Friday, January 26, 2007

The 72 Rule

The "72 Rule" allows to approximate the number of periods an investment takes to double the capital with a calculation so simple that you can do it in your head without calculator, and especially without exponentials or logarithms.

This article was inspired by this fragment of a "Fools" analysis, as far as I know this approach is original.

What is it?: If X is the return of an investment expressed in percents, then the number of periods it takes to double the capital is 72/X. For example, if you have a "long" position in an index that averages 7.2% returns per year, then it would take 72/7.2 = 10 years to double your money.

Is it a good approximation? Let's check it with the example.

If the average return is 7.2% per year, then after a year the investment becomes 1.072 what it was. After two years, it is 1.072*1.072 = 1.149184 the initial amount. After 10 years, it would be 1.07210 » 2.0042. So, the approximation wasn't that bad, 10 periods gives a growth very close to exactly double. Conversely, the exact answer would be Log1.072»9.97. In this case the rule gave an approximation within 0.3% of the correct answer.

The problem to calculate the logarithm in the appropriate base is that only scientific calculators have logarithms at all, and it takes too many repeated multiplications to come to an approximation.

Now, let us see how good the approximation is, and how to find an equivalent rule for other objectives, such as thrice the initial investment or 1.5.

Warning: Math comes ahead.

F is the multiplicative factor per period, from the percentage X, it is simply (X/100)+1

The "72 Rule" asserts that F0.72/(F - 1) » 2 ==>
(2Log2 F)0.72/(F - 1) » 2 ==> (Taking logarithm base 2)
Log2 F * 0.72 / (F - 1) » 1

Let us suppose both sides are equal and try to calculate the specific F for which the rule will work exactly, then:
Log2 F = (F - 1)/0.72
This equation holds if F = 1, but reveals what is the trick for values close to 1: To approximate the logarithm base 2 of F with a straight line. Let's see how close that line resembles the log2:

the approximation is good because the line has about the same slope of the logarithm at X = 1 (derivative of Log2(X) = 1/(X*Ln(2)), when X = 1, that becomes 1/ln(2) » 1.44, close enough to 1/0.72 » 1.39. Interestingly enough, they are different!

Let us recapitulate: So far, we could have used any number not 72 to try the same approximation and the logarithm base 2 would have coincided with the straight line at X = 1, but the resulting line wouldn't have been close to the logarithm. By the way, the constant which leads to the tangent at X = 1 would be Ln 2, about 0.69. Thus, if you feel like it, speak of the "69 Rule".

Now, this is a bit of "hairsplitting" because minute variations in the slope of the approximating line become even minuter with small variations around 1, but for curiosity let's see where the 72 leads us:

The slope coincides when
1/(X*Ln(2) = 1/0.72 ==> X = 0.72/Ln(2) » 1.03874.

And Log21.03874 » 0.0548 and (X - 1)/0.72 » 0.0538

We have:
1) Both approximation and log are equal at X = 1
2) The slope on the log was about 1/0.69 > 1/0.72, thus the log grows "faster" at X = 1
3) by X = 1.03874 the log is still above the line but has the same slope
4) We know that the log slope is going to go down, then the curve will bend lower

a) The log will cross again the line.
b) This point (X = 1.03874) is the maximum error in the approximation

This is the moment when you can use your numerical method of choice. I will use a simple graph that says that such thing happens at about 1.078 (no surprise that it is about as far to 1.03874, the max error point, as the max error is to the other solution to the equation, X = 1).

c) This rule works best between 5% and 9%

A bit of worst case: let's say that the returns are 3.9%, the rule says that it takes 72/3.9 periods to double the capital, or 18.46 periods. The Log1.0392 » 18.11, still close enough for practical purposes.

Now you have it, if you want, let's say, to approx the moment of tripling the investment, you can refine this argument and consider that a 100*Ln 3 rule wouldn't be as good as a rule that has an slightly lower slope (higher number of the rule), and you can use the fraction 0.72/0.69 as a guide (115?).

2.3% per period would triple the investment at 115/2.3 = 50 periods according to the "115 Rule" (the correct answer is slightly more than 48.3, 50 is not a bad approx...)

Another consequence of this analysis is that you can use 70 rather than 72 for very small percentages because if you remember, 69 was the slope at 1, thus, for F values much closer to one, 69 is better than 72, but 70 is even easier to remember. For percentages such as monthly returns... Let's say you have 1% per month, the "70 Rule" approximates the number of months to 70. The Log1.012 is 69.66 something, awesome, right?