dynamic security design and corporate financing

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CHAPTER2DynamicSecurityDesignandCorporateFinancing*YuliySannikovContents1.Introduction712.InformationalProblemsinStaticModels742.1MoralHazard752.2AdverseSelection823.SimpleSecuritiesinDynamicModels874.OptimalDynamicSecurityDesignunderMoralHazard904.1OtherModelsthatInvolveDynamicMoralHazard1025.AsymmetricInformationinDynamicSettings1115.1StaticContractsinDynamicSettings1115.2OptimalDynamicContractswithAdverseSelection117References1211.INTRODUCTIONModiglianiandMiller(1958),hereafterMM,arguethatundercertainidealizedassump-tionsfirmcapitalstructureisirrelevant,i.e.itdoesnotaffectfirmvalue.Thesecondi-tionsinclude:1.Therearenotaxesorbankruptcycosts.2.Therearenoagencyproblems.3.Therearenodifferencesininformationbetweeninsidersandoutsideinvestors.4.Capitalmarketsarefrictionless,i.e.marketparticipantsareperfectlycompetitiveandhavenomarketpower,andthereisnosecuritymispricing.Ofcourse,weknowthattheseidealizedconditionsclearlyfailinpractice.TheimportantmessageoftheModigliani–Millertheoryisthatitrulesoutcertaincom-monlyusedargumentsaboutcapitalstructureasincorrectorincomplete.Thesearguments*IammostgratefultoMiltonHarrisforhisgenerousguidanceandextensivesuggestionsduringthepreparationofthismanuscript.IwouldalsoliketothankNengWang,BrettGreen,BrendanDaley,AlexeiTchistyi,andThomasMariottiforhelpfulcomments.HandbookoftheEconomicsofFinance©2013ElsevierB.V.http://dx.doi.org/10.1016/B978-0-44-453594-8.00002-1Allrightsreserved.71 72YuliySannikovtypicallyfailtotakeintoaccountthattheriskinessofsecuritiesusedincapitalstructure,andthustheirrequiredreturn,dependonthecapitalstructureitself.Herearesometypicalexamples:A.Anargumentagainsthighercapitalrequirementsforbanks:becauseequityhasahigherrequiredreturnthandebt,requiringbankstoholdmoreequityintheircapi-talstructuretoabsorbriskwillmakebankslessprofitable.Thisargumentfailstotakeintoaccountthatadecreaseinleveragemakesequitylessrisky,andthuslowersthecostofequity.1B.Anargumentthatcallsontechnologyfirms,suchasApple,topayouttheircashholdings,becausecashearnsamuchlowerrateofreturnthanthesefirms’costofcapital.Thisargumentfailstotakeintoaccountthatcashislessriskythantherestofthefirmandsotherequiredreturnoncashissignificantlylowerthanthecostofequity.C.Anargumentthatmergingfirmscancreatevaluefortheirshareholdersthrough“corporatediversification”.Thisargumentignoresthatshareholderscandiversifythemselves.Ofcourse,therearevalidcontributionstothesedebatesthatarebasedontheviolationsofModigliani–Millerconditions.Thesefocusontherelationshipbetweenleverage/cashholdingsandincentives,distinctionsbetweeninsideandoutsideequity,andbankruptcycosts.Thisessayexplorestheimplicationsofthesefrictionsoncapitalstructure.TheModigliani–MillerPropositions.Considerafirmthatgeneratesarandomstreamofcashflows:{˜c1,c˜2,...}attimepoints1,2,etc.Firmcapitalstructuredividesthesecashflowsbetweendebtholdersandequityholders,andpossiblyothersecurityholders,insomeway.AccordingtoMM,thevalueofthefirmdoesnotdependonhowthesecashflowsaredividedamongthefirm’sstakeholders.Behindthisresultistheprincipleofarbitrage.Usingfrictionlessfinancialmarkets,participantsshouldbeabletoreplicatethepayoffofanystructuredsecuritybackedbythefirm’scashflows,withzerotransactioncostsandzeromarketimpact.ThemostbasicexamplethatillustratestheMMpropositionsinvolvesaone-periodfirmthatgeneratesasinglerandomcashflowofx˜>0atdate1.SupposethatthefirmhasdebtwithfacevalueDandtherisk-freeinterestrateisr.Thenatdate1,debthold-ersget(1+r)Dandequityholdersgetx˜−(1+r)D,assumingthatx˜isalwayslargeenoughtomakedebtrisk-free.DenotebyEthemarketvalueofthisfirm’sequity.1However,whentaxadvantagesofdebtexist,thentheoverallcostofcapitalcaneasilyincreasewithstricterconstraintsonleverage. DynamicSecurityDesignandCorporateFinancing73MMProposition1Assumethatthereisanidenticalall-equityfirm2,withvalueV.Thatis,firm2alsogeneratesacashflowofx˜>0atdate1.Then:D+E=V.Thatis,thetwofirmswithdifferentcapitalstructurebutthesamecashflowshavethesamevalue.Proof.Theproofusestheprincipleofarbitrage.IfD+E>V,considersellingshortafractionα>0ofequityoffirm1,borrowingαD,andusingαV<αE+αDoftheproceedstobuyafractionαoffirm2.Thispositioncanbeliquidatedatnocostatdate1,sincetheproceedsαx˜fromthestakeinfirm2arejustsufficienttorepurchaseequityoffirm1forα(x˜−(1+r)D)andhave(1+r)αDlefttopaydowndebt.Atthesametime,thistradingstrategygeneratesaninstantaneousprofitofαE+αD−αVatdate0.Thatis,ifD+E>Vthanthereisanarbitrageopportunity.IfD+Er,thenasfirmleverageD/Eincreases,thecostofequityrEincreases.Thatis,asequitybecomesriskierwithgreaterleverage,equityholdersrequireahighercompensationforrisk.MMProposition2isrelatedtoexampleAabouttheamountofequitythatbanksarerequiredtoholdtoabsorblosses.UnderMMassumptions,highercapitalrequire-mentswouldnotmakebankslessprofitablebecauselowerleveragewouldleadtoalowercostofequity.Ofcourse,violationsoftheMMassumptionsimmediatelyenterthedebate.Akeycounterargumentisthatdebthasataxadvantage,andthusleveragecanreducethecostofcapital.Thefocusofthisessayistherelationshipbetweeninformationandcapitalstructure.Insiders,e.g.firmmanagers,mayhaveinformationaboutfirmfundamentalsthatthemarketdoesnotknow.Theremayalsobeaconflictofinterest,i.e.agencyproblems, 74YuliySannikovbetweenfirminsidersandoutsideinvestors.Modelsoftheseinformationalproblemspredictaspecificdivisionofcashflowsbetweeninsidersandoutsiders.Atypicalresultisthatinsidersmustholdanequity-likesecuritybackedbythefirm’sassets.Suchasecurityallowsinsiderstosignalgoodinformationaboutfirmfundamentalsinsettingswithadverseselection,anditgivesinsidersincentivestotakeactionsthatincreasefirmvalueinsettingsofmoralhazard.Whilemodelsofagencyproblemsandasymmetricinformationhaveveryclearimplicationsonthedivisionofcashflowsbetweeninsidersandoutsiders,theytypi-callysayverylittleaboutthedivisionofcashflowsamongoutsiders.Trade-offtheorymodels,suchasthatofLeland(1994),exploretheoptimaldivisionoffirmcashflowsbetweenoutsidedebtandequityholderstakingintoaccounttaxadvantagesofdebtandbankruptcycosts.Therestoftheessayisorganizedasfollows.InSection2wereviewtheoriesofcapitalstructurebasedonstaticmodelsofinformationalasymmetries.InSection3wemoveontoadynamicenvironmentbasedonthemodelofLeland(1994),inwhichweexploretrade-offsbetweenbankruptcycostsandtaxadvantagesofdebt,andincentivepropertiesofsimplecontracts.InSection4weexplorethefull-blownproblemofopti-malcontractsindynamicmoralhazardenvironments.InSection5weexploredynamicadverseselectionandmarketdynamics.2.INFORMATIONALPROBLEMSINSTATICMODELSInthissectionweexplorestaticmodelsofinformationalproblems.Oneclassicearlyref-erenceoncapitalstructureandthescopeofthefirminthepresenceofagencymodelsisJensenandMeckling(1976).Ifafirmmanagerisrisk-neutral,thena100%manage-rialequitystakeinthefirmleadstoanefficientoutcome.Themanagerwillthentakeactionsthatmaximizetheshareholdervalueofthefirm(whichincludesthevalueofnon-pecuniarybenefitsthatthemanagerreceives).Moreover,ifthemanagerhasprivateinformationaboutfirmfundamentals,hedoesnothaveanyincentivetomisrepresentittothemarket.Ifthemanagerisrisk-averse,thentheoptimalsecuritydesignproblembecomesnontrivial.Themostrudimentarywaytocapturerisk-aversioninamodelisbyimpos-ingalimitedliabilityconstraint—themanagercannotconsumenegativeamounts,andamoregeneralwayisbyassumingaconcaveutilityfunction.Ifso,thenitmaybenecessaryandbeneficialforthemanagertosellsomeofhisequitystake,oranothersecuritybackedbythefirm’sassets,toraisefundingforthefirm.Sellingsecuritiestoraisefundscanleadtovariousinefficiencies:reducingthemanager’seffort,requiringcostlymonitoringactionsorinefficientprojectliquidation.Itmayalsobedifficulttosellsecuritiesduetoinformationalasymmetries.Inthissectionweexplorevariousstaticmodelswherethishappens. DynamicSecurityDesignandCorporateFinancing752.1MoralHazardTownsend(1979):Townsend’scostlystateverificationmodel,whichhasbeenadaptedtofinancesettingsbyGaleandHellwig(1985),hasbeenusedwidelyinapplications,includingmacroeconomicsinworkofBernankeandGertler(1989)andBernanke,Gertler,andGilchrist(1999).Thecostlystateverificationmodelcapturesdeadweightcoststhatoutsideinvestorsmightneedtoincurtomonitorthemanager.Amonitoringactioncanberequiredbecausethemanagerprivatelyobservesthefirm’sprofits,whichhemaydivertandrefusetopaybacktoinvestors.Consideranagentwithaprofitableproject,whichneedsaninvestmentofI>0.Theagentdoesnothavethefullamounttoinvest,andneedstoraisesomemoneyfromanoutsideinvestor,theprincipal.Iftheinvestmentismade,theprojecthasarandomgrossreturnx˜,distributedon[0,X]withCDFF.Onlytheagentobservesthetruereturns.However,theprincipalcanverifyreturnsatacostc.Boththeagentandtheprincipalarerisk-neutral,buttheagenthaslimitedliability—hecannotbeforcedtopaybackmorethanwhatheclaimstohave,ormorethanwhatheactuallyhasifverificationtakesplace.Anoptimalcon-tractmaximizestheprincipal’sprofitsubjecttogivingtheagentaspecificexpectedgrosspayoffofW0.ThevalueofW0dependsontheagent’scontributiontowardstheup-frontinvestment,andtherelativebargainingpowersoftheprincipalandtheagent.Assumingthattheprincipalmustcontributeastrictlypositiveamounttoup-frontXinvestment,W0<0xdF(x).Assumingthattheprincipalcanperfectlycommittoanycontract,wecanusetherevelationprincipletoconsideronlytruth-tellingcontracts,inwhichtheagentdirectlyreportsrealizedoutput(seeMyerson,1979).Wecanfocusoncontracts{V,g(x)},whereV⊆[0,X]isthesetofreportsthattheprincipalcommitstoverify,andg(x)≤xisatransferthattheagentisrequiredtomakeifhereportsxandisnotcaughtlying.Iftheagentiscaughtlyingthenwithoutlossofgeneralityweassumethattheprincipaltakesawaytheagent’sentireoutput.Thistransferruleofftheequilibriumpathgivestheagentthemaximalincentivetotellthetruth.Inthisnotation,theoptimalcontractingproblemiswrittenasfollows:max(g(x)−c)dF(x)+g(x)dF(x)v,g(x)V[0,X]VXs.t.(x−g(x))dF(x)=W0,∀x,g(x)x,andthetruth-tellingconstraints.0Thetruth-tellingconstraintsrequirethattheagentbeatleastaswell-offtellingthetruthratherthanlying,afteranyoutputrealization.Ratherthanwritingoutalltruth-tellingconstraintsexplicitly,weprovidealemmathatcharacterizesthesetofallcontractsthatsatisfythetruth-tellingconstraints. 76YuliySannikovLemma1Afeasiblecontractsatisfiesthetruth-tellingconstraintsifandonlyifforsomeconstantD:(A)g(x)=DforxoutsideV.