In this explainer, we will learn how to solve problems involving compound interest.

Financial investments pay interest, usually given as an annual percentage of the amount currently invested.

### Definition: Annual Interest on a Financial Investment

For a financial investment that pays interest compounded annually, the value of the investment after one year is given by
where is the initial value of the investment, also known as the *principal value*, and is the annual interest rate given as a percentage.

To simplify things, the annual percentage interest rate is usually converted to a *decimal* interest rate by dividing by 100,
so . This gives

For investments lasting multiple years, the interest earned is usually reinvested, or *compounded*,
so the value of the investment, and hence the interest paid, increases every year.

Let’s look at an example of how to use this formula to find the return on a financial investment lasting a small number of years.

### Example 1: Solving Word Problems Involving Percentages and Compound Interest

Adel deposited $100 in an account with an annual interest rate of , where the amount of the interest is added to his
account at the end of each year. Given that he did **not** withdraw any money in
3 years, determine the amount of money (in dollars and
cents) in his account at the end of each year.

### Answer

Recall that the value of the return on an investment after one year
with interest compounded annually is given by
where is the value (or *return*) of the investment after one year,
is the principal value of the investment, and is the annual interest rate, given as a decimal.

In this case, the principal value of the investment , and the annual interest rate . Therefore, after the end of the first year, Adel has

We can now repeat this process by substituting the value for the principal value for the next year. So, after the second year, the new value of the investment, will be

And, finally, repeating this process a third time by substituting the value for the principal value, after the third year the new value of the investment, will be

Therefore, Adel has $105.30 in his account after the first year, $110.88 after the second year, and $116.76 after the third year.

The final value is rounded to two decimal places, since the smallest denomination in US currency is .

This approach is sufficient when we need to find the return on an investment after a small number of years but would be very tedious for a problem where we need to find the return after longer periods of time.

So far, we have the value of a financial investment after one year, , given by where is the principal value of the investment and is the annual interest rate, given as a decimal. To find the value of the investment after a second year, , we can substitute the principal value with the value after one year, into the same formula:

We can then substitute in the expression for from the first equation: We can then proceed in this way for , and so on, leading to a formula for the value of a financial investment after a general number of years .

### Definition: Interest after a General Number of Years (Compounded Annually)

For a financial investment that pays interest compounded annually, the value of the investment after a number of years is given by where is the principal value of the investment and is the annual interest rate, given as a decimal.

This formula can be used to find the value of an investment lasting a large number of years far more efficiently. For example, if $5 000 is invested for 7 years with an interest rate of , the value of the investment at the end of the 7 years will be:

Let’s look at an example of how to find the return on an investment after a large number of years.

### Example 2: Creating Exponential Equations and Using Them to Solve Problems

When Maged was born, his grandparents invested $500 in a fund that would mature on his 21st birthday. If the fund earned per year, compounded annually, how much was its value when it matured? Give your answer to the nearest dollar.

### Answer

Recall that the value of the return on an investment with interest compounded annually is given by where is the final value (or *return*) of the investment, is the principal value of the investment, and is the annual interest rate, given as a decimal.

In this case, the principal value of the investment , the annual interest rate , and the number of years . Substituting these values into the formula gives

Many financial options compound interest more frequently than annually. Some compound interest *quarterly*
(4 times per year, or every 3 months), some weekly,
and some daily.

If interest on a financial investment is compounded more frequently than annually, the interest earned after one compounding period is given by the annual rate divided by the number of times per year the interest is compounded, .

For example, if an annual percentage interest rate of is compounded quarterly, then after 3 months the interest paid is of the value of the investment.

If a financial investment of principal value pays interest at an annual rate of compounded times per year, the value of the investment after the first period of years is given by

This is the new value of the investment for the next period. Hence, the value of the investment after the next period is given by

Substituting in the expression ,

This continues, and by the end of the year, after periods, the interest has been compounded times and the value of the investment is given by

This is the value of the investment after one year, which is periods. Extending this to a general number of years , or periods, is simple.

### Definition: Interest after a General Number of Years (Compounded 𝑛 Times per Year)

For a financial investment that pays interest compounded times per year, the value of the investment after a number of years is given by where is the principal value of the investment and is the annual interest rate, given as a decimal.

Let’s look at an example of how to find the value of an investment after a whole number of years when interest is compounded quarterly.

