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[[3.3: Graphs of Exponential Functions | Prior Lesson]] | [[3.5: Continuous Growth | Next Lesson]]
==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===

{{Ex-begin}}

Begin with an example of a bank with an interest rate that is ''compounded annually''. Choose a principal of <math>P=1000</math> dollars, and an interest rate, <math>r=12</math> percent, and have students help to compute the value in the account <math>t</math> years later. A table like the following one may be useful:

{| class="wikitable"
|-
|Years after Initial Deposit
|0
|1
|2
|3
|-
|Computation of Value
| 1000
| 1000(1.12)
| 1000(1.12)(1.12)
| 1000(1.12)(1.12)(1.12)
|-
|Value of Account
| 1000
| 1120
| 1254.4
| 1404.93
|}

Ask students what type of growth this account has, and then ask them to find a formula to describe the value <math>A(t)</math> of the account <math>t</math> years after the initial deposit: <math></math>A(t)=P(1+r)^t.<math></math>

Stress that the bank is paying interest '''on the previously earned interest''' as well as on the principal each year.

{{Ex-end}}

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, <math>r</math>.

*Ask students to identify the nominal rate in Problem 1.

===Compound Interest More Frequently than Annually===

{{Ex-begin}}

Begin with an example of a bank with an interest rate that is ''compounded monthly''. Choose a principal of <math>P=1000</math> dollars, and an annual interest rate, <math>r=12</math> percent, and have students help to compute the value in the account <math>t</math> years later. A table like the following one may be useful:

{| class="wikitable"
|-
|Months after Initial Deposit
|0
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|-
|Computation of Value
| 1000
| 1000(1.01)
| 1000(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
|-
|Value of Account
| 1000
| 1010
| 1020.1
| 1030.3
| 1040.6
| 1051.01
| 1061.52
| 1072.14
| 1082.86
| 1093.69
| 1104.62
| 1115.67
| 1126.83
|}

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 <math>\frac{r}{12}</math>, '''NOT''' <math>r.</math> Describe that this is because they apply the interest rate of <math>r</math> in 12 increments. Fill out the following table:

{{Ex-end}}

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, <math>r</math>, and principal, <math>P</math>, compounding at different frequencies has a big effect on the value of an account. Ask students to find a formula to describe the value <math>A(t)</math> of the account <math>t</math> years after the initial deposit:
<math>A(t)=P\left(1+\frac{r}{12}\right)^{12t}</math>

More generally, if a bank has an interest rate of <math>r</math> that compounds <math>n</math> times per year, the value <math>A(t)</math> of
the account <math>t</math> years after the initial deposit of the principal <math>P</math> is <math>A(t)=P\left(1+\frac{r}{n}\right)^{nt}.</math>
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 <math>r</math> 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.

{{Ex-begin}}
For each of the following two banks, determine what the effective interest rate is.

*'''Bank A''': Pays 12% interest compounded annually

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\%.

*'''Bank B''': Pays 12% interest compounded monthly

The nominal rate is 12%. Using the compound interest formula, we know that after 12 months, our investment would be <math>1000(1.01)^{12}=1126.83</math>. The ''annual'' growth factor is
<math></math>\frac{1126.83}{1000}=1.12683.<math></math>
So, the account effectively earns 12.683\% interest in a year, so its effective interest rate is 12.683\%.

{{Ex-end}}

Remind students that <math>r</math> 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.

==Comments==

3.4: Compound Growth - Revision history

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