(B)g(x)≤Dforx∈V.Proof(⇒)Iftheagentchoosestoreportinthenon-verificationregion,hewillchooseareportthatinvolvesthesmallesttransfer.Therefore,if(A)fails,itisnotincentive-compatibletotellthetruthinthenon-verificationregion.Similarly,if(B)fails,thenthereisx∈Vwithg(x)>D.Butthentheagentwouldprefertoreportsomethinginthenon-verificationregionthantoreportx.WeconcludethatAandBmustholdinatruth-tellingcontract.(⇐)Ifx∈V,thentheagentweaklypreferstotellthetruthandpayatransferofg(x)ratherthanannouncesomethingoutsideVandpayDorannouncesomethingelseinVandpayx.IfxisoutsideV,thentheagentisindifferentamongallannouncementsthatdonottriggerverification,butweaklyprefersthemtoanyannouncementthatdoesleadtoverification.Hence,truth-tellingincentiveshold.Thefollowingtheoremsolvesfortheoptimalcontract,andshowsthatittakestheformofdebt.Theorem1Theoptimalcontractisastandarddebtcontract,asillustratedinFigure1,i.e.∃Ds.t.(1)V=[0,D)and(2)g(x)=xonVandg(x)=DoutsideV.Figure1Functiong(x)andtheverificationregioninthestandarddebtcontract.ProofSincethecontractmustsatisfythetruth-tellingconstraints,thereexistsDsuchthatg(x)≤Dforx∈Vandg(x)=DforxoutsideV.Then,bythefeasibilityconstraints,theentireinterval[0,D)mustbeasubsetofV.IfwehaveacontractthatdoesnotsatisfytheconditionsoutlinedinTheorem1,asillustratedontheleftpanelofFigure2,wecanstrictlyimproveitintwosteps.First,asillustratedonthemiddlepanel,letus(1)moveallpointsinV∩[D,X]tothenon-verificationregion,(2)raiseg(x)toDonV∩[D,X],and(3)raiseg(x)toxon[0,D).Thenthenewcontractsatisfiesthetruth-tellingconstraintsandgenerateshighertotal DynamicSecurityDesignandCorporateFinancing77Figure2TheproofofTheorem1.surplus(sumoftheprincipal’sandagent’spayoffs),butitgeneratesastrictlylowerpayoffthanW0totheagent.Wethentransfervaluefromtheprincipaltoagent,andimprovesurplusfurther,byloweringDinthesecondstep(asshownintherightpanelofFigure2),tothepointwheretheagent’sexpectedpayoffequalsexactlyW0Remark1Amoralhazardproblemexistswhentheagentcantakeanactionthatyieldshimprivatebenefitandatthesametimereducestheoverallvalueoftheproject.Inthissetting,suchanactionishidingoutput.Asolutiontoamoralhazardproblemrequirestheagenttoholdsometypeofequity-likesecurity.Suchasecuritypreventstheagentfromtakingatleastsomeactionsthataredetrimentaltotheoverallvalueoftheproject.Inthissetting,theagent’ssecurityisequityinthenon-verificationregion,i.e.hegetsamar-ginalpayoffof1foreachincrementaldollarofcashflows.Remark2Weassumedthatonlydeterministicverificationisallowed.Iftheprincipalcouldcommittostochasticverification,hecoulddesignabettercontract.Remark3Weassumedthattheprincipalcanfullycommittoanycontract.Thisassumptionprovidesausefulbenchmarkfortheanalysisofcontractingproblems.Theassumptionofcommitmentcanberelaxed.Inthissetting,iftheprincipalcannotcommittoverifytheagent,wecaninsteadconsidercontractsinwhichtheprincipalcanhavearight,butnotanobligation,toverify.Underthisalternativeassumption,therevelationprinciplenolongerholdsbutanequallyefficientoutcomemaybeattainedunderadditionalconditions.Remark4Themoralhazardproblemhasimplicationsontheoptimalamountofinvestmentandthescaleofthefirm.Generally,theoptimalscaleriseswiththeamountofwealththattheagentisabletocontributeinto 78YuliySannikovtheproject.Inthefollowingexample,theprojectisinfeasibleiftheagentcannotcontributetowardsup-frontinvestment.IftheagentcancontributeanamountE>0,theoptimalscaleriseslinearlyinE.Considerascalableproject,whichgeneratesacashflowuniformlydistributedon[0,3I]whenan√investmentofIismade.CashflowscanbeverifiedatthecostofcI,wherec∈(3−6,3).Thentheprin-cipal’spayoffasafunctionofDis:Dy−cI3IDDD3I−DDDdx+dx=−cI+D=3I−cI−.03ID3I23I3I23IThedebtfacevaluethatmaximizesthisexpressionisD=(3−c)I,andsotheprincipal’smaximalpayoffis(3−c)2I/6.Consequently,themaximumamountthefirmcanborrow,i.e.itsdebtcapacity,isalso√(3−c)2I/6.Sincec∈(3−6,3),thentheprojectdebtcapacityislessthanI,andsotheprojectisinfea-sibleunlesstheagentcontributestoup-frontinvestment.Now,supposetheagentisabletocontributeE>0towardsup-frontinvestment.TheninvestmentgreaterthanE/(1−(3−c)2/6)isinfeasible,becauseinthiscase,theamounttheagentmustborrow,I−E,wouldexceedthefirm’sdebtcapacity.Optimalinvestmentisintheinterval(0,E/(1−(3−c)2/6)):itmaxi-mizestheagent’sexpectedpayoff(3I−D)26Isubjecttotheconstraintthattheprincipalbreakseven,i.e.3I−cI−DD=I−E.Duetothescale23Iinvarianceofthisexample,theoptimallevelofinvestmentisincreasinglinearlyinE.2BoltonandSharfstein(1990)presentsasimpletwo-periodmodelofmoralhazard,whichprovidesausefullinkfromstatictofullydynamicinfinite-horizonmodels.Oneoftheimplicationsofthismodelisthatfutureinvestmentandtheprobabilityofcon-tinuingtheprojectcandependonpastperformanceevenwhenfutureNPVisunrelatedtopastperformance.Amanagerhasanopportunitytooperatethefirmfor2periods,butneedsfinanc-ingfromoutsideinvestors.Ineachperiod,ifthefirmgetsoutsidefundingtomakeaninvestmentofI,itgetscashflowsx1≥0withprobabilityθandx2>x1withprobabil-ity1−θ.Cashflowsarei.i.d.overtime,andthereisnodiscountingbetweenperiods.Figure3illustratespossibleoutcomesifinvestmentisalwaysmade.AsinTownsend(1979),theagencyproblemisthatonlyfirmmanager,andnottheoutsiders,observethetruecashflows.Theagentcannotpretendthathegotalowercashflowthanx1≥0,sincethatistheworstcashflowrealization.However,costlystateverificationisnotpossible,sothemanagercanalwaysdiverttheresidualx2−x1forpersonalconsumptionifhereceivesahighcashflow.2ThispointisdevelopedinthedynamicmodelofDeMarzo,Fishman,He,andWang(2011)thatwediscussinSection4.Inthatmodel,thescaleofinvestmentvariesdynamicallywiththeagents’wealth,whichitselfdependsonthefirm’spastperformance. DynamicSecurityDesignandCorporateFinancing79Figure3PossibleoutcomesinthemodelofBoltonandSharfstein(1990).Assumethatx1I.Thatis,ifcashflowswereverifiableitwouldbeprofitabletoinvestineveryperiod.Bytherevelationprinciplewecanrestrictattentiontocontractsinwhichthemanagerreportstruecashflows,andtransfersfromthemanagertotheinvestorsandtheprob-abilityofcontinuedfinancingdependonthemanager’sreport.Becauseinthesecondperiodtheagentwilltellthetruthonlyifheisrequiredtomakethesametransferregardlessofrealizedcashflow,wecanrestrictattentiontocontractsdefinedby:•Ri,thepaymentthemanagermakesattheendofperiod1ifhereportsxi,•βi,theprobabilityofcontinuedfinancinginthesecondperiodifthereportisxi,and•Ri,thepaymentattheendofperiod2ifthemanagerreportsxiinperiod1.Anoptimalcontractmaximizestheprincipal’sprofitsubjecttogivingtheagentanexpectedgrosspayoffofatleastW0.ThevalueofW0isdeterminedbytheamounttheagentcancontributetoup-frontinvestment,andtherelativebargainingpowersoftheprincipalandagent.Formally,wewouldliketosolvemaxθ[R121+β1(R−I)]+(1−θ)[R2+β2(R−I)]Ri,βi,Ris.t.θ[x121−R1+β1(x¯−R)]+(1−θ)[x2−R2+β2(x¯−R)]W0,x212−R2+β2(x¯−R)x2−R1+β1(x¯−R)(IC2),(IC1),xiiRiandx1Rfori=1,2,wheretheobjectivefunctionistheprincipal’sgrossprofit(afterinitialinvestmentinperiod1).Theconstraint(IC2)guaranteesthatiftheagentreceivesahighcashflowofx2,hispayoffifherevealsittruthfully,theleft-handsideof(IC2),isatleastasgoodashispayoffifhereportscashflowx1instead.Notealsothatwedidnotwriteoutexplicitly 80YuliySannikovtheanalogoustruth-tellingconstraint(IC1)forthelowcashflowrealizationinperiod1.Wedonotexpectittobind,asverifiedafterwederivetheoptimalcontract.Theorem2Anoptimalcontractisasfollows:R1=R2=x1,β2=1and(a)ifW0∈[x¯−x1,2(x¯−x1)]thenR1=x1,R2=2x¯−x1−W0andβ1=W0/(x¯−x1)−1;(b)ifW02(x¯−x1),thenβ1=1andR1=R2=2x¯−x1−W0;(c)ifW050.Inthisequilibrium,instate1thefirmforgoesapositive-NPVprojectduetotheasymmetricinformationproblem.Ingeneral,theinvestmentregion(1−α)y˜αx˜+SasillustratedinFigure5.Figure5Thesetofparameters(x,y)forwhichthefirmsellsequity.Assumingthatthejointdistributionof(x,y)over[0,∞)×[0,∞)ischaracterizedbyastrictlypositivedensity,theequationαE[˜x+˜y|M]=Emusthaveatleastonesolution.Indeed,notethattheleft-handsideiscontinuousinα,anditgoesfrom0atα=0toinfinityasα→1andtheslopeofthelowerboundaryofMgetssteeper.Theorem3InequilibriumfirmswithlowassetvaluexmayissueequitytofinanceaprojectwithnegativeNPV,andfirmswithahighassetvaluemayforgoaprojectwithpositiveNPV.ProofFigure6TheProofofTheorem3. 84YuliySannikovFirst,wearguethatinequilibriumαI=E,Iviolatingtheequilibriumdefinition.Sinceα0,somepositive-NPVprojectsarenottakenbyfirmswithhighx.Ofcourse,itisdifficulttoissueequityinthepresenceofinformationalasymme-tries,becauseequityisverysensitivetoprivateinformation.InthemodelofMyersandMajluf(1984),itisassumedthatthefirmcannotraisefundsthroughothersecurities,suchasdebtorhybridsecurities.Wenextdiscussamodelthatinvestigatesoptimalsecu-ritydesignunderasymmetricinformation,usinggeneralsecurities.DeMarzoandDuffie(1999)considerasettinginwhichanissuerownsassetswithrandomcashflows,andwantstoraisecapitalbyissuingasset-backedsecurities.Theissuerreceivessuperiorprivateinformationabouttheassets’cashflowsaftersecuritydesignstagebutbeforesale.WesawintheoverviewofMyersandMajluf(1984)thatitisdifficulttosellequityinthepresenceofprivateinformation,becauseequityisquiteinformation-sensitive.Herewewillallowforgeneralsecurities,andsolveforoptimalsecuritydesign.Thedesiretoissueismotivatedbytheassumptionthattheissuerislesspatientthanthemarket,e.g.becausetheissuerhassuperiorinvestmentopportunities.Thisiscapturedbyassumingthattheissuerdiscountscashflowsattherateofδ<1,whilethemarketdoesnotdiscount.Themodelhasoneperiod,andtheissuer’sassetsgeneratearandomcashflowofx˜∈[0,∞)attheendoftheperiod.Asecurityischaracterizedbyanon-decreasingfunctiong:[0,∞)→[0,∞),whichgivesthepayouttoinvestorsattheendoftheperiodasafunctionofrealizedcashflow,x.Functionghastosatisfythefeasibilitycon-straintg(x)≤x.Formallythetimelineisasfollows:•theissuerdesignsasecurityg;•theissuerlearnsprivateinformationz;•theissuerchoosesthequantityq∈[0,1]ofthesecuritytosell;•themarketpricesthesecuritiessoldatPg(q)depending,onthequantitysold;•cashflowsarerealized,investorsarepaidaccordingtog,andtheissuerreceivestheremainingcashflow.Theissuer’sexpectedpayoffisgivenbyδE[x−g(x)|z]+δ(1−q)E[g(x)|z]+qPg(q)=δE[x|z]+q(Pg(q)−δE[g(x)|z]).Notethattheissuerdoesnotdiscounttheproceedsfromthesaleofthesecurity,butdiscountsatrateδthecashflowsreceiveduponthematurityoftheassets. DynamicSecurityDesignandCorporateFinancing85Theissuerwouldliketodesignasecuritytomaximizeherexanteexpectedpayoff.Formally,givenasecurityg,theperfectBayesianequilibriumconsistsofamapz→qsuchthat(1)theissuerchoosesqtomaximizeherexpectedpayoff,(2)givenq,themarketformsitsbeliefabouttheissuer’stypeaccordingtoBayesrule,and(3)giventhebelief,themarketpricesthesecurityatitsexpectedpayoff.Theissuer’soptimizationproblemwithrespecttoqinequilibriumismax(Pg(q)−δf),qwheref=E[g(x)|z]istheexpectedpayoffofthesecurity.Sincetheissuer’sexpectedpayoffdependsonhertypezonlythroughfafterthesecurityghasbeendesigned,itisconvenienttothinkoffastheissuer’stypeinthesignalingequilibrium,inwhichthequantityqischosen.Assumingthatinequilibrium,quantitysoldq(f)isacontinuousfunctionofthevalueofthesecurityf,thescheduleq(f)canbecharacterizedfromthefirst-orderconditionqP′(q)+Pgg(q)−δf=0.1/q′(f)fineq.UsingthefactthatinequilibriumPg(q)=fistheinverseofq(f),sothatPg′(q)=1/q′(f),wehave:′q(f)1−1=−⇒q(f)=const·f(1−δ).q(1−δ)fTheconstantofintegrationispinneddownbytheconditionthattheworsttypef0sellsq=1,andso:111−δf01−δf0q(f)=andq(Pg(q)−δf)=q(1−δ)f=(1−δ).