Investmentsof UncertainCostbyRobertS. PindyckMIT-CEEPR92-005WPMarch 1992M.I.T.LIBRARIESJUL2 9 19921REGEIVEDI-~-- -mom 1INVESTMENTSOFUNCERTAINCOST*byRobertS.PindyckMassachusettsInstituteofTechnologyCambridge,MA02139Thisdraft:March23,1992AbstractIstudy irreversibleinvestmentdecisionswhen projectstake time to complete,and are subjectto twotypesof uncertaintyoverthecostof completion.The firstistechnicaluncertainty, i.e.,uncertaintyovertheamountoftime,effort,andmaterialsthat willultimatelybe requiredtocompletetheproject,andthatisonlyresolvedas theinvestmenttakesplace.Thesecondisinputcostuncertainty,i.e.,uncertaintyoverthepricesandquantitiesoflabor andmaterialsthatareexpectedtoberequired,andwhichisexternaltothefirm'sinvestmentactivity.Thispaperderivessimpledecisionrulesthatmaximizethefirm'svalue,andthatareeasytoimplement.Ishowhowthesetwotypesofuncertaintyhaveverydifferenteffectsonthedecisiontoinvest,andhowtheyaffectthevalueof theopportunitytoinvest.JELClassificationNumbers:G31,C61,E22Keywords:Investmentunderuncertainty,sunkcosts,optimalstopping,opportunitycost,optionvalue.*ThisresearchwassupportedbytheM.I.TCenterforEnergyPolicyResearch,andbytheNationalScienceFoundationunderGrantNo.SES-8618502.MythankstoAlexandarAngelusandDavidCharitonfortheirexcellentresearchassistance,andtoJohnCox,GeneGrossman,WilliamPounds,andMartinWeitzmanforhelpfulcommentsandsuggestions.9a1.Introduction.Inmoststudiesofinvestmentunderuncertainty,itisthefuturepayoffsfromthein-vestmentthatareuncertain.Thesameistrueformosttextbooktreatmentsofprojectevaluationandcapitalbudgeting,whichshow,forexample,howtheCAPMcanbeusedtodiscountuncertainfuturecashflows.Theemphasisonuncertaintyoverfuturepayoffsalsoappliestothegrowingliteratureonirreversibleinvestment.Muchofthatliteraturestudiesoptimalstoppingrulesforthetimingofsunkcostsofknownmagnitude,inexchangeforcapitalwhosevaluefluctuatesstochastically.1Attimes,however,thecostofaninvestmentismoreuncertainthanthefuturevalueofitspayoff.Thisisoftenthecaseforlargeprojectsthattakeconsiderabletimetobuild.Anexampleisa nuclearpowerplant,wheretotalconstructioncostsareveryhardto predictduetobothengineeringandregulatoryuncertainties.Althoughthefuturevalueof acompletednuclearplantisalsouncertain(becausethedemandforelectricityandcostsofalternativefuelsareuncertain),constructioncostuncertaintyismuchgreater,andhasoftendeterredutilitiesfromundertakingnewplants.Therearemanyotherexamples,rangingfromlargepetrochemicalcomplexes,tothedevelopmentofanewlineofjetaircraft,tomajorurbanconstructionprojects.Also,largesizeisnotarequisiteforcostuncertainty.Many(ifnotmost)R&Dprojectsinvolveconsiderablecostuncertainty;thedevelopmentofanewdrugbyapharmaceuticalcompanyisanexample.Inadditiontotheiruncertaincosts,alloftheinvestmentsmentionedaboveareirre-versible.Expendituresonnuclearpowerplants,petrochemicalcomplexes,thedevelopmentofnewdrugs,andsoonaresunkcoststhatcannotbe recoveredshould theinvestmentturnout,ezpost,tohavebeenabadone.Ineachcase,theinvestmentcouldturnouttobebadbecausedemandfortheproductislessthananticipated,orbecausethecostoftheinvestmentturnsouttobegreaterthananticipated.Whateverthereason,thefirmcannot"disinvest"and