variability.
How is the Rule of 70 used in economics?
– Estimate how long a country’s GDP will take to double using the annual real GDP growth rate.
– Estimate long‑term population doubling if you have a steady population growth rate.
– Provide intuitive communication in public policy and planning to show how small differences in growth rates compound over time.
Difference between the Rule of 70 and the Rules of 69 and 72
– Rule of 69: Closest to the exact continuous‑compounding doubling time because it stems from ln(2) ≈ 0.6931 → 69.31 when using percent units. Use when rates are continuously compounded or when you want the best theoretical accuracy.
– Rule of 70: A convenient round number approximation of the 69.31 value — easy to remember, slightly simpler.
– Rule of 72: Historically popular because 72 has many small integer divisors (2, 3, 4, 6, 8, 9, 12), which makes mental math especially convenient for certain rates (e.g., 6, 8, 9, 12). In practice, 72 gives very good approximations in many common cases.
The bottom line
The Rule of 70 is a fast, useful heuristic for estimating doubling time for investments, populations, GDP, and other exponential growth phenomena. It simplifies the exact logarithmic calculation into a single division: 70 divided by the growth rate (percent). Use it for quick comparisons and rough planning, but recognize its limitations: it assumes a steady growth rate and ignores taxes, fees, volatility, and inflation unless you explicitly adjust for them. For precise planning use the exact formulas and realistic scenarios, and revisit assumptions periodically.
Sources
– Investopedia. “Rule of 70.” (source text provided)
– Worldometers. “United States Population” (population figure referenced in the example)
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.