Rule of 72

In finance, the rule of 72, the rule of 70 and the rule of 69 are methods for estimating an investment's doubling time. The rule number ( e.g., 72 ) is divided by the interest percentage per period to obtain the approximate number of periods (usually years) required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations and when only a basic calculator is available.

These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations. They can also be used for decay to obtain a halving time. The choice of number is mostly a matter of preference, 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible. There are a number of variations to the rules that improve accuracy. For periodic compounding, the exact doubling time for an interest rate of r per period is$T = \frac{\ln(2)}{\ln(1+r)}$,where T is the number of periods required. The formula above can be used for more than calculating the doubling time. If you want to know the tripling time, for example, simply replace the constant 2 in the numerator with 3. As another example, if you want to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5.

CHOICE OF RULE

The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less accurate at higher interest rates.
For continuous compounding, 69 gives accurate results for any rate. This is because ln(2) is about 69.3%; see derivation below. Since daily compounding is close enough to continuous compounding, for most purposes 69, 69.3 or 70 are better than 72 for daily compounding. For lower annual rates than those above, 69.3 would also be more accurate than 72.

RateActual YearsRule of 72Rule of 70Rule of 69.372 adjustedE-M rule
0.25%277.605288.000280.000277.200277.667277.547
0.5%138.976144.000140.000138.600139.000138.947
1%69.66172.00070.00069.30069.66769.648
2%35.00336.00035.00034.65035.00035.000
3%23.45024.00023.33323.10023.44423.452
4%17.67318.00017.50017.32517.66717.679
5%14.20714.40014.00013.86014.20014.215
6%11.89612.00011.66711.55011.88911.907
7%10.24510.28610.0009.90010.23810.259
8%9.0069.0008.7508.6639.0009.023
9%8.0438.0007.7787.7008.0378.062
10%7.2737.2007.0006.9307.2677.295
11%6.6426.5456.3646.3006.6366.667
12%6.1166.0005.8335.7756.1116.144
15%4.9594.8004.6674.6204.9564.995
18%4.1884.0003.8893.8504.1854.231
20%3.8023.6003.5003.4653.8003.850
25%3.1062.8802.8002.7723.1073.168
30%2.6422.4002.3332.3102.6442.718
40%2.0601.8001.7501.7332.0672.166
50%1.7101.4401.4001.3861.7201.848
60%1.4751.2001.1671.1551.4891.650
70%1.3061.0291.0000.9901.3241.523