Exponentials are powerful

Today we also do a short post. This started from a phrase I heard in a podcast, but it also relates to the if you do 1% more every day you will be at 37x at the end of the year quote.

The podcast quote was something along the lines of investing in an instrument with 1% higher return, could result in doubling the expected outcome by the retirement age. Sounds shocking, but it is true.

Consider investing in vehicle A with a rate of return \(r\), over \(k\) years. Then, you’d expect to have a total of \(s(1+r)^k\), where \(s\) is the initial sum.

Now, consider investing in vehicle B, which has a rate of return \(1.01r\), just 1% higher. Then, you’d have \(s(1+1.1r)^k\).

The ratio of the gains would be

\[\left(\frac{1+1.1r}{1+r}\right)^k = \left(1 + \frac{0.1r}{1+r}\right)^k\]

Every term in \(r' = \frac{0.1r}{1+r}\) is positive, so the ratio of the gain is positive, and exponentially higher as \(k\) increases. Thus, for some value of \(k\) it can get to 2, or even higher.

I’m leaving it as an exercise to the user to find the values of the interest rates such that the doubling occurs exactly at the end of a typical person’s work years, to match the podcast quote regarding retirement.

There could be lessons here regarding investment, going above and beyond when working on stuff, and so on. But, also remember that nature seems to abhor never-ending exponentials, just like it abhors a vacuum.