fm sol 5 ,6

5.1 Future value: Chuck Tomkovick is planning to invest $25,000 today in a mutual

fund that will provide a return of 8 percent each year. What will be the value of the

investment in 10 years?

LO 2

Solution:

0 5 years

├────────────────────┤

PV = $25,000 FV = ?

Amount invested today = PV = $25,000

Return expected from investment = i = 8%

Duration of investment = n = 10 years

Value of investment after 10 years = FV10

5.2 Future value: Ted Rogers is investing $7,500 in a bank CD that pays a 6 percent

annual interest. How much will the CD be worth at the end of five years?

LO 2

Solution:

0 5 years

├────────────────────┤

PV = $7,500 FV = ?





5.3 Future value: Your aunt is planning to invest in a bank deposit that will pay 7.5

percent interest semiannually. If she has $5,000 to invest, how much will she have at

the end of four years?

LO 2

Solution:

0 4 years

├────────────────────┤

PV = $5,000 FV = ?


Return expected from investment = i = 7.5%


Frequency of compounding = m = 2


5.4 Future value: Kate Eden received a graduation present of $2,000 that she is planning

on investing in a mutual fund that earns 8.5 percent each year. How much money can

she collect in three years?

LO 2

Solution:

0 3 years

├────────────────────┤

PV = $2,000 FV = ?

Amount Kate invested today = PV = $2,000




5.5 Future value: Your bank pays 5 percent interest semiannually on your savings

account. You don’t expect the current balance of $2,700 to change over the next four

years. How much money can you expect to have at the end of this period?

LO 2

Solution:

0 4 years

├────────────────────┤

PV = $2,700 FV = ?






5.6 Future value: Your birthday is coming up, and instead of any presents, your parents

promised to give you $1,000 in cash. Since you have a part time job and thus don’t

need the cash immediately, you decide to invest the money in a bank CD that pays 5.2

percent quarterly for the next two years. How much money can you expect to gain in

this period of time?

LO 2

Solution:

0 2 years

├────────────────────┤

PV = $1,000 FV = ?






5.7 Multiple compounding periods: Find the future value of an investment of $100,000

made today for five years and paying 8.75 percent for the following compounding

periods:

a. Quarterly

b. Monthly

c. Daily

d. Continuous

LO 2

Solution:

0 5 years

├────────────────────┤

PV = $100,000 FV = ?




a. Frequency of compounding = m = 4


b. Frequency of compounding = m = 12


c. Frequency of compounding = m = 365


d. Frequency of compounding = m = Continuous


5.8 Growth rates: Matt Murton, an outfielder for the Chicago Cubs, is expected to hit 25

home runs in 2008. If his home run hitting ability is expected to grow by 12 percent

every year for the next five years, how many home runs is he expected to hit in 2013?

LO 4

Solution:

0 5 years

├────────────────────┤

PV = 25 FV = ?

Number of home runs hit in 2008 = PV = 25

Expected annual increase in home runs hit = i = 12%

Growth period = n = 5 years

No. of home runs after 5 years = FV5

5.9 Present value: Roy Gross is considering an investment that pays 7.6 percent. How

much will he have to invest today so that the investment will be worth $25,000 in six

years?

LO 3

Solution:

0 6 years

├────────────────────┤

PV = ? FV = $25,000

Value of investment after 6 years = FV5 = $25,000



Amount to be invested today = PV

5.10 Present value: Maria Addai has been offered a future payment of $750 two years

from now. If her opportunity cost is 6.5 percent compounded annually, what should

she pay for this investment today?

LO 3

Solution:

0 2 years

├────────────────────┤

PV = ? FV = $750

Value of investment after 2 years = FV2 = $750




5.11 Present value: Your brother has asked you for a loan and has promised to pay back

$7,750 at the end of three years. If you normally invest to earn 6 percent, how much

will you be willing to lend to your brother?

LO 3

Solution:

0 3 years

├────────────────────┤

PV = ? FV = $7,750

Loan repayment amount after 3 years = FV3 = $7,750




5.12 Present value: Tracy Chapman is saving to buy a house in five years time. She plans

to put down 20 percent down at that time, and she believes that she will need $35,000

for the down payment. If Tracy can invest in a fund that pays 9.25 percent annually,

how much will she need to invest today?

