*Contributor: Lynn Ellis. Lesson ID: 13743*

It takes money to make money, right? How can you make even more money from the money you have? Compound interest is the answer. Explore the math behind compound interest in this lesson.

categories

subject

Math

learning style

Auditory

personality style

Beaver

Grade Level

High School (9-12)

Lesson Type

Quick Query

- How can you take $1,000 and turn it into over $1,800 in 10 years without doing anything but wait?

When you have some money that you would like to save, you have options.

You can put your money in a jar, hide it well, and let it sit there, "saving" your money for a time when you need it.

You can put it in a savings account in a bank or credit union, where it will earn some interest. If you do that, you will have more money when you need to get it back.

- But how much more will you have?

That will depend on the interest rate (called the APR, or annual percentage rate), the amount you are saving, how long you have it in the bank, and something called compounding.

Imagine that you have $100, and you put it in a bank account where you earn 4% interest each year. After one year, you will have the $100 you put into the account PLUS the $4 you made in interest (4% of $100).

Now you have $104 in your bank account.

The next year you will earn 4% of the $104 you now have in the account. That is interest earnings of $4.16 in the second year. Notice that you earn more in the second year than you did in the first year.

After 10 years, if all you did was wait, you would have $148.02 in the account. Not bad!

- What if, instead of putting the interest in your account at the end of each year, the bank puts the interest you earn into your account every month?
- Will that give you any more money after 10 years?
- If so, how much more?

Think about that for a minute before you move on.

Okay, you should have an idea in your mind about what you think would happen. Let's go ahead and break it down.

After one month, you haven't earned the entire 4% - that's the amount that you earn for the whole year. You have made ^{1}/_{12} of the 4% since there are 12 months in a year.

So after one month, the bank puts 33¢ in your account. The next month, you make interest on the $100 you started with AND the 33¢ that has been added. Maybe 33¢ doesn't sound like that much, but it adds up over time.

After 10 years, if all you did was wait, you would have $149.08. It's not a lot more, but it is more money.

If the bank added interest to your account every day, you would have $149.18.

The more often the bank gives you your interest, the more money you have! That's the power of compound interest!

Let's look at the math behind this. We will construct a formula that we can use to calculate the amount in your account at any point.

In our example above, your investment is $100, and the interest rate is 4%.

After the 1st year:

$100 + $100(0.04)

We can factor out the $100 and write this as:

100[1 + 1(0.04)]

100(1 + 0.04)

After the 2nd year, your bank account would have your original deposit, plus the first year's interest, PLUS the new interest on that amount.:

[$100(1 + 0.04)] + [$100(1 + 0.04)](0.04)

If we factor out the common factor again, we will have:

[$100(1 + 0.04)](1 + 0.04)

which simplifies to:

$100(1 + 0.04)^{2}

Keep going for a few more years and see what happens.

- Do you see the pattern?

We can generalize this to determine the amount in the account after *t* years. Take a minute to try that.

- Did you get $100(1 + 0.04)
^{t}?

We can generalize it further to figure out the amount in the account after *t *for any starting amount and any interest rate.

We will call the amount invested initially the PRINCIPAL (represented by P), and we will represent the interest rate with the letter *r*. Finally, the amount in the account at the end of *t* years we will call A.

- What do you think the formula will be?

Write down what you think it is.

- Did you get A = P(1 + r)
^{t}?

That is the formula for the amount in an account if you invest P dollars at an annual interest rate of r for t years, and the interest is compounded annually.

It's exponential, which means it will look something like this:

That's what happens mathematically if the interest is compounded yearly.

- But what if it is compounded monthly, quarterly, or daily?

Take several minutes to see if you can follow what we did above and develop a formula for *n* compounding periods for year. After you have spent some time thinking about it and writing a possible formula, come back and read the rest of the lesson.

Welcome back! I hope you enjoyed considering what the formula could be. The important thing is that you thought about it and tried out your ideas, even if you did not get it "right."

The formula looks like this:

A = P(1 + ^{r}/_{n})^{nt}

- Is that what you got?
- If not, can you figure out where that came from?

If the interest is compounded annually it is only happening once, so n would equal 1 and would be left out of the original formula.

- However, if the interest is compounded monthly, how many times a year would this happen?

Twelve, of course, so in that case n would equal 12!

- What about quarterly?

That would be four times a year or n = 4. I bet you could figure out n if the interest were compounded daily per year.

For a detailed explanation of the derivation of this formula, watch *Derivation of Compound Interest Formula* from Parabola Magic:

Now that you understand the basics of compound interest, go to the *Got It?* section to practice using these formulas.

Make sure you have them written down before you move on.

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