recoverthemoneyitspent.2'Foranoverviewof thatliterature,seeDixit(1992)andPindyck(1991).2Thesecostsaresunkbecausetheyarefirm-orindustry-specific.Apetrochemicalplant,forexample,canonlybeusedto producecertainchemicals.Althoughitcouldbesoldtoanotherchemicalcompany,itsThispaperstudiestheimplicationsofcostuncertaintyforthedecisiontoundertakeanirreversibleinvestment,andforthedecisiontoabandona projectwhenits costturnsouttobelargerthanexpected.Iamconcernedwithprojectsthattaketimetocomplete,sothattwodifferentkindsofuncertaintyarise.The first,whichIcalltechnical uncertainty,relatestothephysicaldifficultyofcompletingaproject:Assumingfactorcostsareknown,howmuchtime,effort,andmaterialswillultimatelyberequired?Animportantcharacteristicofthiskindofuncertaintyisthatitcanonlyberesolvedbyundertakingandcompletingtheproject.3 Onethenobservesactualcosts(andconstructiontime)unfoldastheprojectproceeds.Thesecostsmayfromtimetotimeturnout tobegreaterorlessthan anticipated(asimpedimentsarise orasthework progressesfasterthan planned),butthe totalcostof theinvestmentisonly knownforcertainwhentheprojectiscomplete.Anothercharacteristicofthisuncertaintyisthatitislargelydiversifiable.Itresultsnotfromunpredictablechangesininputprices,butonlyfromtheinabilitytopredicthowdifficultaprojectwillbe,whichislikelytobeindependentoftheoveralleconomy.Thesecondkindofcostuncertaintyrelatestoinputcosts.Forexample,thepricesoflabor,land,andmaterialsrequiredforaprojectarelikelytofluctuateovertime.Also,un-predictablechangesingovernmentregulationscanchangetherequiredquantitiesofoneormoreinputs.For example,newsafetyregulationsmayaddtolaborrequirements,orchang-ingenvironmentalregulationsmayrequiremorecapital.Inputcostsevolvestochasticallywhetheror notthefirmisinvesting,andaremoreuncertainthe fartherinto thefuturetheyareincurred.Henceinputcostuncertaintyisparticularlyimportantforprojectsthatarelikelytotakea longtimeto complete,orthataresubjectto voluntaryorinvoluntarydelays.Inaddition,thiskindofuncertaintymaybepartlynondiversifiable;changesinwagerates,costismostlysunk,particularyif theindustryiscompetitive.Thereasonisthatthevalueof theplantwillbeaboutthesame forall firmsinthe industry, sotherewillbelittle gainedfromselling it.If theplantturnsouttobea"bad"investmentforonecompany,itislikelytobe just asbadforothercompanies.3Thisisasimplification,inthatforsomeprojectscostuncertaintycanbereducedbyfirstundertakingadditionalengineeringstudies.Theinvestmentproblemisthenmorecomplicatedbecauseonehasthreechoicesinsteadof two:startconstructionnow,undertakeanengineeringstudyandthenbeginconstructiononlyif thestudyindicatescostsarelikelytobelow,orabandontheprojectcompletely.AsIdiscussin theconcludingsection,themodeldevelopedinthispapercouldbeextendedalongthis line.materialscosts,etc.,arelikelytobecorrelatedwithoveralleconomicactivity.Thispaperderivesdecisionrulesforirreversibleinvestmentssubjecttobothtypesofcostuncertainty.Forsimplicityandclarity,I assumethat thevalueof