350 Chapter 11 • Compound Interest and present Value
REVIEw ExERCIsEs
For the following investments, find the total number of compounding periods and the
interest rate per period.
Term of
Investment
Nominal
(Annual) Rate
(%)
Interest
Compounded
Compounding
Periods
Rate per
Period
(%)
1. 3 years 13 annually 3 13
ExamplE5
usIng the Compound
Interest formula
Use the compound interest formula to calculate the compound amount of
$5,000
invested
at
10%
interest compounded semiannually for 3 years.
SolutionStrategy
This problem is solved by substituting the investment information into the compound interest
formula. It is important to solve the formula using the sequence of steps outlined above. Note that the
6
tryitexerciSe 5
Use the compound interest formula to calculate the compound amount of
$3,000
invested at
8%
interest compounded quarterly for 5 years.
CHECK YOUR ANSWER WITH THE SOLUTION ON PAGE 363.
11
Section i
Complete worked-out
StepS For solVing the CoMPoUnd interest ForMUlA
SteP 1. Add the
1
and the interest rate per period,
i
.
SteP 2. Raise the sum from Step 1 to the
nth
(number of compounding periods) power
seCtIon I • Compound Interest—the tIme Value of money 351
Manually calculate the compound amount and compound interest for the following
investments.
Principal
Time
Period (years)
Nominal
Rate
(%)
Interest
Compounded
Compound
Amount
Compound
Interest
8. $4,000 2 10 annually $4,840.00 $840.00
Using table 11-1, calculate the compound amount and compound interest for the following
investments.
Principal
Time
Period (years)
Nominal
Rate
(%)
Interest
Compounded
Compound
Amount
Compound
Interest
12. $7,000 4 13 annually $11,413.29 $4,413.29
the following investments require table factors for periods beyond the table. Create the new
table factor, rounded to five places, and calculate the compound amount for each.
Principal
Time
Period (years)
Nominal
Rate
(%)
Interest
Compounded
New Table
Factor
Compound
Amount
19. $13,000 3 12 monthly 1.43077 $18,600.01
For the following investments, compute the amount of compound interest earned in 1 year
and the annual percentage yield (APY).
Principal
Nominal
Rate
(%)
Interest
Compounded
Compound Interest
Earned in 1 Year
Annual Percentage
Yield (APY)
24. $5,000 10 semiannually $512.50 10.25%
25. $2,000 4 annually $80.00 4%
solve the following word problems by using table 11-1.
29. Sherry Smith invested
$3,000
at the Horizon Bank at
6%
interest compounded quarterly.
a. What is the annual percentage yield of this investment?
Complete worked-out solutions for
Exercises 8–28 appear in Appendix B.
85461_ch11_hr_341-368_1.indd 351 9/23/15 4:54 PM
352 Chapter 11 • Compound Interest and present Value
b. What will Sherry’s investment be worth after 6 years?
30. As a savings plan for college, when their son Bob was born, the Wilburs deposited
$10,000
in
an account paying
8%
compounded annually. How much will the account be worth when Bob
is18years old?
31. You are the owner of a UPS Store franchise. You have just deposited
$12,000
in an investment
accountearning
12%
compounded monthly. This account is intended to pay for store improve ments
in
2
1
2
years. At that rate, how much will be available in the account for the project?
Table factor required =1%, 30 periods
32. The First National Bank is offering a 6-year certificate of deposit (CD) at
4%
interest
compounded quarterly; Second National Bank is offering a 6-year CD at
5%
interest
compounded annually.
a. If you were interested in investing
$8,000
in one of these CDs, calculate the compound
amount of each offer.
First National Bank: 1%, 24 periods
b. What is the annual percentage yield of each CD?
First National Bank: 1%, 4 periods
c. (Optional) If Third National Bank has a 6-year CD at
4.5%
interest compounded monthly, use
the compound interest formula to calculate the compound amount of this offer.
33. A certain animal husbandry program has a flock of sheep that increases in size by
15%
every
year. If there are currently
48
sheep, how many sheep are expected to be in the flock in 5 years?
