Principles of Corporate Finance
Questions and Answers
In this document you will find some sample questions about the topics included in the final
exam. Answers are provided. The questions are usually in order of topic but not necessarily
in order of difficulty.
Ch 5: Techniques of project valuation: NPV, IRR, etc
Ch 5
8 Payback Consider the following projects:
.
a. If the opportunity cost of capital is 10%, which projects have a positive NPV?
b. Calculate the payback period for each project.
c. Which project(s) would a firm using the payback rule accept if the cutoff period
were three years?
d. Calculate the discounted payback period for each project.
e. Which project(s) would a firm using the discounted payback rule accept if the
cutoff period were three years?
A8) 8. a.
NPVA   $1000 
$1000
 $90.91
(1.10)
NPVB   $2000 
$1000 $1000 $4000 $1000 $1000




 $4,044.73
(1.10) (1.10) 2 (1.10) 3 (1.10) 4 (1.10)5
NPVC   $3000 
$1000 $1000 $1000 $1000



 $39.47
(1.10) (1.10) 2 (1.10) 4 (1.10) 5
Projects B and C have positive NPVs.
b.
Payback A = one year
Payback B = two years
Payback C = four years
c.
A and B
d.
PVA 
$1000
 $909.09
(1.10)1
The present value of the cash inflows for Project A never recovers the initial
outlay for the project, which is always the case for a negative NPV project.
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The present values of the cash inflows for Project B are shown in the third
row of the table below, and the cumulative net present values are shown in
the fourth row:
C0
C1
-2,000.00
-2,000.00
C2
+1,000.00
909.09
-1,090.91
+1,000.00
826.45
-264.46
C3
+4,000.00
3,005.26
2,740.80
C4
+1,000.00
683.01
3,423.81
C5
+1,000.00
620.92
4,044.73
Since the cumulative NPV turns positive between year 2 and year 3, the
discounted payback period is:
2
264.46
 2.09 years
3,005.26
The present values of the cash inflows for Project C are shown in the third row of the table
below, and the cumulative net present values are shown in the fourth row:
C0
C1
C2
C3
C4
C5
-3,000.00
-3,000.00
+1,000.00
909.09
-2,090.91
+1,000.00
826.45
-1,264.46
0.00
0.00
-1,264.46
+1,000.00
683.01
-581.45
Since the cumulative NPV turns positive between year 4 and year 5, the
discounted payback period is:
4
e.
581.45
 4.94 years
620.92
Using the discounted payback period rule with a cutoff of three years, the
firm would accept only Project B.
Q12 IRR rule Mr. Cyrus Clops, the president of Giant Enterprises, has to make a choice
between two possible investments:
The opportunity cost of capital is 9%. Mr. Clops is tempted to take B, which has the higher
IRR.
a. Explain to Mr. Clops why this is not the correct procedure.
b. Show him how to adapt the IRR rule to choose the best project.
c. Show him that this project also has the higher NPV.
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+1,000.00
620.92
39.47
12.
a.
Because Project A requires a larger capital outlay, it is possible
that Project A has both a lower IRR and a higher NPV than Project B. (In fact,
NPVA is greater than NPVB for all discount rates less than 10%.) Because the
goal is to maximize shareholder wealth, NPV is the correct criterion.
d.
To use the IRR criterion for mutually exclusive projects, calculate the IRR
for the incremental cash flows:
A-B
C0
C1
C2
IRR
−200
+110
+121
10%
Because the IRR for the incremental cash flows exceeds the cost of capital,
the additional investment in A is worthwhile.
c.
NPVA   400 
250
300

 $ 81.86
1.09 (1.09) 2
NPVB   200 
140
179

 $79.10
1.09 (1.09) 2
16. NPV/IRR. Consider projects A and B:
Cash Flows (dollars)
Project
C0
C1
C2
NPV at 10%
A
–30,000
21,000
21,000
+$6,446
B
–50,000
33,000
33,000
+ 7,273
Calculate IRRs for A and B. Which project does the IRR rule suggest is best? Which
project is really best? (LO8-3)
1.
Answer:
IRRA = discount rate (r), which is the solution to the following equation:
1

1
$21,000   
 $30,000  r = IRRA = 25.69%
2
 r r  (1  r ) 
IRRB = discount rate (r), which is the solution to the following equation:
1

