Rule of 72

October 2, 2007

The Rule of 72 is a helpful approximation used to determine the time frame required for an investment to double. It’s a nice trick to remember when a quick calculation is necessary. Simply divide 72 by the prevailing interest rate (expressed as 100*r) and the solution is the resulting time necessary for the investment to double. The derivation is simple for compound and continuous compound problems. The future value of an asset with an interest
rate compounded per annum for t years is given by,
FV=(1+r)^{t}PV.
To determine the time period at which the future value of the asset is doubled, we simply solve the previous equation for t while setting the future value to twice that of the present value.
2PV=(1+r)^{t}PV
2=(1+r)^{t}
ln (2)=t\ln (1+r)
t=\frac{\ln (2)}{\ln (1+r)}

A first order Taylor series approximation for ln(1+r) is simply r, and
easily simplifies our result,
t\approx \frac{\ln (2)}{r}.
Given that the ln(2)=0.693, the rule can be applied as follows%
t\approx\frac{69.3}{100\ast r}
t\approx\frac{72}{100\ast r}.
72 or 70 are commonly used for the convenience of division, and the approximation is relatively accurate. For cases involving continuous compounding the derivation is even more straight forward.
FV=e^{rt}PV
2PV=e^{rt}PV
2=e^{rt}
t=\frac{\ln (2)}{r}
Because interest is compounded continuously a Taylor series approximation isn’t necessary, and the rule applies similarly,
t=\frac{69.3}{100\ast r}
t\approx\frac{72}{100\ast r}.
As an example, assume you want to put $1,000 in an account accruing an annual percentage rate of 8%. According to our rule, it would take around 9 years to double the investment to $2,000. The actual calculation is 8.66 years.

Posted by firstorderconditions
Filed in Econ Theory
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