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Answers and Solutions: 11 - 1
Chapter 11
Cash Flow Estimation
and Risk Analysis
ANSWERS TO END-OF-CHAPTER QUESTIONS
11-1 a. Project cash flow, which is the relevant cash flow for project analysis, represents the
actual flow of cash, which includes investments in capital and working capital, but does
not include interest expenses or noncash charges like depreciation (except to the extent
that depreciation affects taxes). In other words, project cash flow is the free cash flow
generated by the project. Accounting income, on the other hand, reports accounting
data as defined by Generally Accepted Accounting Principles (GAAP).
c. Net operating working capital changes are the increases in current operating assets
resulting from accepting a project less the resulting increases in current operating
liabilities, or accruals and accounts payable. A net operating working capital change
must be financed just as a firm must finance its increases in fixed assets. Salvage value
is the market value of an asset after its useful life. Salvage values and their tax effects
must be included in project cash flow estimation.
Answers and Solutions: 11 - 2
d. Stand-alone risk is the risk a project would have if it was held in isolation. Corporate
(within-firm) risk is the risk that a project contributes to a company after taking into
consideration the cash flows of the company’s other projects; because projects are not
perfectly correlated, corporate risk usually will be less than stand-alone risk. Market
(beta) risk is the risk that a company contributes to a well diversified portfolio.
f. A risk-adjusted discount rate incorporates the risk of the project’s cash flows. The cost
of capital to the firm reflects the average risk of the firm’s existing projects. Thus, new
projects that are riskier than existing projects should have a higher risk-adjusted
discount rate. Conversely, projects with less risk should have a lower risk-adjusted
discount rate. This adjustment process also applies to a firm’s divisions. Risk
differences are difficult to quantify, thus risk adjustments are often subjective in nature.
A project’s cost of capital is its risk-adjusted discount rate for that project.
with strategic issues. Finally, they are also called embedded options because they are a
part of another project.
i. Investment timing options give companies the option to delay a project rather than
implement it immediately. This option to wait allows a company to reduce the
11-2 Only cash can be spent or reinvested, and since accounting profits do not represent cash,
they are of less fundamental importance than cash flows for investment analysis. Recall
that in the stock valuation chapters we focused on dividends and free cash flows, which
represent cash flows, rather than on earnings per share, which represent accounting profits.
11-3 Since the cost of capital includes a premium for expected inflation, failure to adjust cash
flows means that the denominator, but not the numerator, rises with inflation, and this
lowers the calculated NPV.
11-4 Capital budgeting analysis should only include those cash flows which will be affected by
the decision. Sunk costs are unrecoverable and cannot be changed, so they have no bearing
11-5 When a firm takes on a new capital budgeting project, it typically must increase its
investment in receivables and inventories, over and above the increase in payables and
Answers and Solutions: 11 - 4
11-6 Scenario analysis analyzes a limited number of outcomes. Although the base case scenario
may be the most likely, or expected outcome, the bad and good scenarios are frequently
worst case and best case scenarios, that is, when everything goes bad together, or
11-7 The costs associated with financing are reflected in the weighted average cost of capital.
To include interest expense in the capital budgeting analysis would “double count” the
cost of debt financing.
11-8 Daily cash flows would be theoretically best, but they would be costly to estimate and
probably no more accurate than annual estimates because we simply cannot forecast
11-9 In replacement projects, the benefits are generally cost savings, although the new
machinery may also permit additional output. The data for replacement analysis are
generally easier to obtain than for new products, but the analysis itself is somewhat more
complicated because almost all of the cash flows are incremental, found by subtracting
the new cost numbers from the old numbers. Similarly, differences in depreciation and
any other factor that affects cash flows must also be determined.
Answers and Solutions: 11 - 5
11-10 Stand-alone risk is the project’s risk if it is held as a lone asset. It disregards the fact that
it is but one asset within the firm’s portfolio of assets and that the firm is but one stock in
a typical investor’s portfolio of stocks. Stand-alone risk is measured by the variability of
11-11 It is often difficult to quantify market risk. On the other hand, we can usually get a good
idea of a project’s stand-alone risk, and that risk is normally correlated with market risk:
The higher the stand-alone risk, the higher the market risk is likely to be. Therefore,
firms tend to focus on stand-alone risk, then deal with corporate and market risk by
making subjective, judgmental modifications to the calculated stand-alone risk.
