CHAPTER 9RISK ANALYSIS, REAL OPTIONS, AND CAPITAL BUDGETINGAnswers to Concepts Review and Critical Thinking Questions1.Forecasting risk is the risk that a poor decision is made because of errors in projected cash flows.The danger is greatest with a new product because the cash flows are probably harder to predict.2.With a sensitivity analysis, one variable is examined over a broad range of values. With a scenarioanalysis, all variables are examined for a limited range of values.3.It is true that if average revenue is less than average cost, the firm is losing money. This much of thestatement is therefore correct. At the margin, however, accepting a project with marginal revenue in 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 canexceed the accounting and cash break-even points but still be below the financial break-even point.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 thefinancial break-even. For a project with an initial investment and sales after, this ordering will always apply. The cash break-even is achieved first since it excludes depreciation. The accounting break-even is next since it includes depreciation. Finally, the financial break-even, which includes the time value of money, is achieved.6.Traditional NPV analysis is often too conservative because it ignores profitable options such as theability to expand the project if it is profitable, or abandon the project if it is unprofitable. The option to alter a project when it has already been accepted has a value, which increases the NPV of the project.7.The type of option most likely to affect the decision is the option to expand. If the country justliberalized 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.This may make it more difficult for competitors entering the market.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 canpotentially 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 have models to estimate the future growth of a forest depending on its age.10.When the additional analysis has a negative NPV. Since the additional analysis is likely to occuralmost 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 on experience.Solutions to Questions and ProblemsNOTE: 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.Basic1.a. To calculate the accounting breakeven, we first need to find the depreciation for each year. Thedepreciation is:Depreciation = $896,000/8Depreciation = $112,000 per yearAnd the accounting breakeven is:Q A = ($900,000 + 112,000)/($38 – 25)Q A = 77,846 unitsb.We will use the tax shield approach to calculate the OCF. The OCF is:OCF base = [(P – v)Q – FC](1 – t c) + t c DOCF base = [($38 – 25)(100,000) – $900,000](0.65) + 0.35($112,000)OCF base = $299,200Now we can calculate the NPV using our base-case projections. There is no salvage value or NWC, so the NPV is:NPV base = –$896,000 + $299,200(PVIFA15%,8)NPV base = $446,606.60To calculate the sensitivity of the NPV to changes in the quantity sold, we will calculate the NPV at a different quantity. We will use sales of 105,000 units. The NPV at this sales level is:OCF new = [($38 – 25)(105,000) – $900,000](0.65) + 0.35($112,000)OCF new = $341,450And the NPV is:NPV new = –$896,000 + $341,450(PVIFA15%,8)NPV new = $636,195.93So, the change in NPV for every unit change in sales is:∆NPV/∆S = ($636,195.93 – 446,606.60)/(105,000 – 100,000)∆NPV/∆S = +$37.918If sales were to drop by 100 units, then NPV would drop by:NPV drop = $37.918(100) = $3,791.80You may wonder why we chose 105,000 units. Because it doesn’t matter! Whatever sales number we use, when we calculate the change in NPV per unit sold, the ratio will be the same.c.To find out how sensitive OCF is to a change in variable costs, we will compute the OCF at avariable cost of $24. Again, the number we choose to use here is