1
$33,000   
 $50,000  r = IRRB = 20.69%
2
 r r  (1  r ) 
The IRR of project A is 25.69%, and that of B is 20.69%. However, project B has the
higher NPV and therefore is preferred. The incremental cash flows of B over A are
$20,000 at time 0 and +$12,000 at times 1 and 2. The NPV of the incremental cash
flows (discounted at 10%) is $826.45, which is positive and equal to the difference in
the respective project NPVs.
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Q22: Profitability Index versus NPV. Consider projects A and B with the following cash
flows: (LO8-3)
C0
C1
C2
C3
A
–$36
+$20
+$20
+$20
B
– 50
+ 25
+ 25
+ 25
a. Which project has the higher NPV if the discount rate is 10%?
b. Which has the higher profitability index?
c. Which project is most attractive to a firm that can raise an
unlimited amount of funds to pay for its investment projects?
Which project is most attractive to a firm that is limited in the
funds it can raise?
22 a.
NPVA = –$36 + [$20  annuity factor (10%, 3 periods)]
 1

1
= –$36 + $20  

 $13.74
3
 0.10 0.10  (1.10) 
NPVB = –$50 + [$25  annuity factor (10%, 3 periods)]
 1

1

 $12.17
= –$50 + $25  
3
 0.10 0.10  (1.10) 
Thus Project A has the higher NPV if the discount rate is 10%.
b.
Project A has the higher profitability index, as shown in the table below:
Project
A
B
c.
PV of
Cash Flow
$49.74
$62.17
Investment
NPV
$36
$50
$13.74
$12.17
Profitability
Index
0.38
0.24
A firm with a limited amount of funds available should choose Project A
since it has a higher profitability index of 0.38, i.e., a higher “bang for the
buck.” Note that A also has a higher NPV as well.
For a firm with unlimited funds, the possibilities are:
(i) If the projects are independent projects, then the firm should choose
both projects.
(ii) However, if the projects are mutually exclusive, then Project A should
be selected. It has the higher NPV.
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Ch 6
Q11. Real and nominal flows CSC is evaluating a new project to produce encapsulators.
The initial investment in plant and equipment is $500,000. Sales of encapsulators in
year 1 are forecasted at $200,000 and costs at $100,000. Both are expected to
increase by 10% a year in line with inflation. Profits are taxed at 35%. Working capital
in each year consists of inventories of raw materials and is forecasted at 20% of sales
in the following year.
The project will last five years and the equipment at the end of this period will have no
further value. For tax purposes the equipment can be depreciated straight-line over
these five years. If the nominal discount rate is 15%, show that the net present value
of the project is the same whether calculated using real cash flows or nominal flows.
Answer 11)
Revenues
Costs
Depreciation
Pretax Profit
Taxes at 35%
Profit after Tax
Depreciation
Cash Flow from Operations
Change in Working Capital
Capital Investment
Net Cash Flows
Discount Factor @ 15%
Present Value
-40,000
-500,000
-540,000
1.000
-540,000
NPV
-147,510
200,000
100,000
100,000
0
0
0
100,000
100,000
-4,000
220,000
110,000
100,000
10,000
3,500
6,500
100,000
106,500
-4,400
242,000
121,000
100,000
21,000
7,350
13,650
100,000
113,650
-4,840
266,200
133,100
100,000
33,100
11,585
21,515
100,000
121,515
-5,324
292,820
146,410
100,000
46,410
16,244
30,167
100,000
130,167
58,564
96,000
0.870
83,478
102,100
0.756
77,202
108,810
0.658
71,544
116,191
0.572
66,433
188,731
0.497
93,832
Since the nominal rate is 15% and the expected inflation rate is 10%, the real rate is given by
the following:
(1 + rnominal) = (1 + rreal)  (1 + inflation rate)
1.15 = (1 + rreal)  (1.10)
rreal = 0.04545 = 4.545%
Adjusting the cash flows to real dollars and using this real rate gives us the same result for
NPV (with a slight rounding error).
Net Cash Flows (nominal)
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-540,000
1
96,000
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YEAR
2
3
102,100
108,810
4
116,191
5
188,731
Adjustment Factor for Real CF
Net Cash Flows (real)
Discount Factor @ 4.545%
Present Value
1
-540,000
1.000
-540,000
NPV
-147,505
0.909
87,273
0.957
83,479
0.826
84,380
0.915
77,203
0.751
81,751
0.875
71,545
0.683
79,360
0.837
66,434
0.621
117,187
0.801
93,834
15. Project NPV After spending $3 million on research, Better Mousetraps has developed
a new trap. The project requires an initial investment in plant and equipment of $6
million. This investment will be depreciated straight-line over five years to a value of
zero, but, when the project comes to an end in five years, the equipment can in fact be
sold for $500,000. The firm believes that working capital at each date must be
maintained at 10% of next year's forecasted sales. Production costs are estimated at
$1.50 per trap and the traps will be sold for $4 each. (There are no marketing
expenses.) Sales forecasts are given in the following table. The firm pays tax at 35%
and the required return on the project is 12%. What is the NPV?
Q15
Note: This answer assumes that the $3 million initial research costs are sunk and excludes
this from the NPV calculation. It also assumes that working capital needs begin to accrue in
year 0.
Unit Sales
500
600
1,000
1,000
Revenues
2,000
2,400
4,000
4,000
Costs
Depreciation
Pretax Profit (includes salvage in
Year 5)
Taxes at 35%
Profit after Tax
Depreciation
Cash Flow from Operations
Change in Working Capital
Capital Investment
Net Cash Flows
Discount Factor @ 12%
Present Value
-200
-6,000
-6,200
1.000
-6,200
NPV
-181
750
1,200
900
1,200
1,500
1,200
1,500
1,200
900
1,200
50
18
33
1200
1,233
-40
300
105
195
1200
1,395
-160
1,300
455
845
1200
2,045
0
1,300
455
845
1200
2,045
160
1,193
0.893
1,065
1,235
0.797
985
2,045
0.712
1,456
2,205
0.636
1,401
300
105
195
1200
1395
240
325
1,960
0.567
1111
Q18. Project NPV A widget manufacturer currently produces 200,000 units a year. It buys
widget lids from an outside supplier at a price of $2 a lid. The plant manager believes
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600
2,400
that it would be cheaper to make these lids rather than buy them. Direct production
costs are estimated to be only $1.50 a lid. The necessary machinery would cost
$150,000 and would last 10 years. This investment could be written off for tax
purposes using the seven-year tax depreciation schedule. The plant manager
estimates that the operation would require additional working capital of $30,000 but
argues that this sum can be ignored since it is recoverable at the end of the 10 years.
If the company pays tax at a rate of 35% and the opportunity cost of capital is 15%,
would you support the plant manager's proposal? State clearly any additional
assumptions that you need to make.
A18.
Assume the following:
a. The firm will manufacture widgets for at least 10 years.
b. There will be no inflation or technological change.
c. The 15% cost of capital is appropriate for all cash flows and is a real, after-tax
rate of return.
d. All operating cash flows occur at the end of the year.
e. We cannot ignore incremental working capital costs and recovery.
Note: Since purchasing the lids can be considered a one-year ”project,” the two
projects have a common chain life of 10 years.
Compute NPV for each project as follows:
NPV(purchase) = 
($2  200,000)  (1  0 .35)
  $1,304,880
1.15 t
t 1
10