δff1−δFromthisexpression,weseethetrade-offsinvolvedinoptimalsecuritydesign.Equityg(x)=xmaximizesf0sinceitsellsallcashflowsoftheworsttypetoinvestors.However,equityisalsoveryinformationallysensitive.Therefore,asassetfundamentalszimprove,f=E[g(x)|z]risesveryfast,leadingtoverylowpayoffsforissuerswithgoodfundamentals.Itmaybepossibletoraisetheexpectedpayoffofissuerswithgoodfun-damentalsbydesigningalessinformationallysensitivesecurity,thatis,makingg(x)lesssensitivetothecashflowx.However,thathasacost,asthevalueoff0endsuplower.Itfollowsthat,incomparisonwithequity,lessinformationallysensitivesecuritiesgivelowerpayoffstoissuerswithlow-qualityassets,butmaygivesignificantlyhigherpayoffs 86YuliySannikovtoissuerswithhigh-qualityassets.Below,wecharacterizeconditionswhenequityistheoptimalsecurity,andwhenriskydebtis.Intuitively,itisoptimaltosellequityiftheseller’sprivateinformationissmall.Theorem4Pr(x|z)1Ifsupx,z,z′x*,asillustratedinFigure7.WehaveE[g(x)−D(x)|z]=E[(g(x)−D(x))πz(x)|z0]E[(g(x)−D(x))πz(x∗)|z0]=0,sinceforallx,(g(x)−D(x))πz(x)(g(x)−D(x))πz(x∗).Intuitively,debtisoptimalbecauseiftheissuer’sprivateinfochangesfromz0toz,thenstandarddebt,intheclassofmonotonesecurities,isonethatminimizestheincreaseintheissuer’sprivatevaluation,andthusilliquiditycosts.3.SIMPLESECURITIESINDYNAMICMODELSInthissection,weexploresimplesecuritiesinafullydynamicmodel,beforewecon-sideroptimaldynamicsecuritydesigninthenextsection.Inparticular,wefocusonthemodelofLeland(1994),whichbuildsuponthedynamicmodelsofMerton,1974andBlackandCox(1976)toexplorethetrade-offsbetweenthetaxadvantagesofdebtandbankruptcycosts.Merton(1974),BlackandCox(1976),andLeland(1994)investigatethemodelofafirmthatpaysnodividendsandwhoseassetvalue,undertherisk-neutralprobabilitymeasure,followsthegeometricBrownianmotion:dVt/Vt=rdt+σdZt.(1)InEqn(1),ZtisastandardBrownianmotionundertherisk-neutralmeasureandristherisk-freerate.Below,wefirstdescribetheclassicriskydebtvaluationmodelofMerton(1974),inwhichdebthasfixedmaturity,andthendescribeLeland(1994),whichfocusesonperpetualdebt.Importantly,usingbothofthesemodels,wearealsoabletodiscusstheincen-tivepropertiesofaparticularcapitalstructure.Thisdiscussion,whichfocusesonthe 88YuliySannikovsensitivitiesofthevalueofequitytothevalueandvolatilityofassets,servesasaprecur-sortothenextsectionthatconsidersoptimalsecuritydesign.Merton(1974)considersacapitalstructurethatconsistsofzero-coupondebtwithpromisedpaymentDandfixedmaturityT.ThefirmdefaultsattimeTifVT0forcalloptions,thereisassetsubstitution.Giventhefixed-debtcontractinplace,equityholderswouldliketoincreasetheriskofassets,andgraduallywilliftheycan.Leland(1994)explorestrade-offsbetweenthetaxadvantagesofdebtandbank-ruptcycostsinamodelwithperpetualdebt,inwhichthefirm’sassetsalsofollow(1).DebtwithacontinuouscouponrateofCissetattime0.Duetotaxeffects,theeffectivecostofthesecouponpaymentstoequityholdersis(1−τ)C,whereτisthetaxrateandτCistheinteresttaxshield.3Intheversionofthemodelwithanendogenousbankruptcychoice,equityholdersdecideateachmomentoftimewhethertokeeppayingcouponsordefaultandforfeittheassetstodebtholders.EffectivelyequityholdershaveaperpetualAmericanputoptiontosellthefirm’sassetsfor(1−τ)C/r.ItisexercisedonlywhenthevalueofassetsdropstothecriticalboundaryVB,whichwederivebelow.Thestationarynatureofthissettingimpliesthatthepricesofallsecuritiesarefunc-tionsofV.Sincetherequiredrateofreturnundertherisk-neutralmeasureisr,thevalueofanysecurityF(V)mustsatisfytheequationCF/F(V)+µF/F(V)=r,t(2)dividendyieldcapitalgainsrate3Whiletheamountofdebtisfixedattime0inthemodelofLeland(1994),Goldstein,Ju,andLeland(2001)provideanimportantextensioninwhichthefirmhasanoptiontoincreasethelevelofdebt,butcannotdecreaseit. DynamicSecurityDesignandCorporateFinancing89whereCFisthecashflowthatthesecuritypaysandµFtisthedriftofF(V).Theappro-priatevalueofCFisCfordebt,−(1−τ)Cforequity,andτCforthefirmasawhole.Using(1)andIto’slemma,µF=rVF′(V)+1σ2V2F′′(V),andsothesecurity-pricingt2Eqn(2)becomes′122′′rVF(V)+σVF(V)+CF=rF(V).2ForanyvalueofCF,thisequationhasageneralsolutionoftheform−2rF(V)=A0+A1V+A2Vσ2.MatchingtheboundaryconditionsatthebankruptcyboundaryVBandinfinity,equityhasvalue−2r(1−τ)C(1−τ)CVσ2E(V)=V−+−VB.rrVBTofindthevalueofthefirm’sdebtD(V),aswellasthevalueofthewholefirmv(V)=E(V)+D(V),assumethatinbankruptcyafractionαoffirmassetsVBislostduetobankruptcycosts.Then−2rCCVσ2D(V)=+(1−α)VB−andrrVB−2rτCτCVσ2v(V)=V+++αVB.rrVBTheboundaryVB,atwhichequityholdersexercisetheiroptiontoabandontheassetsintheendogenousbankruptcycase,canbedeterminedbythesmooth-pastingconditionE′(VB)=0.Thisleadsto(1−τ)CVB=.(3)r+1σ22GivenCandτ,thepointatwhichequityholdersdefaultisindependentoftheini-tialvalueoftheassetsV0orbankruptcycostsα.VBisproportionaltothedebtburden(1−τ)C,andVBdecreasesasassetvolatilitygoesup. 90YuliySannikovEquityholderschoosetheboundaryVBtomaximizetheirpayoff,withouttakingintoaccountthepayoffofdebtholders.Thisleadstoaconflictofinterest.Whendeter-miningtheoptimalcapitalstructureinendogenousbankruptcycaseexante,marketparticipantsrecognizethatexpostequityholdersdefaultattheboundarygivenby(3).Maximizingthevalueofthefirm−2rτCτCV0σ2(1−τ)Cv(V0)=V0+−+αVBsubjecttoVB=rrVBr+1σ22withrespecttoC,wefindthat1−σ22∗r+2σ2rα2r2rC(V0)=V0a+(1−α)+.1−τσ2τσ2Naturally,theoptimalcouponrateisproportionaltoV0.AsintheMertonmodel,equityDeltawithrespecttoassetsinthismodelislessthan1.Deltadecreasesto0asVdropstowardsVB.Thisphenomenoncancausesevereagencyproblemstoarisewhenequityholdersareunderwaterinarichermodelwhereequityholderscantakecostlyactionsthatmayimprovethevalueofthefirm.Forexample,ifequityholderswerealsocontrollinginvestment,theywouldrefusepositive-NPVprojectswhenVgetsclosetoVB.Underalternativeconditionsdeterminingbankruptcy,suchasinthepresenceofprotectivedebtcovenants,agencyproblemscanbelessseverenearbankruptcy.EquityDeltawithrespecttoassetvaluecouldbelargenearbankruptcyifbankruptcyisforcedbythecontract,ratherthandeterminedbytheequityholders’decisiontowalkawayfromtheassets.Theoptimaldesignofthesecurityheldbythe“agent”—thefirm’sinsiders—isthesubjectofthenextsection.Section4presentsseveralexplicitmodelsthatcapturehowtheagent’sactionscanaffectthefirm’svalueinadynamicsetting.4.OPTIMALDYNAMICSECURITYDESIGNUNDERMORALHAZARDThefoundationofoptimalsecuritydesignunderdynamicmoralhazardliesindynamicagencytheories,whichhavebeenstudiedbyRadner(1985),Rogerson(1985),Green(1987),SpearandSrivastava(1987),Abreu,Pearce,andStacchetti(1990),andPhelanandTownsend(1991).Incorporatefinance,discrete-timemodelsofdynamicfinancingundermoralhazardincludetheworkofAlbuquerqueandHopenhayn(2004),ClementiandHopenhayn(2006)andDeMarzoandFishman(2007a,2007b).Continuous-timeprincipal-agentmodels(seeSannikov,2008),offerparticularlytractablemethodsthat DynamicSecurityDesignandCorporateFinancing91havebeenadoptedincorporatefinance.Biaisetal.(2007)isanimportantpaperthatillustratestherelationshipbetweencontinuous-timeanddiscrete-timemethods.WebeginthissectionbyreviewingtheworkofDeMarzoandSannikov(2006),whichcharacterizesoptimalsecuritydesignunderdynamicmoralhazardusingcontin-uous-timemethods,andthatofDeMarzoetal.(2011),whichinvestigatestherelation-shipbetweenagencyproblemsandinvestmentdynamics.Afterthat,webrieflyreviewanumberofothermodelsthatadoptthedynamicagencyframeworktostudyabroadersetofissues.DeMarzoandSannikov(2006)(hereafterDS)modelthesametypeofmoralhaz-ardasthetwo-periodmodelofBoltonandScharfstein(1990)whichwehavealreadyreviewed,ininfinitehorizon.Theagenthasaprofitableinvestmentopportunitythat,iffundedwithaninitialinvestmentofI>0,wouldgenerateacashflowstreamoftheformdYt=µdt+σdZt,whereZtisastandardBrownianmotion.Theagentwouldliketowriteacontractwiththeprincipal(aninvestor)tofundthisprojectanduseitasacollateraltofundtheagent’sconsumption.Theagentwouldliketoborrowmoneyagainstthisprojectbecausehehasahigherdiscountrateγthanthemarketrater,andalsobecausehemaynothavesufficientfundstofinancetheup-frontinvestment.However,anagencyproblemexistsbecausetheagentprivatelyobservesthetruecashflowsYt.Thus,theagentmaydivertsomeofthecashflowsforpersonalconsump-tionorasperks.TheprincipalonlylearnsaboutcashflowsthatareleftYtaftertheagent’sdiversionaction,possiblythroughtheagent’sreport.AssumethatthecumulativeamountofcashdivertedYt−Ytmustbeacontinuousnon-decreasingprocess(i.e.theagentcannotsendtheprincipalahighercashflowthanrealized).Itisinefficienttodivertcash,sothattheagentisabletoenjoyonlyafractionλ∈(0,1]ofdivertedcash.Inordertosolvetheagencyproblem,thecontractbetweentheprincipalandtheagentmayforceaterminationoftheprojectintheeventthattheobservedcashflowsYtareinsufficient.Intheeventoftermination,theprincipalreceivestheproject’sassets,whichhevaluesatL,andtheagentpursueshisoutsideoptionwithvalueR.Terminationisinefficient,i.e.