LO 3

Solution:

0 5 years

├────────────────────┤

PV = ? FV = $35,000

Amount needed for down payment after 5 years = FV5 = $35,000




5.13 Present value: You want to buy some deep discount bonds that have a value of

$1,000 at the end of seven years. Bonds with similar risk are said to pay 4.5 percent

interest. How much should you pay for them today?

LO 3

Solution:

0 7 years

├────────────────────┤

PV = ? FV = $1,000

Face value of bond at maturity = FV7 = $1,000

Appropriate discount rate = i = 4.5%

Number of years to maturity = n = 7 years.

Present value of bond = PV

5.14 Present value: Elizabeth Sweeney wants to accumulate $12,000 by the end of 12

years. If the interest rate is 7 percent, how much will she have to invest today to

achieve her goal?

LO 3

Solution:

0 12 years

├────────────────────┤

PV = ? FV = $12,000

Amount Ms. Sweeney wants at end of 12 years = FV12 = $12,000

Interest rate on investment = i = 7%

Duration of investment = n = 12 years.

Present value of investment = PV

5.15 Interest rate: You are in desperate need of cash and turn to your uncle who has

offered to lend you some money. You decide to borrow $1,300 and agree to pay back

$1,500 in two years. Alternatively, you could borrow from your bank that is charging

6.5 percent interest. Should you go with your uncle or the bank?

LO 2

Solution:

0 2 years

├────────────────────┤

PV = $1,300 FV = $1,500

Amount to be borrowed = PV = $1,300

Amount to be paid back after 2 years = FV2 = $1,500

Interest rate on investment = i = ?

Duration of investment = n = 2 years.


You should go with the bank borrowing.

5.16 Time to attain goal: You invest $150 in a mutual fund today that pays 9 percent

interest. How long will it take to double your money?

LO 1,2

Solution:

0 n years

├────────────────────┤

PV = $150 FV = $300

Value of investment today = PV = $150

Interest on investment = n = 9%

Future value of investment = FV = $300

Number of years to double investment = n

INTERMEDIATE

5.17 Growth rate: Your Finance textbook sold 53,250 copies in its first year. The

publishing company expects the sales to grow at a rate of 20 percent for the next three

years and by 10 percent in the fourth year. Calculate the total number of copies that

the publisher expects to sell in years 3 and 4. Draw a time line to show the sales level

for each of the next four years.

LO 4

Solution:

Number of copies sold in its first year = PV = 53,250

Expected annual growth in the next 3 years = i = 20%

Number of copies sold after 3 years = FV3 =

Number of copies sold in the fourth year = FV4

0 3 4 years

├───────────┼────────┤

PV = 53,250 92,016 10,218 copies

5.18 Growth rate: CelebNav, Inc., had sales last year of $700,000, and the analysts are

predicting a good year for the start up, with sales growing 20 percent a year for the

next three years. After that, the sales should grow 11 percent per year for another two

years, at which time the owners are planning on selling the company. What are the

projected sales for the last year of the company’s operation?

LO 4

Solution:

0 1 2 3 4 5

years

├───────┼────────┼───────┼────────┼───────┤

g1 = 20% g2 = 11%

PV = $700,000

FV=?

Sales of CelebNav last year = PV = $700,000

Expected annual growth in the next 3 years = g1 = 20%

Expected annual growth in years 4 and 5 = g2= 11%

Sales in year 5 = FV5

5.19 Growth rate: You decide to take advantage of the current online dating craze and

start your own Web site. You know that you have 450 people who will sign up

immediately, and through a careful marketing research and analysis you determine

that membership can grow by 27 percent in the first two years, 22 percent in year 3,

and 18 percent in year 4. How many members do you expect to have at the end of

four years?

LO 4

Solution:

Betty Pessagno, 01/31/12,

Italic for g

0 1 2 3 4 years

├───────┼────────┼───────┼────────┤

g1-2=27% g3=22% g4=18%

PV = 450 FV = ?

