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6
Discounted Chapter Cash Flows and Valuation
LEARNING OBJECTIVES
1.
Explain why cash flows occurring at different times must be discounted to a
common date before they can be compared, and be able to compute the present
value and future value for multiple cash flows.
When making decisions involving cash flows over time, we should first identify the magnitude
and timing of the cash flows and then discount each individual cash flow to its present value. The
process of discounting the cash flows adjusts them for the time value of money, because todays
dollars are not equal in value to dollars in the future. Once all of the cash flows are in present
value terms, the cash flows can be compared to make decisions. Section 6.1 discusses the
computation of present values and future values of multiple cash flows.
2.
Describe how to calculate the present value of an ordinary annuity and how an
ordinary annuity differs from an annuity due.
An ordinary annuity is a series of equally spaced level cash flows over time. The cash flows for
an ordinary annuity are assumed to take place at the ends of the periods. To find the value of an
ordinary annuity, we start by calculating the annuity factor, which is equal to (1 present value
factor)/i. Then, we multiply this factor by the constant future payment. An annuity due is an
annuity in which the cash flows occur at the beginnings of the periods. A lease is an example of
an annuity due. In this case, we are effectively prepaying for the service. To calculate the value
of an annuity due, we multiply the ordinary annuity value times (1 + i). Section 6.2 discusses the
computation of level cash flows (annuities and perpetuities).
1
3.
Explain what a perpetuity is and how it is used in business, and be able to calculate
the value of a perpetuity.
A perpetuity is like an annuity except that the cash flows are perpetualthey never end. British
Treasury Department bonds, called consols, were the first widely used securities of this kind.
The most common example of perpetuity today is preferred stock. The issuer of preferred stock
promises to pay fixed rate dividends forever. The preferred stockholders must be paid before
common stockholders. To calculate the present value of a perpetuity, we simply divide the
promised constant dividend payment (CF) by the interest rate (i). Learn by Doing Application
6.8 in Section 6.2 illustrates an application and calculation of a perpetuity problem found in
business.
4.
Discuss growing annuities and perpetuities, as well as their application in business,
and be able to calculate their value.
Financial managers often need to value cash-flow streams that increase at a constant rate over
time. These cash flow streams are called growing annuities or growing perpetuities. An example
of a growing annuity would be a 10-year lease contract with an annual adjustment for the
expected rate of inflation over the life of the contract. If the cash flows continue to grow at a
constant rate indefinitely, this cash flow stream is called a growing perpetuity. Application and
calculation of cash flows that grow at a constant rate are discussed in Section 6.3.
5.
Discuss why the effective annual interest rate (EAR) is the appropriate way to
annualize interest rates, and be able to calculate EAR.
2
The EAR is the annual growth rate that takes compounding into account. Thus, the EAR is the
true cost of borrowing or lending money. When we need to compare interest rates, we must make
sure that the rates to be compared have the same time and compounding periods. If interest rates
are not comparable, they must be converted into common terms. The easiest way to convert rates
to common terms is to calculate the EAR for each interest rate. The use and calculations of EAR
are discussed in Section 6.4.
I.
True or False Questions
1.
Calculating the present and future values of multiple cash flows is relevant only for
individual investors.
a.
b.
2.
True
False
Calculating the present and future values of multiple cash flows is relevant for businesses
only.
a.
b.
3.
True
False
In computing the present and future value of multiple cash flows, each cash flow is
discounted or compounded at a different rate.
3
a.
b.
4.
True
False
The present value of multiple cash flows is greater than the sum of those cash flows.
a.
b.
5.
True
False
When you pay the same amount every month as your insurance premium for a term life
policy for a period of five years, the stream of cash flows is called a perpetuity.
a.
b.
6.
True
False
When you pay the same amount every month on your car loan for a period of three years,
the stream of cash flows is called an annuity.
a.
b.
7.
True
False
In todays financial markets, the best example of a perpetuity is the common stock issued
by firms.
4
a.
b.
8.
True
False
Since the issuers of preferred stock promise to pay investors a fixed dividend, usually
quarterly, forever, these are the most important perpetuities in the financial markets.
c.
d.
9.
True
False
The present value of a perpetuity is the promised constant cash payment divided by the
interest rate (i).
e.
f.
10.
True
False
In ordinary annuities, cash flows occur at the beginning of each period.
g.
h.
11.
True
False
In an annuity due, cash flows occur at the beginning of each period.
5
i.
j.
12.
True
False
The lease payments by a business on a warehouse rental are an example of an annuity
due.
k.
l.
13.
True
False
The present value of an annuity due is less than the present value of an ordinary annuity.
m.
n.
14.
True
False
The present value of an annuity due is equal to the present value of an ordinary annuity.
o.
p.
2.
True
False
The future value of an annuity due is greater than the future value of an ordinary annuity.
a.
True
b.
False
6
15.
The future value of an annuity due is equal to the future value of an ordinary annuity.
c.
d.
16.
True
False
Cash flow streams that increase at a constant rate over time are called growing annuities
or growing perpetuities.
e.
f.
17.
True
False
A fertilizer manufacturing company enters into a contract with a county parks and
recreation department that calls for the company to sell 10 percent more of its best lawn
feed every year for the next five years. If they also agree to maintain the total price as
constant over the contract period, this growth in revenue is an example of a growing
perpetuity.
g.
True
h.
False
7
18.
You have received news about an inheritance that will pay you $5,000 next year.
Beginning the following year, your inheritance will increase by 5 percent every year
forever. This is a growing perpetuity.
i.
j.
19.
True
False
Trey Hughes opened a pizza place last year. He expects to increase his revenue from last
year by 7 percent every year for the next 10 years. This is an example of a growing
annuity.
k.
l.
