Where does the Rule of 72 come from?

JLP's post on the Rules of 72, 114, and 144 got my math juices flowing. Where does the Rule of 72 come from?

If you're not into math, no biggie, but you might experience a little boredom. 😉

Say I have starting principal P_s with periodic interest rate r (expressed as a decimal). I let my earnings compound for t periods. The principal after t periods is P(t) = P_s(1+r)^t. (If my interest rate is 8% then to figure out the amount of principal I have after 5 years, I multiply the current principal by 1.08, five times.)

The Rule of 72 is a formula that tells me how long it takes to double my money at a given interest rate, so the doubling time T is the time at which P(t) equals 2P_s. So 2P_s = P_s(1+r)^T, or 2 = (1+r)^T.

If we take the natural logarithm of both sides, we have ln 2 = T ln (1+r) ≈ T (r – r²/2 + r³/3 + higher order terms in r) after we take a Taylor expansion of ln (1+r). So, to first order in r,

ln 2 ≈ rT.

But r is the rate of return expressed as a decimal. If we express the rate as R%, then R/100 = r and

RT ≈ 100 ln 2 ≈ 69.3, which is just a little shy of 72.

The reason why this result is a little less than 72 is because we only took the first-order term of the Taylor expansion. The first-order approximation is valid only for small r. As r gets larger compared to 1, the higher-order terms become more important. The next-most-important second-order term is negative (-r²/2) so for a given T, r will have to increase compared to the first-order approximation, and hence the product RT will also increase. This is exactly what JLP showed in the tables in his post, and what we see in the graph below of the actual (exact) product RT. To make this plot, I solved for T for a bunch of different rates r by plugging r into T = ln 2 / ln (1+r), and multiplied 100r by T to get the value on the vertical axis.

Derivation of the Rules of 114 and 144 are left as exercises for the reader. 😉