Answers and Solutions: 11 - 6
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
11-1 a. Equipment $ 17,000,000
NWC Investment 5,000,000
11-2 Operating Cash Flows: t = 1
Sales revenues $18,000,000
Operating costs 9,000,000
11-3 Equipment's original cost $12,000,000
Depreciation (80%) 9,000,000
Book value $ 3,000,000
Answers and Solutions: 11 - 7
11-4 Cash outflow = $40,000.
Increase in annual after-tax cash flows: CF = $9,000.
11-5 a. The MACRS rates are 33.33%, 44.45%, 14.81%, and 7.41%. The first MACRS
depreciation expense is 33.33%($1,700,000) = $566,610. The others are calculated
similarly. The applicable depreciation values are as follows for the two scenarios:
Scenario 1 Scenario 2
Year (Straight Line) (MACRS)
1 $425,000 $566,610
2 425,000 755,650
3 425,000 251,770
4 425,000 125,970
b. To find the difference in net present values under these two methods, we must
determine the difference in incremental cash flows each method provides. The
depreciation expenses cannot simply be subtracted from each other, as there are tax
Answers and Solutions: 11 - 8
cash flows that represent the benefit from depreciation expense and solve for net
present value based upon a WACC of 10%.
CF0 = 0; CF1 = 56644; CF2 = 132260; CF3 = -69292; CF4 = -119612; and I/YR = 10.
Solve for NPV = $27,043.62
So, all else equal the use of the accelerated depreciation method will result in a higher
NPV (by $27,043.62) than would the use of a straight-line depreciation method.
11-6 a. The net cost is $1,118,000:
Price ($1,080,000)
Modification (22,500)
Increase in NWC (15,500)
Cash outlay for new machine ($1,118,000)
Answers and Solutions: 11 - 9
2. The depreciation expense in each year is the depreciable basis, $1,102,500, times
the MACRS allowance percentages of 0.3333, 0.4445, and 0.1481 for Years 1, 2,
and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $367,463,
$490,061, and $163,280. The depreciation tax savings is calculated as the tax rate
(35%) times the depreciation expense in each year.
c. The terminal year cash flow is $473,343:
d. The project has an NPV of $78,790; thus, it should be accepted.
Year Net Cash Flow PV @ 12%
0 ($1,118,000) ($1,118,000)
11-7 a. The net cost is $89,000:
Price ($70,000)
Modification (15,000)
Change in NWC (4,000)
($89,000)
Answers and Solutions: 11 - 10
b. The operating cash flows follow:
Year 1 Year 2 Year 3
After-tax savings $15,000 $15,000 $15,000
c. The additional end-of-project cash flow is $24,519:
Salvage value $30,000
Tax on SV* (9,481)
d. The project has an NPV of -$6,700. Thus, it should not be accepted.
Year Net Cash Flow
0 ($89,000)
1 26,332
2 30,113
3 44,555
With a financial calculator, input the following: CF0 = -89000, CF1 = 26332, CF2 =
Answers and Solutions: 11 - 11
11-8 a. Sales = 1,000($138) $138,000
Cost = 1,000($105) 105,000
Net before tax $ 33,000
Taxes (34%) 11,220
Net after tax $ 21,780
Not considering inflation, NPV is -$4,800. This value is calculated as
After adjusting for expected inflation, we see that the project has a positive NPV and
should be accepted. This demonstrates the bias that inflation can induce into the
capital budgeting process: Inflation is already reflected in the denominator (the cost
of capital), so it must also be reflected in the numerator.