20.
True
False
The APR is the annualized interest rate using compound interest.
m.
n.
21.
True
False
The APR is defined as the simple interest charged per period multiplied by the number of
periods per year.
o.
True
8
p.
22.
False
The correct way to annualize an interest rate is to compute the effective annual interest
rate.
q.
r.
23.
True
False
The correct way to annualize an interest rate is to compute the annual percentage rate
(APR).
s.
t.
24.
True
False
The effective annual interest rate (EAR) is defined as the annual growth rate that takes
compounding into account.
u.
v.
25.
True
False
The EAR is the true cost of borrowing and lending.
w.
True
9
x.
26.
False
The quoted interest rate is by convention a simple annual interest rate, such as the APR.
y.
z.
27.
True
False
The quoted interest rate is by definition a simple annual interest rate, such as the EAR.
aa.
bb.
28.
True
False
The Truth-in-Lending Act and the Truth-in-Savings Act require by law that the APR
be disclosed on all consumer loans and savings plans and that it be prominently displayed
on advertising and contractual documents.
cc.
dd.
29.
True
False
Only the APR or some other quoted rate should be used as the interest rate factor for
present or future value calculations.
ee.
True
10
ff.
False
11
II.
Multiple-Choice Questions
30.
To solve future value problems with multiple cash flows involves which of the following
steps?
a.
First, draw a time line to make sure that each cash flow is placed in the correct
time period.
b.
c.
Third, add up the future values.
d.
31.
Second, calculate the future value of each cash flow for its time period.
All of the above are necessary steps.
Which one of the following steps is NOT involved in solving future value problems?
a.
First, draw a time line to make sure that each cash flow is placed in the correct
time period.
b.
c.
Third, add up the values.
d.
32.
Second, discount each cash flow for its time period.
All of the above are necessary steps.
To solve present value problems with multiple cash flows involves which of the
following steps?
12
a.
First, draw a time line to make sure that each cash flow is placed in the correct
time period.
b.
c.
Third, add up the present values.
d.
33.
Second, calculate the present value of each cash flow for its time period.
All of the above are necessary steps.
Which one of the following steps is NOT involved in solving present value problems?
a.
First, draw a time line to make sure that each cash flow is placed in the correct
time period.
b.
c.
Third, add up the values.
d.
34.
Second, compound each cash flow for its time period.
All of the above are necessary steps.
Calculating the present and future values of multiple cash flows is relevant
a.
for businesses only.
b.
for individuals only
c.
for both individuals and businesses.
d.
none of the above.
13
35.
In computing the present and future value of multiple cash flows,
a.
b.
each cash flow is discounted or compounded at a different rate.
c.
earlier cash flows are discounted at a higher rate.
d.
36.
each cash flow is discounted or compounded at the same rate.
later cash flows are discounted at a higher rate.
In computing the present and future value of multiple cash flows,
a.
b.
each cash flow is discounted or compounded at the same rate.
c.
earlier cash flows are discounted at a higher rate.
d.
37.
earlier cash flows are discounted at a lower rate.
none of the above.
The present value of multiple cash flows is
a.
greater than the sum of the cash flows.
b.
equal to the sum of all the cash flows.
c.
less than the sum of the cash flows.
14
d.
38.
none of the above.
The future value of multiple cash flows is
a.
b.
equal to the sum of all the cash flows.
c.
less than the sum of the cash flows
d.
39.
greater than the sum of the cash flows.
none of the above.
If your investment pays the same amount at the end of each year for a period of six years,
the cash flow stream is called
a.
b.
an ordinary annuity.
c.
an annuity due.
d.
40.
a perpetuity.
none of the above.
If your investment pays the same amount at the beginning of each year for a period of 10
years, the cash flow stream is called
15
a.
b.
an ordinary annuity.
c.
an annuity due.
d.
41.
a perpetuity.
none of the above.
If your investment pays the same amount at the end of each year forever, the cash flow
stream is called
a.
b.
an ordinary annuity.
c.
an annuity due.
d.
42.
a perpetuity.
none of the above.
Cash flows associated with annuities are considered to be
a.
b.
a cash flow stream of the same amount (a constant cash flow stream).
c.
a mix of constant and uneven cash flow streams.
d.
43.
an uneven cash flow stream.
none of the above.
Which ONE of the following statements is true about amortization?
16
a.
Amortization refers to the way the borrowed amount (principal) is paid down over
the life of the loan.
b.
With an amortized loan, each loan payment contains some payment of principal
and an interest payment.
c.
A loan amortization schedule is just a table that shows the loan balance at the
beginning and end of each period, the payment made during that period, and how
much of that payment represents interest and how much represents repayment of
principal.
d.
45.
All of the above are true.
Which one of the following statements is NOT true about amortization?
a.
Amortization refers to the way the borrowed amount (principal) is paid down over
the life of the loan.
b.
With an amortized loan, each loan payment contains some payment of principal
and an interest payment.
c.
With an amortized loan, a smaller proportion of each months payment goes
toward interest in the early periods.
d.
A loan amortization schedule is just a table that shows the loan balance at the
beginning and end of each period, the payment made during that period, and how
17
much of that payment represents interest and how much represents repayment of
principal.
46.
Which one of the following statements is true about amortization?
a.
With an amortized loan, a bigger proportion of each months payment goes
toward interest in the early periods.
b.
With an amortized loan, a bigger proportion of each months payment goes
toward interest in the later periods.
c.
With an amortized loan, a smaller proportion of each months payment goes
toward interest in the early periods.
d.
47.
None of the above.