µ>rL+γR.DSderivetheoptimalcontractbetweentheagentandtheprincipalinthisset-ting,whichmaximizestheprincipal’sprofitsubjecttoasetofconstraints,includingtheincentiveconstraints.Therearenorestrictionsregardingtheformofcontractsallowed.Thatis,contractscanspecifyinthemostgeneralwayhowthetermination 92YuliySannikovtimeτandtheagent’scompensationdCtdependonthehistoryofobservedcashflows{Ys,s∈[0,t]}.Becausethemarginalbenefittotheagentfromdivertingcashisaconstantλ,andbecausetheagentcandivertcashatanunboundedrate,theoptimalcontractshouldnotallowanycashflowdiversion.Intuitively,itischeapertoallowtheagenttoconsumesomeoftheproject’scashflowsdirectly,ratherthanthroughcashflowdiversionbehindtheprincipal’sback.Formally,theoptimalcontractsolvesthefollowingconstrainedoptimizationproblem:τmaxEYe−rt(µdt−dCt)+e−rτL(4)τ,C0τsubjecttoEYe−γtdC−γτt+eR=W0and0τY−γt−γτEe(dCt+λ(dYt−dYt))+eRW00forallstrategies{Yt}oftheagent,wheresuperscriptsYandYovertheexpectationhighlightthattheagent’scompensa-tiondCt,aswellastheterminationtimeτ,dependontheagent’sreports.ParameterW0inthefirstconstraintistheagent’sexpectedpayoff,whichisdeterminedbytherelativebargainingpowersoftheprincipalandagent.Problem(4)isfeasiblewheneverW0≥R.Thesecondsetofconstraintsrestrictscontractstothoseinwhichtheagentpreferstoabstainfromcashflowdiversion.Thissetofcontractscanbecharacterizedviaasimpleconditiononthesensitivityoftheagent’scontinuationpayofftoobservedcashflows{Yt}.Foragivencontract{τ,C},theagent’scontinuationpayoffwhenherefrainsfromcashflowdiversionisgivenbyτWY−γ(s−t)−γ(τ−t)t=EtedCs+eR.tTheorem6Foranycontract{τ,C}thereexistsaprocessβtsuchthatdWt=γWtdt−dCt+βt(dYt−µdt)(5)anditisoptimalfortheagenttorefrainfromcashflowdiversionifandonlyifβt≥λforallt≤τ.Alsocon-versely,ifaprocessWt≤Rfollows(5)untiltimeτ,atwhichWτ=R,andsatisfiesthetransversalityconditionE[1t≤τe−γtWt]→0ast→∞,thenWtistheagent’scontinuationpayoff. DynamicSecurityDesignandCorporateFinancing93ProofSeeDS.Therepresentation(5)canbederivedusingtheMartingaleRepresentationTheorem,byfocusingonthemartingale:tτ−γs−γtYe−γsdC−γτ.Vt=edCs+eWt=Ets+eR00ThetermsγWtdt−dCtinthelawofmotionofWtareduetosimpleaccountingfortheagent’sdiscountrate,andpayoffreceivedbytheagentthroughconsumption.Theincentiveconditionisβt≥λbecausetheagentrefrainsfromcashflowdiversionifhereceivesatleastλinthepresentvalueofhisfuturepayoffforeachdollarofcashflowreportedtotheprincipal.DuetoTheorem6,theprincipal’sproblem(4)canbereducedtoastochasticcontrolproblem.CorollaryThereisaone-to-onecorrespondencebetweencontracts{τ,C}thatsatisfytheconstraintsofproblem(4)andcontrolledprocessesdWt=γWtdt−dCt+βt(dYt−µdt)σdZtthatsatisfythetransversalityconditionundercontrols(Ct,βt≥λ),withτdefinedasthefirsttimewhenWthitsR.Thecontrolproblemistomaximizetheprincipal’sobjective(4)byachoiceofcon-trols(Ct,βt≥λ)thatdrivethestateWtaccordingto(5),subjecttotheconstraintthattimeτoccurswhenWthitsRforthefirsttimeaswellasthetransversalitycondition.Tosolvethecontrolproblem,wecanfollowanintuitivelineofreasoningtoconjectureasolution,andthenverifythatthesolutionisoptimal.Intuitively,contractinginthissettinginvolvesinefficienciesbecausetheagentneedsincentivestoreportcashflowstruthfully.Theprincipalprovidestheseincentivesbytyingtheagent’scontinuationpayofftoperformancethroughthesensitivitycoeffi-cientβt≥λ.Thisintroducesvolatilityintheagent’spayoff.AbadhistoryofcashflowsdrivesWtdowntoR,andnecessitatesinefficienttermination.Duetothisintuition,thechoiceβt=λ,whichminimizesthevolatilityofWtwhilestillsatisfyingtheincentiveconstraints,isoptimal. 94YuliySannikovThechoiceoftheagent’scompensationalsoinvolvesaninterestingtrade-off.Becausetheagent’sdiscountrateγishigherthanr,itisexpensivefortheprincipaltopostponepaymentstotheagent.However,accordingto(5),earlypaymentstotheagentreducetheagent’scontinuationpayoffWtandincreasethechancethatWthitsRduetoacashflowdrawdown.ThisconcernisparticularlypressingwhenWtislow,sothatthechanceofterminationissignificant.Therefore,weconjecturethatthereisacriticallevelofW>RsuchthatitisoptimaltopaynothingtotheagentwhenWtW,thenattime0theagentreceivesalump-sumpaymentofdC0=W0−W.TerminationoccurswhenWthitstheboundaryRforthefirsttime.WestillneedtodeterminetheoptimallevelofW.Wecandothatbyvaluingthesecuritythattheprincipalisholdingunderthiscontract,andthenfindingWthatmaximizesthevalueofthatsecurity.ThevalueofthissecurityisafunctionofcurrentWt.AsinLeland(1994),wecanvaluetheprincipal’ssecuritythroughequationCF/F(W)+µF/F(W)=r,tdividendyieldcapitalgainsrateFwhereCFisthecashflowthatthesecuritypaysandµtisthedriftofF(W).On[R,W),dCt=0andsotheprincipal’sexpectedcashflowisμ.Thus,usingIto’slemma,F(W)mustsatisfy′122′′µ+γWF(W)+λσF(W)=rF(W)2(6)µFtforallW∈[R,W).Thissecond-orderordinarydifferentialequationrequirestwoboundaryconditionstosolve.First,F(R)=L,becausethevalueoftheassetstotheprin-cipalisLwhenterminationoccurs.Second,F′(W)=−1,becauseatthepointwheretheagentgetspaiditcoststheprincipalonedollartogivetheagentanextradollarofutility.ForanychoiceofW,thecorrespondingsolutiongivestheprincipal’svalueofthecontract,inwhichpaymentstotheagentaremadeatpointW. DynamicSecurityDesignandCorporateFinancing95So,whatistheoptimalchoiceofW?Toanswerthisquestion,inFigure8weillus-tratethephasediagramofsolutionstoEqn(6)startingwithboundaryconditionF(R)=L,fordifferentlevelsofF’(R).ForanyW≥R,itcanbeshownthatF’(W)isincreasinginF’(R).4Inthisdiagram,W=W2maximizestheprincipal’sprofit.Thischoicecorrespondstothetopsolution,whichhasasingleinflectionpointW2whereF′(W2)=−1.Thatsolutionisconcaveontheinterval[R,W2],andthenitbecomesconvex.ForanyothervalueofW,theprincipal’ssecurityhaslowervalue.ThebottomsolutioncorrespondstotwodifferentchoicesofW,W1,andW3.Theslopeattheinflectionpoint,whichisbetweenW1andW3,issteeperthan−1.Thebottomsolutiondoesnotcorrespondtotheoptimalcontractbecause,bycontinuityofsolutionstodifferentialequationsinini-tialconditions,asolutionwithaslightlyhigherslopeF’(R)stillreachesslope−1atsomepointW>R,anditisthereforesuperior(i.e.itishigherthanthebottomsolu-tionateverypointW,anditrepresentstheprincipal’svaluefunctionunderthecontract,inwhichtheagentispaidatpointW).5NotealsothatsinceF′(W)=−1andsolutionFhasaninflectionpointatW,i.e.F′′(W)=0,Eqn(6)impliesthatµ−γW=rF(W).Thisequation,whichcorrespondstothedashedlineinFigure8,canbetakenasaconditionthatdeterminestheoptimallevelofW.Itcanbeinterpretedasfollows:itmakessensetopostponepaymentstotheagenttoreducethelikelihoodoftermination,butonlyuptothepointwheretheexpectedcashflowsexhausttherequiredreturnsoftheprincipalandtheagent.Wecanverifythatthiscontract,whichweguessedusingintuitivereasoning,isindeedoptimalusinganargumentthatwesketchbelow:Asketchoftheverificationargument.LetfunctionF,togetherwiththepointW,bedeterminedon[R,W]byEqn(6)andboundaryconditionsF(R)=L,F′(W)=−1andµ−γW=rF(W).LetusextendFbeyondWlinearlywithslope−1.WewillshowthatthereisnocontractwithvaluehigherthanF(W0)totheprincipal.Foranarbitrarycontract{C,τ},inwhichtheagent’scontinuationvaluefollows:dWt=γWtdt−dCt+βtσdZt,t−rs−rtconsidertheprocessGt=0e(µdt−dCt)+eF(Wt).4Ifnot,thentwosolutionsof(6),FandFsuchthatF′(R)>F′(R)wouldhavethesameslopeF′(W)=F′(W)atsomepointW>R.TakethelowestsuchW.ThenF′(W′)>F′(W′)andforallW′∈[R,W)andsoF(W)>F(W).However,thenEq.(6)impliesthatF′′(W)>F′′(W),soF′(W−ε)<F′(W−ε)forsomesmallε>0,acontradiction.5NotealsothatasinglechoiceofWisoptimalforallvaluesofW0. 96YuliySannikovFigure8FunctionsF(W),suchthatF(R)=L,whichsolveEqn(6)withdifferentboundaryconditionsF’(R).WeclaimthatGtisasuper-martingale.DifferentiatingGtwithrespecttot,weget−rt′−rF(W′dGt=e(µdt−dCt(1+F(Wt))t)dt+(γWtF(Wt)0122′′−rt′+βtσF(Wt))dt)+eβtσF(Wt)dZt.2λ2σ2F′′(Wt)UsingthefactthatF"(Wt)≤0,andF’(Wt)≥−1,weseethatGtisasuper-martingalebyfocusingonitsdrift.Therefore,theprincipal’sprofitfromthiscontractis:τEe−rt(µdt−dC−rτt)+eL=E[Gτ]G0=F(W0).0WefinishdiscussingDSbyoutliningoneparticularcapitalstructurethatimplementstheoptimalcontract,andmentioningafewcomparativestaticsresults.Theoptimalcontractcanbeimplementedinmanyways,butoneparticularlyattractiveimplementationinvolvesacreditline.Toconstructtheimplementation,wemapWtintotheoutstandingbalanceMtonacreditline,sothatpointWcorrespondstobalanceMt=0,andpointRcorrespondstothecreditlimit(W−R)/λ.ThenMt=(W−Wt)/λevolvesaccordingtodCtγdMt=γMtdt−dYt++µ−Wdt.λλThisleadstoacapitalstructureconsistingofacreditline,perpetualdebtandequity.Theagentholdsafractionλofequity.Theprincipalholdsperpetualdebt,whichreceives DynamicSecurityDesignandCorporateFinancing97γaflowofpaymentsofµ−W,andthecreditline,whichreceivestheprojectcashflowsλnetoftheperpetualdebtpaymentsanddividends,andthefraction1−λofequity.6TotaldividendsonequityaredCt/λ,andtheyarepaidonlywhenMt=0,i.e.thecreditlineisfullypaidoff.Theinterestrateonthecreditlineequalstotheagent’sdiscountrateγ.Thecontracttriggersterminationwhenthecreditlineisdrawntothelimit.Notethatthecashflowsonthesecuritiesheldbytheprincipal(perpetualdebt,creditline,andequity)arethesameasintheoptimalcontractbasedonWt.Indeed,whenWt=[R,W)thentheprincipalreceivesjustthereportedcashflowsdYt.γPerpetualdebtpaysaflowof(µ−W)dt,andthecreditlinereceivesthepaymentsλγofdYtnetofthecouponpaymentsonperpetualdebt(µ−λW)dt.Theinterestratechargedonthecreditlineincreasesitsbalance,butdoesnotgenerateanactualcashflow.AtpointWt=W,theagentreceivesdCtineachcontract,andtheprincipalreceivestherest,dYt−dCt.DSverifythatunderthisimplementation,theagenthasincentivestorefrainfromcashflowdiversion,andinsteadchoosestousefirmcashflowstopaydownthecreditline,andpaydividendsonlywhenthecreditlineisfullypaidoff.Thecontinuous-timeformulationmakesanalyticcomparativestaticspossibleinthisdynamiccontractingsetting.Forexample,theoptimalmixofcreditlineanddebtdependsonthevolatilityofcashflowsσandtheagent’sdiscountrateγ.Theimple-mentationusesalongercreditlinewhenσislargerorγissmaller.SeeDSfordetails,andothercomparativestaticsresults.Besidescreatingaconvenientmethodologythatisapplicabletostudyarangeofissues,themodelofDShasanumberofimportanteconomicimplications.Theoptimalcontractclearlydividestherisksbetweentheagent,thefirminsider,andoutsideinves-tors.Themodeldoesnothavespecificpredictionsregardingthedivisionofcashflowsamongoutsideinvestors:theModigliani–Millertheoremholdswithrespecttothosecashflows.Theimplementationoftheoptimalcontractintheformofacreditlinesuggestsonewaytodividethesecashflowsbetweenoutsideequityholdersanddebtholders.Certainly,theimplementationisnotunique,e.g.Biaisetal.(2007)provideanalternativeimplementationthatmapstheagent’scontinuationpayoffWtintothefirm’scashbal-ance.Inanycase,itisconvenienttolinkWttosomemeasureofthefirm’sfinancialslack.Inthisinterpretation,themodelhasanumberofimportantpredictions.Pastperfor-manceispositivelyrelatedtothefirm’spayouts,andthefirm’sfinancialslack.Followingpoorperformance,firmsstoppayingdividendsandmaybeliquidatedinefficiently,evenwhenpastperformanceisuncorrelatedwithfutureprofitability.Thefirm’smanagershouldbeexposedtotheriskofthefirm—heshouldbecompensatedwithanon-tradablestakeofthefirm’sequity.6γW<0,thenthecashinflowsthatreducetheagent’screditlinebalancecanbeinterpretedasIfµ−λinterestpaymentsfromacompensatingbalancethattheagentholdswiththeprincipal,inordertoaccessaparticularlylong(inthiscase)creditline. 