Number of Web site memberships at t = 0 = PV = 450

Expected annual growth in the next 2 years = g1-2 = 27%

Expected annual growth in years 3 = g3= 22%

Expected annual growth in years 4 = g4= 18%

Number of members in year 4 = FV4

5.20 Multiple compounding periods: Find the future value of an investment of $2,500

made today for the following rates and periods:

a. 6.25 percent compounded semiannually for 12 years

b. 7.63 percent compounded quarterly for 6 years

c. 8.9 percent compounded monthly for 10 years

d. 10 percent compounded daily for 3 years

e. 8 percent compounded continuously for 2 years

LO 2

Solution:

a.


Italic for g

b.

c.

d.

e.

5.21 Growth rates: Xenix Corp had sales of $353,866 in 2008. If it expects its sales to be

at $476,450 in three years, what is the rate at which the company’s sales are expected

to grow?

LO 4

Solution:

Sales in 2008 = PV = $353,866

Expected sales three years from now = $476,450

To calculate the expected sales growth rate, we set up the future value equation.


Italic for the g

5.22 Growth rate: Infosys Technologies, Inc., an Indian technology company reported a

net income of $419 million this year. Analysts expect the company’s earnings to be

$1.468 billion in five years. What is the company’s expected earnings growth rate?

LO 4

Solution:

Earnings in current year = PV = $419,000,000

Expected earnings five years from now = $1,468,000,000

To calculate the expected earnings growth rate, we set up the future value equation.

5.23 Time to attain goal: Zephyr Sales Company has currently reported sales of $1.125

million. If the company expects its sales to grow at 6.5 percent annually, how long

will it be before the company can double its sales? Use a financial calculator to solve

this problem.

LO 1,2

Solution:

Enter

6.5% -$1.125 $2.250

N i% PMT PV FV

Answer: 11 years


Italic for g

5.24 Time to attain goal: You are able to deposit $850 into a bank CD today, and you will

only withdraw the money once the balance is $1,000. If the bank pays 5 percent

interest, how long will it take you to attain your goal?

LO 1,2

Solution:

Amount invested today = PV = $850

Expected amount in the future = FV = $1,000

Interest rate on CD = i = 5%

To calculate the time needed to reach the target FV, we set up the future value

equation.

5.25 Time to attain goal: Neon Lights Company is a private company with sales of $1.3

million a year. They want to go public but have to wait until the sales reach $2

million. Providing that they are expected to grow at a steady 12 percent annually,

when is the earliest that Neon Lights can start selling their shares?

LO 1,2

Solution:

Current level of sales = PV = $1,300,000

Target sales level in the future = FV = $2,000,000

Projected growth rate = g = 12%


equation.

5.26 Present value: Caroline Weslin needs to decide whether to accept a bonus of $1,900

today or wait two years and receive $2,100 then. She can invest at 6 percent. What

should she do?

LO 3

Solution:

0 2 years

├────────────────────┤

PV = $1,900 FV = ?

Amount to be received in 2 years = FV2 = $2,100



Present value of amount today PV =


Italic for g

Since the amount to be received today ($1,900) is greater than the present value of the

$2,100 to be received in two years, Ms. Weslin should choose to receive the amount

of $1,900 today

5.27 Multiple compounding periods: Find the present value of $3,500 under each of the

following rates and periods.

a. 8.9% compounded monthly for five years.

b. 6.6% compounded quarterly for eight years.

c. 4.3% compounded daily for four years.

d. 5.7% compounded continuously for three years.

LO 2

Solution:

0 n years

├────────────────────┤

PV = ? FV = $3,500

a. Return expected from investment = i = 8.9%



Present value of amount = PV

b. Return expected from investment = i = 6.6%



Present Value of amount = PV

c. Return expected from investment = i = 4.3%




d. Return expected from investment = i = 5.7%


Frequency of compounding = m = Continuous


5.28 Multiple compounding periods: Samantha is looking to invest some money, so that

she can collect $5,500 at the end of three years. Which investment should she make

given the following choices:

a. 4.2% compounded daily

b. 4.9% compounded monthly

c. 5.2% compounded quarterly

d. 5.4% compounded annually

LO 2

Solution:

0 3 years

├────────────────────┤

PV = ? FV = $5,500

a. Return expected from investment = i = 4.2%




Samantha should invest $4,848.92 today to reach her target of $5,500 in three years.

b. Return expected from investment = i = 4.9%





c. Return expected from investment = i = 5.2%





d. Return expected from investment = i = 5.4%





Samantha should invest in choice D.