irrelevant: We will get the same 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:OCF new = [($38 – 24)(100,000) – 900,000](0.65) + 0.35($112,000)OCF new = $364,200So, the change in OCF for a $1 change in variable costs is:∆OCF/∆v = ($299,200 – 364,200)/($25 – 24)∆OCF/∆v = –$65,000If variable costs decrease by $5 then, OCF would increase byOCF increase = $65,000*5 = $325,0002.We will use the tax shield approach to calculate the OCF for the best- and worst-case scenarios. Forthe best-case scenario, the price and quantity increase by 10 percent, so we will multiply the base case numbers by 1.1, a 10 percent increase. The variable and fixed costs both decrease by 10 percent, so we will multiply the base case numbers by .9, a 10 percent decrease. Doing so, we get:OCF best = {[($38)(1.1) – ($25)(0.9)](100K)(1.1) – $900K(0.9)}(0.65) + 0.35($112K)OCF best = $892,650The best-case NPV is:NPV best = –$896,000 + $892,650(PVIFA15%,8)NPV best = $3,109,607.54For the worst-case scenario, the price and quantity decrease by 10 percent, so we will multiply the base case numbers by .9, a 10 percent decrease. The variable and fixed costs both increase by 10 percent, so we will multiply the base case numbers by 1.1, a 10 percent increase. Doing so, we get: OCF worst = {[($38)(0.9) – ($25)(1.1)](100K)(0.9) – $900K(1.1)}(0.65) + 0.35($112K)OCF worst = –212,350The worst-case NPV is:NPV worst = –$896,000 – $212,350(PVIFA15%,8)NPV worst = –$1,848,882.723.We can use the accounting breakeven equation:Q A = (FC + D)/(P – v)to solve for the unknown variable in each case. Doing so, we find:(1): Q A = 130,200 = (€850,000 + D)/(€41 – 30)D = €582,200(2): Q A = 135,000 = (€3.2M + 1.15M)/(P –€56)P = €88.22(3): Q A = 5,478 = (€160,000 + 105,000)/(€105 – v)v = €56.624.When calculating the financial breakeven point, we express the initial investment as an equivalentannual cost (EAC). Dividing the in initial investment by the seven-year annuity factor, discounted at12 percent, the EAC of the initial investment is:EAC = Initial Investment / PVIFA12%,5EAC = £200,000 / 3.60478EAC = £55,481.95Note, this calculation solves for the annuity payment with the initial investment as the present value of the annuity, in other words:PVA = C({1 – [1/(1 + R)]t } / R)£200,000 = C{[1 – (1/1.12)5 ] / .12}C = £55,481.95The annual depreciation is the cost of the equipment divided by the economic life, or:Annual depreciation = £200,000 / 5Annual depreciation = £40,000Now we can calculate the financial breakeven point. The financial breakeven point for this project is: Q F = [EAC + FC(1 – t C) – Depreciation(t C)] / [(P – VC)(1 – t C)]Q F = [£55,481.95 + £350,000(.75) – £40,000(0.25)] / [(£25 – 5) (.25)]Q F = 20,532.13 or about 20,532 units5.If we purchase the machine today, the NPV is the cost plus the present value of the increased cashflows, so:NPV0 = –฿1,500,000 + ฿280,000(PVIFA12%,10)NPV0 = ฿82,062.45We should not purchase the machine today. We would want to purchase the machine when the NPV is the highest. So, we need to calculate the NPV each year. The NPV each year will be the cost plus the present value of the increased cash savings. We must be careful however. In order to make the correct decision, the NPV for each year must be taken to a common date. We will discount all of the NPVs to today. Doing so, we get:Year 1: NPV1 = [–฿1,375,000 + ฿280,000(PVIFA12%,9)] / 1.12NPV1 = ฿104,383.88Year 2: NPV2 = [–฿1,250,000 + ฿280,000(PVIFA12%,8)] / 1.122NPV2 = ฿112,355.82Year 3: NPV3 = [–฿1,125,000 + ฿280,000(PVIFA12%,7)] / 1.123NPV3 = ฿108,796.91Year 4: NPV4 = [–฿1,000,000 + ฿280,000(PVIFA12%,6)] / 1.124NPV4 = ฿96,086.55Year 5: NPV5 = [–฿1,000,000 + ฿280,000(PVIFA12%,5)] / 1.125NPV5 = ฿5,298.26Year 6: NPV6 = [–฿1,000,000 + ฿280,000(PVIFA12%,4)] / 1.126NPV6 = –฿75,762.72The company should purchase the machine two years from now when the NPV is the highest.6.We need to calculate the NPV of the two options, go directly to market now, or utilize test marketingfirst. The NPV of going directly to market now is:NPV = C Success (Prob. of Success) + C Failure (Prob. of Failure)NPV = $20,000,000(0.45) + $5,000,000(0.55)NPV = $11,750,000Now we can calculate the NPV of test marketing first. Test marketing requires a $2 million cashoutlay. 