Chapter 11
Cash Flow Estimation
and Risk Analysis

11-1 a. Cash flow, which is the relevant financial variable, represents the actual flow of cash. Accounting income, on the other hand, reports accounting data as defined by Generally Accepted Accounting Principles (GAAP).

b. Incremental cash flows are those cash flows that arise solely from the asset that is being evaluated. For example, assume an existing machine generates revenues of $1,000 per year and expenses of $600 per year. A machine being considered as a replacement would generate revenues of $1,000 per year and expenses of $400 per year. On an incremental basis, the new machine would not increase revenues at all, but would decrease expenses by $200 per year. Thus, the annual incremental cash flow is a before-tax savings of $200. A sunk cost is one that has already occurred and is not affected by the capital project decision. Sunk costs are not relevant to capital budgeting decisions. Within the context of this chapter, an opportunity cost is a cash flow that a firm must forgo to accept a project. For example, if the project requires the use of a building that could otherwise be sold, the market value of the building is an opportunity cost of the project.

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.

d. The real rate of return (rr), or, for that matter the real cost of capital, contains no adjustment for expected inflation. If net cash flows from a project do not include inflation adjustments, then the cash flows should be discounted at the real cost of capital. In a similar manner, the IRR resulting from real net cash flows should be compared with the real cost of capital. Conversely, the nominal rate of return (rn) does include an inflation adjustment (premium). Thus if nominal rates of return are used in the capital budgeting process, the net cash flows must also be nominal.

e. Sensitivity analysis indicates exactly how much NPV will change in response to a given change in an input variable, other things held constant. Sensitivity analysis is sometimes called “what if” analysis because it answers this type of question. Scenario analysis is a shorter version of simulation analysis that uses only a few outcomes. Often the outcomes considered are optimistic, pessimistic and most likely. Monte Carlo simulation analysis is a risk analysis technique in which a computer is used to simulate probable future events and thus to estimate the profitability and risk of a project.

f. A risk-adjusted discount rate incorporates the riskiness 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.

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 on the capital budgeting decision. Opportunity costs represent the cash flows the firm gives up by investing in this project rather than its next best alternative, and externalities are the cash flows (both positive and negative) to other projects that result from the firm taking on this project. These cash flows occur only because the firm took on the capital budgeting project; therefore, they must be included in the analysis.

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 accruals, thus increasing its net operating working capital. Since this increase must be financed, it is included as an outflow in Year 0 of the analysis. At the end of the project’s life, inventories are depleted and receivables are collected. Thus, there is a decrease in NOWC, which is treated as an inflow.

11-6 Simulation analysis involves working with continuous probability distributions, and the output of a simulation analysis is a distribution of net present values or rates of return. Scenario analysis involves picking several points on the various probability distributions and determining cash flows or rates of return for these points. Sensitivity analysis involves determining the extent to which cash flows change, given a change in one particular input variable. Simulation analysis is expensive. Therefore, it would more than likely be employed in the decision for the $200 million investment in a satellite system than in the decision for the $12,000 truck.


11-1 Equipment $ 9,000,000
NWC Investment 3,000,000
Initial investment outlay $12,000,000

11-2 Operating Cash Flows: t = 1
Sales revenues $10,000,000
Operating costs 7,000,000
Depreciation 2,000,000
Operating income before taxes $ 1,000,000
Taxes (40%) 400,000
Operating income after taxes $ 600,000
Add back depreciation 2,000,000
Operating cash flow $ 2,600,000

11-3 Equipment's original cost $20,000,000
Depreciation (80%) 16,000,000
Book value $ 4,000,000

Gain on sale = $5,000,000 - $4,000,000 = $1,000,000.
Tax on gain = $1,000,000(0.4) = $400,000.
AT net salvage value = $5,000,000 - $400,000 = $4,600,000.

11-4 a. The net cost is $126,000:

Price ($108,000)
Modification (12,500)
Increase in NWC (5,500)
Cash outlay for new machine ($126,000)

b. The operating cash flows follow:

Year 1 Year 2 Year 3
1. After-tax savings $28,600 $28,600 $28,600
2. Depreciation tax savings 13,918 18,979 6,326
Net cash flow $42,518 $47,579 $34,926


1. The after-tax cost savings is $44,000(1 - T) = $44,000(0.65)
= $28,600.

2. The depreciation expense in each year is the depreciable basis, $120,500, times the MACRS allowance percentages of 0.33, 0.45, and 0.15 for Years 1, 2, and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $39,765, $54,225, and $18,075. 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 $50,702:

Salvage value $65,000
Tax on SV* (19,798)
Return of NWC 5,500

BV in Year 4 = $120,500(0.07) = $8,435.
*Tax on SV = ($65,000 - $8,435)(0.35) = $19,798.

d. The project has an NPV of $10,841; thus, it should be accepted.

