1、Chapter 14Project Risk and Uncertainty Origin of Project Risk Methods of Describing Project Risk Probability Concepts fro Investment Decisions Probability Distribution of NPW Decision Trees and Sequential Investment Decisions1Origins of Project Risk Risk:the potential for loss Project Risk:variabili
2、ty in a projects NPW Risk Analysis:The assignment of probabilities to the various outcomes of an investment project2Example 14.1 Improving the Odds All It Takes is$7 Million and a DreamNumber of PrizesPrize CategoryTotal Amount1First Prize$27,007,364228Second Prizes($899 each)204,97210,552Third Priz
3、es($51 each)538,152168,073Fourth Prizes($1 each)168,073Total Winnings$27,918,56115263341393Number of PrizesPrize CategoryWinning Odds1First Prize0.0000001416228Second Prizes 0.000032310,552Third Prizes0.00149168,073Fourth Prizes0.02381C(,)!()!,.4 4 64 464 467 0 5 9 0 5 20 0 0 0 0 0 0 1 4 1 6O d d s
4、fo r F irs t P riz e s:17,0 5 9,0 5 24Methods of Describing Project Risk Sensitivity Analysis:a means of identifying the project variables which,when varied,have the greatest effect on project acceptability.Break-Even Analysis:a means of identifying the value of a particular project variable that ca
5、uses the project to exactly break even.Scenario Analysis:a means of comparing a“base case”to one or more additional scenarios,such as best and worst case,to identify the extreme and most likely project outcomes.5Example 14.2-After-tax Cash Flow for BMCs Transmission-Housings Project “Base Case”01234
6、5Revenues:Unit Price5050505050 Demand(units)2,0002,0002,0002,0002,000 Sales revenue$100,000$100,000$100,000$100,000$100,000Expenses:Unit variable cost$15$15$15$15$15 Variable cost30,00030,00030,00030,00030,000 Fixed cost10,00010,00010,00010,00010,000 Depreciation17,86330,61321,86315,6135,575Taxable
7、Income$42,137$29,387$38,137$44,387$54,425Income taxes(40%)16,85511,75515,25517,75521,770Net Income$25,282$17,632$22,882$26,632$32,6556Cash Flow Statement012345Operating activitiesNet income25,28217,63222,88226,63232,655Depreciation17,86330,61321,86315,6135,575Investment activitiesInvestment125,000Sa
8、lvage40,000Gains tax2,611Net cash flow$125,500$43,145$48,245$44,745$42,245$75,619(Example 14.2,Continued)7Example 14.2-Sensitivity Analysis for Five Key Input Variables Deviation-20%-15%-10%-5%0%5%10%15%20%Unit price$57$9,999$20,055$30,111$40,169$50,225$60,281$70,337$80,393Demand12,01019,04926,08833
9、,13040,16947,20854,24761,28668,325Variable cost52,23649,21946,20243,18640,16937,15234,13531,11828,101Fixed cost44,19143,18542,17941,17540,16939,16338,15737,15136,145Salvage value37,78238,37838,97439,57340,16940,76541,36141,95742,553Base8Sensitivity graph BMCs transmission-housings project(Example 14
10、.2)-20%-15%-10%-5%0%5%10%15%20%$100,00090,00080,00070,00060,00050,00040,00030,00020,00010,0000-10,000BaseUnit PriceDemandSalvage valueFixed costVariable cost9Example 14.3-Sensitivity Analysis for Mutually Exclusive AlternativesElectrical PowerLPGGasolineDiesel FuelLife expectancy7 year7 years7 years
