Realoptions CHAPTER13
RealOptions
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TopicsinChapter
Decisiontrees Applicationoffinancialoptionstoreal options
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Whatisarealoption?
Realoptionsexistwhenmanagerscan influencethesizeandriskofaprojectscash flowsbytakingdifferentactionsduringthe projectslifeinresponsetochangingmarket conditions. Alertmanagersalwayslookforrealoptionsin projects. Smartermanagerstrytocreaterealoptions.
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Whatisthesinglemostimportant characteristicofanoption?
Itdoesnotobligateitsownertotake anyaction.Itmerelygivestheowner therighttobuyorsellanasset.
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Howarerealoptionsdifferent fromfinancialoptions?
Financialoptionshaveanunderlying assetthatistradedusuallyasecurity likeastock. Arealoptionhasanunderlyingasset thatisnotasecurityforexamplea projectoragrowthopportunity,andit isnttraded.
(More...)
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Howarerealoptionsdifferent fromfinancialoptions?
Thepayoffsforfinancialoptionsare specifiedinthecontract. Realoptionsarefoundorcreated insideofprojects.Theirpayoffscanbe varied.
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Whataresometypesof realoptions?
Investmenttimingoptions Growthoptions
Expansionofexistingproductline Newproducts Newgeographicmarkets
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Typesofrealoptions (Continued)
Abandonmentoptions
Flexibilityoptions
Contraction Temporarysuspension
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FiveProceduresfor ValuingRealOptions
1.DCFanalysisofexpectedcashflows, ignoringtheoption. 2.Qualitativeassessmentofthereal optionsvalue. 3.Decisiontreeanalysis. 4.Standardmodelforacorresponding financialoption. 5.Financialengineeringtechniques.
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AnalysisofaRealOption: BasicProject
Initialcost=$70million,CostofCapital= 10%,riskfreerate=6%,cashflowsoccurfor 3years. Probability 30% 40% 30%
Demand High Average Low
Annual cashflow $45 $30 $15
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Approach1:DCFAnalysis
E(CF)=.3($45)+.4($30)+.3($15) =$30. PVofexpectedCFs=($30/1.1)+ ($30/1.12)+($30/1/13) =$74.61million. ExpectedNPV=$74.61$70 =$4.61million.
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InvestmentTimingOption
Ifweimmediatelyproceedwiththe project,itsexpectedNPVis$4.61 million. However,theprojectisveryrisky:
*SeeFM12Ch13MiniCase.xlsfor calculations.
_______________________________
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Ifdemandishigh,NPV=$41.91million.* Ifdemandislow,NPV=$32.70million.*
InvestmentTiming (Continued)
Ifwewaitoneyear,wewillgain additionalinformationregarding demand. Ifdemandislow,wewontimplement project. Ifwewait,theupfrontcostandcash flowswillstaythesame,excepttheywill beshiftedaheadbyayear.
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Procedure2: QualitativeAssessment
Thevalueofanyrealoptionincreases if:
Thisprojectisriskyandhasoneyear beforewemustdecide,sotheoptionto waitisprobablyvaluable.
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theunderlyingprojectisveryrisky thereisalongtimebeforeyoumust exercisetheoption
Procedure3:DecisionTreeAnalysis
(Implementonlyifdemandisnotlow.)
Cost 0 Prob. 1 -$70 $0 30% 40% 30% -$70 $0 Future Cash Flows 2 $45 $30 $0 3 $45 $30 $0 4 $45 $30 $0 NPV this Scenario
a
$35.70 $1.79 $0.00
Discount the cost of the project at the risk-free rate, since the cost is known. Discount the operating cash flows at the cost of capital. Example: $35.70 = -$70/1.06 + $45/1.12 + $45/1.13 + $45/1.13. See FM12 Ch 13 Mini Case.xls for calculations.
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ProjectsExpectedNPVif Wait
E(NPV)= [0.3($35.70)]+[0.4($1.79)]+[0.3($0)]
E(NPV)=$11.42.
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DecisionTreewithOptionto Waitvs.OriginalDCFAnalysis
DecisiontreeNPVishigher($11.42million vs.$4.61). Inotherwords,theoptiontowaitisworth $11.42million.Ifweimplementprojecttoday, wegain$4.61millionbutlosetheoption worth$11.42million. Therefore,weshouldwaitanddecidenext yearwhethertoimplementproject,basedon demand.
