The Rule of 72 – and what does the Swiss National Bank have to do with it

I was listening to an academic talk and someone mentioned the “Rule of 72”. Apparently invented by Einstein, it is a simple numerical approximation that helps you understand the power of compound interest. This, according to legend, became popular when interest rates offered on deposits by the Swiss National Bank dropped to $2-3 \%$ in the 1930s. Only the Germans appeared to be suffering hyperinflation, the Swiss clearly weren’t (though that was before the advent of modern monetary policy, which made the connection between interest rates and inflationary expectations).

Einstein is also touted as the source of a quote on compound interest – “Compound interest is the eighth wonder of the world. He who understands it, earns it…he who doesn’t …pays it”. By the way, I have seen several of the physical wonders of the world and have learnt several of the wonders of theoretical physics. While, for instance, the power of the exponential is to be seen to be believed (read Perelman’s book for how to hold a ship with a few loops of rope around a post, as in the image above), one can see it also in the ability of tiny humans to combine forces to build buildings as big as the Pyramid of Khufu, the Madurai Meenakshi temple, or the Burj Khalifa building in Dubai (below)

I would point to those things, rather than the mere accumulation of interest, as a more picturesque depiction of the power of the exponential. Also, as the website snopes.com seems to indicate, the attribution of the quote to Einstein might be an urban legend.

For what it is worth, the Rule of 72 is as follows. If you want to know how long it will take to double your money at an interest rate of $X \%$ , the number of years is $\frac{72}{X}$ . Obviously, this rule doesn’t apply if the interest rate is negative, as has been the case in some European countries in the last several years.

A quick check on Excel tells you that a better approximation is to use $69$ or $70$ in the numerator. Using $72$ harkens back to a predilection for multiples of 12, something that dates back to Babylonian times.

You can also use this formula to deduce the ruinous effects of inflation.

A complete description, that I found after preparing this article, is to be found in this page. The Rule of 72 actually appears in articles aimed at investors in the NASDAQ market, as well as in bank advertisements.

Credits:

The photograph of the rope at dock: Pratik Panda at Dreamstime.com

The photograph of the pyramid : By Nina – Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=282496

The photograph of the Meenakshi temple: Wikipedia

The photograph of the Burj Khalifa building: Wikipedia