Rule of 72

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.

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

<math> \begin{matrix} FV &=& PV\cdot(1+r)^t \\ 2PV &=& PV\cdot(1+r)^t \\ 2 &=& (1+r)^t \\ \ln(2) &=& t \cdot \ln(1+r) \\ t &=& \frac{\ln(2)}{\ln(1+r)} \\ \\ \textrm{Taylor} & & \textrm{expansion:}\\ \ln(1+r) &=& r - {r^2 \over 2} + \frac{r^3}{3} \cdots & \approx & r \\ \\ t & \approx & \frac{\ln(2)}{r} & \approx & \frac{0.693}{r} \end{matrix} <math>

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, Taylor series