mkl654321
mkl654321
Joined: Aug 8, 2010
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August 19th, 2010 at 10:30:19 AM permalink
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.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
thecesspit
thecesspit
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August 19th, 2010 at 11:21:18 AM permalink
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.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
Wizard
Administrator
Wizard
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August 19th, 2010 at 12:38:51 PM permalink
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.
It's not whether you win or lose; it's whether or not you had a good bet.
pacomartin
pacomartin
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August 19th, 2010 at 2:23:44 PM permalink
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.
kmcd
kmcd
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March 29th, 2012 at 11:18:46 PM permalink
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.
i0r0retardod
i0r0retardod
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March 30th, 2012 at 1:19:15 AM permalink
Aint you one of the mods on ATS.
edward
edward
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March 30th, 2012 at 1:32:10 AM permalink
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..

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