ADVANCED

5.29 You have $2,500 you want to invest in your classmate’s start-up business. You

believe the business idea to be great and hope to get $3,700 back at the end of three

years. If all goes according to the plan, what will be your return on investment?

LO 2,3

Solution:

0 3 years

├────────────────────┤

PV = $2,500 FV = $3,700

Amount invested in project = PV = $2,500

Expected return three years from now = FV =$3,700

To calculate the expected rate of return, we set up the future value equation.

5.30 Patrick Seeley has $2,400 that he is looking to invest. His brother approached him

with an investment opportunity that could double his money in four years. What

interest rate would the investment have to yield in order for Patrick’s brother to

deliver on his promise?

LO 2,3

Solution:

0 4 years

├────────────────────┤

PV = $2,400 FV = $4,800


Expected return three years from now = FV =$4,800

Investment period = n = 4 years

To calculate the expected rate of return, we set up the future value equation.

5.31 You have $12,000 in cash. You can deposit it today in a mutual fund earning 8.2

percent semiannually; or you can wait, enjoy some of it, and invest $11,000 in your

brother’s business in two years. Your brother is promising you a return of at least 10

percent on your investment. Whichever alternative you choose, you will need to cash

in at the end of 10 years. Assume your brother is trustworthy and that both

investments carry the same risk. Which one will you choose?

LO 2,3

Solution:

Option A: Invest in account paying 8.2 percent semiannually for 10 years.

0 10 years

├────────────────────┤

PV = $12,000 FV = ?



Interest earned on investment = i = 8.2%



Option B: Invest in brother’s business to earn 10 percent for eight years.

0 8 years

├────────────────────┤

PV = $11,000 FV = ?



Interest earned on investment = i = 10%



You are better off investing today in the mutual fund and earn 8.2 percent

semiannually for 10 years.

5.32 When you were born, your parents set up a bank account in your name with an initial

investment of $5,000. You are turning 21 in a few days and will have access to all

your funds. The account was earning 7.3 percent for the first seven years, and then the

rates went down to 5.5 percent for six years. The economy was doing well at the end

of 1990s and your account was earning 8.2 percent for three years in a row.

Unfortunately, the next two years you only earned 4.6 percent. Finally, as the

economy recovered, your return jumped to 7.6 percent for the last three years.

a. How much money was in your account before the rates went down drastically

(end of year 16)?

b. How much money is in your account now, end of year 21?

c. What would be the balance now if your parents made another deposit of

$1,200 at the end of year 7?

LO 2,3

Solution:

0 1 7 13 14 15 16 21 years

├───┼∙∙∙∙∙∙∙∙∙∙┼∙∙∙∙∙∙∙∙∙∙∙∙────┼────┼───┼───∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙──┤

PV = $5,000 FV = ?

i1 = 7.3% i2 = 5.5% i3 = 8.2% i4 = 4.6% i5 = 7.6%

a. Initial investment = PV = $5,000

Interest rate for first 7 years = i1 = 7.3%

Interest rate for next 6 years = i2 = 5.5%


Investment value at age 16 years = FV16

b. Interest rate for from age 17 to 18 = i4 = 4.6%


Investment at start of 16th year = PV = $14,300.55

Investment value at age 21 years = FV21

c. Additional investment at start of 8th year = $1,200

Total investment for next 6 years = $8,187.82 + $1,200 = $9,387.82


Interest rate for years 13 to 16 = i3 = 8.2%

Interest rate for from age 17 to 18 = i4 = 4.6%


Investment value at age 21 = FV21

5.33 Cedric Benson, a top five draft pick of the Chicago Bears, and his agent are

evaluating three contract options. In each case, there is a signing bonus and a series of

payments over the life of the contract. He uses a 10.25 percent rate of return to

evaluate the contracts. Given the cash flows for each of the following options, which

one should he choose?