thecompletedprojectisknownwithcertainty,butIshowhowthemodelcanbeextendedsothatthisisalsostochastic.ThedecisionrulesIderiveallowforthepossibilityofabandoningtheprojectmidstream,andmaximizethevalueof the firmina competitivecapitalmarket.Theserulesalso have a particularlysimple form:Investaslong the expectedcostto completetheprojectisbelowacriticalnumber.Hencetheycanbeusedtoevaluateprojects,ratherthansimplycharacterizeinvestmentdecisions.Inaddition,thederivationof thedecisionruleyieldsthevalueof theinvestmentopportunity, i.e.,whatonewouldpayforthe rightto undertaketheproject.Iexplorehowthisvalue,andthecriticalexpectedcosttocompletion,dependonthe levelandcharacteristicsof uncertainty,aswellasotherparameters.Technicalandinputcostuncertaintybothincreasethevalueofthe investmentopportu-nity.Thereasonisthatthe payofffunctionforthe investmentopportunityisconvexinthecostoftheinvestment;lettingKbethecostandVbethevalueofthecompletedproject,thepayofffunctionismax[0,V- K].Notethattheinvestmentopportunityisanalogoustoaputoption,i.e.,itgivestheholdertherighttosellanassetworthanuncertainamountKforafixed"exerciseprice"V.Aswithanyoption,itsvalueisincreasedbyanincreaseinthevarianceof thepriceoftheunderlyingasset.4However,thesetwotypesofuncertaintyaffecttheoptimalinvestmentdecisioninverydifferentways.Technicaluncertaintyraises thecriticalexpectedcosttocompletion.Hencea projectcanhaveanexpectedcostthatmakesitsconventionallymeasuredNPVnegative,but if the varianceof thecost issufficientlyhigh, it canstill beeconomicalto begininvesting.Thereasonisthat investingrevealsinformationaboutcost,andthusaboutthe expectednetpayofffrominvestingfurther.Itthereforehasa shadowvaluebeyondits directcontributiontothecompletionoftheproject,whichlowersthefullexpectedcostoftheinvestment.54Usingput-callparity,wecanalsothinkof thisasacall optionwithastochasticexerciseprice(K)onanassetwithafixedvalue(V).Inmymodel,thefirmhasamorecomplicatedcompoundoption;itcanspendanuncertainamountof moneyinreturnforanoptiontocontinuethepartiallycompletedproject.5Itisanalogoustotheshadowvalueof productionarisingfromalearningcurve,whichlowersthefullAlso,sinceinformationaboutcostarrivesonlywheninvestmentistakingplace,thereisnovaluetowaiting.Asan example,aprojectrequiresafirstphaseinvestmentof$1. Then,withprobability.5theprojectwillbefinished,andwithprobability.5asecondphasecosting$4willberequired.Completionoftheprojectyieldsacertainpayoffof$2.8.Sincetheexpectedcostof theprojectis$3, theconventionallymeasuredNPV isnegative.Butthis ignoresthe valueof the optiontoabandonthe projectshould thesecondphasebe required.The correctNPVis- 1 +(.5)(2.8)=$0.4,sooneshouldproceedwithatleastthefirstphase.Inputcostuncertaintyhastheoppositeeffect- itreducesthecriticalexpectedcost.HenceaprojectcouldhaveaconventionallymeasuredNPVthatispositive,butitmightstill be uneconomicaltobegininvesting.The reasonisthatfluctuationsinfactorcostsoccurwhetherornotinvestmentis takingplace,soaswithmostirreversibleinvestments,thereisa valueof waitingtoseeif those costschangebeforecommitingresources.Also,thiseffectismagnifiedwhenstochasticfluctuationsinfactorcostsarecorrelatedwiththeeconomy,i.e.,inthecontextoftheCAPM,totheextentthatthe"beta"ofcostishigh.Thereasonisthata