# 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 | |
---|---|---|---|

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) |

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