tvm solutions manual ch05

7/23/2019 TVM Solutions Manual Ch05

http://slidepdf.com/reader/full/tvm-solutions-manual-ch05 1/44

1

Chapter 5

The Time Value of Money

Before You Go On Questions and Answers

Section 5.1

1. Why is a dollar today worth more than a dollar one year from now?

A dollar is worth more today than one year from now, due to its potential earning

capacity. If you have the money in your hand today, you have the opportunity to invest it

and earn interest or you can purchase goods and services for your immediate

consumption. Given that people have a positive preference for consumption, time value

of money holds true.

2. What is a time line, and why is it important in financial analysis?

A time line is a horizontal line that starts at time zero (today) and shows cash flows as

they occur over time. It is an important tool used to analyze cash flows over certain time

periods, as timing of each cash flow has a big impact on the final figure, and therefore on

the resulting investment decision.

Section 5.2

1. What is compounding, and how does it affect the future value of an investment?

Compounding is the process that refers to converting the initial (principal) amount into a

future value. In order to obtain the future value of the principal amount, you calculate

what the value at the end of the time period will be assuming the initial investment will

earn interest, which is reinvested and will earn additional interest in the future periods.



2

2. What is the difference between simple interest and compound interest?

The difference is the interest earned on interest.

3. How does changing the compounding period affect the amount of interest earned on an

investment?

The more frequent the compounding schedule, the higher the interest earned. For

example, $100 invested for one year at 10 percent compounded annually will earn you

$10 of interest at the end of the year, but if your bank compounded interest quarterly,

your earnings from interest would increase to $10.38.

Section 5.3

1. What is the present value and when is it used?

Present value is the amount a future sum is worth today given a certain return rate. The

present value concept should be used when calculating how much money you need today

in order to reach your financial goal sometime in the future.

2. What is the discount rate? How does the discount rate differ from the interest rate in the

future value equation?

The discount rate is the compound interest rate used to determine the present value of

future cash flows. Both discount and interest rates essentially represent the same concept.

The only difference is the context in which they are used.

3. What is the relation between the present value factor and the future value factor?

The present value factor is the reverse of the future value factor. To obtain the present

value factor, you divide 1 by the future value factor (1 + i).



3

4. Explain why you would expect the discount factor to become smaller the longer the time

to payment.

The discount factor will become smaller the longer the time to payment due to time value

of money. The longer you have to wait to obtain the money, the less value it will have to

you. Mathematically, the discount factor is calculated as 1/(1 + i)n. The longer the time to

payment, the larger n gets, which will make the discount factor smaller.

Section 5.4

1. What is the difference between the interest rate (i) and the growth rate ( g ) in the future

value equation?

The interest rate and the growth rate in the future value equation essentially represent the

same concept. The growth rate is used when we deal with numerical values such as sales

or change over time. When referring to money being invested, we use the term interest

rate.

Self Study Problems

5.1 Amit Patel is planning to invest $10,000 in a bank certificate of deposit (CD) for five

years. The CD will pay interest of 9 percent. What is the future value of Amit’s

investment?

Solution:

Present value of the investment = PV = $10,000

Interest rate = i = 9%

Number of years = n = 5.



4

0 1 2 3 4 5

├───┼───┼───┼────┼───┤

-$10,000 FV5=?

5.2 Megan Gaumer expects to need $50,000 as a down payment on a house in six years. How

much does she need to invest today in an account paying 7.25 percent?

Solution:

Amount Megan will need in six years = FV6 = $50,000

Number of years = n = 6

Interest rate on investment = i = 7.25%

Amount needed to be invested now = PV = ?

0 1 2 3 4 5 6 Year

├───┼───┼───┼────┼───┼───┤

PV = ? FV6 = $50,000

$32,853.85

6)0725.01(

000,50$

)1(

FVPV

n

n

i

5.3 Kelly Martin has $10,000 that she can deposit into a savings account for five years. Bank

A pays compounds interest annually, Bank B twice a year, and Bank C quarterly. Each

n

5

5

FV PV(1 )

FV $10,000(1 0.09)

ni

$15, 386.24



5

bank has a stated interest rate of 4 percent. What amount would Kelly have at the end of

the fifth year if she left all the interest paid on the deposit in each bank?