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 + {[C Success (Prob. of Success)] + [C Failure (Prob. of Failure)]} / (1 + R)tNPV = –$2,000,000 + {[$20,000,000 (0.75)] + [$5,000,000 (0.25)]} / 1.15NPV = $12,130,434.78The company should not go directly to market with the product since that option has lower expected payoff.7.We need to calculate the NPV of each option, and choose the option with the highest NPV. So, theNPV of going directly to market is:NPV = C Success (Prob. of Success)NPV = Rs.1,200,000 (0.55)NPV = Rs.660,000The NPV of the focus group is:NPV = C0 + C Success (Prob. of Success)NPV = –Rs.120,000 + Rs.1,200,000 (0.70)NPV = Rs.720,000And the NPV of using the consulting firm is:NPV = C0 + C Success (Prob. of Success)NPV = –Rs.400,000 + Rs.1,200,000 (0.90)NPV = Rs.680,000The firm should conduct a focus group since that option has the highest NPV.8.The company should analyze both options, and choose the option with the greatest NPV. So, if thecompany goes to market immediately, the NPV is:NPV = C Success (Prob. of Success) + C Failure (Prob. of Failure)NPV = ₦30,000,000(.55) + ₦3,000,000(.45)NPV = ₦17,850,000.00Customer segment research requires a ₦1 million cash outlay. Choosing the research 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. So, the NPV of the customer segment research is:NPV= C0 + {[C Success (Prob. of Success)] + [C Failure (Prob. of Failure)]} / (1 + R)tNPV = –₦1,000,000 + {[₦30,000,000 (0.70)] + [₦3,000,000 (0.30)]} / 1.15NPV = ₦18,043,478.26Graphically, the decision tree for the project is:₦3 million at t = 0The company should undertake the market segment research since it has the largest NPV.9. a.The accounting breakeven is the aftertax sum of the fixed costs and depreciation charge dividedby the aftertax contribution margin (selling price minus variable cost). So, the accounting breakeven level of sales is:Q A = [(FC + Depreciation)(1 – t C)] / [(P – VC)(1 – t C)]Q A = [($340,000 + $20,000) (1 – 0.35)] / [($2.00 – 0.72) (1 – 0.35)]Q A = 281,250.00b.When calculating the financial breakeven point, we express the initial investment as anequivalent annual cost (EAC). Dividing the in initial investment by the seven-year annuity factor, discounted at 15 percent, the EAC of the initial investment is:EAC = Initial Investment / PVIFA15%,7EAC = $140,000 / 4.1604EAC = $33,650.45Note, this calculation solves for the annuity payment with the initial investment as the presentvalue of the annuity, in other words:PVA = C({1 – [1/(1 + R)]t } / R)$140,000 = C{[1 – (1/1.15)7 ] / .15}C = $33,650.45Now we can calculate the financial breakeven point. The financial breakeven point for this project is:Q F = [EAC + FC(1 – t C) – Depreciation(t C)] / [(P – VC)(1 – t C)]Q F = [$33,650.45 + $340,000(.65) – $20,000(.35)] / [($2 – 0.72) (.65)]Q F = 297,656.79 or about 297,657 units10.When calculating the financial breakeven point, we express the initial investment as an equivalentannual cost (EAC). Dividing the in initial investment by the five-year annuity factor, discounted at 8 percent, the EAC of the initial investment is:EAC = Initial Investment / PVIFA8%,5EAC = ¥300,000 / 3.60478EAC = ¥75,136.94Note, this calculation solves for the annuity payment with the initial investment as the present value of the annuity, in other words:PVA = C({1 – [1/(1 + R)]t } / R)¥300,000 = C{[1 – (1/1.08)5 ] / .08}C = ¥75,136.94The annual depreciation is the cost of the equipment divided by the economic life, or:Annual depreciation = ¥300,000 / 5Annual depreciation = ¥60,000Now we can calculate the financial breakeven point. The financial breakeven point for this project is: Q F = [EAC + FC(1 – t C) – Depreciation(t C)] / [(P – VC)(1 – t C)]Q F = [¥75,136.94 + ¥100,000(.66) – ¥60,000(0.34)] / [(¥60 – 8) (.34)]Q F = 3,517.98 or about 3,518 unitsIntermediate11.a. At the accounting breakeven, the IRR is zero percent since the project recovers the initialinvestment. The payback period is N years, the length of the project since the initial investmentis exactly recovered over the project life. The NPV at the accounting breakeven is:NPV = I [(1/N)(PVIFA R%,N) – 1]b. At the cash breakeven level, the IRR is –100 percent, the payback period is negative, and theNPV is negative and equal to the initial cash outlay.c. The definition of the financial breakeven is where the NPV of the project is zero. If this is true,then the IRR of the project is equal to the required return. It is impossible to state the paybackperiod, except to say that the payback period must be less than the length of the project. Sincethe discounted cash flows are equal to the initial investment, the undiscounted cash flows aregreater than the initial investment, so the payback must be less than the project life.ing the tax shield approach, the OCF at 110,000 units will be:OCF = [(P – v)Q – FC](1 – t C) + t C(D)OCF = [($28 – 19)(110,000) – 150,000](0.66) + 0.34($420,000/4)OCF = $590,100We will calculate the OCF at 111,000 units. The choice of the second level of quantity sold is arbitrary 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:OCF = [($28 – 19)(111,000) – 150,000](0.66) + 0.34($420,000/4)OCF = $596,040The sensitivity of the OCF to changes in the quantity sold is:Sensitivity = ∆OCF/∆Q = ($596,040 – 590,100)/(111,000 – 110,000)∆OCF/∆Q = +$5.94OCF will increase by $5.94 for every additional unit sold.13.a. The base-case, best-case, and worst-case values are shown below. Remember that in the best-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 costsBase 190 元15,000 元225,000Best 209 元13,500 元202,500Worst 171 元16,500 元247,500Using the tax shield approach, the OCF and NPV for the base case estimate is:OCF base = [(元21,000 – 15,000)(190) –元225,000](0.65) + 0.35(元720,000/4)OCF base = 元657,750NPV base = –元720,000 + 元657,750(PVIFA15%,4)NPV base = 元1,157,862.02The OCF and NPV for the worst case estimate are:OCF worst = [(元21,000 – 16,500)(171) –元247,500](0.65) + 0.35(元720,000/4)OCF worst = 元402,300NPV worst = –元720,000 + 元402,300(PVIFA15%,4)NPV worst = +元428,557.80And the OCF and NPV for the best case estimate are:OCF best = [(元21,000 – 13,500)(209) –元202,500](0.65) + 0.35(元720,000/4)OCF best = 元950,250NPV best = –元720,000 + 元950,250(PVIFA15%,4)NPV best = 元1,992,943.19b. To calculate the sensitivity of the NPV to changes in fixed costs we choose another level offixed costs. We will use fixed costs of 元230,000. The OCF using this level of fixed costs and the other base case values with the tax shield approach, we get:OCF = [(元21,000 – 15,000)(190) –元230,000](0.65) + 0.35(元720,000/4)OCF = 元654,500And the NPV is:NPV = –元720,000 + 元654,500(PVIFA15%,4)NPV = 元1,148,583.34The sensitivity of NPV to changes in fixed costs is:∆NPV/∆FC = (元1,157,862.02 – 1,148,583.34)/(元225,000 – 230,000)∆NPV/∆FC = –元1.856For every dollar FC increase, NPV falls by 元1.86.c. The accounting breakeven is:Q A= (FC + D)/(P – v)Q A = [元225,000 + (元720,000/4)]/(元21,000 – 15,000)Q A = 68At the accounting breakeven, the DOL is:DOL = 1 + FC/OCFDOL = 1 + (元225,000/元180,000) = 2.25For each 1% increase in unit sales, OCF will increase by 2.25%.14.The marketing study and the research and development are both sunk costs and should be