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TheOptiontoWaitChanges Risk
Thecashflowsarelessriskyundertheoption towait,sincewecanavoidthelowcash flows.Also,thecosttoimplementmaynotbe riskfree. Giventhechangeinrisk,perhapsweshould usedifferentratestodiscountthecashflows. Butfinancetheorydoesnttellushowto estimatetherightdiscountrates,sowe normallydosensitivityanalysisusingarange ofdifferentrates.
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Procedure4:Usetheexisting modelofafinancialoption.
Theoptiontowaitresemblesafinancial calloptionwegettobuytheproject for$70millioninoneyearifvalueof projectinoneyearisgreaterthan$70 million. Thisislikeacalloptionwithastrike priceof$70millionandanexpiration dateofoneyear.
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InputstoBlackScholesModel forOptiontoWait
X=strikeprice=costtoimplement project=$70million. rRF=riskfreerate=6%. t=timetomaturity=1year. P=currentstockprice=Estimatedon followingslides. 2=varianceofstockreturn= Estimatedonfollowingslides.
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EstimateofP
Forafinancialoption:
Forarealoption:
P=currentpriceofstock=PVofallof stocksexpectedfuturecashflows. Currentpriceisunaffectedbytheexercise costoftheoption. P=PVofallofprojectsfutureexpected cashflows. Pdoesnotincludetheprojectscost.
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Step1:FindthePVoffuture CFsatoptionsexerciseyear.
0 Prob. 1 Future Cash Flows 2 3 4 $45 30% 40% 30% $30 $15 $45 $30 $15 $45 $30 $15 PV at Year 1 $111.91 $74.61 $37.30
Example: $111.91 = $45/1.1 + $45/1.12 + $45/1.13. See FM12 Ch 13 Mini Case.xls for calculations.
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Step2:FindtheexpectedPV atthecurrentdate,Year0.
PVYear 0 PVYear 1 $111.91
High
$67.82
Average Low
$74.61 $37.30
PV2006=PV of Exp. PV2007 = [(0.3* $111.91) +(0.4*$74.61) +(0.3*$37.3)]/1.1 = $67.82. See FM12 Ch 13 Mini Case.xls for calculations.
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TheInputforPinthe BlackScholesModel
Theinputforpriceisthepresentvalue oftheprojectsexpectedfuturecash flows. Basedonthepreviousslides, P=$67.82.
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Estimating2forthe BlackScholesModel
Forafinancialoption,2isthevariance ofthestocksrateofreturn. Forarealoption,2isthevarianceof theprojectsrateofreturn.
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ThreeWaystoEstimate2
Judgment. Thedirectapproach,usingtheresults fromthescenarios. Theindirectapproach,usingthe expecteddistributionoftheprojects value.
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Estimating2withJudgment
Thetypicalstockhas 2ofabout12%. Aprojectshouldberiskierthanthefirm asawhole,sincethefirmisaportfolio ofprojects. Thecompanyinthisexamplehas2= 10%,sowemightexpecttheprojectto have2between12%and19%.
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Estimating2withthe DirectApproach
Usethepreviousscenarioanalysisto estimatethereturnfromthepresent untiltheoptionmustbeexercised.Do thisforeachscenario Findthevarianceofthesereturns,given theprobabilityofeachscenario.
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FindReturnsfromthePresent untiltheOptionExpires
PVYear 0 PVYear 1 $111.91
High
Return 65.0% 10.0% -45.0%
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$67.82
Average Low
$74.61 $37.30
Example: 65.0% = ($111.91- $67.82) / $67.82. See FM12 Ch 13 Mini Case.xls for calculations.
ExpectedReturnandVariance ofReturn.
E(Ret.)=0.3(0.65)+0.4(0.10) +0.3(0.45) E(Ret.)=0.10=10%. 2=0.3(0.650.10)2+0.4(0.100.10)2+ 0.3(0.450.10)2 2=0.182=18.2%.
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Estimating2withthe IndirectApproach
Fromthescenarioanalysis,weknow theprojectsexpectedvalueandthe varianceoftheprojectsexpectedvalue atthetimetheoptionexpires. Thequestionsis:Giventhecurrent valueoftheproject,howriskymustits observedvarianceoftheprojectsvalue expectedreturnbetogeneratethe atthetimetheoptionexpires? 31
TheIndirectApproach(Cont.)
Fromoptionpricingforfinancialoptions, weknowtheprobabilitydistributionfor returns(itislognormal). Thisallowsustospecifyavarianceof therateofreturnthatgivesthevariance oftheprojectsvalueatthetimethe optionexpires.