Solution:

Present value = PV = $10,000



Compound period m:

A = 1

B = 2

C = 4

Amount at the end of 5 years = FV5 = ?

0 1 2 3 4 5 Year

├───┼───┼───┼────┼───┤

-$10,000 FV5 = ?

A: FVn = PV × (1 + i/m)m x n

FV5

= 10,000 × (1 + 0.04/1)1x5

= $12,166.53

B: FV5 = 10,000 × (1 + 0.04/2)2x5

= $12,189.94

C: FV5 = 10,000 × (1 + 0.04/4)4x5

= $12,201.90

5.4 You have an opportunity to invest $2,500 today and receive $3,000 in three years. What

will be the return on your investment?

Solution:



6

Your investment today = PV = $2,500

Amount to be received = FV3= $3,000

Time of investment = n = 3

Return on the investment = i = ?

0 1 2 3 Year

├───┼────┼───┤

-$2,500 $3,000

FV PV (1 )

$3,000 $2,500 (1 )

$3,000(1 )

$2,500

6.27%

n

n

n

n

i

i

i

i =

5.5 Emily Smith deposits $1,200 in her bank today. If the bank pays 4 percent simple

interest, how much money will she have at the end of five years? What if the bank pays

compound interest? How much of the earnings will be interest on interest?

Solution:

Deposit today = PV = $1,200



Amount to be received back = FV5 = ?

0 1 2 3 4 5 Year

├───┼───┼───┼────┼───┤

-$1.200 FV5 = ?

a. Future value with simple interest



7

Simple interest per year = $1,200 × (0.04) = $48

Simple interest for 5 years = $48 × 5 = $240

FV5 = $1,200 + $240 = $1,440

b. Future value with compound interest

FV5 = $1,200 × (1 + 0.04)5

= $1,459.98

Simple interest = ($1,440 – $1,200) × 5 = $240

Interest on interest = $1,459.98 – $1,200 – $240 = $19.98

Critical Thinking Questions

5.1 Explain the phrase ―a dollar today is worth more than a dollar tomorrow.‖

The implication is that if one was to receive a dollar today instead of in the future, the

dollar could be invested and will be worth more than a dollar tomorrow because of the

interest earned during that one day. This makes it more valuable than receiving a dollar

tomorrow.

5.2 Explain the importance of a time line.

Time lines are important tools used to analyze investments that involve cash flow streams

over a period of time. They are horizontal lines that start at time zero (today) and show

cash flows as they occur over time. Because of time value of money, it is crucial to keep

track of not only the size, but also the timing of the cash flows.

5.3 What are the two factors to be considered in time value of money?



8

The factors that are critical in time value of money are the size of the cash flows and the

timing of the cash flows.

5.4 Differentiate future value from present value.

Future value measures what one or more cash flows are worth at the end of a specified

period, while present value measures what one or more cash flows that are to be received

in the future will be worth today (at t = 0).

5.5 Differentiate between compounding and discounting.

The process of converting an amount given at the present time into a future value is

called compounding . It is the process of earning interest over time. Discounting is the

process of converting future cash flows to what its present value is. In other words,

present value is the current value of the future cash flows that are discounted at an

appropriate interest rate.

5.6 Explain how compound interest differs from simple interest.

Suppose I invest $100 for three years at a rate of 10 percent. Simple interest would imply

that I will earn $10 for each of the three years for a total of $30 interest. At the end of

three years I would have $130. Compound interest recognizes that the interest earned in

years 1 and 2 will also earn interest over the remaining period. Thus, the $10 earned in

the first year would earn interest at 10 percent for the next two years, and the $10 earned

in the second year would earn interest for the third year. Thus the total amount that I

would have at the end of three years would be: 10.133$)10.1(100$ 3 . By compounding,

I have earned an additional interest of $3.10. The total interest or compound interest is

the $33.10 earned on the $100 invested, while the simple interest earned is equal to $30.



