In this section we will explore the idea of compound interest, which describes how money grows when it is invested. Compound interest is a common example of exponential growth, which can be described by exponential functions. We’ll learn more about exponential functions in the next section, but for now we’ll just focus on compound interest.
3.6.1 – Use Compound-Interest Formulas
Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest. The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account. The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time [latex]\,t,[/latex] principal [latex]\,P,[/latex] APR [latex]\,r,[/latex] and number of compounding periods in a year [latex]\,n:[/latex]
For example, observe the table below, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases.
| Frequency | Value after 1 year |
|---|---|
| Annually | $1100 |
| Semiannually | $1102.50 |
| Quarterly | $1103.81 |
| Monthly | $1104.71 |
| Daily | $1105.16 |
The Compound Interest Formula
Compound interest can be calculated using the formula
where
- [latex]A\left(t\right)\,[/latex] is the account value,
- [latex]t\,[/latex] is measured in years,
- [latex]P\,[/latex] is the starting amount of the account, often called the principal, or more generally present value,
- [latex]r\,[/latex] is the annual percentage rate (APR) expressed as a decimal, and
- [latex]n\,[/latex] is the number of compounding periods in one year.
Example 1 – Calculating Compound Interest
If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?
Because we are starting with $3,000, [latex]\,P=3000.\,[/latex] Our interest rate is 3%, so [latex]\,r\text{ }=\text{ }0.03.\,[/latex] Because we are compounding quarterly, we are compounding 4 times per year, so [latex]\,n=4.\,[/latex] We want to know the value of the account in 10 years, so we are looking for [latex]\,A\left(10\right),[/latex] the value when [latex]\,t\text{ }=\text{ }10.[/latex]
[latex]$$\begin{array}{lll}A\left(t\right)\hfill & =P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & \text{Use the compound interest formula}.\hfill \\ A\left(10\right)\hfill & =3000{\left(1+\frac{0.03}{4}\right)}^{4\cdot 10}\begin{array}{cccc}& & & \end{array}\hfill & \text{Substitute using given values}.\hfill \\ \hfill & \approx \text{\$}4045.05\hfill & \text{Round to two decimal places}.\hfill \end{array}$$[/latex]
The account will be worth about $4,045.05 in 10 years.
Try It
An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?
Show answer
about $3,644,675.88
Example 2 – Using the Compound Interest Formula to Solve for the Principal
A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?
The nominal interest rate is 6%, so [latex]\,r=0.06.\,[/latex] Interest is compounded twice a year, so [latex]\,k=2.[/latex]
We want to find the initial investment, [latex]\,P,[/latex] needed so that the value of the account will be worth $40,000 in [latex]\,18\,[/latex] years. Substitute the given values into the compound interest formula, and solve for [latex]\,P.[/latex]
[latex]$$\begin{array}{lll}\,\,\,\,\,\,\,A\left(t\right)\hfill & =P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & \text{Use the compound interest formula}.\hfill \\ 40,000\hfill & =P{\left(1+\frac{0.06}{2}\right)}^{2\left(18\right)}\begin{array}{cccc}& & & \end{array}\hfill & \text{Substitute using given values }A\text{, }r, n\text{, and }t.\hfill \\ 40,000\hfill & =P{\left(1.03\right)}^{36}\hfill & \text{Simplify}.\hfill \\ \frac{40,000}{{\left(1.03\right)}^{36}}\hfill & =P\hfill & \text{Isolate }P.\hfill \\ \,\,\,\,\,\,\,\,\,\,P\hfill & \approx \text{\$}13,801\hfill & \text{Divide and round to the nearest dollar}.\hfill \end{array}$$[/latex]
Lily will need to invest $13,801 to have $40,000 in 18 years.]
Try It
Refer to Example 2. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?
Show answer
$13,693
Access these online resources for additional instruction and practice with compound interest.
Key Equations
| compound interest formula | [latex]\begin{array}{l}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt} ,\text{ where}\hfill \\ A\left(t\right)\text{ is the account value at time }t\hfill \\ t\text{ is the number of years}\hfill \\ P\text{ is the initial investment, often called the principal}\hfill \\ r\text{ is the annual percentage rate (APR), or nominal rate}\hfill \\ n\text{ is the number of compounding periods in one year}\hfill \end{array}[/latex] |
Key Concepts
- The value of an account at any time [latex]\,t\,[/latex] can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known. See Example 1.
- The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known. See Example 2.
Glossary
- annual percentage rate (APR)
- the yearly interest rate earned by an investment account, also called nominal rate
- compound interest
- interest earned on the total balance, not just the principal
- nominal rate
- the yearly interest rate earned by an investment account, also called annual percentage rate