Year Net Cash Flow PV @ 12%
0 ($126,000) ($126,000)
1 42,518 37,963
2 47,579 37,930
3 85,628 60,948
NPV = $ 10,841

Alternatively, place the cash flows on a time line:

0 1 2 3
| | | |
-126,000 42,518 47,579 34,926

With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $10,841.

11-5 a. The net cost is $89,000:

Price ($70,000)
Modification (15,000)
Change in NWC (4,000)

b. The operating cash flows follow:

Year 1 Year 2 Year 3
After-tax savings $15,000 $15,000 $15,000
Depreciation shield 11,220 15,300 5,100
Net cash flow $26,220 $30,300 $20,100


1. The after-tax cost savings is $25,000(1 – T) = $25,000(0.6)
= $15,000.
2. The depreciation expense in each year is the depreciable basis, $85,000, times the MACRS allowance percentage of 0.33, 0.45, and 0.15 for Years 1, 2 and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $28,050, $38,250, and $12,750. The depreciation shield is calculated as the tax rate (40%) times the depreciation expense in each year.

c. The additional end-of-project cash flow is $24,380:

Salvage value $30,000
Tax on SV* (9,620)
Return of NWC 4,000

*Tax on SV = ($30,000 - $5,950)(0.4) = $9,620.

Note that the remaining BV in Year 4 = $85,000(0.07) = $5,950.

d. The project has an NPV of -$6,705. Thus, it should not be accepted.

Year Net Cash Flow PV @ 10%
0 ($89,000) ($89,000)
1 26,220 23,836
2 30,300 25,041
3 44,480 33,418
NPV = ($ 6,705)

Alternatively, with a financial calculator, input the following: CF0 = -89000, CF1 = 26220, CF2 = 30300, CF3 = 44480, and I = 10 to solve for NPV = -$6,703.83.

11-6 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

-$150,000 + = -$4,800.

Considering inflation, the real cost of capital is calculated as follows:

(1 + rr)(1 + i) = 1.15
(1 + rr)(1.06) = 1.15
rr = 0.0849.

Thus, the NPV considering inflation is calculated as

-$150,000 + = $106,537.

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.

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 must be replaced, inflation will cause the replacement cost to jump, necessitating a sharp output price increase to cover the now higher depreciation charges.

11-7 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.

?NPV = [0.05(-$70 - $3)2 + 0.20(-$25 - $3)2 + 0.50($12 - $3)2
+ 0.20($20 - $3)2 + 0.05($30 - $3)2]0.5
= $23.622 million.

CV = = 7.874.

11-8 a. Expected annual cash flows:

Project A: Probable
Probability ? Cash Flow = Cash Flow
0.2 $6,000 $1,200
0.6 6,750 4,050
0.2 7,500 1,500
Expected annual cash flow = $6,750

Project B: Probable
Probability ? Cash Flow = Cash Flow
0.2 $ 0 $ 0
0.6 6,750 4,050
0.2 18,000 3,600
Expected annual cash flow = $7,650

Coefficient of variation:

CV =

Project A:
?A =

Project B:
?B =
= $5,797.84.

CVA = $474.34/$6,750 = 0.0703.
CVB = $5,797.84/$7,650 = 0.7579.

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.

Project A: With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 10, and then solve for NPV = $10,036.25.

Project B: With a financial calculator, input the appropriate cash flows into the cash flow register, input I = 12, and then solve for NPV = $11,624.01.
Project B has the higher NPV; therefore, the firm should accept Project B.

c. The portfolio effects from Project B would tend to make it less risky than otherwise. This would tend to reinforce the decision to accept Project B. Again, if Project B were negatively correlated with the GDP (Project B is profitable when the economy is down), then it is less risky and Project B’s acceptance is reinforced.

11-9 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

Sales in dollars $2,700,000
Costs (200 x $6,000) 1,200,000
Earnings before taxes $1,500,000
Taxes (40%) 600,000
Net income $ 900,000 =Cash flow under the assumption used in the problem.
0 = - $4,000,000.

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%.

Assuming complete independence between the distributions, and normality, it would be possible to derive ?IRR statistically. Alternatively, we could employ simulation to develop a distribution of IRRs, hence ?IRR. There is no easy way to get ?IRR.

b. NPV = $900,000(PVIFA15%,8) - $4,000,000.