98YuliySannikovRemarkDSalsopresentavariationofthemodel,inwhichtheagencyprobleminvolvescostlyeffortinputratherthancashflowdiversion.Specifically,theyassumethattheprincipalobservescashflows0iftheagentworks,dYt=dYt−atdt,whereat=Aiftheagentshirks.TheagentgetsaprivatebenefitofB=λAifheshirks.Theyshowthattheoptimalcontractisthesameasinthebaselinemodel(withcashflowdiversion),anditgivestheagentincentivestoworkuntiltheterminationtimeτ,ifandonlyifthefollowingconditionissatisfied:µ−AγλA′minF(w)+−wF(w).(7)rwrγZhu(2011)solvesfortheoptimalcontractwithshirkinginthisvariationoftheDSmodel,whencondi-tion(7)isviolated.WesummarizethefindingsofZhu(2011)inSection4.1.DeMarzoetal.(2011)(hereafterDFHW)investigatehowdynamicagencyproblemsaffectthescaleofthefirm.TheyusetheagencymodelofDS,andaddinvestmentdeci-sionsthatareobservableandcontractible.Thefirm’scapitalstockevolvesaccordingtodKt=(�(ιt)−δ)Ktdt,whereιtisthecostofinvestmentperunitofcapital,andfunctionΦsatisfiesΦ(0)=0,Φ’>0andΦ”≤0.Intheabsenceofinvestment,capitalsimplydepreciatesatrateδ.TheconcavityoffunctionΦreflectsadjustmentcosts.7Afteraccountingforinvestmentandadjustmentcosts,thefirm’scumulativecashflowprocesstakestheformdXt=Kt(dYt−ιtdt),wheredYt=atµdt+σdZt.Theagent’sactionat≤1reducesthemeanofcashflows.Settingat<1isinefficient,butitgivestheagentaprivatebenefitflowofλ(1−at)Kt.Intheeventoftermination,whichmayberequiredtosolvetheagencyproblem,theagentreceivesthepayoffof7DFHWincorporateadjustmentcostsinthefirm’scapitalaccumulationequationinawaythatisdifferentfrom,butequivalentto,theformpresentedhere.Specifically,theymodelthefirm’sinvestmenttechnol-ogyintheformdKt=(It−δKt)dtandthecashflowsintheformdXt=Kt(dYt−c(It/Kt)dt)−Itdt,wherecisaconvexadjustmentcostfunction.Thetwoformulationsareequivalent:theypresenttwowaysofdescribingthefirm’sproductionsetofexpectedoutputandnewcapital,perunitofexistingcapital.Theproductionsetis(atμ−c(i)−i,i−δ)undertheformulationofDFHW,and(atμ−ι,Φ(ι)−δ)undertheformulationweworkwithabove. DynamicSecurityDesignandCorporateFinancing99R=0,andtheprincipalreceivesvalueqKtfromtheproject’sassets,whereq�′(−∞)1becausethefirmcanalwaysliquidatebydisinvesting.Theagent’sdiscountrateγishigherthantheprincipal’sdiscountrater.TherangeofpossiblecontractsspecifyhowthecashflowsthattheagentisallowedtokeepCt,theterminationtimeτ,andtherateofinvestmentιtdependontheentirehistoryoffirmperformance{Ys,s∈[0,t]}.ForthesamereasonsasinDS,weknowthattheoptimalcontracthastocreateincentivesfortheagenttoalwayschooseat=1.Formally,theoptimalcontractingproblemis:ττa=1e−rt(µ−ι−rt−rτmaxEt)Ktdt−edCt+eqKττ,dC,ι00τsubjecttoEa=1e−γtdCt=W0and0τEaˆe−γt(dCt+λ(1−ˆat)Kt)W00forallstrategies{ˆat}oftheagent.Inthissetting,foranarbitrarycontract(τ,C,ι),thelawofmotionoftheagent’scontinuationpayoffwhenhefollowsthestrategy{at=1}isdWt=γWtdt−dCt+βt(dXt+Kt(ιt−µ)dt).KtσdZtTheincentiveconstraintisβt≥λ.Theprincipal’svaluefunctionF(Wt,Kt)undertheoptimalpolicydependsonbothWtandKt.ItmustsatisfytheBellmanequationr=maxCF/F(W,K)+µF/F(W,K).i,dC,βtdividendyieldcapitalgainsrateDuetothescale-invariancepropertiesofthemodel,DFHWconjecturethatthevaluefunctionhastheformF(W,K)=f(w)K,wherew=W/K.Thenwt=Wt/Ktfollows:dCtdwt=(γ−�(ιt)+δ)wtdt−+βtσdZt,KtandtheBellmanequationreducesto′′122′′rf(w)=maxµ−ι−dc(1+f(w))+(�(ι)−δ)f(w)+(γ−�(ι)+δ)wf(w)+βσf(w),ι,dC,β2 100YuliySannikovwheredc=dCt/Kt.ThesolutiontothisequationcanbefoundusingsimilarlogicasweusedinthediscussionofDS.Theorem7Theprincipal’svaluefunctionisoftheformF(W,K)=Kf(W/K).Functionfisconcave,andsatisfiestheequationrf(w)=maxµ−ι+(�(ι)−δ)f(w)+(γ−�(ι)+δ)wf′(w)+1λ2σ2f′′(w)(8)ι2ontheinterval[0,w]determinedbytheboundaryconditionsf(0)=q,f′(w)=−1andf′′(w¯)=0.Intheoptimalcontract,theagent’scompensation,investmentandterminationtimeτaredeterminedbytwostatevariables:KtandWt.ThelawofmotionofWtisdeterminedcontractuallybydWt=γWtdt−dCt+λ(dXt+Kt(ιt−µ)dt),wherepaymentsdCtaresetto0forWtqa(w)qm(w).Empirically,theaverageq(whichiseasytomeasure)isoftenusedempiricallyasaproxyforthemarginalq,followingtheresultsofHayashi(1982).However,eventhoughthemodelofDFHWexhibitsthesamehomogeneitypropertiesasHayashi(1982),herethemarginalandaverageqarenotthesame,duetoagencycosts.Thedifferencebetweenaverageandmarginalq’sdependonw,andthusthehistoryoffirmperformance.Themodelhasthefollowingpredictionsabouttherelationshipbetweeninvestment,Tobin’sq,andthefirm’sfinancialslackw:•Financialslackispositivelyrelatedtopastperformance.•Averageandmarginalq,aswellasinvestment,areincreasingwithfinancialslack.•Theagent’scashcompensationincreaseswithfinancialslack.•Themaximalleveloffinancialslackishigherforfirmswithmorevolatilecashflowsandlowerliquidationvalues.Ingeneral,themodelpredictsthatinvestmentispositivelycorrelatedwithprofits,pastinvestment,financialslack,andmanagerialcompensation,evenwithtime-invariantinvestmentopportunities. 102YuliySannikovDSaswellasDFHWserveasamicrofoundationoffinancingfrictionsthatexistinthepresenceofagencyproblems.Manypapersassumeasetoffinancialfrictions,insteadofderivingthem,andinsteaddevoteattentiontotheimplicationsofthesefrictionsontheissuesofinvestmentandfinancingpoliciesaswellasriskmanagement.ForexampleRampiniandVishwanathan(2010,2012)assumefinancingfrictionsintheformofcol-lateralconstraintsinapartialequilibriumsetting.BrunnermeierandSannikov(2011)andHeandKrishnamurthy(2012)assumeconstraintswithrespecttoequityissuance,togetherwithrestrictionsonhedgingofcertainaggregaterisks,tostudytheimplica-tionsoffrictionsontheissuesfinancialstabilityingeneralequilibriumsettings.ThemodelsofBolton,Chen,andWang(2011,2012)(hereafterBCW)areparticu-larlyclosetoDFHWinhowtheymodelthefirm’sproductiontechnology,buttheyassumefinancialfrictionsdirectlyinsteadofexplicitlymodelinganagencyproblem.Specifically,BCWassumethatthefirm’sproductionandinvestmenttechnologyisgov-ernedbyequationsdKt=(�(ιt)−δ)KtdtanddXt=Kt(dYt−ιtdt),wheredYt=µdt+σdZt.TheseequationsareidenticaltothoseofDFHW,assumingthatthemanagerialcom-pensationcontractinplaceenforcesfulleffort,at=1.Insteadofmodelingtheagent’sincentivesexplicitly,BCWassumefinancialfrictionsthatarerelatedtothefeaturesoftheoptimalcontractthatmotivateeffort.Specifically,theyassumethatthefirmmain-tainsacashbalance(recalltheimplementationsoftheoptimalcontractinDSandBiaisetal.(2007))andthatitiscostlytoissuenewequitywhenthefirmrunsoutofcash.InDS,newequityisissuedonlywhentheoldmanagerisfiredandreplacedwithanewmanager.Inaddition,BCWassumethatitiscostlytokeepcashinsidethefirminsteadofpayingitouttoshareholders,justasinDSitiscostlytopostponepaymentstotheagentastheagentislesspatientthantheprincipal.UndertheoptimalpolicyinBCW,thefirm’sfinancialslackissensitivetothefirm’scashflows,andevolvesbetweenendpointswheredividendpayoutsaremadeaftergoodperformanceandnewequityisissuedunderpoorperformance.Thus,BCWpresentasimpledynamicmodelthatcap-turesmanyofthefeaturesoftheoptimalcontractofDSandDFHWwithoutdelvingintothedetailsofanagencyproblemexplicitly.4.1OtherModelsthatInvolveDynamicMoralHazardAnumberofpapersadaptacontinuous-timedynamicagencyframeworktostudytheinteractionbetweenagencyproblemsandvariousotherissues.Webrieflyreviewseveralofthemhere.Wefocus,forthemostpart,onthetechnicalelementsofthesemodels.PiskorskiandTchistyi(2010),whoconsidertheproblemofoptimalmortgagedesign,haveanumberofimportantcontributions.First,theyadaptamodelsimilartothatofDStothestudyofmortgages.Second,theyinvestigatewhathappensintheoptimalcontractwhenmarketconditionsexogenouslychange.Theyfocusspecifically DynamicSecurityDesignandCorporateFinancing103onchangesofmarketinterestratesandfindthatitisoptimaltotightentheagent’saccesstocreditwheninterestratesrise.ThemarketinterestrateinthemodelofPiskorskiandTchistyi(2010)isaPoissonswitchingprocesswithtwolevels{rL,rH},andtheswitchingintensityfrominterestrateritorj,j≠i,givenbyδ(ri).ThecashflowprocessdYt=µdt+σdZt,isinterpretedastheborrower’sincome,whichisunobservable.Itisassumedthattheborrowercanhideincomewithoutanycost,soparameterλinDSissetto1.Inanyincentive-compatiblecontracttheagent’scontinuationvaluefollows:dW′t=γWtdt−dCt−δ(rt)(Wt−Wt)dt+βt(dYt−µdt),′whereWtistheagent’scontinuationvalueconditionalontheeventthattheinterestrateswitchesattimet,andtheincentiveconstraintisβt≥1.Theprincipal’svaluefunctionFi(Wt)dependsontwostatevariables—theagent’scontinuationvalueWtandthecurrentinterestratert=ri.Itischaracterizedbythesystemoftwoequations:r′′′iFi(W)=maxµ+δ(ri)(Fj(W)−Fi(W))+(γW−δ(ri)(W−W))Fi(W)W′12′′+σFi(W),2fori=L,Handj≠i,withthefamiliarboundaryconditionsF′′′i(R)=L,Fi(Wi)=−1andFi(Wi)=0.Undertheoptimalcontract,theagent’scontinuationvaluejumpsfromWttoWt′whenevertheinterestrateswitches.Akeyequationthatdeterminesthejumpintheagent’scontinuationvalue,whentheinterestratejumpsfromrt−=ritort=rjattimet,isthefirst-ordercondition′(W′)=F′(WFjtit).(9)Wewouldliketoemphasizethatcondition(9)arisesverycommonlyinmodelswhereastateswitchesviaaPoissonprocess.PiskorskiandTchistyi(2010)offerseveralimplementationsoftheoptimalcontract.Inparticular,thevariableWtcanbemappedintothebalanceonthehomeowner’shomeequitylineofcredit.ThechangesinthevalueofWtinresponsetointerestrateshiftscanbelinkedtosomepropertiesofadjustable-ratemortgages.Inparticular,itisafeatureoftheoptimalcontractthattheagent’sdefaultprobabilityriseswhentheinter-estratesincrease. 104YuliySannikovHoffmannandPfeil(2010)consideradynamicagencymodel,inwhichfirmprofit-abilityexperiencesobservableshocks.TheirmodelbuildsuponDS,exceptthattheyallowforPoissonshocksthatchangetheexpectedrateofcashflowsμ.OneofthekeymessagesofHoffmannandPfeil(2010)isthat,despiteconventionalintuition,theoptimalcontractrewardstheagentforluckwhenitiscorrelatedwiththefirm’sfutureprofitability.Herewereviewavariationoftheirmodel.AssumethattheexpectedrateofcashflowsisaPoissonswitchingprocesswithvalues{μL,μH},sothatdYt=µtdt+σdZt,µt=µLorµH.Theswitchingintensityfromstateμitoμj,j≠i,isgivenbyδ(μi).8TheagencyproblemisthesameasinDS:theagentcandivertcashflows,andhereceivesbenefitequaltoafractionλ∈(0,1]ofthedivertedcashflows.Theoptimalcontractdependsonμtandtheagent’scontinuationvalueWt,whichfollows:dW′t=γWtdt−dCt−δ(µt)(Wt−Wt)+λ(dYt−µtdt).Theprincipal’svaluefunctionsolvesthesystemoftwoequationsrF′′′i(W)=maxµi+δ(µi)(Fj(W)−Fi(W))+(γW−δ(µi)(W−W))Fi(W)W′12′′+σFi(W),2fori=L,Handj≠i,withthefamiliarboundaryconditionsF′′′i(R)=L,Fi(Wi)=−1andFi(Wi)=0.Themostimportantpointofthispaperisthatundertheoptimalcontracttheagentisrewardedforluck,i.e.whenthemeanofcashflowsjumpsfromμLtoμH.Thiscon-clusionseemstocontradicttheconventionalwisdomofHolmstromandMilgrom(1991);thatitisoptimaltofilteroutfactorsoutsidethemanger’scontrolwhenevaluat-ingtheagent’sperformance.Anexampleofthisinvolvestheevaluationoffundmanagerperformancerelativeabenchmarkindex,ratherthaninabsoluteterms.However,inthemodelofHoffmannandPfeil(2010)itisoptimaltorewardthemanagerforluckbecauseofthedynamicfeatureofthemodelthatluckispositivelyrelatedtofutureprofitability.98HoffmannandPfeil(2010)assumeinsteadthatthemeanofcashflowsmayexperienceonlyaone-timejumpup(goodluck)ordown(badluck).9Section6inDeMarzoetal.