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IndirectEstimateof2
Hereisaformulaforthevarianceofa stocksreturn,ifyouknowthecoefficient ofvariationoftheexpectedstockprice atsometime,t,inthefuture:
2=
1n(CV2+1) t
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Fromearlierslides,weknowthe valueoftheprojectforeachscenario attheexpirationdate.
PVYear 1 $111.91
High Average Low
$74.61 $37.30
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ExpectedPVand PV
E(PV)=.3($111.91)+.4($74.61) +.3($37.3) E(PV)=$74.61. PV=[.3($111.91$74.61)2+.4($74.61 $74.61)2+.3($37.30$74.61)2]1/2 PV=$28.90.
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ExpectedCoefficientof Variation,CVPV(atthetimethe
optionexpires)
CVPV=$28.90/$74.61=0.39.
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Nowusetheformulato estimate2.
Fromourpreviousscenarioanalysis,we knowtheprojectsCV,0.39,atthetimeit theoptionexpires(t=1year).
2=
1n(0.392+1) 1
=14.2%
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TheEstimateof2
Subjectiveestimate:
Directestimate:
12%to19%. 18.2%. 14.2%
Indirectestimate:
Forthisexample,wechose14.2%,butwe recommenddoingsensitivityanalysisovera rangeof2.
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BlackScholesInputs:P=$67.83;X=$70; rRF=6%;t=1year; 2=0.142.
V = $67.83[N(d1)] - $70e-(0.06)(1)[N(d2)]. d1 = ln($67.83/$70)+[(0.06 + 0.142/2)](1) (0.142)0.5 (1).05 = 0.2641. d2 = d1 - (0.142)0.5 (1).05= d1 - 0.3768 = 0.2641 - 0.3768 = - 0.1127.
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BlackScholesValue
N(d1) = N(0.2641) = 0.6041 N(d2) = N(- 0.1127) = 0.4551 V = $67.83(0.6041) - $70e-0.06(0.4551) V = $40.98 - $70(0.9418)(0.4551) V = $10.98.
Note: Values of N(di) obtained from Excel using NORMSDIST function. See FM12 Ch 13 Mini Case.xls for details.
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Step5:Usefinancial engineeringtechniques.
Althoughtherearemanyexisting modelsforfinancialoptions,sometimes nonecorrespondtotheprojectsreal option. Inthatcase,youmustusefinancial engineeringtechniques,whichare coveredinlaterfinancecourses. Alternatively,youcouldsimplyuse 41 decisiontreeanalysis.
OtherFactorstoConsiderWhen DecidingWhentoInvest
Delayingtheprojectmeansthatcash flowscomelaterratherthansooner. Itmightmakesensetoproceedtodayif thereareimportantadvantagestobeing thefirstcompetitortoenteramarket. Waitingmayallowyoutotake advantageofchangingconditions.
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ANewSituation:Costis$75 Million,NoOptiontoWait
Cost Year 0 Prob. NPV this Future Cash Flows Year 1 Year 2 Year 3 Scenario $45 -$75 30% 40% 30% $30 $15 $45 $30 $15 $45 $30 $15 $36.91 -$0.39 -$37.70
Example: $36.91 = -$75 + $45/1.1 + $45/1.1 + $45/1.1. See FM12 Ch 13 Mini Case.xls for calculations.
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ExpectedNPVofNew Situation
E(NPV)=[0.3($36.91)]+[0.4($0.39)]+ [0.3($37.70)] E(NPV)=$0.39. Theprojectnowlookslikealoser.
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GrowthOption:Canreplicatetheoriginal projectafteritendsin3years.
NPV=NPVOriginal+NPVReplication =$0.39+$0.39/(1+0.10)3 =$0.39+$0.30=$0.69. Stillaloser,butyouwouldimplement Replicationonlyifdemandishigh.
Note: the NPV would be even lower if we separately discounted the $75 million cost of Replication at the risk-free rate.
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DecisionTreeAnalysis
Cost Year 0 Prob. 1 $45 30% -$75 40% 30% $30 $15 Future Cash Flows 2 3 4 $45 $30 $15 -$30 $30 $15 $45 $0 $0 5 $45 $0 $0 6 $45 $0 $0 NPV this Scenario $58.02 -$0.39 -$37.70
Notes: The Year 3 CF includes the cost of the project if it is optimal to replicate. The cost is discounted at the risk-free rate, other cash flows are discounted at the cost of capital. See FM12 Ch 13 Mini Case.xls for all calculations. 46
ExpectedNPVofDecision Tree
E(NPV)=[0.3($58.02)]+[0.4($0.39)] +[0.3($37.70)] E(NPV)=$5.94.