NPV(make) =  $150,000  $30,000 
10

t 1
($1.50  200,000)  (1  0 .35)
1.15 t
 0.1429 0.2449 0.1749 0.1249
 0.35  $150,000  




1
1.15 2
1.15 3
1.15 4
 1.15
0.0893 0.0893 0.0893 0.0445  $30,000




 $1,118,328
1.15 5
1.15 6
1.15 7
1.15 8 
1.15 10
Thus, the widget manufacturer should make the lids.
Q20. Project NPV Marsha Jones has bought a used Mercedes horse transporter for her
Connecticut estate. It cost $35,000. The object is to save on horse transporter rentals.
Marsha had been renting a transporter every other week for $200 per day plus $1.00
per mile. Most of the trips are 80 or 100 miles in total. Marsha usually gives the driver
a $40 tip. With the new transporter she will only have to pay for diesel fuel and
maintenance, at about $.45 per mile. Insurance costs for Marsha's transporter are
$1,200 per year.
The transporter will probably be worth $15,000 (in real terms) after eight years, when
Marsha's horse Nike will be ready to retire. Is the transporter a positive-NPV
investment? Assume a nominal discount rate of 9% and a 3% forecasted inflation rate.
Marsha's transporter is a personal outlay, not a business or financial investment, so
taxes can be ignored.
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A20.
The table below shows the real cash flows. The NPV is computed using the real rate,
which is computed as follows:
(1 + rnominal) = (1 + rreal)  (1 + inflation rate)
1.09 = (1 + rreal)  (1.03)
rreal = 0.0583 = 5.83%
t=0
t=1
Investment
-35,000.0
Savings
8,580.0
Insurance
-1,200.0
Fuel
1,053.0
Net Cash Flow -35,000.0 6,327.0
NPV (at 5.83%) = $14,087.9
t=2
t=3
t=4
t=5
t=6
t=7
8,580.0
-1,200.0
1,053.0
6,327.0
8,580.0
-1,200.0
1,053.0
6,327.0
8,580.0
-1,200.0
1,053.0
6,327.0
8,580.0
-1,200.0
1,053.0
6,327.0
8,580.0
-1,200.0
1,053.0
6,327.0
8,580.0
-1,200.0
1,053.0
6,327.0
Q24. Equivalent annual cash flows As a result of improvements in product engineering,
United Automation is able to sell one of its two milling machines. Both machines
perform the same function but differ in age. The newer machine could be sold today
for $50,000. Its operating costs are $20,000 a year, but in five years the machine will
require a $20,000 overhaul. Thereafter operating costs will be $30,000 until the
machine is finally sold in year 10 for $5,000.
The older machine could be sold today for $25,000. If it is kept, it will need an
immediate $20,000 overhaul. Thereafter operating costs will be $30,000 a year until
the machine is finally sold in year 5 for $5,000.
Both machines are fully depreciated for tax purposes. The company pays tax at 35%.
Cash flows have been forecasted in real terms. The real cost of capital is 12%. Which
machine should United Automation sell? Explain the assumptions underlying your
answer.
24. In order to solve this problem, we calculate the equivalent annual cost for each of the two
alternatives. (All cash flows are in thousands.)
Alternative 1—Sell the new machine: If we sell the new machine, we receive the cash
flow from the sale, pay taxes on the gain, and pay the costs associated with keeping
the old machine. The present value of this alternative is:
PV1  50  [0 .35(50  0)]  20 

