Compound interest is a blessing for saving, but a curse for debt.

The only good thing about debt is that it's connected to one of the most important constants in maths: the number .

To see the connection, suppose you borrow at an annual interest rate of . How much will you owe after three years if you don’t make any repayments? It’s tempting to think that you’ll owe the original plus of , which is , giving a total of .

Unfortunately, though, this calculation is not quite correct. After one year you will owe the original amount, , plus the interest from the first year, which is

This gives a total debt of

To find out what you owe after another year, you need to add the interest from the second year, which is

After the second year the interest is therefore

The interest in the third year will be

so after the third year you will owe a total of

You can see that the amount of interest accrued in each year is more than the amount accrued in the previous year. If you’ve borrowed more than this growth, or *compounding*, of the interest is much more noticeable: it is the curse of debt and, equally, the blessing of saving.

You may have seen a pattern in the expressions above. And indeed, there is a general expression describing how your total debt will grow:

where is the initial amount you borrowed, is the rate of interest (where is written as a decimal number, such as , rather than a percentage, ) and is the number of times the interest is compounded. The more often the interest is compounded, the greater the total, which is where you have to be careful.

To make things simpler, suppose you borrow () at an annual interest rate of (). If the bank just calculates your total amount at the end of the first year, you will owe:

But what if the bank decided to compound the interest quarterly, increasing the total every three months using an interest rate of ? Then after one year you would owe:

Compounding monthly would result in a total of

Daily compounding would give

If the bank compounds times in a year you will owe:

and the more times they compound, the greater the value for , the more you will owe.

If you take things to the limit, calculating this for higher and higher values of , you’ll find the total amount you’d owe converges to

We call this number , the limit of the value for our compounding formula as tends to infinity. The number encapsulates growth, and can often be used to rewrite complicated equations describing growth in a much simpler way.

The person to first catch sight of the number in the context of compound interest was the mathematician Jacob Bernoulli in 1683. Bernoulli tried to find the limit of the expression

as tends to infinity and managed to work out that it was between and . It was the first time in history that a number was defined as a limit.

## Comments

## APPLICATION OF EULER CONSTANT IN CALCULATIONS

Good job in showing the derivation of e. Now with the value of e, how can I compute the compound interest of other percentages say 20%, 30% etc without using the fundamental formula of compound interest(1 + 1/n)^n but using the Euler constant(e) directly.

## using e to find compound interest

Exactly my question. There is a formula to calculate compound interest using e. You never say what that is in this article! Can we please cut through the mumbo jumbo about e and all the theory and just get the formula?

## Why no formula?

Hi, if you want to know the formula using e, which I did, it is A=Pe^rt. This article goes through everything except that. Why I have no idea.

Really, guys. Come back down from 30,000 feet for just a minute.

## Deriving Pe^rt

Start with P(1+r/n)^nt. Let's call x=n/r. This means r/n = 1/x and n=xr.

So P(1+r/n)^nt = P(1+1/x)^xrt = P((1+1/x)^x)rt. The limit of (1+1/x)^x = e so P((1+1/x)^x)rt = Pe^rt.

## cool post

Thanks. I'm in pre-calc and I just screenshotted half of the page for an assignment.

Best regards,

Art