In this section we will explore the idea of compound interest, which describes h
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