11、7 yearsInitial cost$29,739$21,200$20,107$22,263Salvage value$3,000$2,000$2,000$2,200Maximum shifts per year260260260260Fuel consumption/shift31.25 kWh11 gal11.1 gal7.2 galFuel cost/unit$0.05/kWh$1.02/gal$1.20/gal$1.13/galFuel cost/shift$1.56$11.22$13.32$8.14Annual maintenance cost Fixed cost$500$1,0
12、00$1,000$1,000 Variable cost/shift$4.5$7$7$710a)Ownership cost(capital cost):b)Electrical power:CR(10%)=($29,739-$3,000)(A/P,10%,7)c)+(0.10)$3,000=$5,792d)LPG:CR(10%)=($21,200-$2,000)(A/P,10%,7)e)+(0.10)$2,000=$4,144f)Gasoline:CR(10%)=($20,107-$2,000)(A/P,10%,7)g)+(0.10)$2,000=$3,919h)Diesel fuel:CR
13、(10%)=($22,263-$2,200)(A/P,10%,7)i)+(0.1)$2,200=$4,34111b)Annual operating cost as a function of number of shifts per year(M):Electric:$500+(1.56+4.5)M=$500+5.06M LPG:$1,000+(11.22+7)M=$1,000+18.22M Gasoline:$1,000+(13.32+7)M=$1,000+20.32M Diesel fuel:$1,000+(8.14+7)M=$1,000+15.14M12c)Total equivale
14、nt annual cost:Electrical power:AE(10%)=6,292+5.06MLPG:AE(10%)=5,144+18.22MGasoline:AE(10%)=4,919+20.32MDiesel fuel:AE(10%)=5,341+15.14M13Sensitivity Graph for Multiple Alternatives 0 20 40 60 80 100 120 140 160 180 200 220 240 260Electric forklift truck is preferredGasoline forklift truck is prefer
15、red12,00010,00080006000400020000GasolineLPGElectric PowerDieselAnnual Equivalent Cost$Number of shifts(M)14Break-Even Analysiswith unknown Sales Units(X)012345Cash Inflows:Net salvage37,389 X(1-0.4)($50)30X30X30X30X30X 0.4(dep)7,14512,2458,7456,2452,230Cash outflows:Investment-125,000-X(1-0.4)($15)-
16、9X-9X-9X-9X-9X-(0.6)($10,000)-6,000-6,000-6,000-6,000-6,000Net Cash Flow-125,00021X+1,14521X+6,24521X+2,74521X+24521X+33,61715Break-Even Analysis1.PW of cash inflowsPW(15%)Inflow=(PW of after-tax net revenue)+(PW of net salvage value)+(PW of tax savings from depreciation =30X(P/A,15%,5)+$37,389(P/F,
17、15%,5)+$7,145(P/F,15%,1)+$12,245(P/F,15%,2)+$8,745(P/F,15%,3)+$6,245(P/F,15%,4)+$2,230(P/F,15%,5)=30X(P/A,15%,5)+$44,490=100.5650X+$44,490161.2.PW of cash outflows:PW(15%)Outflow=(PW of capital expenditure_+(PW)of after-tax expenses=$125,000+(9X+$6,000)(P/A,15%,5)=30.1694X+$145,1133.The NPW:PW(15%)=
18、100.5650X+$44,490-(30.1694X+$145,113)=70.3956X-$100,623.4.Breakeven volume:PW(15%)=70.3956X-$100,623=0Xb=1,430 units.17DemandPW of inflowPW of OutflowNPWX100.5650X-$44,49030.1694X+$145,11370.3956X-$100,6230$44,490$145,113100,62350094,773160,19865,4251000145,055175,28230,2271429188,197188,22528143018
19、8,298188,255431500195,338190,3674,9702000245,620205,45240,1682500295,903220,53775,36618OutflowBreak-Even Analysis Chart0 300 600 900 1200 1500 1800 2100 2400$350,000300,000250,000200,000150,000100,00050,0000-50,000-100,000ProfitLossBreak-even VolumeXb=1430Annual Sales Units(X)PW(15%)Inflow19Scenario
20、 AnalysisVariable ConsideredWorst-CaseScenarioMost-Likely-CaseScenarioBest-CaseScenarioUnit demand1,6002,0002,400Unit price($)485053Variable cost($)171512Fixed Cost($)11,00010,0008,000Salvage value($)30,00040,00050,000PW(15%)-$5,856$40,169$104,29520Probability Concepts for Investment Decisions Rando