A more straightforward way to calculate the present value without having to calculate
a real required rate of return is to use the constant growth formula, instead. Here, the
present value of all of the future cash flows is:
Answers and Solutions: 11 - 12
b. If part of the costs were fixed, and hence did not rise with inflation, then sales
revenues would rise faster than total costs. However, when the plant wears out and
11-9 First determine the net cash flow at t = 0:
Purchase price ($12,000)
Sale of old machine 4,150
Tax on sale of old machine (240)a
Change in net working capital (2,200)b
Total investment ($10,290)
Answers and Solutions: 11 - 13
Depreciation:
Year 1 2 3 4 5 6
Newa $2,400 $8,840 $2,304 $1,382 $1,382 $691
Old 650 650 650 650 650 325
Change $1,750 $3,190 $1,654 $732 $732 $366
Depreciation tax savingsb $ 700 $ 1,276 $ 662 $293 $293 $ 146
Finally, place all the cash flows on a time line:
0 1 2 3 4 5 6
| | | | | | |
Net investment (10290)
After-tax revenue increase 2,340 2,340 2,340 2,340 2,340 2,340
Depreciation tax savings 700 1,276 662 293 293 146
Working capital recovery 2,200
15%
Answers and Solutions: 11 - 14
11-10 1. Net investment at t = 0:
Cost of new machine $182,500
Net investment outlay (CF0) $182,500
2. After-tax
Year Earnings T(Dep) Annual CFt
1 $28,200 $ 14,600 $42,800
2 28,200 23,360 51,560
Notes:
a. The after-tax earnings are $47,000(1 – T) = $47,000(0.6) = $28,200.
b. Find Dep over Years 1-8:
The old machine was fully depreciated; therefore, Dep = Depreciation on the new machine.
Dep Dep
Year Rate Basis Depreciation
1 0.2000 $182,500 $36,500
3. Now find the NPV of the replacement machine:
Place the cash flows on a time line:
0 1 2 3 4 5 6 7 8
Answers and Solutions: 11 - 15
11-11 E(NPV) = 0.05(-$70) + 0.20(-$25) + 0.50($12) + 0.20($20) + 0.05($30)
= -$3.5 + -$5.0 + $6.0 + $4.0 + $1.5
= $3.0 million.
Answers and Solutions: 11 - 16
11-12 a.
0
1
2
3
4
5
Machine cost
(350,000)
Net working
capital
(35,000)
Return of NWC
35,000
Sale of machine
33,000
Tax on sale
(13,200)
Total CF
(385,000)
112,662
128,230
86,734
76,374
120,800
NPV
15,732
Notes:
a Depreciation Schedule, Basis = $250,000
MACRS Rate
Basis =
Year Beg. Bk. Value MACRS Rate Depreciation Ending BV
Answers and Solutions: 11 - 17
b. If savings increase by 20%, then savings will be (1.2)($110,000) = $132,000.
If savings decrease by 20%, then savings will be (0.8)($110,000) = $88,000.
(1) Savings increase by 20%:
0
1
2
3
4
5
Machine cost
(350,000)
Net working capital
(35,000)
Return of NWC
35,000
Sale of machine
33,000
Tax on sale
(13,200)
Total CF
(385,000)
125,862
141,430
99,934
89,574
134,000
Answers and Solutions: 11 - 18
(2) Savings decrease by 20%:
0
1
2
3
4
5
Machine cost
(350,000)
Net working capital
(35,000)
Return of NWC
35,000
Sale of machine
33,000
Tax on sale
(13,200)
Total CF
(385,000)
99,462
115,030
73,534
63,174
107,600
NPV
(34,307)
c. Worst-case scenario:
0
1
2
3
4
5
Machine cost
(350,000)
Net working capital
(40,000)
Return of NWC
40,000
Sale of machine
28,000
Tax on sale
(11,200)
Total CF
(390,000)
99,462
115,030
73,534
63,174
109,600
NPV
(38,065)
IRR
5.99%
Answers and Solutions: 11 - 20
Best-case scenario:
0
1
2
3
4
5
Machine cost
(350,000)
Net working capital
(30,000)
Return of NWC
30,000
Sale of machine
38,000
Tax on sale
(15,200)
Total CF
(380,000)
125,862
141,430
99,934
89,574
132,000
NPV
69,528
IRR
17.15%
MIRR
13.76%
Cumulative CF
(380,000)
(254,138)
(112,708)
(12,774)
76,800
208,800
Payback
3.14
Answers and Solutions: 11 - 21
11-13 a. Old depreciation = $5,500 per year.