30
30
30
30
30




2
3
4
1.12 1.12
1.12
1.12
1.12 5
5
0.35 (5  0)

 $93.80
5
1.12
1.125
The equivalent annual cost for the five-year period is computed as follows:
PV1 = EAC1  [annuity factor, 5 time periods, 12%]
–93.80 = EAC1  [3.605]
EAC1 = –26.02, or an equivalent annual cost of $26,020
Alternative 2—Sell the old machine: If we sell the old machine, we receive the cash
flow from the sale, pay taxes on the gain, and pay the costs associated with keeping
the new machine. The present value of this alternative is:
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t=8
15,000.0
8,580.0
-1,200.0
1,053.0
21,327.0
20
20
20
20
20




2
3
4
1.12 1.12
1.12
1.12
1.12 5
20
30
30
30
30
30






5
6
7
8
9
1.12
1.12
1.12
1.12
1.12
1.1210
5
0 .35 (5  0)


 $127.51
10
1.12
1.1210
PV2  25  [0.35(25  0)] 
The equivalent annual cost for the 10-year period is computed as follows:
PV2 = EAC2  [annuity factor, 10 time periods, 12%]
–127.51 = EAC2  [5.650]
A Normal Project:
Valuing a new Computer system
Obsolete Technologies is considering the purchase of a new computer system to help handle
its warehouse inventories. The system costs $50,000, is expected to last 4 years, and should
reduce the cost of managing inventories by $22,000 a year. The opportunity cost of capital is
10%. Should Obsolete go ahead?
Answer:
The net present value is
The project has a positive NPV of $19,738. Undertaking it would increase the value of the
firm by that amount.
Investment Timing Problem
Obsolete Technologies is contemplating the purchase of a new computer system. The
proposed investment has a net present value of almost $20,000, so it appears that the cost
savings would easily justify the expense of the system. However, the financial manager is not
persuaded. She reasons that the price of computers is continually falling and therefore
suggests postponing the purchase, arguing that the NPV of the system will be even higher if
the firm waits until the following year. Unfortunately, she has been making the same
argument for 10 years, and the company is steadily losing business to competitors with more
efficient systems. Is there a flaw in her reasoning?
Answer:
TABLE Obsolete Technologies: The gain from purchase of a computer is
rising, but the NPV today is highest if the computer is purchased in year 3
(dollar values in thousands).
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Year of
Purchase
Cost of
Computer
PV
NPV at Year of
NPV
Savings Purchase (r = 10%) Today
0
$50
$70
$20
$20.0
1
45
70
25
22.7
2
40
70
30
24.8
3
36
70
34
25.5
4
33
70
37
25.3
5
31
70
39
24.2
←optimal
purchase date
Long- and Short Lived Equipment --- Equivalent Annual Cost Concept
Low-energy lightbulbs typically cost $3.50, have a life of 9 years, and use about $1.60 of
electricity a year. Conventional lightbulbs are cheaper to buy, for they cost only $.50. On the
other hand, they last only about a year and use about $6.60 of energy. If the discount rate is
5%, which product is cheaper to use?
To answer this question, you need first to convert the initial cost of each bulb to an annual
figure and then to add in the annual energy cost. The following table sets out the
calculations:
Low-Energy
Bulb
Conventional
Bulb
1. Initial cost, $
3.50
0.50
2. Estimated life, years
9
1
3. Annuity factor at 5%
7.1078
.9524
4. Equivalent annual annuity, $,
=(1)/(3)
.49
.52
5. Annual energy cost, $
1.60
6.60
6. Total annual cost, $, = (4) +
(5)
2.09
7.12
Assumption: Energy costs are incurred at the end of each year.
It seems that a low-energy bulb provides an annual saving of about $7.12 – $2.09 = $5.03.
When to Replace an Old Machine
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Our earlier comparison of machines I and J took the life of each machine as fixed. In
practice, the point at which equipment is replaced reflects economics, not physical collapse.
We usually decide when to replace. For example, we usually replace a car not when it finally
breaks down but when it becomes more expensive and troublesome to keep up than a
replacement.
replacement problem:
You are operating an old machine that will last 2 more years before it gives up the ghost. It
costs $12,000 per year to operate. You can replace it now with a new machine that costs
$25,000 but is much more efficient (only $8,000 per year in operating costs) and will last for 5
years. Should you replace the machine now or stick with it for a while longer? The
opportunity cost of capital is 6%.
Costs (thousands of dollars)
Year:
New machine
Equivalent annual
annuity
0
25
1
8
2
8
3
8
4
8
5
8
13.93 13.93 13.93 13.93 13.93
PV at
6%
$58.70
58.70
The cash flows of the new machine are equivalent to an annuity of $13,930 per year. So we
