Continuously Compounded Interest: What's e Got To Do With It?

Lesson Plan - Get It!

Audio:

If you want to maximize the amount of interest you can earn when you put your money into a savings or investment account, you'll want to know all about the number e.

Mathematicians call the number Euler's Constant because Leonhard Euler was the first to formalize an understanding of the number.

His work was built on that of other mathematicians who came before him, including Jacob Bernoulli. Check out the Additional Resources in the right-hand sidebar for opportunities to learn more about these fascinating mathematicians.

You're probably wondering what this has to do with money. Let's dive into that.

If you have not already worked with compound interest, I suggest you familiarize yourself with basic compound interest first because we will be building on that. You can find our lesson on compound interest in the Additional Resources as well.

We know from working with compound interest that to calculate the value of an account with an initial investment of P (principal), an APR of r, with n compounding periods per year, over a total time of t years, we can use the following formula:

A = P(1 + ^r/_n)^nt

If n is 4, interest is being compounded quarterly. If n is 12, interest is being compounded monthly. If n is 365, interest is being compounded daily.

We also know that the more often interest is compounded, the more money we earn in interest.

Jacob Bernoulli, a Swiss mathematician in the late 17th century, started thinking about the effect of increasing the number of compounding periods in a year so that there was no discernible time frame between each instance of compounding.

In other words, what would happen if n got so large that it becomes infinity. Now, a number can't ever become infinity, but math does gives us ways to think about what would happen if that number grows arbitrarily large.

This is what he came up with:

We read this equation as "the limit as n approaches infinity of the quantity 1 plus 1 divided by n, raised to the n power."

Leonhard Euler first proved that this number e is irrational. It is approximately 2.718. Since it is irrational, it is a decimal that will keep on going forever without repeating.

The idea of limits is really an idea from calculus, but there are a few things to notice and understand algebraically here.

• First of all, do you recognize the expression (1 + ¹/_n)ⁿ?
• Doesn't it look suspiciously similar to a part of our compound interest formula?

That happens to be from where it comes.

Secondly, notice that this limit expression does not contain r or t. Clearly, those are important values for calculating interest, so they must figure into the formula for continuously compounded interest somehow.

• Are you ready for the big reveal?

Watch Derivation of Continuously Compounded Interest Formula from Parabola Magic:

As you saw in the video, the formula for continuously compounded interest is:

A = Pe^(rt)

In the Got It? section, you will apply this formula and compare it to other compounding options.