The annuity transformation method is used to transform
a.
a present value annuity to a future value annuity.
b.
a present value annuity to a future value annuity.
c.
an ordinary annuity to an annuity due.
d.
a perpetuity to an annuity.
18
48.
A firm receives a cash flow from an investment that will increase by 10 percent annually
for an infinite number of years. This cash flow stream is called
a.
b.
a growing perpetuity.
c.
an ordinary annuity.
d.
49.
an annuity due.
a growing annuity.
Your investment in a small business venture will produce cash flows that increase by 15
percent every year for the next 25 years. This cash flow stream is called
a.
b.
a growing perpetuity.
c.
an ordinary annuity.
d.
50.
an annuity due.
a growing annuity.
Which one of the following statements is TRUE about the effective annual rate (EAR)?
a.
The effective annual interest rate (EAR) is defined as the annual growth rate that
takes compounding into account.
19
b.
The EAR conversion formula accounts for the number of compounding periods
and, thus, effectively adjusts the annualized interest rate for the time value of
money.
c.
d.
51.
The EAR is the true cost of borrowing and lending.
All of the above are true.
The true cost of borrowing is the
a.
b.
effective annual rate.
c.
quoted interest rate.
d.
52.
annual percentage rate.
periodic rate.
The true cost of lending is the
a.
annual percentage rate.
b.
effective annual rate.
c.
quoted interest rate.
d.
none of the above.
20
53.
Which one of the following statements is NOT true?
a.
b.
The EAR is the appropriate rate to do present and future value calculations.
c.
The EAR is the true cost of borrowing and lending.
d.
54.
The APR is the appropriate rate to do present and future value calculations.
The EAR takes compounding into account.
Which one of the following statements is NOT true?
a.
The Truth-in-Lending Act was passed by Congress to ensure that the true cost of
credit was disclosed to consumers.
b.
The Truth-in-Savings Act was passed to provide consumers an accurate estimate
of the return they would earn on an investment.
c.
The above two pieces of legislation require by law that the APR be disclosed on
all consumer loans and savings plans.
d.
55.
All of the above are true statements.
Which one of the following statements is NOT true?
a.
The correct way to annualize an interest rate is to compute the effective annual
interest rate (EAR).
21
b.
The APR is the annualized interest rate using simple interest.
c.
The correct way to annualize an interest rate is to compute the annual percentage
rate (APR).
d.
You can find the interest rate per period by dividing the quoted annual rate by the
number of compounding periods.
56.
FV of multiple cash flows: Chandler Corp. is expecting a new project to start producing
cash flows, beginning at the end of this year. They expect cash flows to be as follows:
1
$643,547
2
$678,214
3
$775,908
4
5
$778,326 $735,444
If they can reinvest these cash flows to earn a return of 8.2 percent, what is the future
value of this cash flow stream at the end of five years? (Round to the nearest dollar.)
a.
$3,889,256
b.
$4,227,118
c.
$5,214,690
d.
$4, 809,112
22
57.
FV of multiple cash flows: Stiglitz, Inc., is expecting the following cash flows starting at
the end of the year$113,245, $132,709, $141,554, and $180,760. If their opportunity
cost is 9.6 percent, find the future value of these cash flows. (Round to the nearest dollar.)
a.
b.
$732,114
c.
$685,312
d.
58.
$644,406
$900,810
FV of multiple cash flows: Tariq Aziz will receive from his investment cash flows of
$3,125, $3,450, and $3, 800. If he can earn 7.5 percent on any investment that he makes,
what is the future value of his investment cash flows at the end of three years? (Round to
the nearest dollar.)
a.
b.
$10,944
c.
$10,812
d.
59.
$11,120
$12,770
FV of multiple cash flows: Shane Matthews has invested in an investment that will pay
him $6,200, $6,450, $7,225, and $7,500 over the next four years. If his opportunity cost
23
is 10 percent, what is the future value of the cash flows he will receive? (Round to the
nearest dollar.)
a.
b.
$29,900
c.
$30,455
d.
60.
$27,150
$31,504
FV of multiple cash flows: International Shippers, Inc., have forecast earnings of
$1,233,400, $1,345,900, and $1,455,650 for the next three years. What is the future value
of these earnings if the firms opportunity cost is 13 percent? (Round to the nearest
dollar.)
a.
b.
$4,551,446
c.
$3,900,865
d.
61.
$4,214,360
$4,875,212
PV of multiple cash flows: Jack Stuart has loaned money to his brother at an interest rate
of 5.75 percent. He expects to receive $625, $650, $700, and $800 at the end of the next
24
four years as complete repayment of the loan with interest. How much did he loan out to
his brother? (Round to the nearest dollar.)
a.
b.
$2,250
c.
$2,404
d.
62.
$2,713
$2,545
PV of multiple cash flows: Ferris, Inc., has borrowed from their bank at a rate of 8
percent and will repay the loan with interest over the next five years. Their scheduled
payments, starting at the end of the year are as follows$450,000, $560,000, $750,000,
$875,000, and $1,000,000. What is the present value of these payments? (Round to the
nearest dollar.)
a.
b.
$2,615,432
c.
$2431,224
d.
63.
$2,735,200
$2,815,885
PV of multiple cash flows: Hassan Ali has made an investment that will pay him
$11,455, $16,376, and $19,812 at the end of the next three years. His investment was to
25
fetch him a return of 14 percent. What is the present value of these cash flows? (Round to
the nearest dollar.)
a.
b.
$36,022
c.
$41,675
d.
64.
$33,124
$39,208
PV of multiple cash flows: Ajax Corp. is expecting the following cash flows$79,000,
$112,000, $164,000, $84,000, and $242,000over the next five years. If the companys
opportunity cost is 15 percent, what is the present value of these cash flows? (Round to
the nearest dollar.)
a.
b.