(2011)independentlymakesthesamepoint. DynamicSecurityDesignandCorporateFinancing105Wheneverthemeanofcashflowsswitchesfromμitoμj,thejumpintheagent’scontinuationvaluesatisfiesthefirst-orderconditionF′(W′)=F′(Wjtit).WehavealreadyencounteredasimilarconditioninPiskorskiandTchistyi(2010),intheeventthattheinterestratejumps.He(2009)considersadynamicagencymodelinwhichboththeagent’shiddenactionsandshocksaffectthescaleofthefirm,ratherthanthecurrentcashflow.Apartfromthisdistinction,themodelhasmanysimilaritiestothatofDFHW.BelowwepresentanextensionofthemodelofHe(2009),whichincorporatesinvestmentdecisionsthataffectthescaleofthefirm.Thismodelcanbeaconvenientoptionforapplications,asithasdifferentmomentpropertiesfromDFHW(e.g.firmcashflowsarehighlycorrelatedinthemodelofHe(2009)—theyfollowarandomwalk—butuncorrelatedinthemodelofDFHW).Considerafirm,whosecashflowisgivenby(A−ιt)Ktdt,whereιtistheinvestmentrateandKtisthefirm’scapital,bothofwhichareobservable.Thefirm’scapitalfollows:dKt/Kt=(�(ιt)−δ)dt+σdZt,iftheagentworksanddKt/Kt=(�(ιt)−δ)dt+σdZtiftheagentshirks,whereδ>δ.NeithertheshocksZtnortheagent’sactionsareobservable.TheagentgetsaprivatebenefitofBKtfromshirking,whereB≥0.Boththeagentandtheprincipalarerisk-neutral,andtheagent’sdiscountrateisγ≥r,whereristheprincipal’sdiscountrate.10Intheeventoftermination,theagent’soutsideoptionis0,andthevalueoftheassetstotheprincipalisgivenbyqKt,where′A−ι1/�(−∞)≤qr,thentherelevantboundaryconditionsthatdeterminethefunctionf(w)aswellasthepointw¯<λwheretheagentgetspaidaref(0)=q,f′(w¯)=−1andf′′(w¯)=0,asinDFHW.Pointw¯isareflectingboundaryoftheprocesswt.Ifγ=r,thenw¯=λandf(w)mustsatisfyA−ιf(0)=qandf(λ)+λ=max,ιr−�(ι)+δandtheprocesswtbecomesabsorbedwhenithitsw¯.Whenthathappens,inefficiencycompletelydisappears,andthesolutionisfirst-best.Inbothcases,terminationoccurswhenwthits0.Zhu(2011)providesageneralsolutiontothevariationoftheDSmodelwithcostlyeffort.Recallthatinthismodeltheprincipalobservescashflows0iftheagentworks,dYt=dYt−atdt,whereat=Aiftheagentshirks.TheagentgetsaprivatebenefitofB=λAifheshirks.ThismodelissimilartothebaselinecashflowdiversionmodelofDS,exceptthattherate,atwhichtheagentcandivertcashflows,isboundedbyA.HerewereviewtheresultsofZhu(2011),specializingthemtothecasewhentheagent’soutsideoptionisR=0.Theformoftheoptimalcontractdependsonthe DynamicSecurityDesignandCorporateFinancing107regioninthespaceofpayoffpairsoftheagentandprincipal,inwhichthepoint(λA/γ,(μ−A)/r)lies.DenotebyF(W)theprincipal’svaluefunctioninthebaselinemodelofDS.RecallthatthisfunctionsolvesEqn(6)withboundaryconditionsF(0)=L,F′(W)=−1andF′′(W)=0(or,equivalently,µ−γW=rF(W)).Furthermore,let′γ′′′g(w)=minF(w)+(w−w)F(w).w′rFunctiong(w)boundsregionAinFigure10.Figure10Regionsthatdeterminetheformoftheoptimalcontract.TheboundarybetweenregionsBandDisthelocusofpoints,wheresolutionsFtoEqn(6)withboundaryconditionsF(0)=LandF′(0)F′(0)reachslope0.TheboundarybetweenregionsCandDisthelocusofpoints,wheresolutionsFtoEqn(6)withboundaryconditionsF′′′′′(W)=−1andµ−γW=rF(W),forsomeWW,reachslope0.Theprincipal’svaluefunctionintheoptimalcontractcanbefoundasfollows:CaseA:Ifpoint(λA/γ,(μ−A)/r)fallsinregionA,thentheoptimalcontractisidenti-caltothatinthebaselinesettingofDS(asdemonstratedinSectionIIIofDS).CaseB:Ifpoint(λA/γ,(μ−A)/r)fallsinregionB,thentheprincipal’svaluefunc-tionisdeterminedbysolving(6)withboundaryconditionsF(0)=LandF′(0)F′(0)onaninterval[0,W]andequation′(W)=rF(W)µ−A+(γW−λA)F(10)′′′on[W,W],withF(W)=−1.ThetwoportionsofFmustmergeatWinadifferen-tiablemanner.Thesecondderivatiesmustalsomatch(i.e.thesupercontactconditionhastohold)toensurethatFisthelargestfunctionthatsatisfiestheseconditions. 108YuliySannikovCaseC:Ifpoint(λA/γ,(μ−A)/r)fallsinregionC,thentheprincipal’svaluefunc-F′′tionisdeterminedbysolving(6)withboundaryconditions(W)=−1and′′′µ−γW=rF(W)onaninterval[W,W],andEqn(10)forWW.ThetwoportionsofFmergeatWinadifferentiablemanner.CaseD:Ifpoint(λA/γ,(μ−A)/r)fallsinregionA,thentheoptimalcontractdoesnotgivetheagentincentivestoworkinanyregion.InbothcasesBandC,functionFismaximizedatapointwhereitsatisfiesEqn(6).Thus,iftheprincipalcanstartthecontractatpointW0wherehisprofitismaximized,thentheagentwillbeinitiallyputtingeffort,andhiscontinuationvaluewillevolveaccordingto:dWt=γWtdt−dCt+λ(dYt−µdt).IncaseB,theagentisputtingineffortwhileWt∈(0,W);attheleftendpointofthisintervaltheagentisfired,andtherightendpointtheagentisallowedtoshirktemporarily.PointWisastickyreflectingboundaryoftheprocessWt.IncaseC,the′′agentisputtingineffortwhenWt=(W,W).AtpointWtheagentispaidsothatWt′reflectsatW.PointWisastickyreflectingboundary:atthatpointtheagentshirksandconsumesprivatebenefitstemporarily(whichisnotsoattractivefortheagentbecauseinregionC,λA/γislow).Biais,Mariotti,Rochet,andVilleneuve(2010)(hereafterBMRV)adapttheagencyframeworktosettingswheretheagent’saction,e.g.negligence,mayleadtolargelosses.Thesesituationsarecommoninpractice,andoftenthedamageissignificantlygreaterthanwhattheagentcancover.BMRVstudythequestionofoptimalincen-tiveprovisioninthesesettings.Theyfindthatlossesoftenrequirethedownsizingofoperations.Formally,BMRVassumethatperformance-relatedinformationtakestheformofaPoissonprocess,ratherthanaBrownianone.LikeDFHW,theyalsoallowforinvestment(anddisinvestment)thatchangesthescaleoftheproject.TheyassumeaproductiontechnologythatusescapitalKttoproduceacashflowofdXt=Kt(dYt−ιtdt),wheredKt=(�(ιt)−δ)KtdtanddYt=µdt−MdNt.ThetermMdNtintheexpressionfordYtrepresentspossiblelosses(whichtranslatetocashflowlossesofthesizeKtM).AlossarriveswhenthecountingPoissonprocessNtjumpsupby1,andtheintensityoflossesΨtdependsontheagent’sunobservableaction.Theagenthastwoactions:workingandshirking.Theimpactoftheseactionsontheintensityoflosses,aswellastheagent’sprivatebenefit,aresummarizedasfollows:intensityoflossesprivatebenefitwork�t=ψ0shirk�t=ψ+�ψAKt. DynamicSecurityDesignandCorporateFinancing109TheinvestmentfunctionΦisincreasingandconcave.BMRVtakeδ=0andcon-sideraparticularformofΦrepresentedinFigure11.11Thatis,itispossibletocostlesslydestroyarbitraryamountsofcapital,andbuildnewcapitalatcostc,aslongasthegrowthratedoesnotexceedgr,andtheagentcanconsumeonlynon-negativeamounts.Moreover,theagent’soutsideoptionisassumedtobe0asinDFHWandHe(2009).BMRVmainlyfocusonparameters,forwhichitisoptimaltogivetheagentincen-tivestoworkatalltimes,themaximalriskpreventioncase.Wefocusonthiscasehere.Theoptimalcontractisbasedontwostatevariables:theagent’scontinuationpayoffWtandthesizeofthefirmKt.Denotetheprincipal’svaluefunctionbyF(Wt,Kt).Theagent’scontinuationvaluefollows:dWt=γWtdt−dCt+βt(dXt+Kt(ιt−µ)dt+ψMKt).meanzeroTheincentiveconstraintisβt≥λwhereλ=A/(�ψM).Ifthisconstraintholds,thenthebenefitfromshirkingisnotgreaterthanthenegativeimpactofshirkingontheagent’scontinuationpayoff,i.e.AKt≤βtΔψMKt.Naturally,theoptimalcontractsetsβt=λ.Becauseinformationabouttheagent’sperformancearrivesviaaPoissonprocess,onehastotakeintoaccountthatitisimpossibletogivetheagentincentivestoworkwhenWt∈[0,λMKt).Thereasonisthatarequiredpunishmentintheeventofalosswouldreducetheagent’scontinuationpayoffbyatleastλMKt,belowtheagent’soutsideoption.12Asaresult,iftheagent’scontinuationpayoffeverfallsbelowλMKt,theopti-malcontractprescribeseitherrandomizationthatlowersWtto0orboostsittoλMKt,ordownsizingthatreducesthefirm’scapitaltoWt/(λM).Duetothescale-invariancepropertiesofthemodel,F(W,K)=Kf(w),wherew=Wt/Kt.wf(λM).WhenWt≤λMKt,i.e.w≤λM,fisalinearfunctionoftheformf(w)=λMThetwo-dimensionalBellmanequationforF(W,K)canbereducedtoaone-dimensional11NotethatΦcanbenegative,i.e.itispossibletodestroycapital,butcannotbenegative,i.e.thedestruc-tionofcapitaldoesnotgenerateanyliquidatingcashflows.12Thisissuearisesalsoindiscrete-timeprincipal-agentmodels,e.g.seeDeMarzoandFishman(2007a,2007b).ItisabsentfromBrownianmodels,orPoissonmodelsinwhichjumpsare“goodnews”. 110YuliySannikovFigure11TheinvestmentfunctioninthesettingofBMRV.equationbyamethodsimilartothatusedinDFHW.Theequationtakestheformrf(w)=maxµ−ψM−ι+(�(ι)−δ)f(w)ι+(γw+λψM−(�(ι)−δ)w)f′(w)+ψ(f(w−λM)−f(w))ontheinterval[λM,w¯],anditmustsatisfytheboundaryconditionsf′(w¯)=−1,andf′′(w¯)=0.Figure12illustratestheformoffunctionfinthismodel.Theoptimalinvestmentrateιsolvesmax�(ι)(f(w)−wf′(w))−ι,ιasinDFHW.Becausefisaconcavefunction,itfollowsthatf(w)−wf′(w)isincreasinginw;thustherateofinvestmentisincreasinginthefirm’sfinancialslack.Figure12FunctionfinthemodelofBMRV. DynamicSecurityDesignandCorporateFinancing111ForthespecificpiecewiselinearformoffunctionΦ(ι)assumedinBMRV,theopti-malinvestmentrateiscgf(w)−wf′(w)c,ιt=0otherwise.Effectively,investmenthappenswhenwtexceedsacriticallevelofwi,wheref(wι)−wιf′(wι)=c.BMRVhasanimportantimplicationabouthowthecontractgivesincentivestotheagenttoworktopreventlosses.Oncewtreachesw¯,theagentispaidacontinuousstreamofpaymentsuntilthenextloss.Ifalossoccurs,paymentsaresuspendedandtheagenthastowaitafixedamountoftimebeforethepaymentsresume.Ifanotherlossoccurs,theagenthastowaitlonger.Toomanylossesinarowleadtoapartialdownsizingorliquidation.135.ASYMMETRICINFORMATIONINDYNAMICSETTINGSFromthestaticmodelsofSection3,weknowthatwhenasymmetricinformationispresent,itisdifficultforfirmstoraisemoneyforpositive-NPVprojects.Theproblemisparticularlyseverewithequityissuance,asitsvalueishighlysensitivetotheprivateinfor-mationoffirmmanagement.Inequilibrium,firmsraisefinancingfornewinvestmentsbysellingequityiftheirassetsareovervaluedbythemarket,butmayrefrainfrominvestingiftheirassetsareundervalued.Upontheannouncementofissuance,themarketrevisesitsbeliefaboutthefirm’sassetsdown,andthefirm’sstockpricedrops.Theproblemislessseverewithdebt,whichislessinformationallysensitive.Firmswithgoodprivateinfor-mationareabletoraisemoremoneybyissuinglessinformationallysensitivesecurities.5.1StaticContractsinDynamicSettingsAnumberofnewissuescometolightoncewethinkaboutthetimedimension.Belowwereviewseveralmodelsthathighlighttheseissues,startingwithmodelsthatincorpo-ratestaticcontractsindynamicsettings.14LucasandMcDonald(1990)consideramodelinwhichfirmscanraisemoneyforinvestmentopportunitiesonlybyissuingequity.Investmentopportunitiescanbepostponed,andinsideinformationaboutassetsinplaceistimevarying.Inequilibrium,managersofovervaluedfirmsissueequitytofinancethe13ArecentpaperofDeMarzo,Livdan,andTchistyi(2012)buildsuponthemodelingelementsofDSandBMRVtostudyoptimalcontractinginsettingswheremanagerscantakehiddentailriskstoboostperceivedreturns.Theyfindthatintheoptimalcontract,managerstakemoretailrisksafterpoorper-formance,whentheyhaveless“skininthegame”.14Holmstrom,(1999)providesalineardynamicmodelwithstaticwagecontracts,whichillustrateshowsignalingincentivesonthedegreeofuncertainty. 