Year Cash Flow Type Option A Option B Option C

0 Signing Bonus $3,100,000 $4,000,000 $4,250,000

1 Annual Salary $ 650,000 $ 825,000 $ 550,000

2 Annual Salary $ 715,000 $ 850,000 $ 625,000

3 Annual Salary $ 822,250 $ 925,000 $ 800,000

4 Annual Salary $ 975,000 $1,250,000 $ 900,000

5 Annual Salary $1,100,000 $1,000,000

6 Annual Salary $1,250,000

LO 4

Solution:

To decide on the best contract from Mr. Benson’s viewpoint, we need to find the

present value of each option. The contract with the highest present value should be the

one chosen.

Option A:

Discount rate to be used = i= 10.25%

Present value of contract = PVA

Option B:


Present value of contract = PVB

Option C:


Present value of contract = PVC

Option C is the best choice for Mr. Benson.

5.34 Surmec, Inc., had sales of $2.1 million last year. The company’s primary business line

is manufacturing of nuts and bolts. Since this is a mature industry, the analysts are

certain that the sales will grow at a steady rate of 7 percent a year for as far as they

can tell. The company reports net income that represents 23 percent of sales. The

company’s management would like to buy a new fleet of trucks but can do so only

once the profit reaches $620,000 a year. At the end of what year will Surmec be able

to buy the new fleet of trucks? What will the sales and profit be that year?

LO 1,2,3,4

Solution:

Current level of sales for Surmec = PV = $2,100,000

Profit margin = 23%

Net Income for the year = 0.23 x $2,100,000 = $483,000

Target profit level in the future = FV = $620,000

Projected growth rate of sales = g = 7%


equation.

The company achieves its profit target during the fourth year.

Sales level at end of year 4 = FV4

Profit for the year = $2,752,671.62 x 0.23 = $633,114.47

5.35 You are graduating in two years and you start thinking about your future. You know

that you will want to buy a house five years after you graduate and that you will want

to put down $60,000. As of right now, you have $8,000 in your savings account. You

are also fairly certain that once you graduate, you can work in the family business and

earn $32,000 a year, with a 5 percent raise every year. You plan to live with your

parents for the first two years after graduation, which will enable you to minimize

your expenses and put away $10,000 each year. The next three years, you will have to

live out on your own, as your younger sister will be graduating from college and has

already announced her plan to move back into the family house. Thus, you will only

be able to save 13 percent of your annual salary. Assume that you will be able to

invest savings from your salary at 7.2 percent. What is the interest rate at which you

need to invest the current savings account balance in order to achieve your goal?

Hint: Draw a time line that shows all the cash flows for years 0 through 7.

Remember, you want to buy a house seven years from now and your first salary will

be in year 3.

LO 1,2,3,4

Solution:

0 1 2 3 4 5 6 7

├─────┼──────┼─────┼─────┼──────┼─────┼──────┤

$10,000 $10,000

Starting salary in year 3 = $32,000

Annual pay increase = 5%

Savings in first 2 years = $10,000

Savings rate for years 3 to 7 = 13%

Year 1 2 3 4 5 6 7

Salary $0 $0 $32,000 $33,600 $35,280 $37,044 $38,896

Savings $0 $0 $10,000 $10,000 $4,586.40 $4,815.72 $5,056.48

Investment rate = i = 7.2%

Future value of savings from salary = FV7

Target down payment = $60,000

Amount needed to reach target = $60,000 - $41,012.28 = FV = $18,987.72

Current savings balance = PV $8,000

Time to achieve target = n = 7 years.

To solve for the investment rate needed to achieve target, we need to set up the future

value equation:

Sample Test Problems

5.1 Santiago Hernandez is planning to invest $25,000 in a money market account for two

years. The account pays an interest of 5.75 percent compounded on a monthly basis.

How much will Santiago Hernandez have at the end of two years?

LO 2

Solution:

0 2 years

├────────────────────┤

PV = $25,000 FV = ?






5.2 Michael Carter is expecting an inheritance of $1.25 million in four years. If he had the

money today, he could earn interest at an annual rate of 7.35 percent. What is the

present value of this inheritance?

