Contemporary Mathematics

# 6.4Compound Interest

Contemporary Mathematics6.4 Compound Interest

Figure 6.6 The impact of compound interest (credit: "English Money" by Images Money/Flickr, CC BY 2.0)

## Learning Objectives

After completing this section, you should be able to:

1. Compute compound interest.
2. Determine the difference in interest between simple and compound calculations.
3. Understand and compute future value.
4. Compute present value.
5. Compute and interpret effective annual yield.

For a very long time in certain parts of the world, interest was not charged due to religious dictates. Once this restriction was relaxed, loans that earned interest became possible. Initially, such loans had short terms, so only simple interest was applied to the loan. However, when loans began to stretch out for years, it was natural to add the interest at the end of each year, and add the interest to the principal of the loan. After another year, the interest was calculated on the initial principal plus the interest from year 1, or, the interest earned interest. Each year, more interest was added to the money owed, and that interest continued to earn interest.

Since the amount in the account grows each year, more money earns interest, increasing the account faster. This growth follows a geometric series (Geometric Sequences). It is this feature that gives compound interest its power. This module covers the mathematics of compound interest.

## Understand and Compute Compound Interest

As we saw in Simple Interest, an account that pays simple interest only pays based on the original principal and the term of the loan. Accounts offering compound interest pay interest at regular intervals. After each interval, the interest is added to the original principal. Later, interest is calculated on the original principal plus the interest that has been added previously.

After each period, the interest on the account is computed, then added to the account. Then, after the next period, when interest is computed, it is computed based on the original principal AND the interest that was added in the previous periods.

The following example illustrates how compounded interest works.

## Determine the Difference in Interest Between Simple and Compound Calculations

It is natural to ask, does compound interest make much of a difference? To find out, we revisit Abena’s CD.

## Example 6.40

### Comparing Simple to Compound Interest on a 3-Year CD

Abena invested $1,000 in a CD that earned 4% compounded annually, and the CD was worth$1,124.86 after 3 years. Had Abena invested in a CD with simple interest, how much would the CD have been worth after 3 years? How much more did Abena earn using compound interest?

1.
Oksana deposits $5,000 in a CD that earned 3% compounded annually and was worth$5,627.54 after 4 years. Had Oksana invested in a CD with simple interest, how much would the CD have been worth after 4 years? How much more did Oksana earn using compound interest?

## Understand and Compute Future Value

Imagine investing for 30 years and compounding the interest every month. Using the method above, there would be 360 periods to calculate interest for. This is not a reasonable approach. Fortunately, there is a formula for finding the future value of an investment that earns compound interest.

## FORMULA

The future value of an investment, $AA$, when the principal $PP$ is invested at an annual interest rate of $rr$ (in decimal form), compounded $nn$ times per year, for $tt$ years, is found using the formula $A=P(1+rn)ntA=P(1+rn)nt$. This is also referred to as the future value of the investment.

## Checkpoint

Note, sometimes the formula is presented with the total number of periods, $nn$, and the interest rate per period, $rr$. In that case the formula becomes $A=P(1+r)nA=P(1+r)n$.

## Example 6.41

### Computing Future Value for Compound Interest

In the following, compute the future value of the investment with the given conditions.

1. Principal is $5,000, annual interest rate is 3.8%, compounded monthly, for 5 years. 2. Principal is$18,500, annual interest rate is 6.25%, compounded quarterly, for 17 years.

In the following, compute the future value of the investment with the given conditions.
1.
Principal is $7,600, annual interest rate is 4.1%, compounded monthly, for 10 years. 2. Principal is$13,250, annual interest rate is 2.79%, compounded quarterly, for 25 years.

## FORMULA

The money invested in an account bearing an annual interest rate of $rr$ (in decimal form), compounded $nn$ times per year for $tt$ years, is called the present value, $PVPV$, of the account (or of the money) and found using the formula $PV=A(1+rn)n×tPV=A(1+rn)n×t$, where $AA$ is the value of the account at the investment’s end. Always round this value up to the nearest penny.

## Example 6.44

### Computing Present Value

Find the present value of the accounts under the following conditions.

1. $AA$ = $250,000, invested at 6.75 interest, compounded monthly, for 30 years. 2. $AA$ =$500,000, invested at 7.1% interest, compounded quarterly, for 40 years.

Find the present value of the accounts under the following conditions.
1.
$A$ = $1,000,000, invested at 5.75% interest, compounded monthly, for 40 years. 2. $A$ =$175,000, invested at 3.8% interest, compounded quarterly, for 20 years.

