Use E for Compound Interest
This is a response to Daniel Huang’s blog post Stop using e for compound interest
The mathematical constant \(e\) is usually introduced in precalculus classes as the limit of a process of continual compounding. If a bank offers 100% yearly interest, compounded \(n\) times per year at a rate of \(\frac{100\%}{n}\), the limit approaches a constant number:
\[ \lim_{n \to \infty} \left(1 + {1 \over n}\right)^n = e \approx 2.71828... \]Daniel rightly objects that a real bank wouldn’t divide the interest linearly when increasing the number of compounding periods, since this leads to an increased return at the end of the year compared to a single compounding period. But a real bank also wouldn’t offer 100% return! Why use such an artificial example to introduce \(e\)?
How I learned it
The point of learning \(e\) as the limiting process of continual compounding is that it allows you to roughly calculate the return from a long-run investment on a pocket calculator. If your timeline of interest covers enough compounding periods, then it doesn’t matter if you approximate the compounding periods as instantaneous.
Consider an investment of capital \(P\) with an annual rate \(r\), invested over a period of \(t\) years with a compounding rate of \(n\) per year (linearly divided). The return on this investment is clearly
\[ R = P (1 + \frac{r}{n})^{nt} \]To find the approximate return, you can take the limit as the compounding rate approaches infinity. This gives:
\[ R \approx P \lim_{n \to \infty} (1 + \frac{r}{n})^{nt} \]Now execute a change of variables. Let \(u = n / r\). This expression is equal to
\[ R = P \lim_{u \to \infty} (1 + \frac{1}{u})^{urt} = P e^{rt} \]This expression is actually useful! It allows you to approximately calculate returns on bank accounts, the stock market, population growth, and anything else that grows at a constate rate. Better yet, it doesn’t require any calculus or complex numbers, just the concept of limits and some algebraic manipulation. Even if you believe (as I do, and Daniel probably does) that calculus is introduced too late in the high school curriculum, there is still a place for this beautiful and useful introduction of \(e\) in the context of growth rates.