LO 3

Solution:

0 4 years

├────────────────────┤

PV = ? FV = $1,250,000

Amount needed for down payment after 4 years = FV4 = $1,250,000




5.3 What is the future value of an investment of $3,000 for three years compounded at the

following rates and frequencies?

a. 8.75% compounded monthly.

b. 8.625% compounded daily.

c. 8.5% compounded continuously.

LO 2

Solution:

a. Interest rate on investment = i = 8.75%



b. Frequency of compounding = m = 365


c. Frequency of compounding = m = Continuous


5.4. Twenty-five years ago, Amanda Cortez invested $10,000 in an account paying an

annual interest rate of 5.75 percent. What is the value of the investment today? What

is the interest-on-interest earned on this investment?

LO 2,3

Solution:

0 25 years

├────────────────────┤

PV = $10,000 FV = ?






Simple interest on investment = $10,000 x 0.0575 x 25

= $1,150

Interest-on-interest = $40,458.46 – $1,150 = $39,308.46

5.5 You just bought a corporate bond at $863.75 today. In five years the bond will mature

and you will receive $1,000. What is the rate of return on this bond?

LO 2,4

Solution:

0 5 years

├────────────────────┤

PV = $863.75 FV = $1,000

Amount to be borrowed = PV = $863.75

Amount to be paid back after 5 years = FV5 = $1,000

Years to maturity = n = 5 years.

Interest rate on investment = i


The rate of return on this bond is 2.97 percent

CHAPTER SIX

6.1 Future value with multiple cash flows: Konerko, Inc., expects to earn cash flows of

$13,227, $15,611, $18,970, and $19,114 over the next four years. If the company uses

an 8 percent discount rate, what is the future value of these cash flows at the end of

year 4?

Solution:

0 8% 1 2 3 4

├───────┼────────┼───────┼────────┤

$13,227 $15,611 $18,970 $19,114

6.2 Future value with multiple cash flows: Ben Woolmer has an investment that will

pay him the following cash flows over the next five years: $2,350, $2,725, $3,128,

$3,366, and $3,695. If his investments typically earn 7.65 percent, what is the future

value of the investment’s cash flows at the end of five years?

Solution:

0 7.65% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$2,350 $2,725 $3,128 $3,366 $3,695

6.3 Future value with multiple cash flows: You are a freshman in college and are

planning a trip to Europe when you graduate from college at the end of four years.

You plan to save the following amounts starting today: $625, $700, $700, and $750. If

the account pays 5.75 percent annually, how much will you have at the end of four

years?

Solution:

0 5.75% 1 2 3 4

├───────┼────────┼───────┼────────┤

$625 $700 $700 $750

6.4 Present value with multiple cash flows: Saul Cervantes has just purchased some

equipment for his landscaping business. He plans to pay the following amounts at the

end of the next five years: $10,450, $8,500, $9,675, $12,500, and $11,635. If he uses

a discount rate of 10.875 percent, what is the cost of the equipment he purchased

today?

Solution:

0 10.875% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$10,450 $8,500 $9,675 $12,500 $11,635

6.5 Present value with multiple cash flows: Jeremy Fenloch borrowed from his friend a

certain amount and promised to repay him the amounts of $1,225, $1,350, $1,500,

$1,600, and $1,600 over the next five years. If the friend normally discounts

investments at 8 percent annually, how much did Jeremy borrow?

Solution:

0 8% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$1,225 $1,350 $1,500 $1,600 $1,600

6.6 Present value with multiple cash flows: Biogenesis, Inc., expects the following cash

flow stream over the next five years. The company discounts all cash flows at a 23

percent discount rate. What is the present value of this cash flow stream?

Solution:

0 23%

1

2 3 4 5

1 2 3 4 5

-$1,133,676 -$978,452 $275,455 $878,326 $1,835,444

├───────┼────────┼───────┼────────┼───────┤

-$1,133,676 -$978,452 $275,455 $878,326 $1,835,444

6.7 Present value of an ordinary annuity: An investment opportunity requires a

payment of $750 for 12 years, starting a year from today. If your required rate of

return is 8 percent, what is the value of the investment today?

