Learning Outcomes
- Use the simple interest formula.
- Use the compound interest formulas.
- Find the effective rate of interest.
- Find the future value.
Use Simple Interest and Compound Interest Formulas
Interest is money paid for the use of money. The total amount borrowed (whether by an individual from a bank in the form of a loan or by a bank from an individual in the form of a savings account) is called the principal. The rate of interest, expressed as a percent, is the amount charged for the use of the principle for a given period of time, usually on a yearly (that is, per annum) basis.
A General Note: The Simple Interest Formula
Simple interest can be calculated using the formula
where
- I is the amount of interest,
- t is measured in years,
- P is the starting amount of the account, often called the principal, or more generally present value,
- r is the annual percentage rate (APR) expressed as a decimal.
Example 1: Calculating Simple Interest
If we take out a $2,000 loan which charges 4% simple interest for 5 years, how much interest is paid?
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 t, principal P, APR r, and number of compounding periods in a year n:
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 |
A General Note: The Compound Interest Formula
Compound interest can be calculated using the formula
where
- A(t) is the account value,
- t is measured in years,
- P is the starting amount of the account, often called the principal, or more generally present value,
- r is the annual percentage rate (APR) expressed as a decimal, and
- n is the number of compounding periods in one year.
- annually [latex](n=1)[/latex], semiannually [latex](n=2)[/latex], quarterly [latex](n=4)[/latex], monthly [latex](n=12)[/latex], weekly [latex](n=52)[/latex], yearly [latex](n=365)[/latex]
Example 2: 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?
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?
Try It
Continuous Compounding
If the number of compoundings goes to infinity, then we have continuous compounding.
A General Note: The Compounding Continuously Formula
If you are compounding continuously, use the formula
where
- A(t) is the account value,
- t is measured in years,
- P is the starting amount of the account, often called the principal, or more generally present value,
- r is the annual percentage rate (APR) expressed as a decimal.
Example 3: Compounding Continuously
If we invest $3,000 in an investment account paying 3% interest compounded continuously, how much will the account be worth in 10 years?
Try It
An initial investment of $100,000 at 12% interest is compounded continuously. What will the investment be worth in 30 years?
The Effective Rate of Interest
The effective rate of interest is the annual simple interest rate that would yield the same amount as compounding n times per year, or continuously, after one year.
A General Note: The Effective Rate of Interest
The effective rate of interest [latex]r_E[/latex] of an investment earning an annual interest rate r is given by
- Compounding n times per year: [latex]r_E=\left(1+\frac{r}{n}\right)^n-1[/latex]
- Continuous Compounding: [latex]r_E=e^r-1[/latex]
Example 4: Effective Rate of Interest
Find the effective rate of interest for 7% compounded quarterly.
Try It
Find the effective rate of interest for 5% compounded continuously.
Section 5.7 Homework Exercises
For the following exercises, use the compound interest formula, [latex]A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}[/latex].
1. What was the initial deposit made to the account in the previous exercise?
2. How many years had the account from the previous exercise been accumulating interest?
3. An account is opened with an initial deposit of $6,500 and earns 3.6% interest compounded semi-annually. What will the account be worth in 20 years?
4. How much more would the account in the previous exercise have been worth if the interest were compounding weekly?
5. Solve the compound interest formula for the principal, P.
6. Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth $14,472.74 after earning 5.5% interest compounded monthly for 5 years. (Round to the nearest dollar.)
7. How much more would the account in the previous two exercises be worth if it were earning interest for 5 more years?
8. Use properties of rational exponents to solve the compound interest formula for the interest rate, r.
9. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $9,000 and was worth $13,373.53 after 10 years.
10. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $5,500, and was worth $38,455 after 30 years.
11. Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years?
12. How much less would the account from Exercise 11 be worth after 30 years if it were compounded monthly instead?
13. Determine the rate that represents the better deal: 7% compounded semiannually or 6.9% compounded continuously?
14. Determine the rate that represents the better deal: 9% compounded annually or 8.9% compounded continuously?
In problems 15 – 20, find the principle needed now to get each amount; that is, find the present value.
15. To get $100 after 2 years at 6% compounded monthly
16. To get $75 after 3 years at 8% compounded quarterly
17. To get $1500 after [latex]2 \frac{1}{2}[/latex] years at 1.5% compounded daily
18. To get $800 after [latex]3 \frac{1}{2}[/latex] years at 7% compounded monthly
19. To get $750 after 2 years at 2.5% compounded quarterly
20. To get $300 after 4 years at 3% compounded daily
In problems 21 – 24, find the effective rate of interest.
21. For 5% compounded quarterly
22. For 6% compounded monthly
23. For 4% compounded continuously
24. For 6% compounded continuously