Using a financial calculator, input the following: CF0 = -4000000, CF1 = 900000, Nj = 8, and I = 15 to solve for NPV = $38,589.36. Again, there is no easy way to estimate ?NPV.

c. (1) a. Calculate developmental costs. The 44 random number value, coming between 30 and 70, indicates that the costs for this run should be taken to be $4 million.

b. Calculate the project life. The 17, being less than 20, indicates that a 3-year 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.

c. Estimate the cost per unit. The 1 indicates a cost of $5,000.

d. Now estimate the after-tax cash flow for Year 1. It is

[100($13,500) - 100($5,000)](1 - 0.4) = $510,000 = CF1.

(3) Repeat the process for Year 2. Sales will be 200 with a random number of 79; the price will be $13,500 with a random number of 83; and the cost will be $7,000 with a random number of 86:

[200($13,500) - 200($7,000)](0.6) = $780,000 = CF2.

(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.

(5) a. 0 = - $4,000,000
IRR = -31.55%.

Alternatively, with a financial calculator, input the following: CF0 = -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 = 510000, CF2 = 780000, CF3 = 510000, and I = 15 to solve for NPV = -$2,631,396.40.
The results of this run are very bad because the project’s life is so short. Had the life turned out (by chance) to be 13 years, the longest possible life, the IRR would have been about 25%, and the NPV would have been about $1 million.

(6) & (7) The computer would store sNPVs and sIRRs for the different trials, then display them as frequency distributions:

of occurrence

of occurrence


The distribution would be reasonably symmetrical because all the input data were from symmetrical distributions. One often finds, however, that the input and output distributions are badly skewed. The frequency values would also be used to calculate ?NPV and sIRR; these values would be printed out and available for analysis.

11-10 a. The resulting decision tree is:

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)

($500,000) P = 0.5

P = 0.60

100,000 0.12 (376,709) (45,205)

($10,000) P = 0.20

0 0.40 (10,000) (4,000)

P = 0.40 1.00 Exp. NPV = $117,779

The NPV of the top path is:

- - - $10,000 = $881,718.

Using a financial calculator, input the following: CF0 = -10000,
CF1 = -500000, CF2 = -1000000, = 3000000, and I = 12 to solve for NPV = $881,718.29 » $881,718.
The other NPVs were determined in the same manner. If the project is of average risk, it should be accepted because the expected NPV of the total project is positive.

b. ?2NPV = 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.

?NPV = $445,060.

CVNPV = = 3.78.

Since the CV is 3.78 for this project, while the firm’s average project has a CV of 1.0 to 2.0, this project is of high risk.


11-11 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for FM11 Ch 11 P11 Build a Model.xls) and on the instructor’s side of the web site,


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 which 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 four 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 operating working capital would have to increase by an amount equal to 12% of sales revenues. The firm’s tax rate is 40 percent, and its overall weighted average cost of capital is 10 percent.

a. Define “incremental cash flow.”

Answer: This is the firm’s cash flow with the project minus the firm’s cash flow without the project.

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 required by the investors furnishing the capital is already accounted for when we apply the 10 percent cost of capital discount rate, hence including financing flows would be "double counting." Put another way, if we deducted capital costs in the table, and thus reduced the bottom line cash flows, and then discounted those CFS by the cost of capital, we would, in effect, be subtracting capital costs twice.

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 presumably also expensed for tax purposes. Since, it is a sunk cost, it should not be included in the analysis.

a. 3. Now assume that the plant space could be leased out to another firm at $25,000 a 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 project, it would forgo the opportunity to receive $25,000 in annual cash flows. This represents an opportunity cost to the project, and it should be included in the analysis. Note that the opportunity cost cash flow must be net of taxes, so it would be a $25,000(1 - t) = $25,000(0.6) = $15,000 annual outflow.

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" which must be considered in the analysis. If the firm's sales would be reduced by $50,000, then the net cash flow loss would be a cost to the project. Note that this annual loss would not be the full $50,000, because Shrieves would save on cash operating costs if its sales dropped. Note also that externalities can be positive as well as negative.

b. Disregard the assumptions in part a. What is Shrieves’ depreciable basis?

Answer: Get the depreciation rates from table 11-2 in the book. Note that because of the half-year convention, a 3-year project is depreciated over 4 calendar years:

1 0.33 $240 $ 79
2 0.45 240 108
3 0.15 240 36
4 0.07 240 17

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:

Year 1
Year 2
Year 3
Year 4
Unit Price
Unit Cost


The cost of capital is a nominal cost; i.e., it includes a premium for inflation. In other words, it is larger than the real cost of capital. Similarly, nominal cash flows (those that are inflated) are larger than real cash flows. If you discount the low, real cash flows with the high, nominal rate, then the resulting NPV is too low. Therefore, you should always discount nominal cash flows with a nominal rate, and real cash flows with a real rate. In theory, you could do either way and get the correct answer. However, there is no accurate way to convert a nominal cost of capital to a real cost. Therefore, you should inflate cash flows and then discount at the nominal rate.

