Doubling in your head
How long until an investment doubles? The exact answer needs a logarithm, but there is a shortcut that bankers have used for more than five centuries: divide 72 by the annual rate in percent. Luca Pacioli, the Italian mathematician who described double-entry bookkeeping, mentioned it in 1494.
The formulas
Rule of 72: years to double ≈ 72 ÷ rate (%)
Exact, annual compounding: years = ln 2 ÷ ln(1 + r)
With n compounding periods a year, the exact time is ln 2 ÷ [n × ln(1 + r/n)]; with continuous compounding, ln 2 ÷ r, which is where the rule of 69.3 comes from.
Turned around, the rule gives the rate needed to double in a given time: rate ≈ 72 ÷ years. The exact rate is 2^(1/years) − 1.
Worked example
At 8% a year, compounded annually:
- Rule of 72: 72 ÷ 8 = 9.00 years.
- Exact: ln 2 ÷ ln 1.08 = 9.01 years, almost the same.
- Rule of 70: 8.75 years; rule of 69.3: 8.66 years, closer to continuous compounding.
- Tripling takes 14.27 years.
- To double in 10 years, you need 7.18% a year; the rule of 72 says 7.2%.
How accurate is it?
| Rate | Rule of 72 | Exact (annual) |
|---|---|---|
| 2% | 36.0 years | 35.0 years |
| 4% | 18.0 years | 17.7 years |
| 8% | 9.0 years | 9.0 years |
| 12% | 6.0 years | 6.1 years |
| 24% | 3.0 years | 3.2 years |
The rule is nearly exact around 8%, slightly long at low rates and slightly short at high ones. For quick estimates that is more than good enough.
Everyday uses
- Retirement savings: at a 7% average return, money doubles about every 10 years, so $10,000 at age 30 could be about $80,000 by 60, before inflation.
- Inflation: at 3% inflation, prices double, and cash loses half its purchasing power, in about 24 years.
- Debt: a credit card balance at 24% APR doubles in about 3 years if nothing is paid.
- Fees: a 1% annual fee takes about 70 years to eat half your money, but over a 30-year horizon it still costs roughly a quarter of the final balance.
Limits
Real returns are not steady: a stock fund averaging 8% may double in 6 years or in 15. Taxes, fees and inflation reduce the real growth rate, so use an after-tax, after-fee and, if you want buying power, inflation-adjusted rate.
This calculator gives estimates for education and planning. It is not investment, tax or legal advice; for decisions about your own situation, talk to a qualified financial professional.