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 (1 + 10/100) ^ n = 2 1.1 ^ n = 2
From calculus (bear with me here) 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|
|30||2.64||2.4||(note: 10pct error)|
|50||1.71||1.44||(note: 19pct error)|
|100||1.00||0.72||(note: 38pct error)|
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