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:

Here are the annual operating cash flows (in thousands of dollars):

1 2 3 4
Net Revenues $125 $125 $125 $125
Depreciation 79 108 36 17
Before-Tax Income $ 46 $ 17 $ 89 $108
Taxes (40%) 18 7 36 43
Net Income $ 28 $ 10 $ 53 $ 65
Plus Depreciation 79 108 36 17
Net Operating CF $107 $118 $ 89 $ 82

d. Construct annual incremental operating cash flow statements.


Year 1
Year 2
Year 3
Year 4
Taxes (40%)
Net Operating CF

e. Estimate the required net operating working capital for each year, and the cash flow due to investments in net operating working capital.

Answer: The project requires a level of net operating working capital in the amount equal to 12% of the next year’s sales. Any increase in NOWC is a negative cash flow, and any decrease is a positive cash flow.

Year 0
Year 1
Year 2
Year 3
Year 4

NOWC (% of sales)
CF due to NOWC)

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 $25,000. But, since it has been depreciated to a $0 book value, taxes must be paid on the full salvage value. For this project, the after-tax salvage cash flow is:

Salvage Value $25,000
Tax On Salvage Value (10,000)
Net After-Tax Salvage Cash Flow $15,000

g. Calculate the net cash flows for each year? Based on these cash flows, what are the project’s NPV, IRR, MIRR, and payback? Do these indicators suggest that the project should be undertaken?

Answer: The net cash flows are:

Year 0
Year 1
Year 2
Year 3
Year 4

Initial Outlay

Operating Cash Flows



Salvage Cash Flows


Net Cash Flows

Payback =

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, it is possible to look back at historical data and to statistically analyze the riskiness of the investment. This is often true when the investment involves an expansion decision; for example, if Sears were opening a new store, if Citibank were opening a new branch, or if GM were expanding its Chevrolet plant, then past experience could be a useful guide to future risk. Similarly, a company that is considering going into a new business might be able to look at historical data on existing firms in that industry to get an idea about the riskiness of its proposed investment. However, there are times when it is impossible to obtain historical data regarding proposed investments; for example, if GM were considering the development of an electric auto, not much relevant historical data for assessing the riskiness of the project would be available. Rather, GM would have to rely primarily on the judgment of its executives, and they, in turn would have to rely on their experience in developing, manufacturing, and marketing new products. We will try to quantify risk analysis, but you must recognize at the outset that some of the data used in the analysis will necessarily be based on subjective judgments rather than on hard statistical observations.

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-alone risk ignores both the firm’s diversification among projects and investors’ diversification among firms. Stand-alone risk is measured either by the project’s standard deviation of NPV (?NPV) or its coefficient of variation of NPV (CVNPV). Note that other profitability measures, such as IRR and MIRR, can also be used to obtain stand-alone risk estimates.
Within-firm risk is the total riskiness of the project giving consideration to the firm’s other projects, that is, to diversification within the firm. It is the contribution of the project to the firm’s total risk, and it is a function of (a) the project’s standard deviation of NPV and (b) the correlation of the projects’ returns with those of the rest of the firm. Within-firm risk is often called corporate risk, and it is measured by the project’s corporate beta, which is the slope of the regression line formed by plotting returns on the project versus returns on the firm.

Market risk is the riskiness of the project to a well-diversified investor, hence it considers the diversification inherent in stockholders’ portfolios. It is measured by the project’s market beta, which is the slope of the regression line formed by plotting returns on the project versus returns on the market.

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 relevant risk for capital projects is market risk. However, creditors, customers, suppliers, and employees are all affected by a firm’s total risk. Since these parties influence the firm’s profitability, a project’s within-firm risk should not be completely ignored.
Unfortunately, by far the easiest type of risk to measure is a project’s stand-alone risk. Thus, firms often focus on this type of risk when making capital budgeting decisions. However, this focus does not necessarily lead to poor decisions, because most projects that a firm undertakes are in its core business. In this situation, a project’s stand-alone risk is likely to be highly correlated with its within-firm risk, which in turn is likely to be highly correlated with its market risk.