Book value = $55,000 – 5($5,500) = $27,500.
Gain = $35,000 – $27,500 = $7,500.
Basis
120,000
New depreciation
39,996
53,340
17,772
8,892
-
11-14 a. Cost of new machine ($775,000)
Salvage value, old 135,000
Savings due to loss on sale ($450,000 – $135,000) 0.35 110,250
Cash outlay for new machine ($ 529,750)
MACRS Rate
33.33%
44.45%
14.81%
7.41%
0.00%
Answers and Solutions: 11 - 22
shield
c. CFt = (Operating expenses)(1 – T) + (Depreciation)(T).
0
1
2
3
4
5
After tax cost savings
120,250
120,250
120,250
120,250
120,250
Incremental Depreciation tax
22,750
55,300
20,580
(252)
(252)
e. 1. If the expected life of the old machine decreases, the new machine will look better
as cash flows attributable to the new machine would increase. On the other hand, a
serious complication arises: the two projects now have unequal lives, and an
estimate must be made about the action to be taken when the old machine is
scrapped. Will it be replaced, and at what cost and with what savings?
2. The higher capital cost should be used in the analysis.
Answers and Solutions: 11 - 23
11-15 a. Expected annual cash flows:
Project A: Probable
Probability Cash Flow = Cash Flow
0.2 $6,000 $1,200
Coefficient of variation:
CV =
Project A:
σA =
NPV Expected
=
valueExpected
deviation Standard NPV
$474.34. = (0.2)
)
($750 + (0.6)
)
($0 + (0.2)
)
(-$750 222
Answers and Solutions: 11 - 24
b. Project B is the riskier project because it has the greater variability in its probable
cash flows, whether measured by the standard deviation or the coefficient of
variation. Hence, Project B is evaluated at the 12 percent cost of capital, while
Project A requires only a 10 percent cost of capital.
11-16 a. First, note that with symmetric probability distributions, the middle value of each
distribution is the expected value. Therefore,
Expected Values
Sales (units) 200
Sales price $13,500
Using a financial calculator, input the following: CF0 = -4000000, CF1 = 900000, and
Nj = 8, to solve for IRR = 15.29%.
Expected IRR = 15.29% ≈ 15.3%.
=+
1t t
Assuming complete independence between the distributions, and normality, it would
be possible to derive σIRR statistically. Alternatively, we could employ simulation to
develop a distribution of IRRs, hence σIRR. There is no easy way to get σIRR.
life should be used.
(2) a. Estimate unit sales. The 16 indicates sales of 100 units.
b. Estimate the sales price. The 58 indicates a sales price of $13,500.
(4) Repeat the process for Year 3. Sales will be 100 units with a random number of
19; the price will be $13,500 with a random number of 62; and the cost will be
$5,000 with a random number of 6:
[100($13,500) - 100($5,000)](0.6) = $510,000 = CF3.
Answers and Solutions: 11 - 26
4000000, CF1 = 510000, CF2 = 780000, CF3 = 510000, and solve for IRR =
-31.55%.
b. NPV = - $4,000,000.
With a financial calculator, input the following: CF0 = -4000000, CF1 =
321 )15.1(
000,510$
)15.1(
000,780$
)15.1(
000,510$++
Answers and Solutions: 11 - 27
(6) & (7) The computer would store NPVs and IRRs for the different trials, then
display them as frequency distributions:
Probability
of occurrence
X
XX
XXXX
XXXXXXXX
XXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
Answers and Solutions: 11 - 28
11-17 a. The resulting decision tree is:
NPV
t = 0 t = 1 t = 2 t = 3 P NPV Product
$3,000,000 0.24 $881,718 $211,612
($1,000,000) P = 0.5
P = 0.80 1,500,000 0.24 (185,952) (44,628)
The NPV of the top path is:
- - - $10,000 = $881,718.