### Example 3: Interest Compounded Quarterly

Adam invests $3 000 at a interest rate per year, compounded quarterly. Find the balance after 10 years.

### Answer

Recall that the value of the return on an investment with interest compounded times
per year is given by
where is the final value (or *return*) of the investment, is the principal value of the investment,
is the annual interest rate, given as a decimal, is the number of times
per year the interest is compounded, and is the number of
years invested.

In this case, the principal value of the investment , the annual interest rate , the number of times per year the interest is compounded , and the number of years invested . Therefore, we have

Compounding interest more frequently results in a greater return on the investment for the same rate of annual interest because, effectively, the interest earned for each year is reinvested sooner and is itself able to earn interest for that year. For example, if interest is compounded quarterly, then the interest earned after the first 3 months itself earns interest for the remaining 9 months of the year.

Let’s look at an example of how compounding interest more frequently results in a greater return on an investment.

### Example 4: Comparing the Return of Annually and Monthly Compounded Interest

Bank A offers depositors annual interest compounded once per year. Bank B offers per year, compounded monthly. Write an explicit formula for the return after years on a deposit of with both offers. Which bank’s offer is better?

### Answer

**Part 1**

Recall that the value of the return on an investment with interest compounded times
per year is given by
where is the final value (or *return*) of the investment, is the principal value of the investment, is the annual
interest rate, given as a decimal, is the number of times per year the interest is
compounded, and is the number of years invested.

In this question, the principal value is denoted , the return is denoted , and the number of years is denoted . We therefore need to denote the number of times per year the interest is compounded differently, so let this equal . Then, we have

For bank A, and ; therefore,

And for bank B, and ; therefore,

**Part 2**

We can identify which bank has the better offer simply by evaluating the return after one year, since each subsequent year will increase the value of the investment by the same proportion. For bank A,

And for bank B,

Therefore, bank B’s offer is better. We can verify this with some example values. If the principal value of the investment , then the return after one year from bank A would be

And the return after one year from bank B would be

So while the difference is small, bank B’s offer is slightly better.

Let’s look at a final example of how to use these formulae to solve real-world financial problems.

### Example 5: Banking Applications of Exponential Functions

Fady deposits $100 in a savings account that gives him a interest on his savings each month. Nabil has $350 in a cashing account that he withdraws $5 from each month. After how many months do the two have approximately the same bank balance?

### Answer

In this example, we have an investment that returns a large amount of interest every month, and we are asked to find the number of months after which it reaches a specific value.

Recall that the value of the return on an investment with interest compounded annually is given by
where is the final value (or *return*) of the investment, is the principal value of the investment, and
is the annual interest rate, given as a decimal.

For Fady’s savings account, it makes more sense to consider the return on a monthly basis. Since the interest is *monthly*,
and compounded monthly, we can modify this formula slightly to give the return on his investment, , after months:
where is the *monthly* interest rate, given as a decimal.

For Fady’s savings account, we have a principal value and a monthly interest rate , so the return on his savings account, , will be given by

For Nabil’s cashing account, he starts with an investment of $350 and withdraws $5 each month; therefore, the return on Nabil’s account, , after months is given by

We now need to find the integer value of for which . We can find this value of using trial and improvement, evaluating the difference between the two balances after months, , for different integer values of to find which value gives the smallest difference. Let’s start with a sensible guess, say :

So Fady’s account has a lower balance than Nabil’s, but there may be a larger integer value of for which the difference in balance is smaller.

We are looking for the two consecutive integer values of for which the difference between the balances, , changes sign, since the value of for which the difference is smallest must be one of those two values. So Let’s try a larger value of , say

:

We can see that the sign of the difference has changed somewhere between and , so let’s try a value of in between these two values, say

:

changes sign somewhere between and , so let’s try a value of in between these two values, say

:

changes sign somewhere between and . We have one more value to check,

:

So the sign of the difference changes between and . We can see that the difference is smallest for , where ; therefore, the two accounts have approximately the same balance after 8 months.

Let’s finish by recapping some of the key points from this explainer.

### Key Points

- The return, , on a financial investment with principal value and annual interest rate , with interest compounded annually after years, is given by .
- The return, , on a financial investment with principal value and annual interest rate , with interest compounded times per year after years, is given by .
- Compounding interest more regularly will
*always*result in a greater return on the investment given the same annual interest rate .