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 at their expected values except one. This one variable is then changed, often by specified percentages, and the resulting effect on NPV is noted. (One could allow more than one variable to change, but this then merges sensitivity analysis into scenario analysis.)

j. 2. Perform a sensitivity analysis on the unit sales, salvage value, and cost of capital for the project. Assume that each of these variables can vary from its base case, or expected, value by plus and minus 10, 20, and 30 percent. Include a sensitivity diagram, and discuss the results.

Answer: The sensitivity data are given here in tabular form (in thousands of dollars):


NPV Deviation From Base Case



Base Case


We generated these data with a spreadsheet model in the file ch 11 mini case.xls.

A. The sensitivity lines intersect at 0% change and the base case NPV, $81,573. Since all other variables are set at their base case, or expected values, the zero change situation is the base case.

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.

C. The plot of unit sales is much steeper than that for salvage value. This indicates that NPV is more sensitive to changes in unit sales than to changes in salvage value.

D. Steeper sensitivity lines indicate greater risk. Thus, in comparing two projects, the one with the steeper lines is considered to be riskier.

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 possible magnitudes of the forecast errors. Thus, a sensitivity analysis might indicate that a project’s NPV is highly sensitive to the sales forecast, hence that the project is quite risky, but if the project’s sales, hence its revenues, are fixed by a long-term contract, then sales variations may actually contribute little to the project’s risk. It also ignores any relationships between variables, such as unit sales and sales price.
Therefore, in many situations, sensitivity analysis is not a particularly good indicator of risk. However, sensitivity analysis does identify those variables which potentially have the greatest impact on profitability, and this helps management focus its attention on those variables that are probably most important.

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 percent chance of poor acceptance, a 25 percent chance of excellent acceptance, and a 50 percent 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 case, and best case. It provides a range of possible outcomes.

k. 2. What is the worst-case NPV? The best-case NPV?

k. 3. Use the worst-, most likely, 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 (in thousands of dollars), which are summarized below:

Unit Sales
Unit Price

Best Case
Base Case
Worst Case

Expected NPV =

Standard Deviation =

Coefficient Of Variation =
Std Dev / Expected 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 the world is much more complex, and most projects have an almost infinite number of possible outcomes.

Simulation analysis is a type of scenario analysis which uses a relatively powerful financial planning software such as interactive financial planning system (IFPs) or @risk (a spreadsheet add-in). Simple simulations can also be conducted with other spreadsheet add-ins, such as Simtools. Here the uncertain cash flow variables (such as unit sales) are entered as continuous probability distribution parameters rather than as point values. Then, the computer uses a random number generator to select values for the uncertain variables on the basis of their designated distributions. Once all of the variable values have been selected, they are combined, and an NPV is calculated. The process is repeated many times, say 1,000, with new values selected from the distributions for each run. The end result is a probability distribution of NPV based on a sample of 1,000 values. The software can graph the distribution as well as print out summary statistics such as expected NPV and ?NPV. Simulation provides the decision maker with a better idea of the profitability of a project than does scenario analysis because it incorporates many more possible outcomes.

Although simulation analysis is technically refined, its usefulness is limited because managers are often unable to accurately specify the variables’ probability distributions. Further, the correlations among the uncertain variables must be specified, along with the correlations over time. If managers are unable to do this with much confidence, then the results of simulation analyses are of limited value.

Recognize also that neither sensitivity, scenario, nor simulation analysis provides a decision rule--they may indicate that a project is relatively risky, but they do not indicate whether the project’s expected return is sufficient to compensate for its risk.
Finally, remember that sensitivity, scenario, and simulation analyses all focus on stand-alone risk, which is not the most relevant risk in capital budgeting analysis.

m. 1. Assume that Shrieves’ average project has a coefficient of variation in the range of 0.2-0.4. 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 0.57, which is above the average range of 0.2-0.4, so it falls into the high risk category. The CV measures a project’s stand-alone risk-it is merely a measure of the variability of returns (as measured by NPV) about the expected return.

m. 2. Shrieves typically adds or subtracts 3 percentage points to the overall cost of capital to adjust for risk. Should the new furniture line be accepted?

Answer: Since the project is judged to have above-average risk, its differential risk-adjusted, or project, cost of capital would be 13 percent. At this discount rate, its NPV would be $60,541, so it would still be acceptable. If it were a low risk project, its cost of capital would be 7 percent, its NPV would be $104,975, and it would be an even more profitable project on a risk-adjusted basis.

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 the project. If the project has a potential for bringing on harmful lawsuits, then it might be riskier than first assessed. Also, if the project’s assets can be redeployed within the firm or can be easily sold, then, as a result of “abandonment possibilities,” the project may be less risky than the analysis indicates.