Rule of 72--why 72?

The Rule of 72 states that you divide a rate of return (such as interest on a bond) into 72, and that gives you the number of periods it will take to double your money. For instance, an investment that pays 10% annually will take 7.2 years to double in value (the Rule of 72 assumes that interest is compounded).

My somewhat idle question is, why 72, other than the somewhat trite answer "because that's the number that works when you plug it into the equation"? I have two theories:

1. That number has some kind of logarithmic relation to the concept of doubling a componded investment.
2. The number 72, like the magic Babylonian number 60, can be divided by so many different numbers (factors), and that has something to do with its utility in the formula.

Something to do with the natural log of 2 being ~.69?

A quick wiki search gives me : http://en.wikipedia.org/wiki/Rule_of_72 which suggests 72 is used cos it's easy for common divisors as you suggest, 69.3 works for small discrete values of interest and 70 works for continous rates of growth.

First, the "rule of 72" is an approximation of the time needed to double your money. The following table shows the "rule of 72" values and the exact number of years, for various interest rates.

Interest Rate	Rule of 72	Exact	Difference
0.01	72.00	69.66	2.34
0.02	36.00	35.00	1.00
0.03	24.00	23.45	0.55
0.04	18.00	17.67	0.33
0.05	14.40	14.21	0.19
0.06	12.00	11.90	0.10
0.07	10.29	10.24	0.04
0.08	9.00	9.01	-0.01
0.09	8.00	8.04	-0.04
0.10	7.20	7.27	-0.07
0.11	6.55	6.64	-0.10
0.12	6.00	6.12	-0.12
0.13	5.54	5.67	-0.13
0.14	5.14	5.29	-0.15
0.15	4.80	4.96	-0.16
0.16	4.50	4.67	-0.17
0.17	4.24	4.41	-0.18
0.18	4.00	4.19	-0.19
0.19	3.79	3.98	-0.20
0.20	3.60	3.80	-0.20

Why 72? It doesn't have to be exactly 72. That is just the number that works out best for realistic interest rates you're likely to see on an investment. The rule of 72 works out almost exactly right for an interest rate of 7.8469%. The reason something works in the first place is because the graphs of 1/x and 1/ln(1+x) look very similar for small positive values of x.

Usually if you have some Calculus it is regarded as the first approximation to the Taylor Series for the natural logarithm. It is sometimes called the "rule of 70" since that works reasonably well. As the Wizard says it is a natural for 9 years at 8% and 8 years at 9% which used to be a fairly common value.

It makes it easy to calculate when you will have 8 times you present value at 9% interest. Apply the rule three time and you get 27 years.

As the Wizard stated, 72 is usable for realistic interest rates. 72 is used instead of 71 or 73 for example, because of it's easy divisibility. With an abundant supply of calculators, we could just calculate the exact years to double. But what if you're sitting in a meeting or somewhere else sans calculator? Well, 72 is a multiple of 12, and therefore has many factors. For instance it is divisible evenly by all of the following:

2% = 36 years
3% = 24 years
4% = 18 years
6% = 12 years
8% = 9 years
9% = 8 years
12% = 6 years
18% = 4 years
36% = 2 years

Any rate lower than 2% is unrealistic, and any rate higher than 36% goes from payday loan to mafia.

Aint you one of the mods on ATS.

Its very simple to see how many times it takes to double the money.

If you have 1% return, this means 1.01 - using the windows calculator :-)) 1.01 to the power of 72= 2.047099..

Then an approximation for all other interest rates would be to divide 72 by the return. For example for 5% would be 72/5= 14.4
Verification: 1.05^14.4 = 2.0189..