higher"beta" impliesthathighcostoutcomes,andhencelowprojectvalues,are morelikely tobeassociatedwithhighstock marketreturns,sothatthe investmentopportunityisahedgeagainstnondiversifiablerisk.Putanotherway,ahigher"beta"raisestherequiredexpectedreturn,andhencediscountrate,thatmustbeappliedtopossiblefuturecosts.Sincethe payoff fromcompletingtheprojectisknown,this raisesthe valueof the investmentopportunity,aswellasthebenefitfromwaitingratherthaninvestingnow.Forexample,supposeaninvestmentcanbeundertakennoworlater.Thecostisnow$3, butnextperioditwilleitherfallto$2 orriseto$4, eachwithprobability.5, andthenremainat that level.Investing yieldsa certainpayoff of $3.2,andwe willassumethe risk-freerateofinterestiszero.Ifweinvestnow,theprojecthasaconventionallymeasuredNPVof$0.2.Butthisignorestheopportunitycostofclosingouroptiontowaitforabetteroutcome(adropincost).If wewaituntilnextperiod,wewillonlyinvestifthecostfallscostofproduction;seeMajdandPindyck(1989).to$2.TheNPVifwewaitis(.5)(3.2- 2)=$0.6,soitisclearlybettertowait.Also,thevalueofbeingabletowaitis0.6- 0.2=$0.4.Nowsupposethe"beta"ofcostishigh,sothattherisk-adjusteddiscountrateis25percentperperiod.Becausethepayofffromcompletingthe projectiscertain,thisdiscountrateisonlyappliedtocost.HencetheNPVassumingwewaitisnow(.5)[3.2- 2/1.25]=$0.8.Thehigher"beta"increasesthevalueoftheinvestmentopportunityandalsoincreasesthevalueof waiting,becausecostsinthefuturearemoreheavilydiscounted,sothatthepresentvaluesofnetpayoffsarelarger.Thispaperisrelatedtoseveralearlierstudies.ThevalueofinformationgatheringhasbeenexploredbyRobertsandWeitzman(1981),whodevelopedamodelofsequentialin-vestmentsimilartomineinthattheprojectcanbestoppedinmidstream,andtheprocessofinvestingreducesboththeexpectedcostofcompletingthe projectaswellthevarianceofthatcost.Theyderivean optimalstoppingrule,andshowthatitmay paytogoaheadwiththeearlystagesofaninvestmenteventhoughtheNPVoftheentireprojectisnegative.6GrossmanandShapiro(1986)alsostudyinvestmentsforwhichthetotaleffortrequiredtoreachapayoffisunknown.TheymodelthepayoffasaPoissonarrival,withahazardratespecifiedasa functionofthecumulativeeffortexpended.Theyallowtherateofprogresstobea concavefunctionof effort,andfocusonthe rateof investment,ratherthan onwhetheroneshouldproceedornot.Myresultscomplementthoseoftheseauthors,butmymodelismoregeneralin itstreatmentofcostuncertainty,andyieldsrelativelysimpledecisionrules.ThispaperisrelatedaswelltothebasicmodelofirreversibleinvestmentbyMcDonaldandSiegel(1986).Theyconsiderthepaymentof asunkcostIin returnforaprojectworthV,wherebothVandIevolveasgeometricBrownianmotions.The optimalinvestmentruleistowaituntil VII reachesacriticalvaluethat exceeds1, becauseof the opportunitycostofcommitingresourcesratherthanwaitingfornewinformationaboutVand/orI.Also,Majd6Weitzman,Newey,andRabin(1981)usethismodelto evaluatedemonstrationplants forsyntheticfuelproduction,andshowthatlearningaboutcostscould