9

5.7 If you were given a choice of investing in an account that paid quarterly interest and one

that paid monthly interest, which one should you choose if they both offer the same stated

interest rate and why?

The impact of compounding really dictates that one should pick the account that pays

interest more frequently (as long as the interest rates are the same). This allows for the

interest earned in the earlier periods to earn interest and the investment to grow more.

5.8 Compound rates are exponential over time. Explain.

Growth rates, as well as interest rates, are not linear, but rather exponential over time. In

other words, the growth rate of the invested funds is accelerated by the compounding of

interest. Over time, the principal amount you receive interest on will get larger with

compounding, thus generating higher interest payments.

5.9 What is the Rule of 72?

This is a rule of thumb to determine how fast an investment can double. It is a rule that

allows you to closely approximate the time that it would take to double your money. It

works well with interest rates between 5 and 20 percent, but varies more with higher

rates. The Rule of 72 says that the time to double your money (TDM) approximately

equals 72/i, where i is expressed as a percentage:

5.10 You are planning to take a spring break trip to Cancun your senior year. The trip is

exactly two years away, but you want to be prepared and have enough money when the

time comes. Explain how you would determine the amount of money you will have to

save in order to pay for the trip.

First, determine how much money you will need for the trip. Second, check how much

you already have and how it translates into future value cash — how much it will be worth

in two years. Next, determine how much you will have to deposit today, given the bank’s



10

offered interest rate, to ensure that you will have saved up the difference when the time

for your senior spring break comes.

Questions and Problems

BASIC

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 FV5 = ?

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

$53,973.12

10n10 )08.1(000,25$)1(PVFV i

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



11

├────────────────────┤

PV = $7,500 FV5 = ?





$10,036.69

5n5 )06.1(500,7$)1(PVFV i

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 FV4 = ?


Return expected from investment = i = 7.5%


Frequency of compounding = m = 2


mn 2 4

4

8

0.075FV PV 1 $5,000 1

m 2

$5,000 (1.0375)

$6,712.35

i



12

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 FV3 = ?

Amount Kate invested today = PV = $2,000




$2,554.58

3n

3 )085.1(000,2$)1(PVFV i

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 FV4 = ?

Amount invested today = PV = $2,700Return expected from investment = i = 5%






13

$3,289.69

8

42mn

4

)025.1(700,2$

2

05.01700,2$

m1PVFV

i

5.6 Future value: Your birthday is coming up, and instead of other 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 FV2 = ?



Duration of investment = n = 2 yearsFrequency of compounding = m = 4


$1,108.86

8

24mn

2

)013.1(000,1$

4

052.01000,1$

m1PVFV

i

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



14

c. Daily

d. Continuous

LO 2

Solution:

0 5 years

├────────────────────┤

PV = -$100,000 FV5 = ?




a. Frequency of compounding = m = 4


4$154,154.2

20

54mn

5

)021875.1(000,100$

4

0875.01000,100$

m1PVFV

i

b. Frequency of compounding = m = 12


7$154,637.3

60

512mn

5

)00729.1(000,100$

12

0875.01000,100$

m1PVFV

i

c. Frequency of compounding = m = 365


1$154,874.9

1825

536 5mn

5

)00024.1(000,100$

365

0875.01000,100$

m1PVFV

i

d. Frequency of compounding = m = Continuous



15


3$154,883.0

5488303.1000,100$

e000,100$ePVFV 50875.0

5in

5.8 Growth rates: Joe Mauer, a catcher for the Minnesota Twins, is expected to hit 15 home

runs in 2012. 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 2017?

LO 4

Solution:

0 5 years

├────────────────────┤

PV = -15 FV5 = ?