## Compute and Interpret Effective Annual Yield

As we’ve seen, quarterly compounding pays interest 4 times a year or every 3 months; monthly compounding pays 12 times a year; daily compounding pays interest every day, and so on. Effective annual yield allows direct comparisons between simple interest and compound interest by converting compound interest to its equivalent simple interest rate. We can even directly compare different compound interest situations. This gives information that can be used to identify the best investment from a yield perspective.

Using a formula, we can interpret compound interest as simple interest. The effective annual yield formula stems from the compound interest formula and is based on an investment of $1 for 1 year. Effective annual yield is $Y=(1+rn)n-1Y=(1+rn)n-1$ where $YY$ = effective annual yield, $rr$ = interest rate in decimal form, and $nn$ = number of times the interest is compounded in a year. $YY$ is interpreted as the equivalent annual simple interest rate. ## Example 6.46 ### Determine and Interpret Effective Annual Yield for 6% Compounded Quarterly Suppose you have an investment paying a rate of 6% compounded quarterly. Determine and interpret that effective annual yield of the investment. ## Your Turn 6.46 1. Calculate and interpret the effective annual yield for an investment that pays at a 7% interest compounded quarterly. ## Example 6.47 ### Determine and Interpret Effective Annual Yield for 5% Compounded Daily Calculate and interpret the effective annual yield on a deposit earning interest at a rate of 5% compounded daily. ## Your Turn 6.47 1. Calculate and interpret the effective annual yield on a deposit earning 2.5% compounded daily. ## Example 6.48 ### Choosing a Bank Minh has a choice of banks in which he will open a savings account. He will deposit$3,200 and he wants to get the best interest he can. The banks advertise as follows:

Bank Interest Rate
ABC Bank 2.08% compounded monthly
123 Bank 2.09% compounded annually
XYZ Bank 2.05% compounded daily

Which bank offers the best interest?

1.
Isabella decides to deposit $5,500 in a CD but needs to choose between banks that offer CDs. She identifies four banks and finds out the terms of their CDs. Her findings are in the table below. Bank Interest Rate Smith Bank 3.08% compounded quarterly Park Bank 3.11% compounded annually Town Bank 3.09% compounded daily Community Bank 3.10% compounded monthly Which bank has the best yield? ## Check Your Understanding 19. What is compound interest? 20. Which yields more money, simple interest or compound interest? 21. Find the future value after 15 years of$8,560.00 deposited in an account bearing 4.05% interest compounded monthly.
22.
$10,000 is deposited in an account bearing 5.6% interest for 5 years. Find the difference between the future value when the interest is simple interest and when the interest is compounded quarterly. 23. Find the present value of$75,000 after 28 years if money is invested in an account bearing 3.25% interest compounded monthly.
24.
What can be done to compare accounts if the rates and number of compound periods per year are different?
25.
Find the effective annual yield of an account with 4.89% interest compounded quarterly.