Thegrowthoptionhasturnedalosing projectintoawinner!
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FinancialOptionAnalysis: Inputs
X=strikeprice=costofimplement project=$75million. rRF=riskfreerate=6%. t=timetomaturity=3years.
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EstimatingP:First,findthevalue offutureCFsatexerciseyear.
Cost Year 0 Prob. Future Cash Flows 2 3 4 $45 30% 40% 30% $30 $15 PV at Year 3 $111.91 $74.61 $37.30 Prob. x NPV $33.57 $29.84 $11.19
1
5 $45 $30 $15
6 $45 $30 $15
Example: $111.91 = $45/1.1 + $45/1.12 + $45/1.13. See FM12 Ch 13 Mini Case.xls for calculations.
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NowfindtheexpectedPVat thecurrentdate,Year0.
PVYear 0 Year 1 Year 2 PVYear 3 $111.91
High
$56.05
Average Low
$74.61 $37.30
PVYear 0=PV of Exp. PVYear 3 = [(0.3* $111.91) +(0.4*$74.61) +(0.3*$37.3)]/ 1.13 = $56.05. See FM12 Ch 13 Mini Case.xls for calculations.
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TheInputforPintheBlack ScholesModel
Theinputforpriceisthepresentvalue oftheprojectsexpectedfuturecash flows. Basedonthepreviousslides, P=$56.05.
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Estimating2:FindReturnsfromthe PresentuntiltheOptionExpires
PVYear 0 Year 1 Year 2 PVYear 3 $111.91
High
Annual Return 25.9% 10.0% -12.7%
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$56.05
Average Low
$74.61 $37.30
Example: 25.9% = ($111.91/$56.05)(1/3) - 1. See FM12 Ch 13 Mini Case.xls for calculations.
ExpectedReturnandVariance ofReturn
E(Ret.)=0.3(0.259)+0.4(0.10)+0.3( 0.127) E(Ret.)=0.080=8.0%. 2=0.3(0.2590.08)2+0.4(0.100.08)2 +0.3(0.12750.08)2 2=0.023=2.3%.
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Whyis2somuchlowerthanin theinvestmenttimingexample?
2hasfallen,becausethedispersionof cashflowsforreplicationisthesameas fortheoriginalproject,eventhoughit beginsthreeyearslater.Thismeans therateofreturnforthereplicationis lessvolatile. Wewilldosensitivityanalysislater.
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Estimating2withtheIndirect Method
Fromearlierslides,weknowthevalueoftheproject foreachscenarioattheexpirationdate.
PVYear 3 $111.91
High Average Low
$74.61 $37.30
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ProjectsExpectedPVand PV
E(PV)=.3($111.91)+.4($74.61) +.3($37.3) E(PV)=$74.61. PV=[.3($111.91$74.61)2
+.4($74.61$74.61)2 +.3($37.30$74.61)2]1/2 PV=$28.90.
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Nowusetheindirectformula toestimate2.
CVPV=$28.90/$74.61=0.39. Theoptionexpiresin3years,t=3.
2=
1n(0.392+1) 3
=4.7%
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BlackScholesInputs:P=$56.06;X=$75; rRF=6%;t=3years; 2=0.047.
V = $56.06[N(d1)] - $75e-(0.06)(3)[N(d2)]. d1 = ln($56.06/$75)+[(0.06 + 0.047/2)](3) (0.047)0.5 (3).05 = -0.1085. d2 = d1 - (0.047)0.5 (3).05= d1 - 0.3755 = -0.1085 - 0.3755 = - 0.4840.
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BlackScholesValue
N(d1) = N(-0.1085) = 0.4568 N(d2) = N(- 0.4840) = 0.3142 V = $56.06(0.4568) - $75e(-0.06)(3)(0.3142) = $5.92.
Note: Values of N(di) obtained from Excel using NORMSDIST function. See FM12 Ch 13 Mini Case.xls for details.
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TotalValueofProjectwith GrowthOpportunity
Totalvalue= NPVofOriginalProject+Valueof growthoption
=$0.39+$5.92 =$5.5million.
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SensitivityAnalysisontheImpactof Risk(usingtheBlackScholesmodel)
Ifrisk,definedby2,goesup,then valueofgrowthoptiongoesup:
Doesthishelpexplainthehighvalue manydot.comcompanieshadbefore thecrashof2000?
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2=4.7%,OptionValue=$5.92 2=14.2%,OptionValue=$12.10 2=50%,OptionValue=$24.08