3
)12.1(
000,000,3$
2
)12.1(
000,000,1$
1
)12.1(
000,500$
b. σ2NPV = 0.24($881,718 - $117,779)2 + 0.24(-$185,952 - $117,779)2
+ 0.12(-$376,709 - $117,779)2 + 0.4(-$10,000 - $117,779)2
= 198,078,470,853.
Answers and Solutions: 11 - 30
SOLUTION TO SPREADSHEET PROBLEM
11-18 The detailed solution for the problem is available in the file Ch 11 P18 Build a Model
Solution.xls at the textbook’s Web site.
Mini Case: 11 - 31
MINI CASE
Shrieves Casting Company is considering adding a new line to its product mix, and the
capital budgeting analysis is being conducted by Sidney Johnson, a recently graduated MBA.
The production line would be set up in unused space in Shrieves’ main plant. The
machinery’s invoice price would be approximately $200,000, another $10,000 in shipping
charges would be required, and it would cost an additional $30,000 to install the equipment.
The machinery has an economic life of 4 years, and Shrieves has obtained a special tax ruling
that places the equipment in the MACRS 3-year class. The machinery is expected to have a
salvage value of $25,000 after 4 years of use.
The new line would generate incremental sales of 1,250 units per year for 4 years at an
incremental cost of $100 per unit in the first year, excluding depreciation. Each unit can be
sold for $200 in the first year. The sales price and cost are expected to increase by 3% per
year due to inflation. Further, to handle the new line, the firm’s net working capital would
have to increase by an amount equal to 12% of sales revenues. The firm’s tax rate is 40%,
and its overall weighted average cost of capital is 10%.
a. Define “incremental cash flow.”
a. 1. Should you subtract interest expense or dividends when calculating project cash
flow?
Answer: The cash flow statement should not include interest expense or dividends. The return
a. 2. Suppose the firm had spent $100,000 last year to rehabilitate the production line
site. Should this cost be included in the analysis? Explain.
Answer: The $100,000 cost to rehabilitate the production line site was incurred last year, and
Mini Case: 11 - 32
a. 3. Now assume that the plant space could be leased out to another firm at $25,000
per year. Should this be included in the analysis? If so, how?
Answer: If the plant space could be leased out to another firm, then if Shrieves accepts this
a. 4. Finally, assume that the new product line is expected to decrease sales of the
firm’s other lines by $50,000 per year. Should this be considered in the analysis?
If so, how?
Answer: If a project affects the cash flows of another project, this is an “externality” that must be
b. Disregard the assumptions in part a. What is Shrieves’ depreciable basis? What
are the annual depreciation expenses?
Answer: The asset’s depreciable basis includes shipping and installation costs. Thus, the asset’s
Mini Case: 11 - 33
c. Calculate the annual sales revenues and costs (other than depreciation). Why is it
important to include inflation when estimating cash flows?
Answer: With an inflation rate of 3%, the annual revenues and costs are:
d. Construct annual incremental operating cash flow statements.
Answer:
Year 1
Year 2
Year 3
Year 4
Sales
$250,000
$257,500
$265,225
$273,188
Mini Case: 11 - 34
e. Estimate the required net working capital for each year, and the cash flow due to
investments in net working capital.
Answer: The project requires a level of net working capital in the amount equal to 12% of the
next year’s sales. Any increase in NWC is a negative cash flow, and any decrease is a
f. Calculate the after-tax salvage cash flow.
Answer: When the project is terminated at the end of Year 4, the equipment can be sold for
g. Calculate the net cash flows for each year. Based on these cash flows, what are the
project’s NPV, IRR, MIRR, PI, payback, and discounted payback? Do these
indicators suggest the project should be undertaken?
Answer: The net cash flows are:
Year 0
Year 1
Year 2
Year 3
Year 4
Mini Case: 11 - 35
h. What does the term “risk” mean in the context of capital budgeting; to what
extent can risk be quantified; and when risk is quantified, is the quantification
based primarily on statistical analysis of historical data or on subjective,
judgmental estimates?
