The Hanover Collaboration

Seeing how it is fast becoming job interview season, I thought I’d share a parlor trick with you guys. Business-type people are very easily impressed if you can do compounding interest in your head.
It boils down to being able to estimate natural logarithms. Here’s how:

Recall the Taylor series of the natural logarithm

ln(1+x) = x – x^2/2… – (-x)^n/n + ...

For small x we can just use the first term.

Next, memorize the following two constants.

ln(2) = .7, and ln(1.5) = .4

To estimate ln(x) for large x we will interpolate between a power of two and a power of two times 1.5 . Find the n such that

case 1) 2^n < x < 2^n*1.5. ln(x) is interpolated between .7*n and .7*n+.4, or

case 2) 2^n*1.5 < x < x ^(n+1) ln(x) is interpolated between .7*n + .4 and .7*(n+1).

This interpolation is rather easy since in Case 1 we add .1 for each quarter of the way and in Case 2 we add .1 for each third.

Here’s two examples of how to use this stuff

Ex1) how long will it take to triple my money at 6% interest?

Thought process: 1.06^x = 3, x*ln(1.06) = ln(3) = .06*x = ln(2) + ln(1.5) = .4 + .7, x = 1.1/.06 = 10*(2 – 1/6) = 18.5

Ex2) Some fund increased in value by 80% in 7 years. What was it’s average yearly return?

x^7 = 1.8, 7*ln(x) = ln(1.8)

Since 1.8 is close to two-thirds from (1.5,.4) to (2,.7) we estimate that ln(1.8) is .6. Similarly, if we needed to find ln(3.5) in Ex1, we would note that 3.5 if halfway between 3 and 4 and would have have interpolated between (1.1, 3) and (1.4,4) and get 1.25

ln(x) = .6/7, x = 1+ .6/7 = 1.09

Answer: 9%