Learn what the compound interest formula is, how it’s related to exponents, and how it can help you make money.

Two articles ago, we talked about how you can use the rule of 72 to figure out how long it will take for the value of an investment to double. In the last article, we talked about a fundamental idea in math called exponentiation. Today we’re going to put these two ideas together to discover something called the compound interest formula. You can use it to calculate exactly how much the value of your investments will grow over time.

## Review: What Are Exponents?

In response to the last article, math fan Edward pointed out that the way I worded my description of exponentiation could possibly lead to some confusion. In the article, I said that you square a number by multiplying it by itself twice, cube a number by multiplying it by itself three times, and so on. But actually, when you square a number—as in 3^2 = 3 x 3—there’s only one multiplication performed. So instead of saying that squaring a number is multiplying it by itself twice, I could have been clearer and said that squaring a number is multiplying two copies of the number together using one multiplication operation, and so on for higher powers. Make sense? With that out of the way, let’s talk about compound interest.

## What Is Compound Interest?

Our goal when talking about the rule of 72 was to figure out how long it takes to double money in a compound interest earning investment. But what does that mean?

Well, if you have $1000 invested in an account that earns 1% interest each year, then since 1% of $1000 is $10, at the end of the first year your account will contain $1000 + $10 = $1010. If you leave all of that in the account and continue to earn interest on it, then since 1% of $1010 is $10.10, at the end of the second year the account will contain $1010 + $10.10 = $1020.10. Year after year, you can use this method to see how much your account will grow. And, importantly, you’ll see that not only is the original $1000 that you put into the account earning interest, but the interest that you earn year after year is itself earning interest! And that’s what we mean by compound interest.

## The Compound Interest Formula for One Year

Instead of going through this whole procedure of calculating and adding 1% of the account’s value to its balance year after year, wouldn’t it be nice if there was a way for you to figure out in one fell swoop exactly how much the account will be worth some number of years later? Well, there is! To figure out how it works, let’s think about what we’re actually doing when we calculate the interest earned by our account in one year. As you can check, the value of the account at the end of the first year can be calculated using the formula:

FV = PV x (1 + rate)

Wait a minute! What’s all that? Well, the symbol “PV” represents the present value of the account (the value at the beginning of the year), the symbol “FV” represents the future value of the account (the value at the end of the year), and the “rate” is the annual interest rate written as a decimal. For example, 1% is 0.01, 5% is 0.05, 10% is 0.1, and so on.