$414,322
c.
$480,906
d.
65.
$429,560
$477,235
PV of multiple cash flows: Pam Gregg is expecting cash flows of $50,000, $75,000,
$125,000, and $250,000 from an inheritance over the next four years. If she can earn 11
26
percent on any investment that she makes, what is the present value of her inheritance?
(Round to the nearest dollar.)
a.
b.
$309,432
c.
$412,372
d.
66.
$361,998
$434,599
Present value of an annuity: Transit Insurance Company has made an investment in
another company that will guarantee it a cash flow of $37,250 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? (Round to the nearest dollar.)
a.
b.
$124,868
c.
$251,154
d.
67.
$101,766
$186,250
Present value of an annuity: Herm Mueller has invested in a fund that will provide him
a cash flow of $11,700 for the next 20 years. If his opportunity cost is 8.5 percent, what is
the present value of this cash flow stream? (Round to the nearest dollar.)
27
a.
b.
$132,455
c.
$110,721
d.
68.
$234,000
$167,884
Present value of an annuity: Myers, Inc., will be making lease payments of $3,895.50
for a 10-year period, starting at the end of this year. If the firm uses a 9 percent discount
rate, what is the present value of this annuity? (Round to the nearest dollar.)
a.
b.
$29,000
c.
$25,000
d.
69.
$23,250
$20,000
Present value of an annuity: Lorraine Jackson won a lottery. She will have a choice of
receiving $25,000 at the end of each year for the next 30 years, or a lump sum today. If
she can earn a return of 10 percent on any investment she makes, what is the minimum
amount she should be willing to accept today as a lump-sum payment? (Round to the
nearest hundred dollars.)
a.
$750,000
28
b.
c.
$212,400
d.
70.
$334,600
$235,700
Present value of an annuity: Craymore Tech is expecting cash flows of $67,000 at the
end of each year for the next five years. If the firms discount rate is 17 percent, what is
the present value of this annuity? (Round to the nearest dollar.)
a.
b.
$241,653
c.
$278,900
d.
71.
$214,356
$197,776
Future value of an annuity: Carlos Menendez is planning to invest $3,500 every year
for the next six years in an investment paying 12 percent annually. What will be the
amount he will have at the end of the six years? (Round to the nearest dollar.)
a.
$21,000
b.
$28,403
c.
$24,670
d.
$26,124
29
72.
Future of value an annuity: Jayadev Athreya has started on his first job. He plans to
start saving for retirement early. He will invest $5,000 at the end of each year for the next
45 years in a fund that will earn a return of 10 percent. How much will Jayadev have at
the end of 45 years? (Round to the nearest dollar.)
a.
b.
$3,594,524
c.
$1,745,600
d.
73.
$2,667,904
$5,233,442
Future value of an annuity: You plan to save $1,250 at the end of each of the next three
years to pay for a vacation. If you can invest it at 7 percent, how much will you have at
the end of three years? (Round to the nearest dollar.)
a.
$3,750
b.
$3,918
c.
$4,019
d.
$4,589
30
74.
Future value of an annuity: Zhijie Jiang is saving to buy a new car in four years. She
will save $5,500 at the end of each of the next four years. If she invests her savings at
6.75 percent, how much will she have after four years? (Round to the nearest dollar.)
a.
b.
$23,345
c.
$27,556
d.
75.
$22,000
$24,329
Future value of an annuity: Terri Garner will invest $3,000 in an IRA for the next 30
years. The investment will earn 13 percent annually. How much will she have at the end
of 30 years? (Round to the nearest dollar.)
a.
b.
$912,334
c.
$748,212
d.
76.
$897,598
$1,233,450
Computing annuity payment: Maricela Sanchez needs to have $25,000 in five years. If
she can earn 8 percent on any investment, what is the amount that she will have to invest
every year for the next five years? (Round to the nearest dollar.)
31
a.
b.
$4,261
c.
$4,640
d.
77.
$5,000
$4,445
Computing annuity payment: Jane Ogden wants to save for a trip to Australia. She will
need $12,000 at the end of four years. She can invest a certain amount at the beginning of
each of the next four years in a bank account that will pay her 6.8 percent annually. How
much will she have to invest annually to reach her target? (Round to the nearest dollar.)
a.
b.
$2,980
c.
$2,538
d.
78.
$3,000
$2,711
Computing annuity payment: Jackson Electricals has borrowed $27,850 from its bank
at an annual rate of 8.5 percent. It plans to repay the loan in eight equal installments,
beginning at the end of next year. What is its annual loan payment? (Round to the nearest
dollar.)
32
a.
b.
$5,134
c.
$4,939
d.
79.
$4,708
$4,748
Computing annuity payment: John Harper has borrowed $17,400 to pay for his new
truck. The annual interest rate on the loan is 9.4 percent, and the loan needs to be repaid
in four years. What will be his annual payment if he begins his payment beginning now?
(Round to the nearest dollar.)
a.
b.
$5,450
c.
$4,850
d.
80.
$5,229
$4,953
Computing annuity payment: Trevor Smith wants to have a million dollars at
retirement, which is 15 years away. He already has $200,000 in an IRA earning 8 percent
annually. How much does he need to save each year, beginning at the end of this year to
reach his target? Assume he could earn 8 percent on any investment he makes. (Round to
the nearest dollar.)
33
a.
b.
$14,273
c.
$10,900
d.
81.
$13,464
$16,110
Perpetuity: Your father is 60 years old and wants to set up a cash flow stream that would
be forever. He would like to receive $20,000 every year, beginning at the end of this year.
