Rule of Drum

Kevin Drum brings us a little math problem:

Via Alex Tabarrok, a pair of researchers asked people how big the economy would be if it grew 5% a year for 25 years:

Only around 10–15% of the participants gave estimations between 50% less and 100% more than the true value...furthermore, the majority of the false estimations were systematically below the true value ...which was underestimated by 88.9–92.1% of the participants.

As Kevin points out, this is not exactly one of those "look how many people think the Earth is flat" examples. It's not very easy to do the arithmetic to get the correct answer. 5% interest, compounded yearly, yields 1.05^25 (1.05 to the 25th power). This is not a calculation that I can do in my head. If one resorts to the binomial theorem (the first resort of the physics student!), we have (1+0.05)^25 which expands into the series 1+25/20+25*24/(2*20^2)+ ... This series converges slowly - the first three terms yield 3 compared to the actual value of 3.38. Adding the fourth term gets us to 3.28 and the fifth to 3.36 - pretty good, but a heck of a lot of arithmetic too hard to do in my head - thanks Google.

The revelation for Kevin was that he learned the rule of thumb called the rule of 70.

But Alex made this worth my time by teaching me a new rule of thumb I hadn't heard of before:

A good way of approximating is to use the rule of 70. If x is the growth rate then the doubling time is approximately 70/x. Thus, with a growth rate of 5% we expect a doubling (100% increase) in 14 years and a quadrupling in 28 years so a bit more than a tripling in 25 years (200% increase) is a good guess.

I love good rules of thumb, and this one makes me slightly more knowledgable than I was five minutes ago. Thanks, blogosphere!

When I hear a rule of thumb, I want to know how, why, and when it works. Given a fractional interest rate I, how many years y does it take to double the principal?

That equation is (1+I)^y = 2. To solve for y, we take the (natural) logarithm of both sides, getting:
y*ln(1+I)=ln(2) or y = ln(2)/ln(1+I). The rule of 70 amounts to saying y = 70/100*I or 0.7/I. Now ln(2) = 0.693 and, for small values of I, ln(1+y) is approximately y, so the rule of 70 amounts to approximating the ln(2) by 0.7 and ln(1+y) by y. For I = .05 = 5% the rule of 70 gives y=70/5 = 14, whereas the more precise value is ln(2)/ln(1.05) = 14.21 - which is not too bad. For higher interest rates the error get larger. For 35% interest (think credit card) the rule of 70 give y =70/35 = 2, but the more precise value ln(2)/ln(1.35) is 2.31.

If bankers and economists did things right, of course, they would compound interest continually so that the proportional increase would be e^(I*y), where e is the base of the natural logarithms, or about 2.7183. For I = 5% and y = 25 years, I*y = 1.25 so that the accumulated value would be e^1.25 = 3.49. Not that that is so easy to calculate in your head either.

If you have Google or a calculator handy, of course, the arithmetic is not a problem.