112YuliySannikovinvestmentrightaway,whilemanagersofundervaluedfirmswaittoissueuntiltheirprivateinformationbecomespublic.Stockpricestendtodropupontheannouncementofissuance,andgeneratehighabnormalreturnspriortoissuance.ThemodelofLucasandMcDonald(1990)motivatesustothinkaboutfurtherissuesthatarelikelytoariseindynamicsettingswithadverseselection.First,whileinthemodelofLucasandMcDonald(1990)managerscarryprivateinformationforonlyoneperiodbeforeitbecomespublic,inmoreelaboratemodelsmanagerscansignalthequalityoftheirinformationbywaiting.Second,thedistributionofprivateinformationcanchangeovertime:ifmanagersofundervaluedfirmswaittoissue,theirproportioninthemarketcanriseovertime,alleviatingtheproblemofasymmetricinformation.Thiscanleadtohotmarketswhenthedilutionproblemduetoasymmetricinformationislesssevereasmorehigh-qualityfirmssellsecurities,andcoldmarkets,duringwhichhigh-qualityfirmswaitandonlylow-qualityfirms,orfirmswithdirecapitalneeds,issue.Third,firmscanissuesecuritiesotherthanequityandcanbuildupfinancialslackduringtimeswhentheinformationalproblemislesssevere.ThemodelofDaleyandGreen(2012)shedslightonsomeoftheseissues.Specifically,intheirsettingtheseller/issuer,whohaspersistentprivateinformationaboutassetqual-ity,cansignalbywaiting.Marketbeliefabouttheseller’sprivateinformationtendstoriseduringtheperiodsofnon-issuance.Animprovementofmarketbeliefaboutassetqualityleadstoahotmarket,inwhichsellersissueregardlessofprivateinformation.Interestingly,theequilibriumalsofeaturesaregimewherethemarketfreezes,i.e.issu-ancebecomessuspendeduntilfurtherarrivalofnews.ThemodelsofLucasandMcDonald(1990)andDaleyandGreen(2012)onlygiveaflavorofmanyinterestingissuesthatariseindynamicadverseselectionmodelswithsimplecontracts.Anumberofotherpaperslookattheseissuesbothempiricallyandtheoretically,includingBakerandWurgler(2002)andHennessy,Livdan,andMiranda(2008).Marketdynamicswithadverseselectionisalsoafruitfulareaforfutureresearch,asalotofinterestingquestionshavenotbeenanswered.Itturnsoutthatmanyadverseselectionfrictionsbecomesignificantlylesssevereoncedynamiccontractsareallowed.WefinishthissectionbyquicklyrevisitingLucasandMcDonald(1990)andDaleyandGreen(2012)withtheperspectiveofdynamiccontracts,andreviewseveralinsightsaboutoptimaldynamiccontractingunderadverseselection.LucasandMcDonald(1990)consideraninfinite-horizonmodelinwhichfirmshaveassetsinplaceandmayhaveinvestmentopportunities.Forsimplicity,therisk-freerateisassumedtobe0.Ineachperiodt,themarketperceivesthatthevalueofthefirm’sassetsinplaceisAt.Atthesametime,themanagerlearnsthenextperiod’svalueofthefirm’sassets,whichisuAtwithprobabilityp,At+1=dAtwithprobability1−p. DynamicSecurityDesignandCorporateFinancing113Itisassumedthatu>d,sothatifAt+1=uAt,thefirmisundervaluedonthemarket,andifAt+1=dAt,thefirmisovervalued.Inaddition,themanagerknowswhetherthefirmhasaninvestmentopportunityinthatperiod.Ifanopportunityexists,itrequiresaninvestmentofKAt,andgeneratesvalueβAt+1+KAtinthenextperiod(netoftheunderwritingfees).Itisassumedthatafirmcanraisemoneyforinvestmentonlybyissuingequity.AsinMyersandMajluf(1984),themanagercaresaboutthefirm’soldshareholders.Inaddition,thefollowingassumptionsaremade.Betweenperiodst−1andtthefirmisliquidatedforexogenousreasonswithprobability1−ρ,andgeneratesapayoffofAttoshareholders.Investmentopportunitiesarerandom:ifthefirmhasaninvestmentopportunityinperiodtthatitdoesnottake,itstillhasaninvestmentopportunityinperiodt+1.Ifthefirmtookitsopportunityinperiodt,ordidnothaveonealtogether,itgetsaninvestmentopportunityinperiodt+1withprobabilityq.LucasandMcDonald(1990)focusonanequilibrium,inwhichfirmswithaninvest-mentopportunityissueequitytofinanceitonlyiftheyareovervalued,andunderval-uedfirmspostponetheopportunity.Parameterrestrictionsareimposedtoensurethatovervaluedfirmswithoutaninvestmentprojectandundervaluedfirmsdonotwanttoissue(e.g.duetohighunderwritingcosts).Thenuponissuance,newinvestorswillinferthatAt+1=dAtandthatthefirmhasaninvestmentopportunity.Basedonthisbelief,investorsthendemandinexchangeforcapitalKAtanappropriatefractionofthefirms,suchthattheybreakeven.Duetoscaleinvariance,themanager’svaluationofthefirmisoftheformV(a,b)At,wherea=uorddependingonthevalueofAt+1,andb=0orβ,dependingonwhetherthefirmhasaninvestmentopportunityornot.Thevaluefunctionsatisfiesthefollowingrecursiveequationsinequilibrium:V(a,0)=a{1−ρ+ρ{(1−q)[pV(u,0)+(1−p)V(d,0)]+q[pV(u,β)+(1−p)V(d,β)]}}fora=u,dV(u,β)=u{1−ρ+ρ[pV(u,β)+(1−p)V(d,β)]},andV(d,β)=(1−s)(dβ+K+V(d,0))=dβ+V(d,0),sinceK=s(dβ+K+V(d,β))toensurethatthefirm’snewinvestorsbreakeven.WhileV(a,b)Atisthemanager’svaluationofthefirmbasedonhisprivateinforma-tion,marketvaluationofthefirmwilldependonitsbeliefaboutmanager’sinformation.Inparticular,inperiodt,priortoissueannouncement,marketwillbelievethata=uwithprobabilitypanddwithprobability1−p.Inaddition,theprobabilitythatthefirmhasaprojectisincreasinginthenumberofperiodsnitsstockhasgoneup(andthusthefirmhadbeenundervaluedinthepriorperiod,andunabletoraisefundingfortheproject).Theprobabilitythatthefirmhasaprojectis:qn+1n=1−(1−q) 114YuliySannikovandsomarketvaluationofthefirm’sstockpriortoissueannouncementis:P(n)=p[qnV(u,β)+(1−qn)V(u,0)]+(1−p)[qnV(d,β)+(1−qn)V(d,0)].SinceV(a,β)>V(a,0),thevalueofthefirmisincreasinginthenumberofperiodsitsassetvaluehasgoneup.Ifthefirmdecidestoissue,itsmarketvalueimmediatelychangestoV(d,β).Thedecisiontoissuerevealstwopiecesofnewstothemarket:badnewsthatthefirm’sassetvaluewillgodowninthenextperiod,andgoodnewsthatthefirmhasaninvestmentopportunity.Ifnislarge,thenP(n)≈pV(u,β)+(1−p)V(d,β)>V(d,β),andthepriceofthefirmdropsforsureuponannouncement.Inthiscase,thegoodnewsthatthefirmhasaprojectisnotreallynews.Ifnissmall,thenstockpricereactiontoequityissueannouncementmaybeambiguous.Themodelpredictsthatthefirmgeneratesapositiveabnormalreturnpriortoissue.Thishappensbecausethedecisiontonotissueisrelatedtogoodprivateinformationaboutthefirm’sassetvalue.Themodelalsopredictsthat,usually,thefirm’ssharepriceshoulddropupontheannouncementofissuance,particularlyafterlongperiodsofstockpriceincreases.DaleyandGreen(2012)consideradynamicassetmarketwithasymmetricinfor-mation.Theyinvestigatehowtradepatternschangeovertimeasinformationisgradu-allyrevealedtobuyers,whocontinuouslyupdatetheirbeliefsaboutassetquality.Theequilibriumtheyderivehasaveryinterestingfeatureofmarketbreakdown:aperiodwhentradestopseventhoughtherearebeneficialtradingopportunities,andsellershaveincentivestostrategicallywaitforimprovedmarketconditions.Themodelhasoneselleroftheassetandacontinuumofcompetitivebuyers.Thesellerhasoneassetwhosequalityθ∈{L,H}isprivatelyknowntotheseller.ThesellerderivesapayoffflowofKθfromtheassetuntilatradeoccurs,andanybuyerwhopur-chasestheassetisabletoderiveahigherpayoffflowofVθ>Kθafterthetimeofthetrade.QualityHgeneratesmorevalue,i.e.VH>VLandKH>KL.Buyerscontinuouslymakeofferstotheselleruntilasaleoccurs.Afterthesale,thepurchaserholdstheassetinperpetuity.15DenotebyWt/rthehighestofferthatthebuyersmaketothesellerattimetinequilibrium,whereristhecommondiscountrateofthesellerandallbuyers.BuyersinitiallybelievethattheselleristypeHwithprobabilityπ0∈(0,1).TheyupdatetheirbelieffromthesignalXtthatfollowsdXt=µθdt+σdZt,15DaleyandGreen(2012b)relaxthisassumptionandendogenizethevalueoftheassettothepurchaser,asthepurchasermayneedtotradetheassetinthefutureduetoliquidityshocks. DynamicSecurityDesignandCorporateFinancing115whereμL<μH,aswellasfromtheseller’sdecisions.Ignoringtheinformationabouttheseller’sdecisions,Bayesruleimpliesthatthebuyers’beliefabouttheseller’stypewouldevolveaccordingto16π2222t=exp(zt)/(1+exp(zt)),wheredzt=−(µH−µL)/(2σ)dt+(µH−µL)/σdXt.Thevariablezt∈(−∞,∞)ismoreconvenienttoworkwiththanthebeliefπt∈(0,1).Furthermore,DaleyandGreen(2012)showthatifwetakeintoaccounttheseller’sdecisions,thentheabsenceoftradeisalwaysgoodinformationabouttheseller’stype.Thatis,typeHwouldnevertrade—i.e.hestrictlypreferstowaitforfutureoffers—whenevertypeLpreferstowaitorisindifferent.Intuitively,thisistruebecausetypeHgetsahigherpayoffflowandmorefavorablenewsthantypeLfromholdingontotheasset.Ifnotradeisgoodnews,thenconditionalontheabsenceoftrades,theprocessztthatdeterminesthebuyers’beliefsabouttheseller’stypefollows:dz2−µ2)/(2σ2)dt+(µ2(11)t=−(µHLH−µL)/σdXt+dQt,whereQtisanon-decreasingprocess.Itturnsoutthattheequilibriumischaracterizedbythreeregimes:•z≤α,wheretypeHdoesnottradeandtypeLacceptstheofferofVLwithprob-abilitythatislessthan1,suchthattheposteriorztjumpsuptoα;•z∈(α,β),whereneithertypetrades,and•z≥β,wherebothtypesforsureacceptthepriceofWt=(1−πt)VL+πtVH.Therefore,theprocessQtissuchthatαisareflectingboundaryofthesystem,anddQt=0ontheinterval(α,β).Forexpositionalreasons,wewillassumethattheequilibriumtakesthisformandderiveequationsthatdeterminetheboundariesαandβ,aswellasthevaluefunctionsofbothtypesoftheseller.Ontheinterval[α,β],thevaluefunctionsofthetwotypesofsellerssatisfytheHJBequation(seeFootnote15):rF2/(2σ2)(F′′(z)±F′(z)),(12)θ(z)=rKθ+(µH−µL)θθwiththe“+”signifθ=H,andthe“−”signifθ=L.ThevaluefunctionFL(z)oftypeLreachesthelevelofVLatpointz=αandstaysatVLontheentireinterval(−∞,α).Moreover,bothfunctionsFH(z)andFL(z)reachthelevelof�(z)≡(1−π)VL+πVH=(VL+exp(z)VH)/(1+exp(z))(13)16Thus,dzt=(μH−μL)2/(2σ2)dt+(μH−μL)/σdZtconditionalontypeH,anddzt=−(μH−μL)2/(2σ2)dt+(μH−μL)/σdZtconditionalontypeL. 