BM4.10 – Continuous Compounding | For the Love of Maths

Chapter 4, Lesson 10

In this lesson students will:

understand how the Euler’s number naturally arises from maximizing investment potential over the course of one year
use the continuous compounding formula to determine the future value or principle amount, given other information about the investment

Continuous Compounding

A Maximization Problem

Consider the following question: What is the maximal amount of money you can have at the end of a one-year investment if you are only permitted a principle amount of $\$1$ ? This question was first explored by Jacob Bernoulli in 1683, to which he assume a few things. First, he assumed that for the maximum amount at the end of the year, we should have a $100\%$ compounding interest rate. Substituting $t=1$ (one year), $P=1$ (one dollar principle amount), and $r=1$ ( $100\%$ interest rate), we obtain the following compound interest formula:

$\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{t \times f}\\ FV &= \left( 1 + \frac{1}{f} \right)^{f}. \end{aligned}$

Next, he noticed that the future value depended on the frequency number $f$ . If $f=1$ , at the end of one year we would have $FV = (1+1)^{1} = 2$ , or $\$2$ in our account. What happens if we are allowed to increase the frequency of compounds throughout the year? Well, remember that compound interest allows all interest earned in one compound period to earn interest in the subsequent compounding periods. This means if we increase $f$ we have the capacity to earn more money! The following chart summarizes how much money our investment will have at the end of one year for a few different frequency numbers, rounded to five decimal places.

$\begin{tabular}{l|l} Frequency Number & Future Value\\ \hline 1 (yearly compounding) & \$2\\ \hline 2 (semi-yearly compounding) & \$2.25\\ \hline 4 (quarterly compounding) & \$2.44141\\ \hline 12 (monthly compounding) & \$2.61304\\ \hline 52 (weekly compounding) & \$2.69260 \\ \hline 365 (daily compounding) & \$2.71457\\ \hline 1000 & \$2.71692\\ \hline 10,000 & \$2.71815\\ \hline 100,000 & \$2.71827 \end{tabular}$

Notice that, even though we continue to increase the number of compounds way past daily compounding, the overall future value at the end of one year doesn’t fluctuate much as our frequency number gets into the ten-thousands. It seems as though the maximum amount of money we can earn over the course of one year is around $\$2.72$ .

You may notice that the future value in the final row of our chart above is similar to a number we have seen in Lesson 4.3. The compound interest problem discussed above is exactly where Euler’s number $e \approx 2.7182$ was first discovered. Despite knowing this fact in the 17th century, the use of the letter $e$ to denote this number was not common until Leonhard Euler used it in 1736.

The Continuous Compounding Formula

For the purpose of our course we will need the following two facts: the first is that $\left( 1 + \frac{1}{f} \right)^{f}$ tends towards $e$ as $f$ gets large, and the second is that $\left( 1 + \frac{r}{f} \right)^{f}$ tends towards $e^{r}$ as $f$ gets large. Knowing this, we can rearrange our future value formula for compound interest in such a way as to maximize the amount of revenue we obtain over a period of time.

$\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{t \times f}\\ FV &= P \left[ \left(1+\frac{r}{f}\right)^{f} \right]^{t} \end{aligned}$

If we now allow $f$ to get large, we can make the substitution of $e^{r}$ for $\left(1+\frac{r}{f}\right)^{f}$ .

$\begin{aligned} FV &= P \left[ \left(1+\frac{r}{f}\right)^{f} \right]^{t} \\ FV &= P \left[ e^{r} \right]^{t} \\ FV &= Pe^{rt} \end{aligned}$

The formula $FV = Pe^{rt}$ is known as the continuous compounding formula, and will tell you the maximum amount of money your principle amount $P$ can earn with a specific rate $r$ over a period of $t$ years.

Let’s finish off this lesson by considering the following two questions:

What is the future value of a $\$5500$ investment after two years of continuous compounding at a rate of $5\%$ ?
Determine the principle amount of an investment that grew to $\$7000$ over the course of $70$ weeks at a rate of $2.5\%$ compounded continuously.