If he could invest in account earning 9 percent, how much would he have to invest today
to receive his perpetual cash flow? (Round to the nearest dollar.)
a.
b.
$200,000
c.
$189,000
d.
82.
$222,222
$235,200
Perpetuity: A lottery winner was given a perpetual payment of $11, 444. She could
invest the cash flows at 7 percent. What is the present value of this perpetuity? (Round to
the nearest dollar.)
a.
$112,344
b.
$163,486
34
c.
d.
83.
$191,708
$201,356
Perpetuity: Roger Barkley wants to set up a scholarship at his alma mater. He is willing
to invest $500,000 in an account earning 10 percent. What will be the annual scholarship
that can be given from this investment? (Round to the nearest dollar.)
a.
b.
$500,000
c.
$50,000
d.
84.
$5,000
None of the above
Perpetuity: Chris Collinge has funded a retirement investment with $250,000 earning a
return of 5.75 percent. What is the value of the payment that he can receive in perpetuity?
(Round to the nearest dollar.)
a.
$12,150
b.
$15,250
c.
$14,375
d.
$14,900
35
85.
Perpetuity: Jeff Conway wants to receive $25,000 in perpetuity and will invest his
money in an investment that will earn a return of 13.5 percent annually. What is the value
of the investment that he needs to make today to receive his perpetual cash flow stream?
(Round to the nearest dollar.)
a.
b.
$252,325
c.
$144,350
d.
86.
$640,225
$185,185
Annuity due: You plan to save $1,400 for the next four years, beginning now, to pay for
a vacation. If you can invest it at 6 percent, how much will you have at the end of four
years? Round to the nearest dollar.
a.
$6,124
b.
$5,618
c.
$4,019
d.
$6,492
36
87.
Annuity due: Mark Holcomb has a five-year loan on which he will make annual
payments of $2,235, beginning now. If the interest rate on the loan is 8.3 percent, what is
the present value of this annuity? (Round to the nearest dollar.)
a.
b.
$8,854
c.
$8,612
d.
88.
$9,588
$9,122
Annuity due: Jenny Abel is investing $2,500 today and will do so at the beginning of
another six years for a total of seven payments. If her investment can earn 12 percent,
how much will she have at the end of seven years? (Round to the nearest dollar.)
a.
b.
$28,249
c.
$31,127
d.
89.
$25,223
$29,460
Annuity due: Your inheritance will pay you $100,000 a year for five years beginning
now. You can invest it in a CD that will pay 7.75 percent annually. What is the present
value of your inheritance? (Round to the nearest dollar.)
37
a.
b.
$401,916
c.
$433,064
d.
90.
$399,356
$467,812
Growing perpetuity: Jack Benny is planning to invest in an insurance company product.
The product will pay $10,000 at the end of this year. Thereafter, the payments will grow
annually at a 3 percent rate forever. Jack will be able to invest his cash flows at a rate of
6.5 percent. What is the present value of this investment cash flow stream? (Round to the
nearest dollar.)
a.
b.
$312,766
c.
$285,714
d.
91.
$326,908
$258,133
Growing perpetuity: Norwood Investments is putting out a new product. The product
will pay out $25,000 in the first year, and after that the payouts will grow by an annual
rate of 2.5 percent forever. If you can invest the cash flows at 7.5 percent, how much will
you be willing to pay for this perpetuity? (Round to the nearest dollar.)
38
a.
b.
$233,000
c.
$250,000
d.
92.
$312,000
$500,000
Growing annuity: Hill Enterprises is expecting tremendous growth from its newest
boutique store. Next year the store is expected to bring in net cash flows of $675,000.
The company expects its earnings to grow annually at a rate of 13 percent for the next 15
years. What is the present value of this growing annuity if the firm uses a discount rate of
18 percent on its investments? (Round to the nearest dollar.)
a.
b.
$6,750,000
c.
$7,115,449
d.
93.
$6,448,519
$5,478,320
Growing annuity: Wilbon Corp. is evaluating whether it should take over the lease of an
ethnic restaurant in Manhattan. The current owner had originally signed a 25-year lease,
of which 16 years still remain. The restaurant has been growing steadily at a 7 percent
growth for the last several years. Wilbon Corp. expects the restaurant to continue to grow
39
at the same rate for the remaining lease term. Last year, the restaurant brought in net cash
flows of $310,000. If the firm evaluates similar investments at 15 percent, what is the
present value of this investment? (Round to the nearest dollar.)
a.
b.
$2,838,182
c.
$3,109,460
d.
94.
$2,966.350
$2,709,124
Effective annual rate: Desire Cosmetics borrowed $152,300 from a bank for three years.
If the quoted rate (APR) is 11.75 percent, and the compounding is daily, what is the
effective annual rate (EAR)? (Round to one decimal place.)
a.
b.
14.3%
c.
12.5%
d.
95.
11.75%
11.6%
Effective annual rate: Largent Supplies Corp. has borrowed to invest in a project. The
loan calls for a payment of $17,384 every month for three years. The lender quoted
Largent a rate of 8.40 percent with monthly compounding. At what rate would you
40
discount the payments to find amount borrowed by Largent? (Round to two decimal
places.)
a.
8.40%
b.
8.73%
c.
8.95%
d.
None of the above.
41
III. Essay Questions
96.
How is an annuity due different from the ordinary annuity?
Answer: When constant cash flows are received or paid at the end of each period for a
length of time, we have an ordinary annuity. If the same cash flows happen at the
beginning of each period, we call it an annuity due. Cash flows received at the beginning
of each period earn interest for an extra period compared to cash flows received at the
end of each period for an investment of the same time frame. Thus, annuity dues have
higher values than ordinary annuities.
97.
The annual percentage rate (APR) is not the appropriate rate to do present or future value
calculations. Explain this statement.