justifytheseearlyinvestments.Also,MacKie-Mason(1991)extendstheRobertsandWeitzmananalysisbyaccountingforthefactthatinvestors(whopayforthecostofaproject)andmanagers(whodecidewhethertocontinueorabandontheproject)mayhaveconflictinginterestsandasymmetricinformation.Heshowsthatasymmetriclearningaboutcostthenleadstoinefficientoverabandonmentofprojects.Finally,Zeira(1987)developedamodelwithadjustmentcostsinwhicha firmlearnsaboutitspayofffunctionasitaccumulatescapital.andPindyck(1987)studysequentialinvestmentwhena firmcaninvestatsomemaximumrate(soit takes timeto completea project),the projectcan be abandonedbeforecompletion,andthe value of theproject,receivedonlyuponcompletion,evolvesasa geometricBrownianmotion.Inthispaperthefirmcanalsoinvestatamaximumrate,butitisthecostratherthanthevalueof thecompletedprojectthatisuncertain.7Inthenextsection,amodelof investmentisdevelopedthatincludesbothtechnologicalandfactorpriceuncertainty,andthatisbasedonthemaximizationofthefirm'smarketvalue.InSection3,numericalsolutionsareusedtoshowhowthevalueoftheinvestmentopportunityandtheoptimalinvestmentruledependonthesourceandamountofuncer-tainty,aswellasotherparameters.Section4discussessomeextensionsof thebasicmodel,andSection5concludes.2.TheBasicModel.Consideran investmentina projectwhoseactual costof completionisa random variable,K, and whoseexpectedcost isK= E(K).The projecttakestime to complete;the maximumrateatwhichthefirmcan(productively)investisk.Uponcompletion,the firmreceivesanasset(e.g.,afactoryornewdrug)whosevalue,V,isknownwithcertainty.Iftherewerenouncertaintyoverthetotalcost,valuingtheinvestmentopportunityanddeterminingtheoptimalinvestmentrulewouldbestraightforward.TheprojectwilltaketimeT= K/ktocomplete,sotheopportunitytoinvestisworth:F(K)=maxVe-K/kK/kke-"dt,O]=max[(V+k/r)e-rK/k- k/r, 0](1)whereristhe(risk-free)rateofinterest.Also,theoptimalinvestmentruleistoproceed7In relatedwork,Baldwin(1982)analyzessequentialinvestmentdecisionswhen investmentopportunitiesarriverandomlyandthefirm haslimitedresources.Shevaluesthesequenceof opportunitiesandshowsthata simpleNPVruleleadsto overinvestment,i.e.,thereis avalueto waitingforbetteropportunities.Likewise,if costevolvesstochastically,itmay paytowait forcostto fall.Also,MyersandMajd(1984)determinethevalueof afirm'soptiontoabandonaprojectin returnforascrapvalue,S,whenthevalueof theproject,V,evolvesasageometricBrownianmotion(thefirmhasaputoptiontosellaprojectworthVforapriceS),andshowhowthisabandonmentvalueaffectsthedecisiontoinvestintheproject.withthe projectaslongasF(K) >0, i.e.,aslongasKislessthanacriticalK*,givenby:K* = (k/r)log(1+ rV/k).Figure1showsF(K) forV=10,k=2,andr=0,.1,and.2.Notethatifr=0,F(K) =V - K,andK* =V.But if r>0,F(K) 0,g0_i0,andgK_ 0.Eqn.(2)saysthattheexpectedcosttogodeclineswithongoinginvestment,butalsochangesstochastically.StochasticchangesinKmightbedue totechnicaluncertainty,inwhichcaseg(0,K) =0andgi>0,toinputcostuncertainty,inwhichcaseg(0, K) > 0,ortoboth.8I will againassume that thereis a maximum rate of investmentk.LetF(K) =F(K; V, k)bethevalueof theinvestmentopportunity.ThenF(K) satisfies:F(K) =max Eo[Ve-"- I(t)e-t dt],(3)subjecttoeqn.(2),0 0foranyfiniteK.When- >0,eqn.