e, the fundamental constant of compound interest with our fictional friend John

In the early 1700s, the Bernoulli brothers (famous in other areas, like for their Principle, which forms the foundation for the design of wings on planes) were corresponding with another preeminent mathematician of their day, the Swiss mathematician Leonhard Euler (whose name is tossed around in conversations about who the greatest mathematician has ever been) about compound interest, and how much they'd receive if they-- pardon the calculus pun-- took compound interest to its limits. 

Here is the equation that governs compound interest: 

In this equation:

• A is the amount you'll have after compounding
• P is the amount you have before compounding
• r is the interest rate 
• n is the number of times the compounding occurs per unit of time
• t is the number of units of time through which the compounding takes place

For example: 

If John knows that he wants to see how much money will be in his savings account when he starts with $5000, it compounds monthly at a 4.4% interest rate, and he wants to wait 10 years, then substituting in, the equation now looks like this:

• We don't know A yet
• We know P is 5000
• You get to keep all your money, plus you get returns, each time the money is compounded, so that explains the term in parentheses 
• John's interest rate is 4.4% per year, so that explains the 0.044
• There are 12 months in a year so that 4.4% will be divided evenly throughout the year in 12 installments 
• We want to look 10 years into the future, so the compounding process-- which is a multiplication each time-- needs to happen 12*10 = 120 times, justifying the exponent

As it turns out, if John just lets the money compound, then he'll be getting about a 0.367% interest rate every month, 12 times a year, for 120 compoundings in a decade, and he'll have all his money, plus some interest, in his account 10 years from now, totaling $7,757.29. 

For illustrative purposes, to arrive for ourselves at the conclusion of the correspondence between the Bernoullis and Euler, let's keep the starting amount ($5,000), the interest rate (4.4% on an annual basis), and the length of time (10 years), the same but let's vary how often we compound per year. 

Refer to this table:

How many times we compound per year	How much money we have in 10 years		Interest rate per compounding period
1	$   7,690.86	once a year	4.40%
2	$   7,726.59	every 6 months	2.20%
3	$   7,738.77	every 4 months	1.47%
4	$   7,744.91	every 3 months	1.10%
6	$   7,751.08	every 2 months	0.73%
12	$   7,757.29	every month	0.37%
365	$   7,763.33	every day	0.0120547945205%
8760	$   7,763.53	every hour	0.0005022831050%

 We see here that it is-- maybe counterintuitively-- advantageous to take a smaller interest rate each time you compound but to compound more frequently. 

Taking 4.4% interest once a year is okay, but taking 2.2% twice a year works in John's favor by about $36; 1.47% 3 times a year gives him about $13 on top of that; and so on until we seem to be getting closer and closer to a value. 

There's a special number hidden in this formula, and let's now extract it. Of course, 10% can be represented as 1/10, so what happens if, for instance, we take (1 + 1/10) to the power of 10? Or do the same for 1% = 1/100?

1 divided by a very large number is a very small number. 1 plus a very small number is barely more than 1. 1 exactly, to any power, is always 1, but any number even just barely bigger than 1, to some power, will grow. 

So then what happens when we let d get as big as we want, such that a number just barely above 1 is raised to an enormous power?

Look at this table:

d	Our Magic Number
1	2
2	2.25
5	2.48832
10	2.59374246
100	2.704813829
1000	2.716923932
10000	2.718145927
100000	2.718268237
1000000	2.718280469
10000000	2.718281694
1000000000	2.718282031

 That last equation should look familiar: it's very similar to what we multiply the current value by in order to get the future value. 

This table is asking: what happens if you give me my investment, plus 100% interest, once? (Your investment exactly doubles). How about 50% interest twice? (Your investment is 9/4 times as big). How about 20% interest, 5 times? (Your investment is about 5/2 times as big). 

As we let d get arbitrarily big-- take it to infinity, calculus doesn't care! -- notice how similar the numbers are starting to look. 

There's a concept in calculus called a "limit." A limit is the value of an expression when a variable in that expression gets arbitrarily close to some number you've defined, which, because this is calculus, is allowed to be (and is, in our case) infinity. The closer your number is to the bound, the closer the number you get will be to the true value of the expression. 

This number we've gotten, by raising (1 + a fraction) to the power of the denominator of that fraction can be subjected to this process. Namely, what's the limit of this value when the bit added to 1 is allowed to get as small as we want, and consequently, the exponent becomes arbitrarily big?

Clearly, it's 2.71828.... something something something... 

This number, which does in fact start 2.71828182845904523536... (and goes on forever without ever repeating) is one of the fundamental constants of our whole system of mathematics. 

If you've had any calculus at all, you'll surely recognize that this number is the number e, also called Euler's number after one of the participants in that exchange of letters precisely about this problem, 300 years ago. 

The fact that this limit exists (proving its existence is a matter for someone who wants to learn calculus; take it for granted) is what allows us to see a pattern in the amount of money John earns by specific frequencies of compounding: as he compounds more and more frequently, the percentage of interest he gets with each compounding gets smaller and smaller, but the number of compoundings gets bigger and bigger. 

This means that, if someone had $1, and then they got 100% interest per period, divided up into infinitely small chunks, compounded instantly, they'd end up with about $2.72 at the end of the first period-- that is, with $e. 

So because we now know about e, we can rewrite the formula for compound interest like this:

If John could keep shrinking his compounding interval to the point that it became an infinitely-small instant, this would be his formula to find out how much money he'd have in his account. This is called continuous compounding, and there are ways to prove that this formula is equivalent to the first one presented. 

When John changes his compounding to become continuous-- the most generous his bank could possibly be-- then he realizes his absolute ceiling for how much money he can get (it's absolute because you cannot get any bigger than a limit like the one that defines e) is $7,763.54, just one cent better than the most generous compounding we calculated in the chart. (In the real world, banks would most likely give him the monthly compounding option, which if you'll recall, would have given him $7,757.29)

This number, and the kind of (exponential) equations it gives rise to are some of the most important in all of math, especially in a financial context. But the notions of limits and infinity, and the weirdness of e can make understanding compound interest a difficult and intimidating prospect. But play around with some numbers, and I promise you-- the best way to get comfortable with how this equation behaves is to play with it. Change variables one at a time, and see just how sensitive (or not) John's balance is to the change you made. 

When you do, you'll have gone-- yourself-- on the same journey of discovery as Euler and the Bernoulli brothers did, 300 years ago.