Solution:

0 8% 1 2 3 11 12

├───────┼────────┼───────┼………………┼───────┤

$750 $750 $750 $750 $750

Annual payment = PMT = $750

No. of payments = n = 12

Required rate of return = 8%

Present value of investment = PVA12

6.8 Present value of an ordinary annuity: Dynamics Telecommunications Corp. has

made an investment in another company that will guarantee it a cash flow of $22,500

each year for the next five years. If the company uses a discount rate of 15 percent on

its investments, what is the present value of this investment?

Solution:

0 15% 1 2 3 4 5

├───────┼────────┼───────┼────────┼───────┤

$22,500 $22,500 $22,500 $22,500 $22,500

Annual payment = PMT = $22,500


Required rate of return = 15%

Present value of investment = PVA5

6.9 Future value of an ordinary annuity: Robert Hobbes plans to invest $25,000 a year

for the next seven years in an investment that will pay him a rate of return of 11.4

percent. He will invest at the end of each year. What is the amount that Mr. Hobbes

will have at the end of seven years?

Solution:

0 11.4% 1 2 3 6 7

├───────┼────────┼───────┼………………┼───────┤

$25,000 $25,000 $25,000 $25,000 $25,000

Annual investment = PMT = $25,000


Investment rate of return = 11.4%

Future value of investment = FVA7

6.10 Future value of an ordinary annuity: Cecelia Thomas is a sales executive at a

Baltimore firm. She is 25 years old and plans to invest $3,000 every year in an IRA

account, beginning at the end of this year until she turns 65 years old. If the IRA

investment will earn 9.75 percent annually, how much will she have in 40 years when

she turns 65 years old?

Solution:

0 9.75% 1 2 3 39 40

├───────┼────────┼───────┼………………┼───────┤

$3,000 $3,000 $3,000 $3,000 $3,000





6.11 Future value of an annuity. Refer to Problem 6.10. If Cecelia Thomas starts saving

at the beginning of each year, how much will she have at age 65?

Solution:

0 9.75% 1 2 3 39 40

├───────┼────────┼───────┼………………┼───────┤

$3,000 $3,000 $3,000 $3,000 $3,000



Type of annuity = Annuity due



6.12 Computing annuity payment: Kevin Winthrop is saving for an Australian vacation

in three years. He estimates that he will need $5,000 to cover his airfare and all other

expenses for a week-long holiday in Australia. If he can invest his money in an S&P

500 equity index fund that is expected to earn an average return of 10.3 percent over

the next three years, how much will he have to save every year, starting at the end of

this year?

Solution:

0 10.3% 1 2 3

├───────┼────────┼───────┤

PMT PMT PMT

FVAn = $5,000

Future value of annuity = FVA = $5,000

Return on investment = i = 10.3%

Payment required to meet target = PMT

Using the FVA equation:

Kevin has to save $1,506.20 every year for the next three years to reach his target of

$5,000.

6.13 Computing annuity payment: The Elkridge Bar & Grill has a seven-year loan of

$23,500 with Bank of America. It plans to repay the loan by paying in seven equal

installments starting today. If the rate of interest is 8.4 percent, how much will each

payment be worth?

0 1 2 3 6 7

├───────┼────────┼───────┼………………┼───────┤

PMT PMT PMT PMT PMT PMT

PVAn = $23,500 n = 7; i = 8.4%

Present value of annuity = PVA = $23,500

Return on investment = i = 8.4%

Payment required to meet target = PMT

Type of annuity = Annuity due

Using the PVA equation:

Each payment made by Elkridge Bar & Grill will be $4,221.07, starting today.

6.14 Perpetuity: Your grandfather is retiring at the end of next year. Heould like to

receive a payment of $10,000 a year forever, starting when he retires. If he can invest

at 6.5 percent, how much does need to invest to receive the desired cash flow?

Solution:

Annual payment needed = PMT = $10,000

Investment rate of return = i = 6.5%

Term of payment = Perpetuity

Present value of investment needed = PV

6.15 Perpetuity: Calculate the perpetuity payments for each of the following cases:

a. $250,000 invested at 6%

b. $50,000 invested at 12%

c. $100,000 invested at 10%

Solution:

a. Annual payment = PMT

Investment rate of return = i = 6%


Present value of investment needed = PV = $250,000

b. Annual payment = PMT




c. Annual payment = PMT




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