Compound interest is a great way to have your money work for you. In this lesson, find out the formula for calculating compound interest and practice using the formula with several examples.
Compounding Interest Formula
Did you ever wonder how banks and credit card companies make so much money while seemingly doing so little? Their secret? Compound interest. This is when interest is calculated on both the principal and accrued interest at scheduled intervals.Let’s imagine three brothers. Each starts with $10,000.
First, there’s Joe. Joe keeps his money in a shoebox under his bed. It’s always close at hand, but it also doesn’t do much.
After 15 years, Joe still has $10,000. And it smells more than a little bit like feet.Then there’s John. John puts his money in an account that earns simple interest at a rate of 5%. That means the interest is added in all at once at the end of the period. After 15 years, he has $17,500.
That’s great! And no feet smell.Finally, there’s Jim. Jim puts his money in an account with compound interest. It has the same 5% rate as John’s account, but it’s compounded monthly. After 15 years, he has $21,137. Whoa! He more than doubled his money. And, again, no feet smell.
So how did Jim do it? We need to understand the compound interest formula: A = P(1 + r/n)^nt. A stands for the amount of money that has accumulated. P is the principal; that’s the amount you start with. The r is the interest rate. This is a decimal; in other words, if the interest rate is 9%, we use .09 in the equation. The n is the number of times the interest compounds each year.
Finally, t is the time in years of the deposit or borrowed money.This is like a big bowl of alphabet soup. Let’s quickly review the letters. A for ending amount. P for principal, or starting amount. R for interest rate.
n for number of times the interest compounds. t for time, in years, the money sits.
Practice Problem #1
Let’s try a practice problem: Will deposits $1,000 in an account that earns 4% interest, compounded quarterly. Rounding to the nearest dollar, what will the balance be after 3 years?First of all, good for Will for depositing some money and leaving it to earn interest for 3 years. OK, to solve this, let’s figure out what we know.
We know the starting principal is $1,000. That’s our P.We also know the interest rate is 4%. If we convert that to a decimal, it’s .
04. So that’s our r. We know the total time, or t, is 3.
Remember, t is the time in years.What about the n? That’s the number of times the interest compounds in a year. We know Will’s account compounds quarterly. So that’s 4 times each year. Therefore, our n is 4.
Let’s set up our equation. We start with A = P(1 + r/n)^nt. We’re trying to find A, the account balance at the end of 3 years. So A = 1,000(1 + .04/4)^(4*3).
When solving an equation like this, the order of operations is critical. Remember PEMDAS. So, do the stuff inside the parentheses first.
Then the exponents, then multiplication and division. Then addition and subtraction, though we won’t need that here. I guess you could say PEMD, but that’s not really a word. OK, PEMDAS isn’t a real word either, but it sounds like one.Anyway, let’s start inside the parentheses. .
04/4 is .01. If we add 1, we have 1.01. Now, 4 * 3 is 12, so we need to solve 1.
01 with an exponent of 12. That’s 1.1268…and some more stuff. With that still on our calculator, let’s multiply by 1,000.
We get 1126.83. Rounding to the dollar, that’s $1,127.So after 3 years, Will has earned an extra $127 on top of his original $1,000. And all he had to do was leave his money alone.
Practice Problem #2
As with most things in life, with compound interest, the more money you have, the more you can do. Let’s try a practice problem with a bit more money involved: Sarah deposits $25,000 in an account that earns 6.
5% interest, compounded monthly. Rounding to the nearest dollar, what will the balance be after 8 years?Sarah’s a bit more of a high roller than Will. Not only is she depositing more, she found herself a great 6.5% interest rate and an account that compounds monthly. Compounding frequency is a big deal. Think about how compound interest works.
It takes your interest and adds it to the principal. The more often it does this, the bigger your balance is, and the more interest you earn each period. So there’s a snowball effect.Frequent compoundings can be a huge help. Conversely, rare compoundings kind of defeat the purpose of compound interest.
If your account compounded interest once every 10 years, well, then you’d have to wait 10 years to see any benefit from the compounding.Anyway, Sarah’s account compounds monthly, or 12 times a year. That’s our n, and it’s a great one. We know our P is a whopping $25,000. The interest rate is 6.
5%. As a decimal, that’s .065. Be careful with that decimal point. We always move it two places to the left. Finally, the t is 8, for 8 years.
Sarah’s going to let this money sit for two presidential terms, or two Winter Olympics. Or maybe she deposits it on February 29 and waits a couple of leap years.Let’s go to the formula: A = P(1 + r/n)^nt. So that’s A = 25,000(1 + .065/12)^(12*8).
Let’s be careful with these big numbers. .065/12 is .00541666 repeating. If we add 1 and then raise that to the 96th power, we get 1.
6797 and change. That seems like a relatively small number. I mean, it’s less than 2. But let’s see what happens when we multiply it by 25,000. It’s 41,991.72! To the nearest dollar, that’s $41,992.
So Sarah made almost $17,000 in interest. Wow.
To summarize, we learned about compound interest. This is interest that is calculated on both the principal and accrued interest at scheduled intervals.
The formula we use to find compound interest is A = P(1 + r/n)^nt. In this formula, A stands for the total amount that accumulates. P is the original principal; that’s the money we start with.The r is the interest rate. We convert the percent to a decimal for this one. Then there’s n, which is the number of times the interest compounds in a year. If it’s quarterly, n is 4.
If it’s monthly, n is 12. And t is the time, in years, that the interest is building.
Thoroughly studying this lesson could prepare you to do the following:
- Provide the meaning of compound interest
- Compare compound interest to simple interest
- Write the compound interest formula
- Solve problems with this formula using the order of operations