Number of home runs hit in 2012 = PV = 15

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

Growth period = n = 5 years

Expected home runs in 2017 = FV5

n 55FV PV (1 g) 15 (1.12)

26.4 26 home runs

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?

LO3

Solution:

0 6 years

├────────────────────┤

PV = ? FV6 = $25,000



16

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



Amount to be invested today = PV

$16,108.92

6n

n

)076.1(

000,25$

)1(

FVPV

i

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

now. If she can earn 6.5 percent compounded annually on her investment, what should

she pay for this investment today?

LO3

Solution:

0 2 years

├────────────────────┤

PV = ? FV2 = $750

Value of investment after 2 years = FV2 = $750




$661.24

2n

n

)065.1(

750$

1

FVPV

i

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 per year, how

much will you be willing to lend to your brother?

LO3



17

Solution:

0 3 years

├────────────────────┤

PV = ? FV3 = $7,750

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




$6,507.05

3n

n

)06.1(

750,7$

1

FVPV

i

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?

LO3

Solution:

0 5 years

├────────────────────┤

PV = ? FV5 = $35,000

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






18

$22,488.52

5n

n

)0925.1(

000,35$

1

FVPV

i

5.13 Present value: You want to buy some bonds that will have a value of $1,000 at the end

of seven years. The bonds pay 4.5 percent interest annually. How much should you pay

for them today?

LO3

Solution:

0 7 years

├────────────────────┤

PV = ? FV7 = $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

$734.83

7nn

)045.1(000,1$

1FVPV

i

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

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

her goal?

LO3

Solution:

0 12 years

├────────────────────┤

PV = ? FV12 = $12,000



19

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

$5,328.14

12n

n

)07.1(

000,12$

1

FVPV

i

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 percentinterest annually. Should you go with your uncle or the bank?

LO 2, LO3

Solution:

0 2 years

├────────────────────┤

PV = -$1,300 FV2

= $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.




20

7.42%

i

11538.1

1538.1$1,300$1,500)i1(

)1(

500,1$300,1$

1

FVPV

2

2

n

n

i

i

i

You should go with the bank borrowing.

5.16 Number of periods: You invest $150 in a mutual fund today that pays 9 percent interest

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

LO 2, LO3

Solution:

0 n years

├────────────────────┤

PV = -$150 FVn = $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

n

n

n

n

FV PV (1 )

$300 $150 (1.09)

(1.09) $300 150 2.00

n ln(1.09) ln(2.00)

ln(2.00)n

ln(1.09)

i

8 years



21

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 each year 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 = g = 20%

Number of copies sold after 3 years = FV3 =

n

3

FV = PV (1 )

53,250 (1.20)

n g

92,016 copies

Number of copies sold in the fourth year = FV4

n

nFV PV (1 ) 92,016 (1.10) g

101,218 copies

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 to sell the company. What are the projected sales for

the last year before the sale?

LO 4



22

Solution:

0 1 2 3 4 5 years

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

g1 = 20% g2 = 11%

PV = -$700,000 FV5=?

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

Expected annual growth in the next 3 years = g 1 = 20%

Expected annual growth in years 4 and 5 = g 2= 11%

Sales in year 5 = FV5

3 2 3 2

5 1 2FV PV (1 g ) (1 g ) $700,000 (1.20) (1.11)

$1,490, 348.16

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:

0 1 2 3 4 years

├───────┼────────┼───────┼────────┤

g 1-2=27% g 3=22% g 4=18%

PV = -450 FV4 = ?

Number of Web site memberships at t = 0 = PV = 450Expected annual growth in the next 2 years = g 1-2 = 27%

Expected annual growth in years 3 = g 3= 22%

Expected annual growth in years 4 = g 4= 18%

Number of members in year 4 = FV4



23

2 2

4 1 3 4FV PV(1 g ) (1 g ) (1 g ) 450 (1.27) (1.22) (1.18)

1,045 members

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

$5,232.09

)0928.2(500,2$2

0625.01PVFV

122

12

b.