Answer: The APR is the annualized interest rate using simple interest. In other words,
the APR is the simple interest charged per period multiplied by the number of periods per
year. However, the APR ignores the impact of compounding on cash flows. This makes it
an inappropriate discount rate for doing present and future value calculations. An
appropriate rate for such calculations is the effective annual rate (EAR).
98.
What was the purpose behind the passage of the two consumer protection acts discussed
in this chapter?
42
Answer: In 1968, Congress passed the Truth-in-Lending Act to ensure that all
borrowers receive meaningful information about the cost of credit so they can make
intelligent economic decisions. The act applies to all lenders that extend credit to
consumers, and it covers credit card loans, auto loans, home mortgage loans, home equity
loans, home improvement loans, and some small business loans. Similar legislation, the
so-called Truth-in-Savings Act, applies to consumer savings vehicles such as consumer
certificates of deposits (CDs). These two pieces of legislation require by law that the
APR be disclosed on all consumer loans and savings plans and that it be prominently
displayed on advertising and contractual documents.
43
IV. Answers to True or False Questions
1.
False
2.
False
3.
False
4.
False
5.
False
6.
True
7.
False
8.
True
9.
True
10.
False
11.
True
12.
True
13.
False
14.
False
15.
True
16.
False
17.
True
18.
False
19.
True
20.
True
21.
False
22.
True
44
23.
True
24.
False
25.
True
26.
True
27.
True
28.
False
29.
True
30.
False
45
V.
Answers to Multiple-Choice Questions
31.
d
32.
b
33.
d
34.
b
35.
c
36.
a
37.
b
38.
c
39.
a
40.
b
41.
c
42.
a
43.
b
44.
d
45.
c
46.
a
47.
c
48.
b
49.
d
50.
d
51.
b
52.
b
46
53.
a
54.
d
55.
c
56.
b
57.
a
58.
a
59.
d
60.
b
61.
c
62.
d
63.
b
64.
a
65.
a
66.
b
67.
c
68.
c
69.
d
70.
a
71.
b
72.
b
73.
c
74.
d
75.
a
47
76.
b
77.
c
78.
c
79.
d
80.
a
81.
a
82.
b
83.
c
84.
c
85.
d
86.
d
87.
a
88.
b
89.
c
90.
c
91.
d
92.
a
93.
b
94.
c
95.
b
48
VI. Solutions to Multiple-Choice Questions
56.
Solution:
0
1
2
3
4
5
$643,547
n = 5;
$678,214
$775,908
$778,326
$735,444
i = 8.2%
FV5 = $643,547(1.082) 4 + $678,214(1.082) 3 + $775,908(1.082) 2 + $778,326(1.082)1
+ $735,444
= $882,042.10 + $859,109.52 + $908,374.12 + $842,148.73 + $735,444
= $4,227,118.47
57.
Solution:
0
1
2
3
4
$113,245
n = 4;
$132,709
$141,554
$180,760
i = 9.6%
FV4 = $113,245(1.096) 3 + $132,709(1.096) 2 + $141,554(1.096)1 + $180,760
= $149,090.75 + $159,412.17 + $155,143.18 + $180,760
= $644,406.10
58.
Solution:
0
1
2
3
49
$3,125
n = 3;
$3,450
$3, 800
i = 7.5%
FV3 = $3,125(1.075) 2 + $3,450(1.075)1 + $3,800
= $3,611.33 + $3,708.75 + $3,800
= $11,120.08
59.
Solution:
0
1
2
3
4
$6,200
n = 4;
$6,450
$7,225
$7,500
i = 10%
FV4 = $6,200(1.10) 3 + $6,450(1.10) 2 + $7,225(1.10)1 + $7,500
= $8,252.20 + $7,804.50 + $7,947.50 + $7,500
= $31,504.20
60.
Solution:
0
1
2
3
$1, 233,400
n = 3;
$1,345,900
$1,455,650
i = 13%
50
FV3 = $1,233,400(1.13) 2 + $1,345,900(1.13)1 + $1,455,650
= $1,574,928.46 + $1,520,867 + $1,455,650
= $4,551,445.46
61.
Solution:
0
1
2
3
4
$1625
n = 4;
$650
$700
$800
i = 5.75%
$625
$650
$700
$800
+
+
+
2
3
(1.0575) (1.0575)
(1.0575)
(1.0575) 4
= $591.02 + $581.24 + $591.91 + $639.69
PV =
= $2,403.85
62.
Solution:
0
1
2
3
4
5
$450,000
n = 5;
$560,000
$750,000
$875,000
$1,000,000
i = 8%
$450,000 $560,000 $750,000 $875,000 $1,000,000
+
+
+
+
(1.08)
(1.08) 2
(1.08) 3
(1.08) 4
(1.08) 5
= $416,666.67 + $480,109.74 + $595,374.18 + $643,151.12 + $680,583.20
= $2,815,884.91
PV =
51
63.
Solution:
0
1
2
3
$11,455
$16,376
n = 3;
$19,812
i = 14%
$11,455 $16,376 $19,812
+
+
(1.14)
(1.14) 2
(1.14) 3
= $10,048.25 + $12,600.80 + $13,372.54
= $36,021.58
PV =
64.
Solution:
0
1
2
3
4
5
$79,000
n = 5;
$112,000
$164,000
$84,000
$242,000
i = 15%
$79,000 $112,000 $164,000 $84,000 $242,000
+
+
+
+
(1.15)
(1.15) 2
(1.15) 3
(1.15) 4
(1.15) 5
= $68,695.65 + $84,688.09 + $107,832.66 + $48,027.27 + $120,316.77
PV =
= $429,560.45
65.
