Rule of 72
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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
- <math> FV = PV \cdot (1+r)^t, <math>
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
- <math> 2 = (1+r)^t. <math>
This equation is easily solved for t:
- <math> t = \frac{\ln 2}{\ln(1+r)}. <math>
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
- <math> t = \frac{0.693}{r}. <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
External links
- Exponentialist article "The Scales Of 70" (http://members.optusnet.com.au/exponentialist/The_Scales_Of_70.htm), which extends the rule of 72 beyond fixed-rate growth to variable rate compound growth including positive and negative rates.