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CHAPTER 9
RISK ANALYSIS, REAL OPTIONS, AND
CAPITAL BUDGETING
Answers to Concept Questions
1. Forecasting risk is the risk that a poor decision is made because of errors in projected cash flows. The
2. With a sensitivity analysis, one variable is examined over a broad range of values. With a scenario
3. It is true that if average revenue is less than average cost, the firm is losing money. This much of the
excess of its marginal cost clearly acts to increase operating cash flow.
4. From the shareholder perspective, the financial break-even point is the most important. A project can
This causes a reduction in shareholder (your) wealth.
5. The project will reach the cash break-even first, the accounting break-even next and finally the
financial break-even. For a project with an initial investment and sales after, this ordering will always
of money, is achieved.
6. Traditional NPV analysis is often too conservative because it ignores profitable options such as the
ability to expand the project if it is profitable, or abandon the project if it is unprofitable. The option
7. The type of option most likely to affect the decision is the option to expand. If the country just
liberalized its markets, there is likely the potential for growth. First entry into a market, whether an
entirely new market, or with a new product, can give a company name recognition and market share.
8. Sensitivity analysis can determine how the financial break-even point changes when some factors
(such as fixed costs, variable costs, or revenue) change.
9. There are two sources of value with this decision to wait. Potentially, the price of the timber can
increase, and the amount of timber will almost definitely increase, barring a natural catastrophe or
forest fire. The option to wait for a logging company is quite valuable, and companies in the industry
10. When the additional analysis has a negative NPV. Since the additional analysis is likely to occur almost
immediately, this means when the benefits of the additional analysis outweigh the costs. The benefits
of the additional analysis are the reduction in the possibility of making a bad decision. Of course, the
additional benefits are often difficult, if not impossible, to measure, so much of this decision is based
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this solutions
manual, rounding may appear to have occurred. However, the final answer for each problem is found
without rounding during any step in the problem.
Basic
1. a. To calculate the accounting breakeven, we first need to find the depreciation for each year. The
depreciation is:
Depreciation = $780,000/8
Depreciation = $97,500 per year
b. We will use the tax shield approach to calculate the OCF. The OCF is:
OCFbase = [(P – v)Q – FC](1 – tc) + tcD
OCFbase = [($37 – 23)(92,000) – $875,000](.65) + .35($97,500)
Now we can calculate the NPV using our base-case projections. There is no salvage value or
NWC, so the NPV is:
NPVbase = –$780,000 + $302,575(PVIFA15%,8)
NPVbase = $577,751.31
NPVnew = –$780,000 + $375,375(PVIFA15%,8)
NPVnew = $904,428.31
NPV/S = +$40.83
NPV change = $40.83(–500)
NPV change = –$20,417.31
ratio of OCF to a one dollar change in variable cost no matter what variable cost we use. So,
using the tax shield approach, the OCF at a variable cost of $24 is:
OCFnew = [($43 – 24)(92,000) – 875,000](.65) + .35($97,500)
OCFnew = $242,775
OCF/v = ($302,575 – 242,775)/($23 – 24)
OCF/v = –$59,800
numbers by 1.10, a 10 percent increase. The variable and fixed costs both decrease by 10 percent, so
we will multiply the base case numbers by .90, a 10 percent decrease. Doing so, we get:
OCFbest = {[($37)(1.10) – ($23)(.90)](93,000)(1.10) – $875,000(.90)}(.65) + .35($97,500)
OCFbest = $837,850
NPVbest = –$780,000 + $837,850(PVIFA15%,8)
NPVbest = $2,979,702.33
NPVworst = –$780,000 – $160,940(PVIFA15%,8)
NPVworst = –$1,502,189.52
QA = (FC + D)/(P – v)
annual cost (EAC). Dividing the initial investment by the five-year annuity factor, discounted at 10
percent, the EAC of the initial investment is:
EAC = Initial Investment/PVIFA10%,5
EAC = $483,000/PVIFA10%,5
EAC = $127,414.18
PVA = C({1 – [1/(1 + R)]t }/R)
$483,000 = C{[1 – (1/1.10)5 ]/.10}
C = $127,414.18
Annual depreciation = $483,000/5
Annual depreciation = $96,600
QF = [EAC + FC(1 – tC) – Depreciation(tC)]/[(P – VC)(1 – tC)]
QF = [$127,414.18 + $365,000(1 – .35) – $96,600(.35)]/[($34 – 9)(1 – .35)]
QF = 20,360.26 or about 20,360 units
NPV0 = –$2,300,000 + $435,000(PVIFA12%,10)
NPV0 = $157,847.02
We should not purchase the machine today since the NPV is negative, but the NPV could be positive
Year 1: NPV1 = [–$2,130,000 + $435,000(PVIFA12%,9)]/1.12
NPV1 = $167,668.45
6. We need to calculate the NPV of the two options, go directly to market now, or utilize test marketing
NPV = CSuccess (Prob. of Success) + CFailure (Prob. of Failure)
NPV = $29,000,000(.50) + $8,500,000(.50)
NPV = $19,775,000
outlay. Choosing the test marketing option will also delay the launch of the product by one year. Thus,
the expected payoff is delayed by one year and must be discounted back to Year 0.
