The Math Constant e

In my prior discussion of log returns, I made an offhand reference to the math constant e. Do you remember that kid who memorized 100 digits of \pi because that’s what he thought smart people did? He should have memorized 100 digits of e, because it’s far cooler than \pi. Actually, he should have practiced programming and memorized nothing, but I digress.

I think it’s worth taking the time to explain e in detail. There are other ways to get to e, but for aspiring financiers, this is usually how e approaches you.

Say you had $100 in a bank account that paid 100% interest over a period of time t. How much money would you have at the end of the period? Using simple interest theory:

FV = PV (1 + r) = 100 (1 + 1.00) = 200

Suppose that, instead that instead of receiving 100% once, you received 50% interest twice over the same period. Due to the nature of compounding, this would produce a slightly higher return.

FV = PV (1 +\frac{r}{2})^2 = 100 (1 + \frac{1.00}{2})^2 = 225

Instead receiving 33.33% interest three times over the same period would produce an even higher return.

FV = PV (1 +\frac{r}{3})^3 = 100 (1 + \frac{1.00}{3})^3 \approx 237

Generalizing, let’s assume that you receive \frac{1}{n} interest n times per period. The future value is given by the following relationship:

FV = PV ( 1 + \frac{r}{n})^n

Initially, the benefit from increasing n will greatly increase FV, but the marginal return from increasing n will quickly diminish until the final return is roughly equal to $271.83 which, happens to be e * PV. In pictures:

In math, e is equal to:

e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n

In finance settings, e is seen in continuous growth rates. If r is the continuous growth rate and ln is the logarithm in base e (ln(x)answers the question “e to what power is equal to x?”), this formula provides the continuous growth:

FV = PV e^{rt}

And this formula provides the exponential growth:

r =\frac{ ln(\frac{FV}{PV})}{t}

There are some other neat facts about e that don’t directly intersect with finance or business valuation but are still cool. For instance, the derivative of e^x equals e^x, and e shows up in some surprising places in probability theory, but that’s beyond the scope of this engagement.