can equally well ask whether you would want to replace your old machine, which costs
$12,000 a year to run, with a new one costing $13,930 a year.
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CAsh Flow Estimation, Capital Budgeting Decisions
Idea: Use the incremental cash flows only.
Self-Test
A firm is considering an investment in a new manufacturing plant. The site already is
owned by the company, but existing buildings would need to be demolished. Which of
the following should be treated as incremental cash flows?
a.
The market value of the site.
b. The market value of the existing buildings.
c. Demolition costs and site clearance.
d. The cost of a new access road put in last year.
e. Lost cash flows on other projects due to executive time spent on the new facility.
f. Future depreciation of the new plant.
Answer:
a,b. The site and buildings could have been sold or put to another use.
Their values are opportunity costs, which should be treated as
incremental cash outflows.
c.
Demolition costs are incremental cash outflows.
d.
The cost of the access road is sunk and not incremental.
e.
Lost cash flows from other projects are incremental cash outflows.
f.
Depreciation is not a cash expense and should not be included,
except as it affects taxes. (Taxes are discussed later in this chapter.)
Discount Nominal Cash Flows by the Nominal Cost of Capital
It should go without saying that you cannot mix and match real and nominal
quantities. Real cash flows must be discounted at a real discount rate, nominal cash
flows at a nominal rate. Discounting real cash flows at a nominal rate is a big mistake.
Separate Investment and Financing Decisions
Suppose you finance a project partly with debt. How should you treat the proceeds from the
debt issue and the interest and principal payments on the debt? The probably surprising
answer: Regardless of the actual financing, you should view the project as if it were all-
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equity-financed, treating all cash outflows required for the project as coming from
stockholders and all cash inflows as going to them
Project Cash Flows
It is helpful to think of a project's cash flow as composed of three elements:
Discussion points: Change in working capital, salvage value, etc
An Example:
As the newly appointed financial manager of Blooper Industries, you are about to analyse a
proposal for mining and selling a small deposit of high-grade magnesium ore.6 You are
given the forecasts shown in the spreadsheet.
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Financial projections for Blooper's magnesium mine (dollar values in thousands)
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We can now see how Blooper arrives at its forecast of working capital:
0
1. Receivables (2/12 ×
revenues)
1
2
3
4
5
6
$0 $2,500 $2,625 $2,756 $2,894 $3,039
0
2. Inventories (.15 ×
1,500
following year's expenses)
1,575
1,654
1,736
1,823
0
0
3. Working capital (1 + 2)
4,075
4,279
4,493
4,717
3,039
0
1,500
Note: Columns may not sum due to rounding.
Project Evaluation.
PC Shopping Network may upgrade its modem pool. It last upgraded 2 years ago, when it
spent $115 million on equipment with an assumed life of 5 years and an assumed salvage
value of $15 million for tax purposes. The firm uses straight-line depreciation. The old
equipment can be sold today for $80 million. A new modem pool can be installed today for
$150 million. This will have a 3-year life and will be depreciated to zero using straight-line
depreciation. The new equipment will enable the firm to increase sales by $25 million per
year and decrease operating costs by $10 million per year. At the end of 3 years, the new
equipment will be worthless. Assume the firm's tax rate is 35% and the discount rate for
projects of this sort is 10%. (LO9-2)
a.
Page 294
What is the net cash flow at time 0 if the old equipment is replaced?
b. What are the incremental cash flows in years 1, 2, and 3?
c. What are the NPV and IRR of the replacement project?
Answer:
2.
a. Annual depreciation is ($115  $15)/5 = $20 million.
Book value at the time of sale is $115  (2  $20) = $75 million.
Sales price = $80 million, so net-of-tax proceeds from the sale are:
$80  (0.35  $5) = $78.25 million
Therefore, the net cash outlay at time 0 is $150  $78.25 = $71.75 million.
b.
The project saves $10 million in operating costs and increases sales by $25
million. Depreciation expense for the new machine would be $50 million
per year. Therefore, including the depreciation tax shield, operating cash
flow increases by:
($25 + $10)  (1  0.35) + ($50  0.35) = $40.25 million per year
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c.
NPV = $71.75 + [$40.25  annuity factor (10%, 3 years)]
 1