Round to the nearest whole sheep.
34. The rate of bacteria growth in a laboratory experiment was measured at
16%
per hour. If this
experiment is repeated and begins with
5
grams of bacteria, how much bacteria should be
expected after 12 hours? Round to the nearest tenth of a gram.
Point out to students how this
exercise demonstrates that higher
rates have a more significant effect
UPs store franchises have consistently
services UPS Stores provide include
packing and shipping services,
mailbox and postal services,
SG cityscapes/Alamy
Learning Tip
85461_ch11_hr_341-368_1.indd 352 9/23/15 4:54 PM
seCtIon I • Compound Interest—the tIme Value of money 353
solve the following exercises and word problems by using the compound interest
formula.
Principal
Time Period
(years)
Nominal
Rate (%)
Interest
Compounded
Compound
Amount
Compound
Interest
35. $5,000
4
4.2 semiannually $5,904.40 $904.40
39. Gabriel Hopen, a 32-year-old commercial artist, has just signed a contract with an advertising
agency. Gabriel’s starting salary is
$47,800
. The agency has agreed to increase his salary
by
8.5%
annually. How much will Gabriel’s salary be after 5 years? Round to the nearest
wholedollar.
40. The FernRod Motorcycle Company invested
$250,000
at
4.5%
compounded monthly to be used
for the expansion of their manufacturing facilities. How much money will be available for the
project in
3
1
years?
Complete worked-out solutions for
Exercises 35–38 appear in Appendix B.
busInEss DECIsIon: DAILy ComPounDIng
41. As an incentive to attract savings deposits, most financial institutions today offer daily and even
continuous compounding. This means that savings, or passbook, accounts, as well as CDs,
earn interest compounded each day or even more frequently, such as every hour or even every
minute. (Continuous compounding, in which compounding occurs every instant, involves a
different formula that is derived from the formula we’ve been using.) Let’s take a look at daily
compounding.
To calculate the compound amount,
A
, of an investment with daily compounding, use the
compound interest formula modified as follows:
•
R
ate per period (daily) =
i
365
(nominal interest rate,
i
, divided by
365
)
• Number of periods (days),
n
,
=
number of days of the investment.
A
=P
a
1+i
365bn
Calculator Sequence: 1 i 365 n P A
a. On April 19, Thomas Ash deposited
$2,700
in a passbook savings account at
3.5%
interest
compounded daily. What is the compound amount of his account on August 5?
b. Using daily compounding, recalculate the compound amount for each of the three certificates
of deposit in Exercise 32.
85461_ch11_hr_341-368_1.indd 353 9/23/15 4:54 PM
8
358 Chapter 11 • Compound Interest and present Value
REviEw ExERcisEs
For the following investments, calculate the present value (principal) and the compound
interest. Use table 11-2. round your answers to the nearest cent.
Compound
Amount
Term of
Investment
Nominal
Rate (%)
Interest
Compounded
Present
Value
Compound
Interest
1.
$6,000
3 years
9
annually
$4,633.08
$1,366.92
2.
$24,000
6 years
4
semiannually
$18,923.76
$5,076.24
3.
$650
5 years
8
quarterly
$437.43
$212.57
the following investments require table factors for periods beyond the table. Create the new
table factor rounded to five places and calculate the present value for each.
Compound
Amount
Term of
Investment (years)
Nominal
Rate (%)
Interest
Compounded
New Table
Factor Present Value
11.
$12,000
10
16
quarterly
.20829
$2,499.48
12.
$33,000
38
7
annually
.07646
$2,523.18
$1,400
6
$685.10
$1,000
45
3
$264.44
$110,000
8
solve the following word problems by using table 11-2.
16. How much must be invested today at
6%
compounded quarterly to have
$8,000
in 3 years?
tryitexerciSe 8
Sam and Rosa Alonso want to accumulate
$30,000
, 17 years from now as a college fund for their
baby son, Michael. Use the present value formula to calculate how much they must invest now at
8%
$30,000
Section ii
11
Complete worked-out solutions for
Exercises 1–15 appear in Appendix B.