21、m variable:variable that can have more than one possible value Discrete random variables:Any random variables that take on only isolated values Continuous random variables:any random variables can have any value in a certain interval Probability distribution:the assessment of probability for each ra
22、ndom event21Triangular Probability Distribution0 L Mo H x2(H-L)f(x)(a)Probability function Mo-L(H-L)F(x)10 L Mo H x(b)Cumulative probability distribution22Uniform Probability Distribution0 L H x1(H-L)f(x)(a)Probability functionF(x)10 L H x(b)Cumulative probability distribution23Probability Distribut
23、ions for Unit Demand(X)and Unit Price(Y)for BMCs ProjectProduct Demand(X)Unit Sale Price(Y)Units(x)P(X=x)Unit price(y)P(Y=y)1,6000.20$480.302,0000.60500.502,4000.20530.2024Cumulative DistributionF xP Xxpjjj()()1(for a discrete random variable)(for a continuous random variable)f(x)dx25Cumulative Prob
24、ability Distribution for XUnit Demand(X)Probability P(X=x)1,6000.22,0000.62,4000.2F xP Xxxxx()().,.,.,021600082 000102 400 261200 1400 1600 1800 2000 2200 2400 26001.00.80.60.40.20Annual Sales Units(X)1200 1400 1600 1800 2000 2200 2400 26001.00.80.60.40.20Annual Sales Units(X)Cumulative ProbabilityP
25、robabilityProbability that annual sales units will be less than equal to x2740 42 44 46 48 50 52 54 561.00.80.60.40.20Unit Price($),Y1.00.80.60.40.20Unit Price($),YCumulative ProbabilityProbabilityProbability that annual sales units will be less than equal to y40 42 44 46 48 50 52 54 5628Measure of
26、ExpectationE Xp xjjjj()1(discrete case)(continuous case)xf(x)dx29Measure of VariationVar xxpxjjjj221()()xV a rXVar Xp xp xjjjj22()E XE X22()30XjPjXj(Pj)(Xj-EX)(Xj-EX)2 (Pj)1,6000.20320(-400)232,0002,0000.601,200002,4000.20480(400)232,000EX=2,000VarX=64,000 252,98YjPjYj(Pj)Yj-EY2(Yj-EY)2 (Pj)$480.30$
27、14.40(-2)21.20500.5025.00(0)0530.2010.60(3)21.80EY=50.00VarY=3.00 1.7331Joint and Conditional ProbabilitiesP x yP x Py(,)()()P x yP(,)(,$48)1 600P xyP y(,$48($48)(.)(.).1600010 030003P x yP XxYy P Yy(,)()()32Assessments of Conditional and Joint ProbabilitiesUnit Price YMarginalProbabilityConditional
28、Unit Sales XConditionalProbabilityJointProbability1,6000.100.03$480.302,0000.400.122,4000.500.151,6000.100.05500.502,0000.640.322,4000.260.131,6000.500.10530.202,0000.400.082,4000.100.0233Marginal Distribution for XXj1,600P(1,600,$48)+P(1,600,$50)+P(1,600,$53)=0.182,000P(2,000,$48)+P(2,000,$50)+P(2,
29、000,$53)=0.522,400P(2,400,$48)+P(2,400,$50)+P(2,400,$53)=0.30P xP x yy()(,)34After-Tax Cash Flow as a Function of Unknown Unit Demand(X)and Unit Price(Y)Item 012345Cash inflow:Net salvage X(1-0.4)Y0.6XY0.6XY0.6XY0.6XY0.6XY 0.4(dep)7,14512,2458,7456,2452,230Cash outflow:Investment-125,000 -X(1-0.4)($
30、15)-9X-9X-9X-9X-9X -(1-0.4)($10,000)-6,000-6,000-6,000-6,000-6,000Net Cash Flow-125,0000.6X(Y-15)+1,1450.6X(Y-15)+6,2450.6X(Y-15)+2,7450.6X(Y-15)+2450.6X(Y-15)+33,61735NPW Function1.Cash Inflow:PW(15%)=0.6XY(P/A,15%,5)+$44,490 =2.0113XY+$44,4902.Cash Outflow:PW(15%)=$125,000+(9X+$6,000)(P/A,15%,5)=3
