Invest FAQ: Analysis: Rule of 72


	Articles
		index previous next full text
	Browse
		the bookshelf
	Join
		the email list
	Surf
		invest links
	Tour
		for beginners for investors for traders
	About
		what is it? what's new? contributors contact
	Search
		Enter your search terms Submit search form

Subject: Analysis - Rule of 72

Last-Revised: 19 Feb 1998
Contributed-By: Chuck Cilek (ccilek at nyx10.nyx.net), Chris Lott (contact me), Richard Alpert

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)

Previous article is Analysis: Return on Equity ver..
Next article is Analysis: Same-Store Sales Category is Analysis
Index of all articles