$2,000
6
$983.86
$1,016.14
$50,000
$3,680.50
$46,319.50
$14,500
4
$13,663.64
$836.36
$9,800
$100,000
4
$67,556.00
$32,444.00
$250
6
$235.48
$14.52
$4,000
8
$3,347.04
$652.96
seCtIon II • present Value 359
17. Samantha Wimberly is planning a vacation in Europe in 4 years, after graduation. She estimates
that she will need
$3,500
for the trip.
a. If her bank is offering 4-year certificates of deposit with
8%
interest compounded quarterly,
how much must Samantha invest now to have the money for the trip?
b. How much compound interest will be earned on the investment?
18. Pinnacle Homes, a real estate development company, is planning to build five homes, each
costing
$125,000
, in
2
1
2
years. The Galaxy Bank pays
6%
interest compounded semiannually.
How much should the company invest now to have sufficient funds to build the homes in
thefuture?
19. Tri-Star Airlines intends to pay off a
$20,000,000
corporate bond issue that comes due in 4 years.
How much must the company set aside now at
6%
interest compounded monthly to accumulate
the required amount of money?
20. Stuart Daniels estimates that he will need
$25,000
to set up a small business in 7 years.
a. How much must Stuart invest now at
8%
interest compounded quarterly to achieve his goal?
b. How much compound interest will he earn on the investment?
21. Summertime songbird population within the Mid-America flyway is predicted to increase over
the next 8 years at the rate of
2%
per year. If the songbird population is predicted to reach
55 million
in 8 years, how many songbirds are there today? Round to the nearest million.
22. The requirement for computer server capacity at Acme Industries is expected to increase at a rate
of
15%
per year for the next 5 years. If the server capacity is expected to be
1,400
gigabytes in
5years, how many gigabytes of capacity are there today? Round to the nearest whole gigabyte.
(1 +.15)
5=696.047 =696 Gigabytes
finance company modernization and
expansion programs.
When you buy a bond, you are
lending money to the corporation
© Ford Motor Company
Learning Tip
360 Chapter 11 • Compound Interest and present Value
solve the following exercises and word problems by using the present value formula
Compound
Amount
Term of
Investment
Nominal
Rate (%)
Interest
Compounded
Present
Value
Compound
Interest
23.
$4,500
7 years
3.8
annually
$3,466.02
$1,033.98
27. Alana and Eva Rodriguez are planning a cross-country road trip in 3 years. They estimate
$6,000
will be needed to cover expenses. The National Bank of Pinecrest is offering a 3-year CD paying
3.62%
interest compounded quarterly.
a. How much should they set aside now to achieve their goal? Round to the nearest whole dollar.
b. How much interest will Alana and Eva earn on the CD?
28. Mike Gioulis would like to have
$25,000
in 4 years to pay off a balloon payment on his business
mortgage. His money market account is paying
1.825%
compounded daily. Disregarding leap
years, how much money must Mike put in his account now to achieve his goal? Round to the
nearest whole dollar.
businEss DEcision: thE inflation factoR
29. You are the finance manager for Olympia Industries. The company plans to purchase
$1,000,000
in new assembly line machinery in 5 years.
a. How much must be set aside now at
6%
interest compounded semiannually to accumulate the
$1,000,000
in 5 years?
b. If the inflation rate on this type of equipment is
4%
per year, what will be the cost of the
equipment in 5 years, adjusted for inflation?
c. Use the inflation-adjusted cost of the equipment to calculate how much must be set aside now.
d. Use the present value formula to calculate how much would be required now if you found a
bank that offered
6%
interest compounded daily.
Inflation should be taken into account
when making financial plans that cover
time periods longer than a year.
Dollars
85461_ch11_hr_341-368_2.indd 360 9/23/15 4:55 PM
$15,000
4.5
$18,900
2.7