In finance, the rule of 72 is a simple method of calculating the approximate number of periods over which a quantity will double. If you divide 72 by the expected growth rate, expressed as a percentage, the answer is approximately the number of periods to double the original quantity. For instance, if you were to invest $100 at 9% per annum, then your investment would be worth $200 after 8.0432 years, using an exact calculation. The rule of 72 gives 72/9=8 years, which is close to the exact answer. See time value of money.

On the other hand if you were to leave $100 uninvested when inflation was 9% per annum, the purchasing power of your $100 would have halved after 8 (72/9) years.

Derivation

The future value is given by

where PV is the present value, t is the number of time periods, and r stands for the discount rate per time period.

Now, suppose that the money has doubled, then FV = 2 PV. Substituting this in the above formula and cancelling the factor PV on both side yields

2 = (1 + r)^t.

This equation is easily solved for t:

If r is small, then ln(1+r) approximately equals r (this is the first term in the Taylor series). Together with the approximation ln 2 ≈ 0.693, this gives

So for very small rates, 69.3 would be more accurate than 72. For higher rates, a bigger numerator would be better (e.g. for 20%, using 76 to get 3.8 years would be more accurate than 3.6). 72 is reasonable approximation across this range and is easily divisible by many numbers.

See also

• exponential growth

External links

• Exponentialist article "The Scales Of 70", which extends the rule of 72 beyond fixed-rate growth to variable rate compound growth including positive and negative rates.