1
= – $71.75  $40.25  

 $28.35, or $28.35 million
3
 0.10 0.10  (1.10) 
To find the internal rate of return, set the PV of the annuity to $71.75 and
solve for the discount rate (r):
1

1
$40.25   
 $71.75  r  IRR  31.33%
3
 r r  (1  r ) 
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Ch 10
Sensitivity Analysis
Sensitivity Analysis. Emperor's Clothes Fashions can invest $5 million in a new plant for
producing invisible makeup. The plant has an expected life of 5 years, and expected sales
are 6 million jars of makeup a year. Fixed costs are $2 million a year, and variable costs are
$1 per jar. The product will be priced at $2 per jar. The plant will be depreciated straight-line
over 5 years to a salvage value of zero. The opportunity cost of capital is 10%, and the tax
rate is 40%. (LO10-2)
a. What is project NPV under these base-case assumptions?
b. What is NPV if variable costs turn out to be $1.20 per jar?
c. What is NPV if fixed costs turn out to be $1.5 million per year?
d. At what price per jar would project NPV equal zero?
3.
Revenue = price  quantity = $2  6 million = $12 million
Expense = variable cost + fixed cost = ($1  6 million) + $2 million = $8 million
Depreciation expense = $5 million/5 years = $1 million per year
Cash flow = (1  T)  (revenue – expenses) + (T  depreciation)
= [0.60  ($12 million – $8 million)] + (0.4  $1 million) = $2.8 million
a.
NPV = –$5 million + [$2.8 million  annuity factor (10%, 5 years)]
 1

1

 $5.61 million
= –$5 million + $2.8 million  
5
 0.10 0.10  (1.10) 
b.
If variable cost = $1.20, then expenses increase to:
($1.20  6 million) + $2 million = $9.2 million
CF = [0.60  ($12 million – $9.2 million)] + (0.4  $1 million) = $2.08 million
NPV = –$5 million + [$2.08 million  annuity factor (10%, 5 years)]
 1

1

 $2.88 million
= –$5 million + $2.08 million  
5
 0.10 0.10  (1.10) 
c.
If fixed costs = $1.5 million, expenses fall to:
($1  6 million) + $1.5 million = $7.5 million
Cash flow = [0.60  ($12 million – $7.5 million)] + (0.4  $1 million) = $3.1 million
NPV = –$5 million + [$3.1 million  annuity factor (10%, 5 years)]
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 1

1
= –$5 million + $3.1 million  

 $6.75 million
5
 0.10 0.10  (1.10) 
d.
Call P the price per jar. Then:
Revenue = P  6 million
Expense = ($1  6 million) + $2 million = $8 million
Cash flow = [(1 – 0.40)  (6P – 8)] + (0.40  1) = 3.6P – 4.4
NPV = –5 + [(3.6P – 4.4)  annuity factor (10%, 5 years)]
 1

1
= –5 + [(3.6P – 4.4)  

5
 0.10 0.10  (1.10) 
= –5 + [(3.6P – 4.4)  3.7908] = –21.6795 + 13.6469P = 0  P = $1.59 per
jar
Scenario Analysis.
The most likely outcomes for a particular project are estimated as follows:

Unit price: $50

Variable cost: $30

Fixed cost: $300,000

Expected sales: 30,000 units per year
However, you recognize that some of these estimates are subject to error. Suppose that
each variable may turn out to be either 10% higher or 10% lower than the initial estimate.
The project will last for 10 years and requires an initial investment of $1 million, which will be
depreciated straight-line over the project life to a final value of zero. The firm's tax rate is
35%, and the required rate of return is 12%. (LO10-2)
a. What is project NPV in the best-case scenario, that is, assuming all variables take on the
best possible value?
b. What about the worst-case scenario?
Price
Variable cost
Fixed cost
Sales
Most Likely
$50
$30
$300,000
30,000 units
Best Case
$55
$27
$270,000
33,000 units
Worst Case
$45
$33
$330,000
27,000 units
Cash flow = [(1 – T)  (revenue – cash expenses)] + (T  depreciation)
Depreciation expense = $1 million/10 years = $100,000 per year
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Best-case CF = 0.65  [33,000  ($55 – $27) – $270,000] + (0.35  $100,000) =
$460,100
Worst-case CF = 0.65  [27,000  ($45 – $33) – $330,000] + (0.35  $100,000) =
$31,100
 1

1
12%, 10-year annuity factor = 

 5.65022
10 
 0.12 0.12  (1.12) 
Best-case NPV = (5.65022  $460,100) – $1,000,000 = $1,599,666
Worst-case NPV = (5.65022  $31,100) – $1,000,000 = –$824,278
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CH 7 Introducton to Risk and Return
Q5 Diversification In which of the following situations would you get the largest reduction in risk
by spreading your investment across two stocks?
a. The two shares are perfectly correlated.
b. There is no correlation.
c. There is modest negative correlation.
d. There is perfect negative correlation.
5.
(d)
This strategy does the most to reduce risks because the stocks move in opposite
directions. When one goes up, the other goes down, and vice versa. This does the most to reduce risk
in a portfolio.
Q6 Portfolio risk To calculate the variance of a three-stock portfolio, you need to add nine boxes:
Use the same symbols that we used in this chapter; for example, x1 = proportion invested in stock
1 and σ12 = covariance between stocks 1 and 2. Now complete the nine boxes.
Q7 Portfolio risk Suppose the standard deviation of the market return is 20%.
e. What is the standard deviation of returns on a well-diversified portfolio with a beta of
1.3?
f. What is the standard deviation of returns on a well-diversified portfolio with a beta of
0?
g. A well-diversified portfolio has a standard deviation of 15%. What is its beta?
h. A poorly diversified portfolio has a standard deviation of 20%. What can you say about
its beta?
7.
a.
26%
b.
Zero
c.
.75
d.
Less than 1.0 (the portfolio’s risk is the same as the
market, but some of this risk is unique risk)
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Q8 Portfolio beta A portfolio contains equal investments in 10 stocks. Five have a beta
of 1.2; the remainder have a beta of 1.4. What is the portfolio beta?
i. 1.3.
j. Greater than 1.3 because the portfolio is not completely diversified.
k. Less than 1.3 because diversification reduces beta.
8.
a. 1.3 (Diversification does not affect market risk.) This can be
found by finding the average of all of the betas: (5 x 1.4) + (5 x 1.2) / 10 = 1.3.
Q13 Risk and diversification Lonesome Gulch Mines has a standard deviation of 42%
per year and a beta of + .10. Amalgamated Copper has a standard deviation of 31%
a year and a beta of + .66. Explain why Lonesome Gulch is the safer investment for a
diversified investor.
A13. In the context of a well-diversified portfolio, the only risk characteristic of a
single security that matters is the security’s contribution to the overall portfolio risk.
This contribution is measured by beta. Lonesome Gulch is the safer investment for a
diversified investor because its beta (+0.10) is lower than the beta of Amalgamated
Copper (+0.66). For a diversified investor, the standard deviations are irrelevant.
CH8
CAPM
9. True/false True or false? Explain or qualify as necessary.
a. Investors demand higher expected rates of return on stocks with more variable
rates of return.
b. The CAPM predicts that a security with a beta of 0 will offer a zero expected
return.
c. An investor who puts $10,000 in Treasury bills and $20,000 in the market portfolio
will have a beta of 2.0.
d. Investors demand higher expected rates of return from stocks with returns that are
highly exposed to macroeconomic risks.
e. Investors demand higher expected rates of return from stocks with returns that are
very sensitive to fluctuations in the stock market.
A9 9.
a.
False. investors demand higher expected rates of return on stocks with
more nondiversifiable risk.
b.
