The rule of 72: compounding in doublings
Compounding is hard in your head and easy in doublings. Money at r percent doubles in about 72 ÷ r years; once you count doublings, 8% for 27 years is three of them, so eight times. Know where the rule is off before you quote it.
- Estimate a doubling time from a rate, and a rate from a doubling time, and know the exact versions
- Estimate a future value by counting doublings
- Apply the rule to inflation: the halving time of purchasing power
- Say which way the rule errs at low and high rates
The intuition
At 8% a year money doubles in about 72 ÷ 8 = 9 years; the exact answer is 9.01. At 6% the rule says 12 years and the exact is 11.9. Run it backwards: to double in six years you need about 72 ÷ 6 = 12% a year, exactly 12.25%. The rule works because ln 2 is 0.693 and ln(1 + r) is a little under r, and 72 rounds the gap while dividing cleanly by 2, 3, 4, 6, 8, 9 and 12.
The power of the rule is counting. $10,000 at 8% for 27 years is 27 ÷ 9 = three doublings, so about $80,000; the exact figure is $79,900. Half a doubling is ×1.41, a quarter ×1.19, so fractions of a doubling are easy too. Inflation at 3% halves the purchasing power of cash in about 24 years: after 24 years a dollar buys what 49 cents buys today. The rule is best near 8%; it runs slightly long at low rates (36 years at 2%, against 35.0) and slightly short at high rates (3 years at 24%, against 3.2).
Doubling time ≈ 72 ÷ rate%; exact = ln 2 ÷ ln(1 + r). Rate to double in T years ≈ 72 ÷ T; exact = 2^(1/T) − 1. Years to grow k-fold ≈ log₂(k) × 72 ÷ rate%. Value after n years ≈ P × 2^(n ÷ doubling time). Purchasing power after n years = 1 ÷ (1 + i)ⁿ, halving in about 72 ÷ i% years.
Why it works
- The conventions here: annual compounding throughout. Rule of 72: doubling time ≈ 72 ÷ the rate in percent. Exact doubling time = ln 2 ÷ ln(1 + r). Estimates that use the rule count doublings. Inflation erodes purchasing power as 1 ÷ (1 + i)ⁿ, and the halving time uses the same rule.
- Where 72 comes from. Doubling means (1 + r)ᵗ = 2, so t = ln 2 ÷ ln(1 + r) = 0.693 ÷ ln(1 + r). For small r, ln(1 + r) ≈ r − r²/2, a little under r, so the true divisor is a little over 69.3 and rises with the rate. 72 is close to it around 8% and has many divisors.
- Why it runs long at low rates and short at high ones. The true divisor is about 70 at 2% and about 75 at 24%. Below roughly 7.85% the rule's 72 is too big; above it, too small. For very low rates 70 or 69 is the better divisor.
- Doublings add, values multiply. Two doublings is ×4, three is ×8; k-fold is log₂(k) doublings. Split fractions of a doubling: half is √2 ≈ 1.41, a quarter ≈ 1.19.
- The error scales with the doublings. Each doubling carries the same small error, so three doublings carry three times as much. Check the exact figure when it matters.
- Rates are more sensitive when low. 72 ÷ 2 is 36 years and 72 ÷ 3 is 24: one extra point saves twelve years. 72 ÷ 12 is 6 and 72 ÷ 13 is 5.5: the same point saves half a year.
| Doubling at 8%: 72 ÷ 8 | 9 years (exact 9.01) |
| Doubling at 6%: 72 ÷ 6 | 12 years (exact 11.90) |
| Rate to double in 6 years: 72 ÷ 6 | 12% (exact 12.25%) |
| $10,000 at 8% for 27 years: 3 doublings | $80,000 (exact $79,881) |
| Inflation at 3%: halving time 72 ÷ 3 | 24 years; 49.2% of purchasing power left |
| At 2%: rule 36 years, exact 35.0 | the rule runs long |
| At 24%: rule 3 years, exact 3.22 | the rule runs short |
The formulas
The rule, and what it approximates.
The rule run backwards.
Count the doublings, then multiply by the doubling time.
Doublings add; values multiply.
The same rule on inflation.
Worked example
Years divided by the doubling time is the number of doublings; two to that power times the start is the estimate, then the exact figure. The follow-up asks what to do with a fraction of a doubling.
See it move
Same drill. Change the rate, the years, the starting amount, the target for doubling, and the inflation rate.
- Raise the rate. Both doubling times fall, the doublings over the years rise, and both ending values rise. Watch which of the two doubling-time bars is longer as you cross about 8%.
- Add years. The doublings and both ending values rise; the purchasing power left falls.
- Raise the starting amount. Only the ending values move, in proportion.
- Shorten the years in which to double. The rate needed rises, by the rule and exactly.
- Raise inflation. The halving time falls and less purchasing power is left.
Run it backwards
A client wants to double their money in a stated number of years. What return do they need, by the rule and exactly?
Run the rule backwards: 72 ÷ years. Exactly, 2 to the power 1 ÷ years, minus one. The rule's answer is a little high for long horizons and a little low for short ones, the mirror of its error on doubling times.
Then say whether the number is realistic. Above about 10% a year is more than a diversified equity portfolio has returned over long runs; it needs risk or a longer horizon.
Traps
Say it in the interview
“How long does money take to double at 8%, and what is $10,000 worth after 27 years?”
Check yourself
6 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Doubling time ≈ 72 ÷ rate%; exact = ln 2 ÷ ln(1 + r). Backwards: 72 ÷ years.
- Count doublings: k-fold is log₂(k) of them; half a doubling is ×1.41.
- The rule runs long below about 7.85% and short above; at 2% use 70.
- Inflation halves cash's purchasing power every 72 ÷ inflation years.