# 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.