31、0.1694X+$145,113.3.Net Cash Flow:PW(15%)=2.0113X(Y-$15)-$100,62336The NPW Probability Distribution with Independent Random Variables Event No.xyPx,yCumulativeJoint ProbabilityNPW11,600$48.000.060.06$5,57421,60050.000.100.1612,01031,60053.000.040.2021,66442,00048.000.180.3832,12352,00050.000.300.6840
32、,16862,00053.000.120.8052,23672,40048.000.060.8658,67282,40050.000.100.9668,32692,40053.000.041.0082,80837NPW probability distributions:When X and Y are independent0 20 40 60 80 1000.50.40.30.20.10NPW(Thousand DollarsProbability38Calculation of the Mean of NPW Distribution Event No.xyPx,yCumulativeJ
33、oint ProbabilityNPWWeightedNPW11,600$48.000.060.06$5,574$33421,60050.000.100.1612,0101,20131,60053.000.040.2021,66486742,00048.000.180.3832,1235,78252,00050.000.300.6840,16812,05062,00053.000.120.8052,2366,26872,40048.000.060.8658,6723,52082,40050.000.100.9668,3266,83392,40053.000.041.0082,8083,312E
34、NPW=$40,16839Calculation of the Variance of NPW Distribution EventNo.xyPx,yNPW(NPW-ENPW)2Weighted(NPW-ENPW)11,600$48.000.06$5,5741,196,769,744$71,806,18521,60050.000.1012,010792,884,22779,228,42331,60053.000.0421,664342,396,53613,695,86142,00048.000.1832,12364,725,24311,650,54452,00050.000.3040,1680
35、062,00053.000.1252,236145,631,79717,475,81672,40048.000.0658,672342,396,53620,543,79282,40050.000.1068,326792,884,22779,288,42392,40053.000.0482,8081,818,132,07772,725,283VarNPW=366,474,326$19,14440Comparing Risky Mutually Exclusive ProjectsEvent(NPW)(unit:thousands)ProbabilitiesModel 1Model 2Model
36、3 Model 41,0000.350.100.400.201,50000.4500.402,0000.4000.2502,50000.3500.303,0000.2000.2003,50000004,0000.0500.1504,50000.1000.1041Comparison Rule If EA EB and VA VB,select A.If EA=EB and VA VB,select A.If EA EB and VA VB,Not clear.Model Type E(NPW)Var(NPW)Model 1$1,950747,500Model 22,100915,000Mode
37、l 32,1001,190,000Model 42,0001,000,000Model 2 vs.Model 3 Model 2 Model 3Model 2 vs.Model 4 Model 2 Model 4Model 2 vs.Model 1 Cant decide42PNPW1 NPW2Model 1Model 2Joint Probability$1,000No event(0.35)(0.00)=0.0002,0001,0001,500(0.40)(0.10)=0.040(0.40)(0.45)=0.1803,0001,0001,5002,500(0.20)(0.10)=0.020
38、(0.20)(0.45)=0.190(0.20)(0.35)=0.0704,0001,0001,5002,500(0.05)(0.10)=0.005(0.05)(0.45)=0.230(0.05)(0.35)=0.018 0.44543Decision Tree Analysis A graphical tool for describing (1)the actions available to the decision-maker,(2)the events that can occur,and (3)the relationship between the actions and eve
39、nts.44Structure of a Typical Decision TreeSeekadvice(e )Do not seekadvice(e )No professionalhelpStock(d )Band(d )(7.5%return)C,Low(-30%)B,Medium(9%)A,High(50%)o1U(e ,d ,A)12o1Decision NodeChanceNodeConditionalProfitAdding complexity to the decision problem by seeking additional information45Bills In
40、vestment Decision ProblemOption 1:1)Period 0:(-$50,000-$100)=-$50,100Period 1:(+$75,000-$100)=0.20($24,800)=$69,940PW(5%)=-$50,100+$69,940(P/F,5%,1)=$16,5102)Period 0:(-$50,000-$100)=-$50,100Period 1:(+$54,000-$100)-(0.20)($4,200)=$16,510PW(5%)=-$50,100+$53,540(P/F,5%,1)=$8903)Period 0:(-$50,000-$10