False. a security with a beta of zero will offer the risk-free rate of return.
c.
False. Treasury bills have a beta of 0 and the market has a beta of 1.
Therefore, with 1/3 of the investor’s money in T bills and 2/3 of his or
her money in the market, the beta will be: (1/3  0) + (2/3  1) = 0.67.
d.
True
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e.
True
15. CAPM The Treasury bill rate is 4%, and the expected return on the market
portfolio is 12%. Using the capital asset pricing model:
a.
b.
c.
d.
Draw a graph similar to Figure 8.6 showing how the expected return varies with beta.
What is the risk premium on the market?
What is the required return on an investment with a beta of 1.5?
If an investment with a beta of .8 offers an expected return of 9.8%, does it have a positive
NPV?
e. If the market expects a return of 11.2% from stock X, what is its beta?
a.
Expected Return
15.
0
0
0.5
1
1.5
2
Beta
b.
Market risk premium = rm – rf = 0.12 – 0.04 = 0.08 = 8.0%.
c.
Use the security market line:
r = rf + (rm – rf)
r = 0.04 + [1.5  (0.12 – 0.04)] = 0.16 = 16.0%
d.
For any investment, we can find the opportunity cost of capital using the
security market line. With  = 0.8, the opportunity cost of capital is:
r = rf + (rm – rf)
r = 0.04 + [0.8  (0.12 – 0.04)] = 0.104 = 10.4%
The opportunity cost of capital is 10.4% and the investment is expected
to earn 9.8%. Therefore, the investment has a negative NPV.
17. Cost of capital Epsilon Corp. is evaluating an expansion of its business. The cash-flow
forecasts for the project are as follows:
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Page 214
The firm's existing assets have a beta of 1.4. The risk-free interest rate is 4% and the
expected return on the market portfolio is 12%. What is the project's NPV?
17.
First calculate the required rate of return (assuming the expansion assets bear
the same level of risk as historical assets):
r = rf + (rm – rf)
r = 0.04 + [1.4  (0.12 – 0.04)] = 0.152 = 15.2%
The use this to discount future cash flows; NPV = -25.29
Year
0
1
2
3
4
5
6
7
8
9
10
Cash Flow
-100
15
15
15
15
15
15
15
15
15
15
Discount
Factor
1
0.868
0.754
0.654
0.568
0.493
0.428
0.371
0.322
0.280
0.243
PV
-100.00
13.02
11.30
9.81
8.52
7.39
6.42
5.57
4.84
4.20
3.64
NPV
-25.29
Ch 9
COST OF CAPITAL
Think for a moment what the cost of capital for a project means. It is the rate of return that
shareholders could expect to earn if they invested in equally risky securities. So one way to
estimate the cost of capital is to find securities that have the same risk as the project and
then estimate the expected rate of return on these securities.
11. Cost of capital The total market value of the common stock of the Okefenokee Real
Estate Company is $6 million, and the total value of its debt is $4 million. The treasurer
estimates that the beta of the stock is currently 1.5 and that the expected risk premium
on the market is 6%. The Treasury bill rate is 4%. Assume for simplicity that Okefenokee
debt is risk-free and the company does not pay tax.
a. What is the required return on Okefenokee stock?
b. Estimate the company cost of capital.
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c. What is the discount rate for an expansion of the company's present business?
d. Suppose the company wants to diversify into the manufacture of rose-colored
spectacles. The beta of unleveraged optical manufacturers is 1.2. Estimate the
required return on Okefenokee's new venture.
11. a.
b.
requity = rf +   (rm – rf) = 0.04 + (1.5  0.06) = 0.13 = 13%.
rassets 
D
E
 $4 million
  $6 million

rdebt  requity  
 0.04   
 0.13  .
V
V
$10
million
$10
million

 

rassets = 0.094 = 9.4%.
c.
The cost of capital depends on the risk of the project being evaluated. If the risk of the
project is similar to the risk of the other assets of the company, then the appropriate
rate of return is the company cost of capital. Here, the appropriate discount rate is
9.4%.
d.
requity = rf +   (rm – rf) = 0.04 + (1.2  0.06) = 0.112 = 11.2%.
rassets 
D
E
 $4 million
  $6 million

rdebt  requity  
 0.04   
 0.112  .
V
V
 $10 million
  $10 million

rassets = 0.0832 = 8.32%.
14. Company cost of capital
Financial:
You are given the following information for Golden Fleece
Calculate Golden Fleece's company cost of capital. Ignore taxes.
14.
The total market value of outstanding debt is $300,000. The cost of debt capital is
8%. For the common stock, the outstanding market value is:
$50  10,000 = $500,000. The cost of equity capital is 15%. Thus, Golden Fleece’s
company cost of capital is:




300,000
500,000
  0.08  
  0.15
rassets  
 300,000  500,000 
 300,000  500,000 
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rassets = 0.124 = 12.4%
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