ignored.We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:SalesNew clubs €700 ⨯ 55,000 = €38,500,000Exp. clubs €1,100 ⨯ (–13,000) = –14,300,000Cheap clubs €400 ⨯ 10,000 = 4,000,000€28,200,000For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:Var. costsNew clubs –€320 ⨯ 55,000 = –€17,600,000Exp. clubs –€600 ⨯ (–13,000) = 7,800,000Cheap clubs –€180 ⨯ 10,000 = –1,800,000–€11,600,000The pro forma income statement will be:Sales €28,200,000Variable costs 11,600,000Costs 7,500,000Depreciation 2,600,000EBT 6,500,000Taxes 2,600,000Net income € 3,900,000Using the bottom up OCF calculation, we get:OCF = NI + Depreciation = €3,900,000 + 2,600,000OCF = €6,500,000So, the payback period is:Payback period = 2 + €6.15M/€6.5MPayback period = 2.946 yearsThe NPV is:NPV = –€18.2M – .95M + €6.5M(PVIFA14%,7) + €0.95M/1.147NPV = €9,103,636.91And the IRR is:IRR = –€18.2M – .95M + €6.5M(PVIFA IRR%,7) + €0.95M/IRR7IRR = 28.24%15.The upper and lower bounds for the variables are:Base Case Lower Bound Upper Bound Unit sales (new) 55,000 49,500 60,500Price (new) €700 €630 €770VC (new) €320 €288 €352Fixed costs €7,500,000 €6,750,000 €8,250,000Sales lost (expensive) 13,000 11,700 14,300Sales gained (cheap) 10,000 9,000 11,000 Best-caseWe will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:SalesNew clubs €770 ⨯ 60,500 = €46,585,000Exp. clubs €1,100 ⨯ (–11,700) = – 12,870,000Cheap clubs €400 ⨯ 11,000 = 4,400,000€38,115,000For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:Var. costsNew clubs €288 ⨯ 60,500 = €17,424,000Exp. clubs €600 ⨯ (–11,700) = – 7,020,000Cheap clubs €180 ⨯ 11,000 = 1,980,000€12,384,000Sales €38,115,000Variable costs 12,384,000Costs 6,750,000Depreciation 2,600,000EBT 16,381,000Taxes 6,552,400Net income €9,828,600Using the bottom up OCF calculation, we get:OCF = Net income + Depreciation = €9,828,600 + 2,600,000OCF = €12,428,600And the best-case NPV is:NPV = –€18.2M – .95M + €12,428,600(PVIFA14%,7) + .95M/1.147NPV = €34,527,280.98Worst-caseWe will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:SalesNew clubs €630 ⨯ 49,500 = €31,185,000Exp. clubs €1,100 ⨯ (– 14,300) = – 15,730,000Cheap clubs €400 ⨯ 9,000 = 3,600,000€19,055,000For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:Var. costsNew clubs €352 ⨯ 49,500 = €17,424,000Exp. clubs €600 ⨯ (– 14,300) = – 8,580,000Cheap clubs €180 ⨯ 9,000 = 1,620,000€10,464,000Sales €19,055,000Variable costs 10,464,000Costs 8,250,000Depreciation 2,600,000EBT – 2,259,000Taxes 903,600 *assumes a tax creditNet income –€1,355,400Using the bottom up OCF calculation, we get:OCF = NI + Depreciation = –€1,355,400 + 2,600,000OCF = €1,244,600And the worst-case NPV is:NPV = –€18.2M – .95M + €1,244,600(PVIFA14%,7) + .95M/1.147NPV = –€13,433,120.3416.To calculate the sensitivity of the NPV to changes in the price of the new club, we simply need tochange the price of the new club. We will choose €750, but the choice is irrelevant as the sensitivity will be the same no matter what price we choose.We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:SalesNew clubs €750 ⨯ 55,000 = €41,250,000Exp. clubs €1,100 ⨯ (– 13,000) = –14,300,000Cheap clubs €400 ⨯ 10,000 = 4,000,000€30,950,000For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:Var. costsNew clubs €320 ⨯ 55,000 = €17,600,000Exp. clubs €600 ⨯ (–13,000) = –7,800,000Cheap clubs €180 ⨯ 10,000 = 1,800,000€11,600,000Sales €30,950,000Variable costs 11,600,000Costs 7,500,000Depreciation 2,600,000EBT 9,250,000Taxes 3,700,000Net income € 5,550,000Using the bottom up OCF calculation, we get:OCF = NI + Depreciation = €5,550,000 + 2,600,000OCF = €8,150,000And the NPV is:NPV = –€18.2M – 0.95M + €8.15M(PVIFA14%,7) + .95M/1.147NPV = €16,179,339.89So, the sensitivity of the NPV to changes in the price of the new club is:∆NPV/∆P = (€16,179,339.89 – 9,103,636.91)/(€750 – 