As a kid I heard about the Banker’s rule of 72 for determining the doubling period for investments. Divide 72 by whatever your interest rate is, and that’s how many years (compounding periods) it will take for your investment to double. But despite having the tools to understand it for years, I only discovered in the past couple of years why it works.
Trying the Rule of 72
4% should take 18 years to double, at 8% it should take 9 years to double, at 12% it should take 6 years to double, and at 18% it predicts 4 years to double. (All decimal numbers below are approximate)
1.04^18 = 2.0258
1.08^9 = 1.999
1.12^6 = 1.973
1.18^4 = 1.939
What you may notice is that it’s an approximation. It overestimates for lower percentages and underestimates for larger percentages. We could fudge the number 72 one way or the other to skew the error, but the fact is that it’s pretty close. 72 was likely chosen because of the large number of small prime factors – 2*2*2*3*3. Thus, 72 is evenly divisible by 2, 3, 4, 6, 8, 12, 18 and of course 1 (1.01 ^ 72 = 2.047).
As a high-school kid, I understood a little about logarithms, and I wanted to understand where this rule came from, so I started scratching some math out. The equation that I was looking for was this:
(1 + percent) ^ x = 2
…or perhaps more succinctly:
(1.p) ^ x = 2
So taking the log of both sides, simplifying and solving for x:
log(1.p^x) = log(2)
x * log(1.p) = log(2)
x = log(2)/log(1.p)
Of course I knew that log(2) was a constant and that the percentage ( p ) was the variable, but there it stayed because of that pesky log function. I never revisited this until years later when Wikipedia turned the light on (actually, it completed it by giving me the answer to my burning “why?” question.) Mind you, I had numerical analysis in college, so I had possessed what I needed all along.
The big reveal
A good approximation for the natural-log of small numbers is the number itself. As the input numbers get bigger, this approximation gets worse, but for small numbers it’s pretty good. This was revealed by the first term of the Taylor series expansion for approximating functions given the value of their derivatives. This sounds scary, but it’s really not bad at all. It’s one of the earliest things taught in numerical analysis.
Instead of showing the Taylor Series for the natural log function, let’s just test it out:
p = 0, ln 1.00 = 0, difference = 1.00
p = 1, ln 1.01 = 0.00995, difference = 1.00005
p = 5, ln 1.05 = 0.0488, difference = 1.0012
p = 8, ln 1.08 = 0.0770, difference = 1.003
p = 18, ln 1.18 = 0.1655, difference = 1.0145
The next step toward the rule
Let’s plug in the constant for the ln(2) and our rule for ln(1+p) = p, defining p as p/100 (that is p as a percent) we get:
x = ln(2)/(p/100)
(multiplying numerator and denominator by 100 to get the (p/100) part out of the denominator)
x = 100 * ln(2) / p
100*ln(2) = 69.31472 (approx)
The “rule of 69.3” isn’t as easy to use (remember the number of prime factors 72 has) and has an error that underestimates the doubling period for 1 percent and continues to underestimate the larger percentages by more and more – so shifting 69.3 a little larger gets us to the point where we overestimate some (by a small margin), moving the point where we underestimate to the larger percentages (that is, the error is in the interest rates nobody wants to pay but everyone would like to earn.)
Factorization of numbers from 69 to 76
69 = 23*3
70 = 2*5*7
71 = 71 is prime
72 = 2*2*2*3*3
73 = 73 is prime
74 = 2*37
75 = 3*5*5
76 = 2*2*19
So, 72 fits the bill for both shifting the error a little to the right – marginally underestimating some and marginally overestimating others while also being easy to use for lots of whole-number percentage interest rates.
The Sweet Spot
What we might notice is that shifting the “rule number” upward shifts the point where the rule goes from overestimating to underestimating the doubling period. There should be a percentage where the rule is exactly right. I’ve looked at the math and decided against working the formula out or using an algorithm like bisection or newton’s method and opted for a more trial-and-error approach.
The sweet spot for 72
We saw the numbers for 4 and 8 rendered results that were under for 4 and over for 8. So 6 is halfway between, the prediction for 6 is going to be 72/6 = 12 years. 1.06^12 = 2.012. That’s still a little high, so let’s go between 6 and 8, opting for 7% (this gets ugly since 72/7 = roughly 10.29), which renders 2.006. Let’s try 7.2% which we predict should double in 10 years – 1.072^10 = 2.004, closer but still a little high, so choose a little bigger 7.5? Skipping a few, the sweet spot for 72 is between 7.8 and 7.9 percent – the prediction for 7.8 is too high and the prediction for 7.9 is too low – the prediction for 7.85 which renders is too short a doubling period, so it’s between 7.8 and 7.85, we could try 7.825 which is is too long, so the answer is between 7.825 and 7.85, we could try the midpoint 7.8375 which renders too high a doubling period, so the answer is between 7.8375 and 7.85… I’m tempted to just say 7.84 is close enough, but the true sweet spot for the 72 rule is almost certainly an irrational number.