Solution:
0
1
2
3
4
$50,000
n = 4;
$75,000
$125,000
i = 11%
52
$250,000
$50,000 $75,000 $125,000 $250,000
+
+
+
(1.11)
(1.11) 2
(1.11) 3
(1.11) 4
= $45,045.05 + 60,871.68 + $91,398.92 + $164,682.74
= $361,998.39
PV =
66.
Solution:
0
1
2
3
4
5
$37,250
n = 5;
$37,250
$37,250
i = 15%
Annual payment = PMT = $37,250
No. of payments = n = 5
Required rate of return = 15%
Present value of investment = PVA5
1
1 (1 + i ) n
PVA n = PMT
i
1
1 (1.15) 5
= $37,250
= $37,250 3.3522
0.15
= $124,867.78
53
$37,250
$37,250
67.
Solution:
0
1
2
3
19
20
$11,700
$11,700
n = 20;
$11,700
$11,700
$11,700
9
10
i = 8.5%
1
1 (1 + i ) n
PVA n = PMT
i
1
1 (1.085) 20
= $11,700
= $11,700 9.4633
0.085
= $110,721.04
68.
Solution:
0
1
2
3
$3,895.50
n = 10;
$3,895.50
$3,895.50
i = 9%
54
$3,895.50
$3,895.50
1
1 (1 + i ) n
PVA n = PMT
i
1
1 (1.09)10
= $3,895.50
= $3,895.50 6.4177
0.09
= $24,999.99
55
69.
Solution:
0
1
2
3
29
30
$25,000
$25,000
n = 30;
$25,000
$25,000
4
5
i = 10%
1
1 (1 + i ) n
PVA n = PMT
i
1
1 (1.10) 30
= $25,000
0.10
= $235,672.86
70.
$25,000
= $25,000 9.4269
Solution:
0
1
2
3
$67,000
n = 5;
$67,000
$67,000
i = 17%
56
$67,000
$67,000
1
1 (1 + i ) n
PVA n = PMT
i
1
1 (1.17) 5
= $67,000
= $67,000 3.1993
0.17
= $214,356.19
57
71.
Solution:
0
1
2
3
4
5
6
$3,500
n = 6;
$3,500
$3,500
$3,500
$3,500
$3,500
i = 12%
(1 + i ) n 1
FVA n = PMT
i
6
(1.12) 1
= $3,500
= $3,500 8.1152
0.12
= $28,403.16
72.
Solution:
0
1
2
3
44
45
$5,000
n = 45;
$5,000
$5,000
i = 10%
(1 + i ) n 1
FVA n = PMT
i
45
(1.10) 1
= $5,000
= $5,000 718.9048
0.10
= $3,594.524.18
73.
Solution:
0
1
2
3
58
$5,000
$5,000
$1,250
n = 3;
$1,250
$1,250
i = 7%
(1 + i ) n 1
FVA n = PMT
i
3
(1.07) 1
= $1,250
= $1,250 3.2149
0.07
= $4,018.63
74.
Solution:
0
1
2
3
4
$5,500
$5,500
n = 4;
$5,500
$5,500
i = 6.75%
(1 + i ) n 1
FVA n = PMT
i
(1.0675) 4 1
= $5,500
= $5,500 4.4235
0.0675
= $24,329.43
75.
Solution:
0
1
2
3
29
30
$3,000
$3,000
$3,000
59
$3,000
$3,000
n = 30;
i = 13%
(1 + i ) n 1
FVA n = PMT
i
30
(1.13) 1
= $3,000
= $3,000 293.1992
0.13
= $879,597.65
60
76.
Solution:
0
1
2
3
4
5
PMT
n = 5;
PMT
PMT
i = 8%
PMT
PMT
FVA = $25,000
(1 + i ) n 1
FVA n = PMT
i
5
(1.08) 1
$25,000 = PMT
0.08
$25,000
$25,000
PMT =
=
5
(1.08) 1 5.8666
0.08
= $4,261.41
77.
Solution:
0
1
2
3
4
PMT
PMT
n = 4;
PMT
PMT
i = 6.8%
FVA = $12,000
61
(1 + i ) n 1
FVA n = PMT
(1 + i )
i
$12,000
PMT =
(1.068) 4 1
0.068 (1.068)
= $2,538.16
62
78.
Solution:
0
1
2
3
7
8
PMT
PVAn = $27,850
PMT
PMT
n = 8;
PMT
PMT
i = 8.5%
Present value of annuity = PVA = $27,850
Return on investment = i = 8.5%
Payment required to meet target = PMT
Using the PVA equation:
1
1 (1 + i ) n
PVA n = PMT
i
$27,850
$27,850
PMT =
=
1
5.6392
1 (1.085) 8
0.085
= $4,938.66
Each payment made by Jackson Electricals will be $4,938.66, starting at the end of next
year.
79.
Solution:
0
1
2
3
4
63
PMT
PMT
PVAn = $17,400
PMT
PMT
n = 4;
i = 9.4%
Present value of annuity = PVA = $17,400
Return on investment = i = 9.4%
Payment required to meet target = PMT
Type of annuity = Annuity due
Using the PVA equation:
1
1 (1 + i ) n
PVA n = PMT
(1 + i )
i
$17,400
$17,400
PMT =
=
1
3.2115 1.094
1 (1.094) 4
(1.094)
0.094
= $4,952.49
Each payment made by John Harper will be $5,927.36, starting at the end of next year.
80.