Answer: Risk throughout finance relates to uncertainty about future events, and in capital
budgeting, this means the future profitability of a project. For certain types of projects,
Mini Case: 11 - 36
i. 1. What are the three types of risk that are relevant in capital budgeting?
2. How is each of these risk types measured, and how do they relate to one another?
Answer: Here are the three types of project risk:
• Stand-alone risk is the project’s total risk if it were operated independently. Stand-
i. 3. How is each type of risk used in the capital budgeting process?
Answer: Because management’s primary goal is shareholder wealth maximization, the most
Mini Case: 11 - 37
j. 1. What is sensitivity analysis?
Answer: Sensitivity analysis measures the effect of changes in a particular variable, say
revenues, on a project’s NPV. To perform a sensitivity analysis, all variables are fixed
j. 2. Perform a sensitivity analysis on the unit sales, salvage value, and cost of capital
for the project. Assume each of these variables can vary from its base-case, or
expected, value by 10%, 20%, and 30%. Include a sensitivity diagram, and
discuss the results.
Answer: The sensitivity data are given here in tabular form:
Mini Case: 11 - 38
A. The sensitivity lines intersect at 0% change and the base-case NPV, $88,030. Since
all other variables are set at their base-case, or expected, values the zero change
situation is the base case and gives the base-case NPV, $88,030.
B. The plots for unit sales and salvage value are upward sloping, indicating that higher
variable values lead to higher NPVs. Conversely, the plot for cost of capital is
downward sloping, because a higher cost of capital leads to a lower NPV.
Mini Case: 11 - 39
j. 3. What is the primary weakness of sensitivity analysis? What is its primary
usefulness?
Answer: The two primary disadvantages of sensitivity analysis are (1) that it does not reflect the
effects of diversification and (2) that it does not incorporate any information about the
k. Assume that Sidney Johnson is confident of her estimates of all the variables that
affect the project’s cash flows except unit sales and sales price. If product
acceptance is poor, unit sales would be only 900 units a year and the unit price
would only be $160; a strong consumer response would produce sales of 1,600
units and a unit price of $240. Sidney believes that there is a 25% chance of poor
acceptance, a 25% chance of excellent acceptance, and a 50% chance of average
acceptance (the base case).
k. 1. What is scenario analysis?
Answer: Scenario analysis examines several possible situations, usually worst case, most likely
Mini Case: 11 - 40
k. 2. What is the worst-case NPV? The best-case NPV?
k. 3. Use the worst-, base-, and best-case NPVs and probabilities of occurrence to find
the project’s expected NPV, standard deviation, and coefficient of variation.
Answer: We used a spreadsheet model to develop the scenarios, which are summarized below:
Scenario
Probability
Unit Sales
Unit Price
NPV
l. Are there problems with scenario analysis? Define simulation analysis, and
discuss its principal advantages and disadvantages.
Answer: Scenario analysis examines several possible scenarios, usually worst case, most likely
case, and best case. Thus, it usually considers only 3 possible outcomes. Obviously
Mini Case: 11 - 41
m. 1. Assume that Shrieves’ average project has a coefficient of variation in the range of
0.2 to 0.4. Would the new line be classified as high risk, average risk, or low risk?
What type of risk is being measured here?
Answer: The project has a CV of 1.15, which is above the average range of 0.2 to 0.4, so it falls
m. 2. Shrieves typically adds or subtracts 3 percentage points to the overall cost of
capital to adjust for risk. Should the new line be accepted?
Answer: Since the project is judged to have above-average risk, its differential risk-adjusted, or
m. 3. Are there any subjective risk factors that should be considered before the final
decision is made?
Answer: A numerical analysis such as this one may not capture all of the risk factors inherent in
Mini Case: 11 - 42
n. What is a real option? What are some types of real options?
Answer: Real options exist when managers can influence the size and risk of a project’s cash
flows by taking different actions during the project’s life in response to changing
