Rule of 72

Analysis - Rule of 72


The "Rule of 72" is a rule of thumb that can help you compute when your money will double at a given interest rate. It's called the rule of 72 because at 10%, money will double every 7.2 years.
To use this simple rule, you just divide the annual interest into 72. For example, if you get 6% on an investment and that rate stays constant, your money will double in 72 / 6 = 12 years. Of course you can also compute an interest rate if you are told that your money will double in so-and-so many years. For example, if your money has to double in two years so that you can buy your significant other that Mazda Miata, you'll need 72 / 2 = 36% rate of return on your stash.
Like any rule of thumb, this rule is only good for approximations. Next we give a derivation of the exact number for the case of an interest rate of 10%.

We want to know how long it takes a given principal P to double given either the interest rate r (in percent per year) or the number of years n. So, we are solving this equation:

P * (1 + r/100) ^ n = 2P

Note that the symbol '^' is used to denote exponentiation (2 ^ 3 = 8). Since we said we'll try the case of r = 10%, we're solving this:

P * (1 + 10/100) ^ n = 2P

We cancel the P's to get:

(1 + r/100) ^ n = 2

Continuing:

(1 + 10/100) ^ n = 2
1.1 ^ n = 2

From calculus we know that the natural logarithm ("ln") has the following property:

ln (a ^ b) = b * ln ( a )

So we'll use this as follows:

n * ln(1.1) = ln(2)
n * (0.09531) = 0.693147

Finally leaving us with:

n = 7.2725527

Which means that at 10%, your money doubles in about 7.3 years. So the rule of 72 is pretty darned close.
You can solve the equation for other values of r to see how rough of an approximation this rule provides. Here's a table that shows the actual number of years required to double your money based on different interest rates, along with the number that the rule of 72 gives you.

% Rate	Actual	Rule 72
1	69.66	72
2	35.00	36
3	23.45	24
4	17.67	18
5	14.21	14.4
6	11.90	12
7	10.24	10.29
8	9.01	9
9	8.04	8
10	7.27	7.2
..	..	..
15	4.96	4.8
20	3.80	3.6
25	3.11	2.88
30	2.64	2.4	(note: 10pct error)
40	2.06	1.8
50	1.71	1.44	(note: 19pct error)
75	1.24	0.96
100	1.00	0.72	(note: 38pct error)