e is one of those things in mathematics that appears mysterious at first glance. It has a seemingly random value and occasionally pops up in various contexts. Today, let’s take a closer look at this important number and understand it in the context of one of the most powerful forces known to man: compound interest.
e is essentially the growth multiple of something increasing at 100% over one time unit when compounding continuously. That may sound a bit confusing so let’s break it down.
Consider a scenario where we could double our money every year and receive our new profit on the last day of the year. Starting with $1, we would have $2 at the end of year one. At the end of the second year, we would have $4. At the end of the third year, $8.
This scenario could be modeled with the following formula:
1 * (1 + 1) ^ 3 = 8
starting amount, constant used when calculating the new total at a given rate, rate (1 = 100%), number of years, final value
In this scenario, compounding only occurs once per year i.e. we only receive our new profit once per year. What if we kept the rate the same but increased the number of compounding periods per year?
Consider another example where our money compounded monthly instead of yearly. Instead of one compounding period per time unit, we would have 12 compounding periods. The new profit received at each compounding period would contribute to future compounding! Starting again with $1, at the end of the first month we would have $1.08. This is modeled by the following:
1 * (1 + (1/12)) = 1.08
starting amount, constant, rate (1 i.e. 100%) divided by the compounding periods (12), final value
Now during the second month, that extra $0.08 will contribute to the compounding. At the end of month two we will have $1.17. This continues until we end up with $2.61 at the end of the year. This is modeled by the following:
1 * (1 + (1/12)) ^ 12 = 2.61
formula 2, number of compounding periods, final value
By the end of the three years, we will have $17.84. Pretty big difference from our original $8! This is the magic of compound interest and is modeled by the following:
1 * (1 + (1/12)) ^ (12 * 3) = 17.84
formula 3, total time units (3 years), final value
Now let’s look back at the result of formula 3 for a moment. We went from increasing our money by a factor of 2 to a factor of 2.61 just by increasing the compounding periods per time unit. Does that number look familiar? What happens if we further increase the compounding period? Let’s take a look:
1 * (1 + (1/12)) ^ 12 = 2.61 ← compounding monthly
1 * (1 + (1/52)) ^ 52 = 2.69 ← compounding weekly
1 * (1 + (1/365)) ^ 365 = 2.71 ← compounding daily
Woah, 2.71 is very close to the value of
e (2.7182…)! As we reduce the compounding period, we approach the value of
e. This is because
e is calculated from this same formula except that there are an infinite number of compounding periods per time unit, i.e. continuous (analog) compounding.
For more information, check out this great article from the BetterExplained site.