(16)hasnosolutionwhenr=0,becausethentherewouldbenoreasontoeverinvest.Onewouldalwaysbebetter offwaitinguntilKfellcloseto0sothatthenetpayofffrominvestingislarger.Itwouldnotmatterthatsubstantialtime mighthavetopassforthistohappen,becausenetpayoffswouldnotbediscounted.IfI=0,kislognormallydistributed.Then-ycanbeinterpretedasthestandarddeviationofpercentagechangesperperiod(inthiscase,ayear)inK.Determininga valuefor-7 thatisreasonabledependsonthemakeupof cost.Forexample,a valueof.20wouldbehighforwagerates,butwouldbelowforcommodityinputssuchascopper,steel,oroil.Figure5showsnumericalsolutionsofeqn.(16)fory = 0,.2and.4.(Ineachcase,V= 10,k=2,r=.05,and0=0.)Observethatevenwhenyis.2,thereisasubstantialeffectonthevalueoftheinvestmentopportunity(particularywhenKislarge),andonthecriticalcutoffK*.Wheny =.2,K* isabouthalfof whatitiswhen-y =0,sothatacorrectnetpresentvaluerulewouldrequirethepayofffromtheinvestmenttobeabouttwiceaslargeastheexpectedcostbeforetheinvestmentisundertaken.ThisissimilartothekindsofnumericalresultsobtainedbyMcDonaldandSiegel(1986)andMajdandPindyck(1987)foruncertaintyoverthepayofftoaninvestment,andshowsthattheeffectsofinputcostuncertaintycanalsobequantitativelyimportant.Figure6showsthedependenceofF(K)andK*on0,i.e.,ontheextenttowhichfluctuationsinKarecorrelatedwiththeeconomyandthestockmarket.Recallthat=OPxm=OPKm.Areasonablevalue for0, the marketpriceofrisk,is0.4,so wewould expecttobelessthanthis,perhapsonthe orderof.1 to.3.Figure6 showsF(K) for0=0,.3,andforillustrativepurposes,.6.Asisclearfromthisfigure,a valueof ontheorderof.1willhaveonlyaneglibleeffectonF(K) andK*.Foravalueof.3,however,theeffectislarge,andreducesK* byaround25percentcomparedto0=0.Thusinputcostuncertaintywithalargesystematiccomponentcanhavea substantialimpactonthedecisiontoinvest.TheGeneralCase.Thevalueofthe investmentopportunityandthecriticalexpectedcostK* canbe foundforanycombinationof/,7,and0bynumericallysolvingeqn.(6)anditsassociatedboundaryconditions.Sinceincreasesin/andy (orq)haveoppositeeffectsonK*,itisusefultodeterminetheneteffectforcombinationsofthese parameters.Figure7andTable1showK*asafunctionofbothPandy, for0=0,V=10,k=2,andr=.05.NotethatK*decreaseswith- andincreaseswith/,butismuch6moresensitivetochangesin7.WhateverthevalueofP,a7of0.5reducesK* toaboutafifthofthevalueithaswhen7= 0.Also,thisdropinK* wouldbeevenlargeriftherewerea systematiccomponentto theinputcostuncertainty.Thusformany investments,andparticularlyforlargeindustrialprojectswhereinputcostsfluctuate,increasinguncertaintyislikelytodepressinvestment.Theoppositewillbe thecaseonly forinvestmentslikeR&Dprograms,wheretechnicaluncertaintyisfarmoreimportantand/couldeasilyexceed1.Figure8andTable2showF(K; /,7)asafunctionofPand7forK=8.92,whichisthevalueofK* when#=- =0.Thusthegraphshowsthe"premium"inthevalueoftheinvestmentopportunitythatresultsfrom thetwosourcesof costuncertainty.Notethatthispremiumisincreasinginboth/andy, butisagainmoresensitiveto7.Also,ifyislarge(say,0.5),thispremiumchangesverylittlewhen/isincreased.ApplicationoftheModel.Tousethismodelforinvestmentdecisions,onemustarriveatestimatesofthe/and7thatapplytoaproject'scost,and,secondarily,anestimateoforPKm.Inpractice,thisTable1 - CriticalK*asaFunctionofPand 7.