3.4: Compound Growth
Contents
Objectives:
- Introduce (annual) compound interest
- Compound interest more frequently than annually
- Understand the difference between nominal and effective interest rate
Important Items
Definitions:
compound interest, nominal rate, effective rate
Lesson Guide
Warm-Up
Begin by asking your students if any of them have a bank account, and maybe ask if they know what interest rate they earn. In this section, we will explore different types of interest rates, and how they are computed (and how knowing which is which could help you make better investments!)
Have students do Problems 1 and 2.
You should expect that student will still be confused by effective annual percent rate. Make sure to look up the term ahead of time so that you can give them an accurate definition when they ask.
Introduce (Annual) Compound Interest
|
Instructor Example |
Suggested Example | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
This is a sentence that is in place to generate space in the table |
Begin with an example of a bank with an interest rate that is compounded annually. Choose a principal of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=1000} dollars, and an interest rate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=12} percent, and have students help to compute the value in the account Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} years later. A table like the following one may be useful:
Stress that the bank is paying interest on the previously earned interest as well as on the principal each year. |
For many instructors these are terms you may have not heard before. Make sure to look up these terms in the book so that you know what they are!
* The term compound interest refers to interest that is applied not only to the principal but also to previously earned interest. * The nominal rate of an investment is the given interest rate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} .
- Ask students to identify the nominal rate in Problem 1.
Compound Interest More Frequently than Annually
|
Instructor Example |
Suggested Example | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
This is a sentence that is in place to generate space in the table |
Begin with an example of a bank with an interest rate that is compounded monthly. Choose a principal of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=1000} dollars, and an annual interest rate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=12} percent, and have students help to compute the value in the account Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} years later. A table like the following one may be useful:
Emphasize to students that when a bank compounds interest monthly, for example, the bank will apply interest to the account, but it will be at a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{r}{12}} , NOT Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r.} Describe that this is because they apply the interest rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} in 12 increments. Fill out the following table: |
Highlight that after 12 months, the account will have reached the amount it has after one total year of compounding. Ask the students if the amounts that the two accounts have at the end of one year are the same. Discuss how even with the same nominal rate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
, and principal, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}
, compounding at different frequencies has a big effect on the value of an account. Ask students to find a formula to describe the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)}
of the account Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}
years after the initial deposit:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=P\left(1+\frac{r}{12}\right)^{12t}}
More generally, if a bank has an interest rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
that compounds Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}
times per year, the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)}
of
the account Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}
years after the initial deposit of the principal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=P\left(1+\frac{r}{n}\right)^{nt}.}
If you wish, you may refer to the above formula as the compound interest formula.
- Have students do Problem 3.
- Discuss with your students how changing the frequency of compounding affects the value of the account. I.e., if Bank A and Bank B have the same interest rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} but Bank A compounds less frequently than Bank B, which bank yields more interest?
Understand the difference between nominal and effective interest rate
When the interest is compounded more frequently than once a year, the account effectively earns more than the nominal rate, and so we distinguish between the nominal rate and the effective rate.
Suppose, for example, that an interest rate is 12\% compounded monthly (as in the example above). Explain that we refer to the 12\% as the nominal rate. When the interest is compounded more frequently than once a year, the account effectively earns more than the nominal rate, and so we distinguish between the nominal rate and the effective rate. The effective rate tells you how much interest the investment actually earns. This is sometimes called the APY (annual percentage yield) in the U.S.
|
Instructor Example |
Suggested Example |
|---|---|
|
This is a sentence that is in place to generate space in the table |
For each of the following two banks, determine what the effective interest rate is.
Since an account paying 12% annual interest, compounded annually, grows by exactly 12% in one year, we have that the nominal rate is the same as its effective rate: both are 12\%.
The nominal rate is 12%. Using the compound interest formula, we know that after 12 months, our investment would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1000(1.01)^{12}=1126.83} . The annual growth factor is \frac{1126.83}{1000}=1.12683. So, the account effectively earns 12.683\% interest in a year, so its effective interest rate is 12.683\%. |
Remind students that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} in the compound interest formula is the nominal rate, not the effective rate.
The effective annual rate of an investment tells you how much interest the investment actually earns per year. This is sometimes called the APY (annual percentage yield) in the U.S.
Make sure that students understand that if Bank A and Bank B have the same nominal interest rate, but Bank A compounds less frequently than Bank B, then the effective annual rate of Bank A will be less than the effective annual rate of Bank B.
Use Problem 3 on Worksheet 3.4 to compute the effective annual rate for each compounding frequency. Compare this with the nominal rate.
Have students do Problems 4-5.
Have students do Problems 6-7.
Remind students to refer to the compound interest rate formula (p. 157). Be sure to work your way around to each group. There are more problems listed here than most groups will finish in class, so you might want to encourage them to finish problems outside of class that their groups don't complete in class.
To bring closure to the above problems, as you work your way around the class, ask individuals to write the problems up on the board (you might find it best this time to check answers first so that students are sure about their answer going up on the board). Allow at least 3-5 minutes to go through the solutions together as a class. Depending on your time availability, go through a select few of the problems, and others only if you have time.