$3,934.48

)4768.2(500,2$4

0763.0

1PVFV

64

12

c.

$6,067.86

)4271.2(500,2$12

089.01PVFV

1012

12

d.

$3,374.51

)3498.1(500,2$

365

010.01PVFV

336 5

12

e. $2,933.78

1735.1500,2$

e000,3$ePVFV 208.0in

3



24

5.21 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.

LO3

Solution:

0 n years

├────────────────────┤

PV = ? FVn = $3,500

a. Return expected from investment = i = 8.9%



Present value of amount = PV

$2,246.57

5579.1

500,3$

12

089.01

500,3$

m1

FVPV512mn

5

i

b. Return expected from investment = i = 6.6%


Frequency of compounding = m = 4Present Value of amount = PV



25

$2,073.16

6882.1

500,3$

4

066.01

500,3$

m1

FVPV

84mn

8

i

c. Return expected from investment = i = 4.3%



Present Value of amount = PV

$2,946.96

1877.1

500,3$

365

043.0

1

500,3$

m1

FVPV

436 5mn

4

i

d. Return expected from investment = i = 5.7%


Frequency of compounding = m = Continuous


3

0.057 3

FV $3,500PV

e e

$3,500

1.1865

in

$2,949.88

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

has $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

LO2



26

Solution:

0 3 years

├────────────────────┤

PV = ? FV3 = $5,500

a. Return expected from investment = i = 4.2%




$4,848.92

1343.1

500,5$

365

042.01

500,5$

m1

FVPV336 5mn

3

i

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%




$4,749.54

5579.1

500,5$

12

049.01

500,5$

m1

FVPV

312mn

3

i


c. Return expected from investment = i = 5.2%





27

Present Value of amount = PV

$4,710.31

1677.1500,5$

4

052.01

500,5$

m1

FVPV

34mn

3

i


d. Return expected from investment = i = 5.4%




$4,697.22

33

3

)054.1(

500,5$

)1(

FVPV

i


Samantha should invest in choice D.

5.23 Time to grow: Zephyr Sales Company has 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 4

Solution:

Enter

6.5% -$1.125 $2.250

N g % PMT PV FV

Answer: 11 years



28

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

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

how long will it take for the balance to reach $1,000?

LO 2, LO3

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.

years3.3

)05.1ln(

)1764.1ln(

)1764.1ln()05.1ln(

1764.1850$

000,1$)05.1(

)05.1(850$000,1$

)1(PVFV

n

n

n

n

n

n

i

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

year. Management wants to go public but has to wait until the sales reach $2 million. If

the sales are expected to grow 12 percent annually, when is the earliest that Neon Lights

can go public?

LO 4

Solution:

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

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

Projected growth rate = g = 12%To calculate the time needed to reach the target FV, we set up the future value equation.



29

years3.8

)12.1ln(

)5385.1ln()5385.1ln()12.1ln(

5385.100,300,1$

000,000,2$)12.1(

)12.1(000,300,1$000,000,2$

)g1(PVFV

n

n

n

3

n

n

5.26 Time to grow: You have just inherited $550,000. You plan to save this money and

continue to live off the money that you are earning in your current job. If the $550,000 is

everything that you have other than an old car and some beat-up furniture, and you can

invest the money in a bond that pays 4.6 percent interest annually. How long will it be

before you are a millionaire?

LO 2, LO3

Solution:

n

n

n

n

FV PV (1 )

$1,000,000 $550,000 (1.046)

$1,000,000(1.0046)

$550,000

$1,000,000ln ln(1.046)

$550,000

$1,000,000ln

$550,000

ln(1.046)

0.59784n0.04497

n 13.29 years

i

n

n



30

5.27 Growth rates: Xenix Corp had sales of $353,866 in 2011. If management expects its

sales to be $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.