(Note:V=10,k=2,r= .05,andq= 0.)000.10.20.30.40.508.92576.61134.94633.75242.88572.25590.18.98446.65044.97563.77202.90162.26810.29.13096.75785.05373.83302.94682.30320.39.37506.93855.18553.93073.02252.36080.49.71687.18755.37114.06743.12742.44380.510.1567.50985.61044.24803.26172.54880.610.6937.90535.89844.46293.42772.67580.711.3288.36916.24024.71683.62302.82710.812.0518.89656.63095.01463.84773.00050.912.8619.50207.08015.34674.10163.19821.013.77010.1667.57325.71784.38483.4180requiresestimatingconfidenceintervalsTobreaktotalcostuncertaintydownaroundprojectedcostforeachsourceof uncertainty.intotechnicalandinputcostcomponents,onecanutilizethe factthatthefirstisindependentoftime,whereasthevarianceof costduetothesecondcomponentgrowslinearlywiththetimehorizon.Forexample,a value fory can be basedon an estimate of a one- or two-standarddeviationconfidenceintervalforcostT yearsintothefutureassumingnoinvestmenttakesplacepriortothattime.TheestimatedT-periodstandarddeviation,&T, wouldcomefrommanagers'experiencewithinputcosts,orcouldbederivedfromanaccountingmodelofcostcombinedwithvarianceestimatesfortheevolutionofindividualfactorinputs.Then,- =T1/VT.Forconsistency,onewouldcheckthatestimatesofUTbasedondifferentvalueofTleadtoroughlythesamevaluefory.Likewise,usingeqn.(15)andaninitialestimateofexpectedcost,K(0),avaluefor/canbebasedonanestimateofthetime-independentstandardTable2- F(K) asaFunctionof/andy.(EvaluatedatK* correspondingto/= 7=0)7/300.10.20.30.40.5001.08772.15533.15884.05354.83450.1.13841.09152.15963.15994.05654.83710.2.20261.09832.16423.16704.06064.84090.3.24281.11492.17533.17474.06924.84560.4.39241.14342.19563.18784.08104.85950.5.51991.19182.22773.21464.09744.87460.6.74991.26502.26973.24404.12404.89200.7.90671.36522.32803.28374.15724.91840.81.16641.49422.39983.34014.19784.94870.91.36061.68482.49393.40244.24604.98841.01.60341.87242.59963.47644.30215.0323deviationofIK. Thatstandarddeviationwouldsummarizemanagers'confidenceintervalsforeachstageoftheproject.4.Extensions.Thissectionshowshowthemodelcanbeextendedtoaccountforuncertaintyoverthefuturevalueofthecompletedproject,andtoallowformoregeneralprocessesforK(t).UncertaintyovertheValueof theCompletedProject.Asbefore,wewilllettheevolutionofKbegivenbyeqn.(4),butwewillalsoassumethatVevolvesstochastically:dV=aVdt +aVdzv, (17)wheretheWienerprocessdzvisassumedtobeuncorrelatedwithdzordw.ThusfuturevaluesofVarelognormallydistributed,andsincetheprojecttakestimetocomplete,thepayoffisnecessarilyuncertain.Forsimplicity,wewillassumethatthereisnosystematiccomponenttoanyoftheWienerprocesses.Thenwecanfindtheoptimalinvestmentruleusingdynamicprogramming,discountingwiththerisk-freerateof interest.Thevalueoftheinvestmentopportunityisagaingivenbyeqn.(3),butwithVnowstochastic,andhencereplacedbyV(T).TheBellmanequationforthisproblemis:rF =max (-I(t) - IF-+KFKK +2K2FKK +o'VFv+,V2Fvv(18)I(t)ThisislinearinI,andequation(7)againapplies.TheoptimalruleistoinvestwheneverKK*(V),sothatI=0,eqn.(18)hasthefollowinganalyticalsolution:F(K, V)= m(K/V)",(23)whereS+a1 +- 1r 2(24)22y2 (2 +2a-a,)2TofindF(K, V)forK