41、0)=-$50,100Period 1:(+$35,000-$100)(0.20)(-$14,800)=$37,940PW(5%)=-$50,100+$37,940(P/F,5%,1)=-$13,967Option 2:Period 0:(-$50,000-$150)=-$50,150Period 1:(+$53,750-$150)=$53,600PW(5%)=-$50,150+$53,600(P/F,5%,1)=$898EMV=$898Or,prior optimaldecision is Option 246Decision tree for Bills investment proble
42、m$75,000-$100$54,500-$100$45,000-$100$53,750-$150$16,510$890$-13,967$898A,High(50%)B,Medium(9%)C,Low(-30%)(7.5%return)Stock(d )1Bond(d )2-$50,000-$100-$150(a)0.250.400.35ABCd1d2$898$898(b)(a)Relevant cash flows before(b)taxes(b)Net present worth forEach decision path47Expected Value of Perfect Infor
43、mation(EVPI)What is EVPI?This is equivalent to asking yourself how much you can improve your decision if you had perfect information.Mathematical Relationship:EVPI=EPPI EMV=EOLwhere EPPI(Expected profit with perfect information)is the expected profit you could obtain if you had perfect information,a
44、nd EMV(Expected monetary value)is the expected profit you could obtain based on your own judgment.This is equivalent to expected opportunity loss(EOL).48EOL=(0.25)($15,612)+(0.40)(0)+(0.35)(0)=$3,903Expected Value of Perfect InformationPotential Return LevelProbabilityDecision OptionOptimal Choice w
45、ith Perfect InformationOpportunity Loss Associated with Investing in BondsOption1:Invest in StockOption 2:Invest in BondsHigh(A)0.25$16,510$898Stock$15,612Medium(B)0.40890898Bond0Low(C)0.35-13,967898Bond0 EMV -$405$898$3,903Prior optimal decision,Option 2EPPI=(0.25)($16,510)+(0.40)($898)+(0.35)($898
46、)=$4,801EVPI=EPPI EV =$4,801-$898 =$3,90349Revised Cash flows after Paying Fee to Receive Advice(Fee=$200)1.Stock Investment OptionPeriod 0:(-$50,000-$100-$200)=-$50,300Period 1:(+$75,000-$100)(0.20)($25,000-$400)=$69,980PW(5%)=$50,300+$69,980(P/F,5%,1)=$16,3482.Bond Investment OptionPeriod 0(-$50,0
47、00-$150-$200)=-$50,350Period 1:(+$53,750-$150)=$53,600PW(5%)=-$50,350+$53,600(P/F,5%,1)=$69850Decision tree for Bills investment problem with an option having a professional advice$16,510$890-$13,967$898ABC(d )1(d )2$16,348$728-$14,129$698ABC(d )1(d )2-$405-$898$16,348$728-$14,129$698ABC(d )1(d )2No
48、 professionalhelpFavorable(F)Unfavorable(UF)Do not seekAdvice(e )Seek Advice(e )1o-$89851With Imperfect Sample InformationGiven Level of Stock PerformanceWhat the Report Will SayHigh(A)Medium(B)Low(C)Favorable(F)0.800.650.20Unfavorable(UF)0.200.350.8052Natures TreeP(A)P(F/A)=0.20P(B)P(UF/B)=0.26FUFP
49、(F/A)=0.80P(UF/A)=0.20P(A)P(F/A)=0.20P(B)P(UF/B)=0.14FUFP(F/B)=0.65P(UF/B)=0.35P(C)P(F/C)=0.07P(C)P(UF/C)=0.28FUFP(F/C)=0.20P(UF/C)=0.80ABCP(A)=0.25P(B)=0.40P(C)=0.35F=FavorableUF=UnfavorableP(F)=0.20+0.26+0.07 =0.53P(UF)=1-P(F)=1-0.53 =0.4753Joint and Marginal ProbabilitiesP(A,F)=P(F/A)P(A)=(0.80)(
50、0.25)=0.20P(A,UF/A)P(A)=(0.20)(0.25)=0.05P(B,F)=P(F/B)P(B)=(0.65)(0.40)=0.26P(B,UF)=P(UF/B)P(B)=(0.35)(0.40)=0.1454Joint as Well as Marginal ProbabilitiesWhat Report Will SayJoint ProbabilitiesWhen Potential Level of Return is GivenFavorable(F)Unfavorable(UF)Marginal Probabilities of Return LevelHig