NPV= C0 + {[CSuccess (Prob. of Success)] + [CFailure (Prob. of Failure)]}/(1 + R)t
7. We need to calculate the NPV of each option, and choose the option with the highest NPV. So, the
NPV = CSuccess (Prob. of Success)
NPV = $1,550,000(.40)
NPV = C0 + CSuccess (Prob. of Success)
NPV = –$75,000 + $1,550,000(.60)
NPV = C0 + CSuccess (Prob. of Success)
NPV = –$345,000 + $1,550,000(.85)
NPV = $972,500
NPV = CSuccess (Prob. of Success) + CFailure (Prob. of Failure)
NPV = $26,000,000(.60) + $4,200,000(.40)
must be discounted back to Year 0. So, the NPV of the customer segment research is:
NPV= C0 + {[CSuccess (Prob. of Success)] + [CFailure (Prob. of Failure)]}/(1 + R)t
NPV = –$550,000 + {[$26,000,000(.75)] + [$4,200,000(.25)]}/1.12
QA = [($625,000 + $58,000)(1 – .35)]/[($47.50 – 13.14) (1 – .35)]
QA = 19,877.76
Research
Success
$26 million at t = 1
at 13 percent, the EAC of the initial investment is:
value of the annuity. In other words:
PVA = C({1 – [1/(1 + R)]t}/R)
$406,000 = C{[1 – (1/1.13)7 ]/.13}
C = $91,800.99
is:
QF = [EAC + FC(1 – tC) – Depreciation(tC)]/[(P – VC)(1 – tC)]
QF = [$91,800.99 + $625,000(.65) – $58,000(.35)]/[($47.50 – 13.14)(.65)]
QF = 21,391.20
percent, the EAC of the initial investment is:
EAC = Initial Investment/PVIFA12%,6
EAC = $609,000/PVIFA12%,6
EAC = $148,124.46
PVA = C({1 – [1/(1 + R)]t }/R)
$609,000 = C{[1 – (1/1.12)6 ]/.12}
C = $148,124.46
Annual depreciation = $609,000/6
Annual depreciation = $101,500
QF = [$148,124.46 + $360,000(1 – .34) – $101,500(.34)]/[($51 – 18)(1 – .34)]
QF = 16,125.55
11. a. At the accounting breakeven, the IRR is zero percent since the project recovers the initial
investment. The payback period is N years, the length of the project since the initial investment
is exactly recovered over the project life. The NPV at the accounting breakeven is:
b. At the cash breakeven level, the IRR is –100 percent, the payback period is negative, and the
c. The definition of the financial breakeven is where the NPV of the project is zero. If this is true,
period, except to say that the payback period must be less than the length of the project. Since
the discounted cash flows are equal to the initial investment, the undiscounted cash flows are
greater than the initial investment, so the payback must be less than the project life.
OCF = [(P – v)Q – FC](1 – tC) + tC(D)
OCF = [($27 – 19)(76,000) – 345,000](.66) + .34($496,000/4)
OCF = $215,740
and irrelevant. No matter what level of units sold we choose, we will still get the same sensitivity. So,
the OCF at this level of sales is:
Sensitivity = OCF/Q = ($215,740 – 221,020)/(76,000 – 77,000)
OCF/Q = +$5.28
case, sales and price increase, while costs decrease. In the worst-case, sales and price decrease,
and costs increase.
Scenario Unit sales Variable cost Fixed costs
Base 200 $12,600 $315,000
OCFbest = [($17,100 – 11,340)(220) – $283,500](.65) + .35($790,000/4)
OCFbest = $708,530
NPV = –$790,000 + $442,875(PVIFA15%,4)
NPV = $474,398.54
NPV/FC = ($492,955.90 – 474,398.54)/($315,000 – 325,000)
NPV/FC = –$1.86
QA = (FC + D)/(P – v)
QA = [$315,000 + ($790,000/4)]/($17,100 – 12,600)