3

3

3

3

13

FV PV (1 g)

$476,450 $353,866 (1 g)

$476,450(1 g) 1.3464$353,866

g (1.3464) 1

10.42%

5.28 Growth rate: Infosys Technologies, Inc., an Indian technology company reported net

income of $419 million this year. Analysts expect the company’s earnings to be $1.468

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

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

5

5

5

15

FV PV (1 g)

$1,468,000,000 $419,000,000 (1 g)

$1,468,000,000(1 g) 3.5036

$419,000,000

g (3.5036) 1

8.5

2 %



31

5.29 Present value: Caroline Weslin needs to decide whether to accept a bonus of $1,820

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

she do?

LO3

Solution:

0 2 years

├────────────────────┤

PV = -$1,820 FV2 = ?

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



Present value of amount today PV =

$1,868.99

2n

2

)06.1(

100,2$

)1(

FVPV

i

Since $1869 is greater than $1,820, Caroline should wait two years unless she needs the

money before then.

5.30. Present value: Congress and the President have decided to increase the Federal tax rate

in an effort to reduce the budget deficit. Suppose that Caroline Weslin will pay 35 percent

of her bonus to the Federal government for taxes if she accepts the bonus today and 40

percent if she receives her bonus in two years. Will the increase in tax rates affect her

decision?

LO3



32

Solution:

Yes. It will affect her decision.

If Caroline accepts the bonus today, after paying the taxes, she will have $1,820 ×

(1 - 0.35) = $1,183.00 left over.

If she waits two years and pays the high tax rate, the present value of what she

will have left over is only $1,869 × (1 - 0.40) = $1,121.40.

ADVANCED

5.31 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, LO3

Solution:

0 3 years

├────────────────────┤

PV = -$2,500 FV3 = $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.

13.96%

1396.01)4800.1(

4800.1500,2$

700,3$)1(

)1(500,2$700,3$)1(PVFV

31

3

3

3

3

i

i

ii



33

5.32 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, LO3

Solution:

0 4 years

├────────────────────┤

PV = -$2,400 FV4 = $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.

4

4

4

4

14

FV PV (1 )

$4,800 $2,400 (1 )

$4,800(1 ) 1.4800

$2,400

(2.000) 1 0.1892

i

i

i

i

18.92%

5.33 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?



34

LO 2

Solution:

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

0 10 years

├────────────────────┤

PV = -$12,000 FV10 = ?



Interest earned on investment = i = 8.2%



$26,803.77

)23365.2(000,12$2

082.01PVFV

102

10

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

0 8 years├────────────────────┤

PV = -$11,000 FV8 = ?



Interest earned on investment = i = 10%



$23,579.48

)14359.2(000,11$10.01PVFV8

8



35

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

for 10 years.

5.34 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 in the early 2000s

and your account earned 8.2 percent 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?

Solution:

0 1 7 13 14 15 16 21 years

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

PV = -$5,000 FV21

= ?

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

$14,300.55

)2667.1()3788.1()6376.1(000,5$

)082.1()055.1(073.01000,5$

)1()1()1(PVFV

367

3

3

6

2

7

116 iii



36

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

$19,492.38

))2458.1()0941.1(55.300,14$

)076.1(046.0155.300,14$

)1()1(FVFV

32

3

5

2

41621 ii

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

6 3 2 3

21 7 2 3 4 5

26 3 3

FV FV (1 ) (1 ) (1 ) (1 )

$9,387.82 (1.055) (1.082) 1.046 (1.076)

$9387.82 (1.3788) (1.2667) (1.0941) (1.2458)

i i i i

$22,349.16

LO 2

5.35 Sam Bradford, a number 1 draft pick of the St. Louis Rams, and his agent are evaluating

three contract options. Each option offers a signing bonus and a series of payments over

the life of the contract. Bradford 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



37

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

LO3

Solution:

To decide on the best contract from Sam Bradford’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

647,922,6$

047,696$305,675$918,659$576,613$232,588$569,589$000,100,3$

)1025.1(

000,250,1$

)1025.1(

000,100,1$

)1025.1(

000,975$

)1025.1(

250,822$

)1025.1(

000,715$

)1025.1(

000,650$000,100,3$PV

654321A

Option B:


