## Rule of 72

Posted: December 27, 2018 in Mathematics

People love money. Specially when the money gets double. So, coming up with an estimate of the time it takes to double the money helps an average person to manage their investment. However, the math to calculate the amount of time to double the money from an investment that yields a fix rate of return could be quite complex for some individuals. Fortunately, the Rule of 72 comes in handy for this purposes.

$\text{Time for Investment to Double} = \dfrac{72}{\text{Rate of Return}}$

For example, if a person invested a certain amount of money at a fixed annual interest rate of 9%, the Rule of 72 will give around 72 / 9 = 8 years for the investment to be doubled. Where the actual calculation should be log(2) / log(1.09) = 8.043, but it’s pretty damn close.

Let’s look at another example, you deposit $100 into a saving account with 4% p.a., the Rule of 72 says it will take about 72 / 4 = 18 years for your money to become$200. The actual calculation should be log(2) / log(1.04) = 17.67, again it’s close enough.

The Rule of 72 is very useful in situation like when calculators aren’t around because there is no way I can calculate log(2) / log(1.04) in my head but I can manage to compute 72 / 4 mentally.

The reason that this rule works is because of the following:

First, we are trying to find the linear approximation of $\log(1+r)$.

Let $f(x) = \log(x)$, then first order taylor series around $a=1$ would be

$f(x) \approx f(1) + f'(1)(x-1) = x-1$

hence,

$\log(x) \approx x-1, \quad x\approx 1$

or

$\log(1+r) \approx r, \quad r\approx0$

So, $2 = \left(1+\frac{r}{100}\right)^t$ becomes

$t = \dfrac{\log 2}{\log(1+r/100)} = \dfrac{\log 2}{r/100}\cdot\dfrac{r/100}{\log(1+r/100)} \approx \dfrac{100 \log 2}{r} \approx \dfrac{69.31}{r} \approx \dfrac{72}{r}$

The number 72 is chosen because in fact it is close to 69.31 and 72 has 12 divisors that is considered a lot. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 divide 72.