Irreversible investment under interest rate variab

IrreversibleInvestmentunderInterestRateVariability:SomeGeneralizationsLuisH.R.Alvarez1andErkkiKoskela2May27,20031DepartmentofEconomics,QuantitativeMethodsinManagement,TurkuSchoolofEconomicsandBusinessAdministration,FIN-20500Turku,Finland,e-mail:luis.alvarez@tukkk.fi2DepartmentofEconomics,UniversityofHelsinki,FIN-00014UniversityofHelsinki,Finland,andtheResearchDepartmentoftheBankofFinland,P.O.Box160,00101Helsinki,Finland,e-mail:erkki.koskela@helsinki.fiAbstractThecurrentliteratureonirreversibleinvestmentdecisionsusuallymakestheas-sumptionofconstantinterestrate.Westudytheimpactofinterestrateandrevenuevariabilityonthedecisiontocarryoutanirreversibleinvestmentproject.Giventhegeneralityoftheconsideredvaluationproblem,wefirstprovideathoroughmathemati-calcharacterizationofthetwo-dimensionaloptimalstoppingproblemanddevelopsomenewresults.Weestablishthatinterestratevariabilityhasaprofounddeceleratingoracceleratingimpactoninvestmentdemanddependingonwhetherthecurrentinterestrateisbeloworabovethelongrunsteadystateinterestrateandthatitsquantitativesizemaybeverylarge.Allowingforinterestrateuncertaintyisshowntodeceleraterationalinvestmentdemandbyraisingboththerequiredexercisepremiumoftheirre-versibleinvestmentopportunityandthevalueofwaiting.Finally,wedemonstratethatincreasedrevenuevolatilitystrengthensthenegativeimpactofinterestrateuncertaintyandviceversa.Keywords:Irreversibleinvestment,variableinterestrates,freeboundaryproblems.JELSubjectClassification:Q23,G31,C61AMSSubjectClassification:91B76,49K15,49J151IntroductionMostmajorinvestmentsareatleastpartlyirreversibleinthesensethatfirmscannotdisinvest.Thisisbecausemostcapitalisindustry-orfirm-specificsothatitcannotbeusedinadifferentindustryorbyadifferentfirm.Eventhoughinvestmentwouldnotbefirm-orindustry-specific,theystillcouldbepartlyirreversiblebecauseofthe”lemons”problemmeaningthattheirresalevalueisoftenbelowtheirpurchasecost(cf.DixitandPindyck1994,pp.8-9).SincetheseminalworkbyArrow1968andNickell1974,1978,whoanalyzedirreversibleinvestmentsundercertainty,decisionsaboutirreversibleinvestmentsinthepresenceofvarioustypesofuncertaintieshavebeenstudiedextensively(seee.g.AbelandEberly1996,BaldurssonandKaratzas1997,Baldwin1982,BertolaandCaballero1994,Bertola1998,Caballero1991and1999,Demers1991,HartmanandHendrickson2002,Henry1974,HuandØksendal1998,Kobila1993,McDonaldandSiegel1986,Øksendal2001,andPindyck,1998,1991andSarkar2000).Inthesestudiesoptionpricingtechniqueshavebeenusedtoshowthatinthepresenceofuncertaintyandsunkcoststheirreversibleinvestmentisundertakenwhenthenetpresentvalueis”sufficientlyhigh”comparedwiththeopportunitycost.Bernanke1983andCukierman1980havedevelopedrelatedmodels,wherefirmshaveanincentivetopostponeirreversibleinvestmentbecausedoingthistheycanwaitfornewinformationtoarrive.ThevariousapproachesandapplicationsareexcellentlyreviewedandextendedintheseminalbookbyDixitandPindyck1994.Thestudiesmentionedabove,whichdealwiththeimpactofirreversibilityinavari-etyofproblemsanddifferenttypesofframeworks,haveusedtheassumptionofconstantinterestrate.Amotivationforthisassumptionhasbeentoarguethatinterestratesaretypicallymorestableandconsequentlylessimportantthantherevenuedynamics.AsDixitandPindyck1994state:”Onceweunderstandwhyandhowfirmsshouldbecautiouswhendecidingwhethertoexercisetheirinvestmentoptions,wecanalsounderstandwhyinterestratesseemtohavesolittleeffectoninvestment.(p.13)””Second,1ifanobjectiveofpublicpolicyistostimulateinvestment,thestabilityofinterestratesmaybemoreimportantthanthelevelofinterestrates.(p.50)”Althoughthisargumentationisundoubtedlycorrecttoshort-livedinvestmentprojects,manyrealinvestmentopportunitieshaveconsiderablylongplanningandexerciseperi-ods,whichimpliesthattheassumedconstancyoftheinterestrateisproblematic.Thisobservationraisesseveralquestions:Doesinterestratevariabilitymatterand,ifso,inwhatdirectionandhowmuch?Whatistheroleofstochasticinterestratevolatilityfromthepointofviewofexercisinginvestmentopportunities?IngersollandRoss1992havestudiedtheroleofvariabilityandstochasticityofin-terestrateoninvestmentdecisions.Whiletheyalsodiscussamoregeneralcase,intheirmodelthey,however,emphasizetheroleofinterestrateuncertaintyandconsequentlyspecifytheinterestrateprocessasamartingale,i.e.asaprocesswithnodrift.Itisknownonthebasisofextensiveempiricalresearchboththatinterestratesfluctuatealotovertimeandthatinthelongruninterestratesfollowamoregeneralmean-revertingprocess(foranup-to-datetheoreticalandempiricalsurveysinthefield,seee.g.Bj¨ork1998,ch17,andCochrane2001,ch19).Sincevariabilityofinterestratesmaybedeterministicand/orstochastic,weimmediatelyobservethatinterestratevari-abilitycaningeneralbeimportantfromthepointofviewofexercisingrealinvestmentopportunities.Motivatedbythisargumentationfromthepointofviewoflong-livedinvestments,wegeneralizetheimportantfindingsbyIngersollandRoss1992inthefollowingrespects.First,weallowforstochasticinterestrateofamean-revertingtypeandsecond,weexploretheinteractionbetweenstochasticinterestrateandstochasticrevenuedynamicsintermsofthevalueandtheoptimalexercisepo