Present value of contract = PVB

894,983,6$

049,846$249,690$297,699$299,748$000,000,4$

)1025.1(

000,250,1$

)1025.1(

000,925$

)1025.1(

000,850$

)1025.1(

000,825$000,000,4$PV

4321B

Option C:


Present value of contract = PVC



38

$7,083,096

913,613$155,609$972,596$189,514$866,498$000,250,4$

)1025.1(

000,000,1$

)1025.1(

000,900$

)1025.1(

000,800$

)1025.1(

000,625$

)1025.1(

000,550$000,250,4$PV

54321C

Option C is the best choice for Sam Bradford.

5.36 Surmec, Inc., reported earnings of $2.1 million last year. The company’s primary

business line is the manufacture of nuts and bolts. Since this is a mature industry, the

analysts are confident that the sales will grow at a steady rate of 7 percent a year for as

far as they can tell. The company’s net income equals 23 percent of sales. Management

would like to buy a new fleet of trucks but can only do so once the profit reaches

$620,000 a year. At the end of what year will they be able to buy the trucks? What will

sales and net income be in that year?

LO 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%

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

years3.7

)12.1ln(

)2836.1ln(

)2836.1ln()07.1ln(

2836.100,483$

000,620$)07.1(

)07.1(000,483$000,620$

)g1(PVFV

n

n

nn

n

n

The company achieves its profit target during the fourth year.

Sales level at end of year 4 = FV4

.62$2,752,671

4

n

n

)07.1(000,100,2$

)g1(PVFV



39

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

5.37 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 2

Solution:0 1 2 3 4 5 6 7

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

$10,000 $10,000 $4,586.40 $4,815,72 $5,056.48

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



40

Investment rate = i = 7.2%

Future value of savings from salary = FV7

28.012,41$

48.056,5$45.162,5$86.267,5$25.319,12$24.206,13$

)072.1(48.056,5$)072.1(72.815,4$

)072.1(40.586,4$)072.1(000,10$)072.1(000,10$0$0$FV

01

2347

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:

13.14%

11314.1

1)3735.2(

3735.2000,8$

72.987,18$)1(

)1(000,8$72.987,18$

)1(PVFV

71

7

7

7

i

i

i

i

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 = ?



41






$28,039.13

24

212mn

2

)1216.1(000,25$

12

0575.01000,25$

m1PVFV

i

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?

LO3

Solution:

0 4 years

├────────────────────┤

PV = ? FV = $1,250,000

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




3$941,243.1

4n

n

)0735.1(

000,250,1$

1

FVPV

i

5.3 What is the future value of an investment of $3,000 after three years with compounding

at the following rates and frequencies?

a. 8.75% compounded monthly.



42

b. 8.625% compounded daily.

c. 8.5% compounded continuously.

LO 2

Solution:

a. Interest rate on investment = i = 8.75%



mn 12 3

3

36

0.0875FV PV 1 $3,000 1

m 12

$3,000 (1.00729)

i

$3,896.82

b. Frequency of compounding = m = 365


$3,885.81

1095

336 5mn

3

)000236.1(000,3$

365

08625.01000,3$

m1PVFV

i

c. Frequency of compounding = m = Continuous


$3,871.38

29046.1000,3$

e000,3$ePVFV 308 5.03

in

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?

LO3

Solution:

0 25 years

├────────────────────┤



43

PV = -$10,000 FV = ?






46.458,40$

)0575.1(000,10$)1(PVFV 252525

i

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

= $14,375

FV25 = $24,375

Interest-on-interest = $40,458.46 – $24,375 = $16,083.46”

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, LO3

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




2.97%i

1)1577.1(

1577.1$863.75$1,000)1(

)1(

000,1$75.863$

1

FVPV

51

5

5

n

n

i

i

i

i

The rate of return on this bond is 2.97 percent

tvm solutions manual ch05

Documents