The rule and the exact formula
The Rule of 72 says money growing at r% a year doubles in about 72 ÷ r years. It comes from the exact doubling condition (1 + r)^t = 2, which gives t = ln 2 ÷ ln(1 + r).
For continuous compounding the exact answer is t = ln 2 ÷ r ≈ 69.3 ÷ r%, which is why the Rule of 69.3 is exact in that case. 72 is used because it is close to the annual-compounding answer at typical rates and divides evenly by 2, 3, 4, 6, 8, 9 and 12. The Rule of 70 sits between the two.
In reverse, the rate needed to double in t years is about 72 ÷ t, against the exact r = 2^(1/t) − 1.
Worked example and error
At 8% a year the Rule of 72 gives 72 ÷ 8 = 9 years. The exact answer with annual compounding is ln 2 ÷ ln 1.08 = 9.0065 years, so the rule is off by less than a week. With continuous compounding the time is 0.6931 ÷ 0.08 = 8.66 years.
At 36% the rule gives 2 years but the exact answer is 2.2542 years, an error of about 11%. At 2% the rule gives 36 years against an exact 35.0028 years. To double in 10 years you need 7.2% by the rule and 7.1773% exactly.
| Rate | Rule of 72 (years) | Exact (years) |
|---|---|---|
| 2% | 36.00 | 35.00 |
| 8% | 9.00 | 9.01 |
| 36% | 2.00 | 2.25 |
Limits
The rule assumes one constant rate, reinvested growth and no withdrawals, fees or taxes. Real investment returns vary from year to year and are not guaranteed. The same arithmetic works for anything growing at a steady rate, such as prices rising with inflation.