RatedrillRule of 72
The rule of 72 says money doubles in about 72 / r years, where r is the annual return in percent: at 6% it doubles in roughly 12 years, at 9% in 8. The exact answer is ln(2) / ln(1 + r). The rule is accurate to within months for rates from 4% to 12%.
Syntax
years to double = 72 / rate (in %) exact: n = ln(2) / ln(1 + r)| Part | What it is |
|---|---|
rate | Annual growth rate in percent (use 6, not 0.06) |
Examples
Rate = 6%
72 / 6Result: 12 years (exact: ln 2 / ln 1.06 = 11.90 years)
Excel exact: =NPER(6%, 0, -1, 2)
Rate = 9%
72 / 9Result: 8 years (exact: 8.04 years)
Reverse: what rate doubles money in 4 years?
72 / 4Result: 18% (exact: 2^(1/4) - 1 = 18.92%)
Common mistakes
- Entering the rate as a decimal (72 / 0.06 = 1,200) instead of a percent; the rule uses the whole-number percentage.
- Using the rule for very high rates: at 36% the rule says 2 years but the true answer is 2.25 years; for rates above 20% use the exact formula.
- Applying it to simple interest; the rule assumes compounding. With simple interest, doubling takes 100 / r years.
Practice Rule of 72
These drills check your answer on live data, free, no account needed.
- Exact time to doubleAt 2.5% per year, compounded annually, exactly how many years does it take for money to double? Give the exact answer to 2 decimals, not a rule of thumb.
- Years to reach a goalYou invest $2,000 at 10% per year, compounded annually. How many years will it take to grow to $3,000? Give the exact figure to 2 decimals.
Questions
Why 72 and not 69 or 70?
The exact constant for continuous compounding is 100 x ln(2) = 69.3. Seventy-two is used because it is close enough for typical rates and divides evenly by 2, 3, 4, 6, 8, 9 and 12.
How long does it take to triple money?
Use the rule of 114: years to triple is about 114 / r. At 7% that is 16.3 years (exact 16.24). For quadrupling use 144 / r, which is simply two doublings.
Related
- Compound interestThe compound interest formula A = P(1 + r/n)^(nt), what each symbol means, examples for yearly and monthly compounding, the Excel FV function and errors.
- Future valueHow to calculate the future value of a lump sum with FV = PV x (1 + r)^n, step-by-step examples, the Excel FV function with its sign convention, and errors.
- CAGRCAGR = (ending value / beginning value)^(1/years) - 1. Examples for investments and revenue growth, Excel RRI, and why CAGR differs from the average return.
- Real vs nominal returnThe real return strips inflation out of a nominal return: real = (1 + nominal) / (1 + inflation) - 1. Examples, the quick approximation and its errors.