Solution:
Retirement investment target in 15 years = $1,000,000
Amount invested in IRA account now = PV = $200,000
Return earned by investment = i = 8%
Value of current investment in 15 years = FV15
FV15 = PV(1 + i )15 = $200,000(1.08)15
= $634,433.82
64
Balance of money needed to buy car = $1,000,000 -$634,433.82 =$365,566.18 = FVA
Payment needed to reach target = PMT
(1 + i ) n 1
FVA = PMT
i
FVA
$365,566.18 $365,566.18
PMT =
=
=
n
27.1521
1 (1 + i ) (1.08)15 1
0.08
i
= $13,463.64
81.
Solution:
Annual payment needed = PMT = $20,000
Investment rate of return = i = 9%
Term of payment = Perpetuity
Present value of investment needed = PV
PMT $20,000
=
i
0.09
= $222,222.22
PV of Perpetuity =
82.
Solution:
Annual payment needed = PMT = $11,444
Investment rate of return = i = 7%
Term of payment = Perpetuity
Present value of investment needed = PV
PMT $11,444
=
i
0.07
= $163,485.71
PV of Perpetuity =
65
83.
Solution:
Annual payment needed = PMT
Present value of investment = PVA = $500,000
Investment rate of return = i = 10%
Term of payment = Perpetuity
PMT
i
PMT = PV of Perpetuity i
= $500,000 0.10
= $5,000
PV of Perpetuity =
84.
Solution:
Annual payment needed = PMT
Present value of investment = PVA = $250,000
Investment rate of return = i = 5.75%
Term of payment = Perpetuity
PMT
i
PMT = PV of Perpetuity i
= $250,000 0.0575
= $14,375
PV of Perpetuity =
85.
Solution:
Annual Payment needed = PMT = $25,000
Investment rate of return = i = 13.5%
Term of payment = Perpetuity
66
Present value of investment needed = PV
PMT $25,000
=
i
0.135
= $185,185.19
PV of Perpetuity =
86.
Solution:
0
1
2
3
4
$1,400
$1,400
n = 4;
$1,400
$1,400
i = 6%
(1 + i ) n
FVA = PMT
(1 + i )
i
(1.06) 4 1
= $1,400
(1.06)
0.06
= $1,400 4.3746 1.06
= $6,491.93
67
87.
Solution:
0
1
2
3
4
5
$2,235
$2,235
n = 5;
$2,235
$2,235
$2,235
3
6
i = 8.3%
Annual payment = PMT = $2,235
No. of payments = n = 5
Required rate of return = 8.3%
Present value of investment = PVA5
1
1 (1 + i ) n
PVA = PMT
i
(1 + i )
1
1 (1.083) 5
= $2,235
(1.083)
0.083
= $2,235 3.9613 1.083
= $9,588.44
88.
Solution:
0
1
2
7
PMT
n = 7;
PMT
PMT
PMT
i = 12%
68
PMT
Present value of annuity = PVA Return on investment = i = 9.4%
Payment required to meet target = $2,500
Type of annuity = Annuity due
(1 + i ) n
FVA = PMT
(1 + i )
i
(1.12) 7 1
= $2,500
(1.12)
0.12
= $2,500 10.0890 1.12
= $28,249.23
89.
Solution:
0
1
2
3
4
5
$100,000
n = 5;
$100,000
$100,000
$100,000
i = 7.75%
Annual payment = PMT = $100,000
No. of payments = n = 5
Required rate of return = 7.75%
Present value of investment = PVA5
69
$100,000
1
1 (1 + i ) n
PVA = PMT
i
(1 + i )
1
1 (1.0775) 5
= $100,000
(1.0775)
0.0775
= $100,000 4.0192 1.0775
= $433,064.19
90.
Solution:
Cash flow at t=1 = CF1 = $10,000
Annual growth rate = g = 3%
Discount rate = i = 6.5%
Present value of growing perpetuity = PVA
CF1
$10,000
=
(i g) (0.065 0.03)
= $285,714.29
PVA =
91.
Solution:
Cash flow at t=1 = CF1 = $25,000
Annual growth rate = g = 2.5%
Discount rate = i = 7.5%
Present value of growing perpetuity = PVA
CF1
$25,000
=
(i g) (0.075 0.025)
= $500,000
PVA =
70
92.
Solution:
Time of growth = n = 15 years
Next years expected net cash flow = CF1 = $675,000
Expected annual growth rate = g = 13%
Firms required rate of return = i = 18%
Present value of growing annuity = PVAn
1 + g n
1.13 15
CF1
$675,000
PVA n =
1
1
=
(i g ) 1 + i (0.18 0.13) 1.18
= $13,500,000 477668
= $6,448,519.47
93.
Solution:
Time for lease to expire = n = 16 years
Last years net cash flow = CF0 = $310,000
Expected annual growth rate = g = 7%
Firms required rate of return = i = 15%
Expected cash flow next year = CF1 = $310,000(1 + g) = $310,000(1.07) = $331,700
Present value of growing annuity = PVAn
PVA n =
1 + g n
1.07 16
CF1
$331,700
1
=
1
(i g ) 1 + i (0.15 0.07) 1.15
= $4,146,250 0.684518
= $2,838,181.52
94.
Solution:
71
Loan amount = PV = $152,300
Interest rate on loan = i = 11.75%
Frequency of compounding = m = 365
Effective annual rate = EAR
m1
i
0.1175
EAR = 1 + 1 = 1 +
365
m
= 1.12455 1 = 12.46%
95.
365
1
Solution:
Loan amount = PV
Interest rate on loan = i = 8.4%
Frequency of compounding = m = 12
Effective annual rate = EAR
m1
12
i
0.084
EAR = 1 + 1 = 1 +
1
12
m
= 1.0873 1 = 8.73%
To discount present or future value of cash flows, the most appropriate rate is the EAR,
that is, 8.73 percent.
72