## Section 6.4 Exercises

1 .
What is the difference between simple interest and compound interest?
2 .
What is a direct way to compare accounts with different interest rates and number of compounding periods?
3 .
Which type of account grows in value faster, one with simple interest or one with compound interest?
How many periods are there if interest is compounded?
4 .
Daily
5 .
Weekly
6 .
Monthly
7 .
Quarterly
8 .
Semi-annually
In the following exercises, compute the future value of the investment with the given conditions.
9 .
Principal = $15,000, annual interest rate = 4.25%, compounded annually, for 5 years 10 . Principal =$27,500, annual interest rate = 3.75%, compounded annually, for 10 years
11 .
Principal = $13,800, annual interest rate = 2.55%, compounded quarterly, for 18 years 12 . Principal =$150,000, annual interest rate = 2.95%, compounded quarterly, for 30 years
13 .
Principal = $3,500, annual interest rate = 2.9%, compounded monthly, for 7 years 14 . Principal =$1,500, annual interest rate = 3.23%, compounded monthly, for 30 years
15 .
Principal = $16,000, annual interest rate = 3.64%, compounded daily, for 13 years 16 . Principal =$9,450, annual interest rate = 3.99%, compounded daily, for 25 years
In the following exercises, compute the present value of the accounts with the given conditions.
17 .
Future value = $250,000, annual interest rate = 3.45%, compounded annually, for 25 years 18 . Future value =$300,000, annual interest rate = 3.99%, compounded annually, for 15 years
19 .
Future value = $1,500,000, annual interest rate = 4.81%, compounded quarterly, for 35 years 20 . Future value =$750,000, annual interest rate = 3.95%, compounded quarterly, for 10 years
21 .
Future value = $600,000, annual interest rate = 3.79%, compounded monthly, for 17 years 22 . Future value =$800,000, annual interest rate = 4.23%, compounded monthly, for 35 years
23 .
Future value = $890,000, annual interest rate = 2.77%, compounded daily, for 25 years 24 . Future value =$345,000, annual interest rate = 2.99%, compounded daily, for 19 years
In the following exercises, compute the effective annual yield for accounts with the given interest rate and number of compounding periods. Round to three decimal places.
25 .
Annual interest rate = 2.75%, compounded monthly
26 .
Annual interest rate = 3.44%, compounded monthly
27 .
Annual interest rate = 5.18%, compounded quarterly
28 .
Annual interest rate = 2.56%, compounded quarterly
29 .
Annual interest rate = 4.11%, compounded daily
30 .
Annual interest rate = 6.5%, compounded daily
The following exercises explore what happens when a person deposits money in an account earning compound interest.
31 .
Find the present value of $500,000 in an account that earns 3.85% compounded quarterly for the indicated number of years. 1. 40 years 2. 35 years 3. 30 years 4. 25 years 5. 20 years 6. 15 years 32 . Find the present value of$1,000,000 in an account that earns 6.15% compounded monthly for the indicated number of years.
1. 40 years
2. 35 years
3. 30 years
4. 25 years
5. 20 years
6. 15 years
33 .
In the following exercises, the number of years can reflect delaying depositing money. 40 years would be depositing money at the start of a 40-year career. 35 years would be waiting 5 years before depositing the money. Thirty years would be waiting 10 years before depositing the money, and so on. What do you notice happens if you delay depositing money?
34 .
For each 5-year gap for exercise 32, compute the difference between the present values. Do these differences remain the same for each of the 5-year gaps, or do they differ? How do they differ? What conclusion can you draw?
35 .
Daria invests $2,500 in a CD that yields 3.5% compounded quarterly for 5 years. How much is the CD worth after those 5 years? 36 . Maurice deposits$4,200 in a CD that yields 3.8% compounded annually for 3 years. How much is the CD worth after those 3 years?
37 .
Georgita is shopping for an account to invest her money in. She wants the account to grow to $400,000 in 30 years. She finds an account that earns 4.75% compounded monthly. How much does she need to deposit to reach her goal? 38 . Zak wants to create a nest egg for himself. He wants the account to be valued at$600,000 in 25 years. He finds an account that earns 4.05% interest compounded quarterly. How much does Zak need to deposit in the account to reach his goal of $600,000? 39 . Eli wants to compare two accounts for their money. They find one account that earns 4.26% interest compounded monthly. They find another account that earns 4.31% interest compounded quarterly. Which account will grow to Eli’s goal the fastest? 40 . Heath is planning to retire in 40 years. He’d like his account to be worth$250,000 when he does retire. He wants to deposit money now. How much does he need to deposit in an account yielding 5.71% interest compounded semi-annually to reach his goal?
41 .
Jo and Kim want to set aside some money for a down payment on a new car. They have 6 years to let the money grow. If they want to make a $15,000 down payment on the car, how much should they deposit now in an account that earns 4.36% interest compounded monthly? 42 . A newspaper’s business section runs an article about savings at various banks in the city. They find six that offer accounts that offer compound interest. Bank A offers 3.76% compounded daily. Bank B offers 3.85% compounded annually. Bank C offers 3.77% compounded weekly. Bank D offers 3.74% compounded daily. Bank E offers 3.81% compounded semi-annually. To earn the most interest on a deposit, which bank should a person choose? 43 . Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a principal of$100,000 that are allowed to grow for 20 years. What is the difference in the future values of the account with the highest effective annual yield, and the account with the second highest effective annual yield?
44 .
Paola reads the newspaper article from exercise 32. She really wants to know how different they are in terms of dollars, not effective annual yield. She decides to compute the future value for accounts at each bank based on a principal of $100,000 that are allowed to grow for 20 years. What is the difference in the future values of the account with the highest effective annual yield, and the account with the lowest effective annual yield? 45 . Jesse and Lila need to decide if they want to deposit money this year. If they do, they can deposit$17,400 and allow the money to grow for 35 years. However, they could wait 12 years before making the deposit. At that time, they’d be able to collect $31,700 but the money would only grow for 23 years. Their account earns 4.63% interest compounded monthly. Which plan will result in the most money, depositing$17,400 now or depositing $31,700 in 12 years? 46 . Veronica and Jose are debating if they should deposit$15,000 now in an account or if they should wait 10 years and deposit $25,000. If they deposit money now, the money will grow for 35 years. If they wait 10 years, it will grow for 25 years. Their account earns 5.25% interest compounded weekly. Which plan will result in the most money, depositing$15,000 now or depositing \$25,000 in 10 years?
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