## Sunday, July 21, 2013

### Financial Mathematics: The Rule of 72

At some time or another you've probably heard of the "rule of 72". I'm going to give a derivation of this result and show how good of an approximation it is.

The rule of 72 can be used to answer one of two questions:
1. How long does it take to double my money given a particular rate of return (interest)?
2. What rate of return (interest rate) is required to double my money within a particular period of time?
To begin with, we'll need  the formula we derived for continuous compounding interest:
$$A(t)=A(0)e^{it}$$
Now we need to double our money so we need $a(t)=\frac{A(t)}{A(0)}=2$:
$$e^{it}=2$$
Next we'll take the natural log of both sides:
$$it = \ln 2 \approx .693147$$
We'll let $I=100 \times i$ and substitute that in and then we shall have derived . . .
$$It \approx 69.3147$$
. . . the rule of 69.3147 ???

OK, so we need one more mathematical technique which is formally called bull-shitting:
$$\require{cancel} It \cancel{\approx}= \cancel{69.3147} 72$$
Why do that?

For starters, "The Rule of 72" has a much catchier name than "The Rule of the Natural Log of 2" or "The Rule of $69.3147 . . .$". But the main reason is that 72 is just a more convenient number to use as it is divisible by 2, 3, 4, 6, 8, 9, 12, 18, 24 and 36. Those are all pretty common interest rates and/or periods of time that you might consider.

The way to use is this: if you want to, say, double your money in 8 years, you'll need to earn a $9\%$ rate of return ($9 \times 8 = 72$).

So how accurate is it? Here's a quick comparison (using the dividers of 72):

So it's not exact but it's pretty close. It's good enough to give you a quick and dirty approximation if you don't happen to have a calculator handy.