116YuliySannikovatpointz=β,andsatisfyEqn(13)forallz∈[β,∞).Inaddition,becauseαisareflect-ingboundaryofthesystem,FL′(α)=FH′(α)=0.FunctionsFH(z)andFL(z)thatsolvethesecond-orderordinarydifferentialequation(12),togetherwiththetwoboundariesαandβ,canbefullydeterminedthroughsixboundaryconditions.Inadditiontothefiveconditionsthatwehavealreadyidentified,FL(α)=VL,FL′(α)=FH′(α)=0andFH(β)=FL(β)=Ψ(β),DaleyandGreen(2012)addasixthconditionofFH′(β)=�′(β),whichismotivatedbytheoff-equilibriumbehavior.First,foranyz∈(α,β),ithastobethecasethatonlyatypeLselleracceptsanypricelessthanΨ(z).Otherwise,buyershaveaprofitabledeviationforz∈(α,β):offeringapricelessthanΨ(z)thatbothsellertypeswouldaccept,andgettingastrictlypositivepayoff.Therefore,wecannothaveF′(β)>�′(β),sinceotherwisesuchanHofferwouldexistatz=β−εforsufficientlysmallε.Second,ifztreachesβ,theselleroftypeHshouldnotbeabletobenefitbywaitingamoment,insteadofacceptingthebestofferimmediately.IfFH′(β)<�′(β)andifztkeepsfollowingEqn(11)ifthesellerdoesnotaccept,forsomenon-decreasingprocessQt,thentheselleroftypeHbenefitsfromwaiting.17Thus,wemusthaveF′(β)=�′(β).NotethattheassumptionthatztHkeepsfollowingEqn(11)conditionalontheabsenceoftradeevenifzt≤βisanoff-equilibriumpathbeliefassumption.Figure13,reproducedfromDaleyandGreen(2012),illustratesthevaluefunc-tionsFH(z)andFL(z)inequilibrium,andcomparesthemwiththeaveragepriceΨ(z)atwhichthetransactionwouldtakeplaceintheabsenceoftheinformationalproblem.Theequilibriumhasseveralnotablefeatures.First,sellersoftypeHsignalassetqual-itybywaiting.Indeed,duetothenon-negativetermdQtin(11),theabsenceoftradeisgoodnewsaboutquality.Second,thepatternoftradedependsonthedistributionofprivateinformationthatthesellermayhave,whichiscapturedbythestatevariablezt.Theequilibriumfeaturesahotmarket,forzt≥β,wherebothtypesofthesellertrade,andacoldmarketforzt≤α,whereonlythelow-qualitysellertrades.Third,theequi-libriumhasaninterestingregionof(α,β)wherethemarketfreezes,i.e.tradestopsuntilnewinformationaboutassetqualityisrevealed.ItturnsoutthattheconclusionsofLucasandMcDonald(1990)aswellasDaleyandGreen(2012)changedrasticallyifitispossibletowritedynamiccontracts.Itisastrikingobservation(although,ofcourse,therearefrictionsinpracticethatmaymakedynamiccontractingdifficult).Followingthisobservation,wereviewseveralcommonconclusionsthattheliteratureonoptimaldynamiccontractswithadverseselectiondelivers.17Recallthatontheequilibriumpath,processQtisnon-decreasingbecausenotradeisgoodnewsaboutthetypeoftheseller. DynamicSecurityDesignandCorporateFinancing117Figure13ThevaluefunctionsoftypesHandLinthemodelofDaleyandGreen(2012).5.2OptimalDynamicContractswithAdverseSelectionInbothofthemodelsthatwejustdiscussed,dynamiccontractscanrestorefulleffi-ciency.ConsiderfirstthesettingofLucasandMcDonald(1990).Despiteasymmetricinformation,themanagercouldalwaysraisefundsforinvestmentbyannouncingwhetherAt+1=uAtorAt+1=dAtandissuingasecuritywiththefollowingcharacteristics:(1)itgrantstheholderaclaimtoafractionsofthefirm’sequityattimetand(2)attimet+1grantstheholderaclaimtoalltheremainingequityifthemanager’sannouncementofthevalueofAt+1turnedoutthebeincorrect(orifitturnsoutthatthefirmdidnothaveaninvestmentopportunityattimet).Thissecuritygivesthemanagerincentivestotellthetruth,andfullysolvestheadverseselectionproblem.LikewiseinthesettingofDaleyandGreen(2012),ifweassumethatthesignalXtrevealspublicinformationaboutthequalityoftheassetevenafterthetransferofownership,thentheasymmetricinformationproblemcanalsobesolvedcompletelythroughadynamicsecurity.Indeed,thesellercantransfertheassettothebuyerattime0inexchangeforapaymentthatiscontingentonthefutureobservationofXt.ThepaymentcanbeeasilydesignedinsuchawaythatitsexpectedvalueisVH/rconditionalontheseller’sassetbeingoftypeH,andVL/rconditionalontheseller’sassetbeingoftypeL.Whendynamiccontractsarepossible,thereisacrucialdifferencebetweenasym-metricinformationthatexistsupfrontattime0,andthat,whicharisesinthefuture.Speakingloosely,onlytheformercreatesdistortions.Theproblemoffutureasymmet-ricinformationcanbesolvedbyacontractsignedbeforeasymmetricinformationmaterializes. 118YuliySannikovOnegeneraltakeawayfromtheliteraturethatinvestigatesdynamiccontractsinset-tingswithadverseselectionisthatdistortionsthatexistattime0decaygraduallyovertime.Forexample,Pavan,Segal,andToikka(2009)andGarrettandPavan(2009)focusontheconceptofimpulseresponses,whichcapturestheextenttowhichprivateinforma-tionattime0remainsrelevantatafuturetimet.Intheirsettings,distortionsintheoptimalmechanismdisappearasimpulseresponsesdecayto0.Toillustratehowdistor-tionsduetoadverseselectiongraduallydisappear,wefocusbelowonthesettingofSannikov(2007),asitconvenientlybuildsonamodelofDeMarzoandSannikov(2006)thatwealreadydiscussedinSection4.18Sannikov(2007)studiesadverseselectionasettingsimilartothatofDeMarzoandSannikov(2006)(DS).Thekeynewassumptionisthattheagenthasprivateinforma-tionaboutthemeanofcashflows,μHorμL.OnlytheprojectwithcashflowμH>μLhaspositiveNPV.Theprincipalwouldliketodesignacontractthatmaximizesprofits,subjectto(1)givingtheagentwithagoodprojectadesiredexpectedpayoffofW0,(2)givingtheagentwithagoodprojectincentivestorevealcashflowstruthfully,and(3)screeningoutbadprojectswiththemeanofcashflowsμL.Theagentparticipatesinup-frontinvestment,andisabletoreceiveanoutsidevalueofR∈(0,W0)fromtheresourcescontributed.Iftheprojectisfundedandterminated,theagent’soutsideoptionis0.Inaddition,itisassumedthat(1)theagentandtheprincipalhaveacommondis-countrater,(2)theprojecthasafinitetimehorizonT,butmaybeterminatedearlyforunderperformance,(3)theagenthasaccesstoasecretsavingstechnology,witharateofreturnr,and(4)theagentwithabadprojectmayhavelargesavingstoexaggeratetheproject’scashflowsintheshortrun.AsinDS,theagentprivatelyobservesthecashflowsdYt=µθdt+σdZt,µθ=µLorµH.Theagentmaydivertcashflowsorusehisprivatesavingstoboostcashflows.Moreover,theagentcapturesthefullvalueofdivertedcashflows,i.e.theparameterλinthesettingofDSissetto1.Apartfromadverseselection,thesettingdiffersfromthatofDSinminorways.Therefore,weknowtheformthattheoptimalcontractwouldtakeintheabsenceofadverseselection.Withmoralhazardalone,theoptimalcontractisbasedontheagent’scontinuationvalue,whichfollows:dWt=rWtdt+dYt−µHdt.18Ingeneral,adverseselectionproblemsalonecanbesolvedveryeffectivelyviastate-contingentdynamiccontracts,andinterestingfrictionsariseonlywhenmoralhazardisalsopresent.Inpractice,itisdifficulttoimagineasituationwhereadverseselectionexistsbyitselfwithoutmoralhazard. DynamicSecurityDesignandCorporateFinancing119Becausetheagentisaspatientastheprincipal,itisoptimaltopostponepaymentstotheagentuntiltimeT.IfWthitszerobeforetimeT,thentheprojectisterminatedearly,andotherwisetheagentreceivesthepaymentofWTattimeT.19ThecontractcanbeimplementedthroughacreditlinewithbalanceMt=μH/r−WtandacreditlimitofM=µH/r.Thenthebalanceevolvesaccordingto:dMt=rMtdt−dYt.Ifthecreditlimitisreached,theprojectisterminated.However,unlikeinDS,inthisimplementationtheagentsavesexcesscashflowsifthecreditlineisfullypaidoff,i.e.theagentisnotpaiduntiltimeT.Remarkably,itturnsoutthattheoptimalcontractwithadverseselectionisanaturalmodificationofthiscontract.Thesimplicityoftheoptimalcontractisinformativeabouttheimpactofadverseselection.However,whilethefinalproductiscleanontheoutside,itrequiressophisticatedengineering.Infact,theproofofoptimalityofthecontractdescribedbelowrequiressignificantlymorecomplexargumentsthanthoseusedintheanalysisoftheDSmodel,whichwediscussedinSection4.Thoseargumentscanbefoundinthepaper.Observethattheoptimalcontractunderpuremoralhazard“doesnotwork”withadverseselection.Indeed,thecreditavailabletotheagentattime0,W0=M−M0,exceedsthepayoffofRthattheagentwithabadprojectcanobtainelsewhereifhedoesnotpretendtohaveagoodproject.Therefore,toscreenoutbadprojects,thecreditlimitattime0mustberestrictedtoM0=M0+R.Infact,thepresentvalueoftheexpectedcashflowsreceivedbytheagentwithabadproject,aswellasfundsdrawnfromthecreditline,cannotexceedRatanytimeinthefuture,i.e.weneedte−rsµLds+e−rtMt−M0R(14)0foranyhistoryofreportedcashflows.Otherwiseanagentwithabadproject,withasufficientflexibilitytogeneratehighshort-termcashflows,cangamethesystemandgetapayoffhigherthanR.Equation(14)givesanupperboundonthemaximalamountofcreditthatcanbemadeavailabletotheagent,tMrtrst=R+eM0−eµLds.019Iftheagentisrisk-neutralandhasthesamediscountrateastheprincipal,itisoptimaltopostponepaymentstotheagentindefinitely,oruntilthetimewhenthecontractisfirst-best.WehavealreadyencounteredthisfactinthemodelofHe(2009). 120YuliySannikovFigure14Optimalcreditlimit,asafunctionoftime.AslongasM0>μL/r,isincreasingintandreachesthelevelofMatsomefuturetimeT*.Itturnsoutthattheoptimalcontractwithadverseselectiontakestheformofacreditline,inwhichthecreditlimitisMtuntiltimeT,anditisMfromtimeTtotimeT.Itisaremarkablysimplecontract,asthecreditlimitdependsdeterministicallyontime,thatis,itdoesnotdependontheagent’sbehavior.Figure14illustratestheformofthecreditlimit.Thismodelillustrateshowthedistortionsduetoadverseselectiondisappearovertime.Thecreditavailabletotheagentisinitiallyrestrictedduetoadverseselection,butthecreditlimitrisesovertime,andfromtimeTonwardstheoptimalcontractlooksasifadverseselectionwereneveraproblem.Theagentisinitiallyina“hotseat”.Ashortercreditlimitisunforgivingaboutlosses.Asaresult,iftheagentcouldaffectthecashflowswitheffort,hewouldworkharderupfronttodifferentiatehimselffromtypeswithabadproject.ThesolutionofGarrettandPavan(2009)illustratesasimilarpatternusingamodelthatisdifferentinmanyways.OnemajordifferencefromSannikov(2007)isthattheabsenceofadverseselectionleadstoafirst-bestoutcomeinthemodelofGarrettandPavan(2009),astheagentisrisk-neutralanddoesnothavelimitedliability.Thus,theformoftheoptimalcontractconvergestofirst-bestovertime.Severalotherpapershaveexploreddynamiccontractsinsettingswithasymmetricinformation.Forexample,Tchistyi(2006)andDeMarzoandSannikov(2010)investi-gatesettingswherethefirm’scashflowsarecorrelatedovertime.Thus,theagenthasprivateinformationaboutthedistributionoffuturecashflows(atleastofftheequi-libriumpath).Williams(2011)andKwon(2012)alsoaddresstheissuesofpersistence,andGolosov,Troshkin,andTsyvinsky(2010)aswellasFarhiandWerning(2011)focusontheseissuesinthecontextofpublicfinance.Whileseveralcommonthemesemerge,ingeneralthereisnounifiedwaytoanalyzesettingsofdynamicadverseselectionandmoralhazard,andthisareaisripeforfutureresearch. 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