# 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.

% RateActualRule 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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