700)∆NPV/∆P = €141,514.06For every euro increase (decrease) in the price of the clubs, the NPV increases (decreases) by €141,514.06.To calculate the sensitivity of the NPV to changes in the quantity sold of the new club, we simply need to change the quantity sold. We will choose 60,000 units, but the choice is irrelevant as the sensitivity will be the same no matter what quantity we choose.We will calculate the sales and variable costs first. Since we will lose sales of the expensive clubs and gain sales of the cheap clubs, these must be accounted for as erosion. The total sales for the new project will be:SalesNew clubs €700 ⨯ 60,000 = €42,000,000Exp. clubs €1,100 ⨯ (– 13,000) = –14,300,000Cheap clubs €400 ⨯ 10,000 = 4,000,000€31,700,000For the variable costs, we must include the units gained or lost from the existing clubs. Note that the variable costs of the expensive clubs are an inflow. If we are not producing the sets anymore, we will save these variable costs, which is an inflow. So:Var. costsNew clubs €320 ⨯ 60,000 = €19,200,000Exp. clubs €600 ⨯ (–13,000) = –7,800,000Cheap clubs €180 ⨯ 10,000 = 1,800,000€13,200,000The pro forma income statement will be:Sales €31,700,000Variable costs 13,200,000Costs 7,500,000Depreciation 2,600,000EBT 8,400,000Taxes 3,360,000Net income € 5,040,000Using the bottom up OCF calculation, we get:OCF = NI + Depreciation = €5,040,000 + 2,600,000OCF = €7,640,000The NPV at this quantity is:NPV = –€18.2M –€0.95M + €7.64(PVIFA14%,7) + €0.95M/1.147NPV = €13,992,304.43So, the sensitivity of the NPV to changes in the quantity sold is:∆NPV/∆Q = (€13,992,304.43 – 9,103,636.91)/(60,000 – 55,000)∆NPV/∆Q = €977.73For an increase (decrease) of one set of clubs sold per year, the NPV increases (decreases) by€977.73.17.a. The base-case NPV is:NPV = –£1,750,000 + £420,000(PVIFA16%,10)NPV = £279,955.54b.We would abandon the project if the cash flow from selling the equipment is greater than thepresent value of the future cash flows. We need to find the sale quantity where the two are equal, so:£1,500,000 = (£60)Q(PVIFA16%,9)Q = £1,500,000/[£60(4.6065)]Q = 5,427.11Abandon the project if Q < 5,428 units, because the NPV of abandoning the project is greater than the NPV of the future cash flows.c.The £1,500,000 is the market value of the project. If you continue with the project in one year,you forego the £1,500,000 that could have been used for something else.18. a.If the project is a success, present value of the future cash flows will be:PV future CFs = £60(9,000)(PVIFA16%,9)PV future CFs = £2,487,533.69From the previous question, if the quantity sold is 4,000, we would abandon the project, and the cash flow would be £1,500,000. Since the project has an equal likelihood of success or failure in one year, the expected value of the project in one year is the average of the success and failure cash flows, plus the cash flow in one year, so:Expected value of project at year 1 = [(£2,487,533.69 + £1,500,000)/2] + £420,000Expected value of project at year 1 = £2,413,766.85The NPV is the present value of the expected value in one year plus the cost of the equipment, so:NPV = –£1,750,000 + (£2,413,766.85)/1.16NPV = £330,833.49b. If we couldn’t abandon the project, the present value of the fut ure cash flows when the quantityis 4,000 will be:PV future CFs = £60(4,000)(PVIFA16%,9)PV future CFs = £1,105,570.53The gain from the option to abandon is the abandonment value minus the present value of the cash flows if we cannot abandon the project, so:Gain from option to abandon = £1,500,000 – 1,105,570.53Gain from option to abandon = £394,429.47We need to find the value of the option to abandon times the likelihood of abandonment. So, the value of the option to abandon today is:Option value = (.50)(£394,429.47)/1.16Option value = £170,012.70。
