Arbitrage-Pricing-Theory-(APT)

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MScAccounting&FinanceMScFinanceFoundationsofFinancialAnalysisLecture7:ArbitragePricingTheory(APT)andConsumptionCAPM(CCAPM)AlexandraDiasUniversityofLeicesterSchoolofManagementOverviewoftheLectureWewilllookat:Introduction:EquilibriumversusarbitragevaluationFactormodelsThe(one-factor)marketmodelMulti-factormodelsandtheAPTExamplefromcreditriskmanagementStochasticdiscountfactorsTheConsumptionCAPMUniversityofLeicesterSchoolofManagement1IntroductionBeforewelookedatequilibriumvaluationtheory:CAPM,state-preference(orArrow-Debreu)wheregeneralequilibriumimpliesno-arbitrageButno-arbitragedoesnotimplyequilibriumWithArbitragePricingTheory(APT)weexplorethelessrestrictivehypothesisofno-arbitrageParallelbetweenAPTandState-preferencetheory:State-preferenceisbasedonfundamentalsecuritiespayingexclusivelyinagivenstateofnature(diculttomodelinpractice)APTassumestheexistenceofasetofcommonfactorsdeterminantforallassetreturnsInAPTprimitivesecuritiesarede nedasthosewhoseriskisdeterminedbyonesinglespeci criskfactorUniversityofLeicesterSchoolofManagement2FactormodelsIThemainbuildingblockofAPTisafactormodelAone-factormodel:TheMarketModelAssetex-postreturnsdependontheirownspeci cstochasticcomponentandonitscommonassociationwithasinglefactor.IntheCAPMworldthiscommonfactoristhereturnonthemarketportfolioSinglefactormodel:thereturnonassetjfollowstheprocess:~rj= j+ j~rM+~jwithE(~j)=0,cov(~rM;~j)=0andcov(~j;~k)=0,8j6=k.UniversityofLeicesterSchoolofManagement3FactormodelsIIComponentsinreturnsofonefactormodel:~rj= j+ j~rM+~jwithE(~j)=0,cov(~rM;~j)=0andcov(~j;~k)=0,8j6=k.1.anasset-speci cconstant j2.acommoninuence,uniquefactorreturnonthemarket. jmeasuresthesensitivityofasset'sjreturntouctuationsinthemarketreturn3.anasset-speci cstochasticterm~j.Containsallotherstochasticcomponentsof~rjthatareuniquetoassetjUniversityofLeicesterSchoolofManagement4FactormodelsIIIThehypothesiscov(~rM;~j)=0intheonefactormodel:~rj= j+ j~rM+~jwithE(~j)=0,cov(~rM;~j)=0andcov(~j;~k)=0,8j6=k.Addingcov(~rM;~j)=0,8j6=ksigni esthatallreturncharacteristicscommontodi erentassetsareexplainedbytheirlinkwiththemarketreturnIfthiswereempiricallysupported,theCAPMwouldbe\theassetpricingmodelEmpiricalworkpointstotheneedofmorethanonefactorAPTallowsformorefactors.Hopefullyareasonablenumberoffactorswillsuce.UniversityofLeicesterSchoolofManagement5ThemarketmodelLaterwewillgeneralizethemodeltomulti-factorThesingle-factormodelisusefultoestimatethebetasfortheCAPMaslongasweassumestationarityoftherelationbetweenassetreturnandmarketreturnModernPortfolioTheoryrequiresthecomputationofthematrixofvarianceandcovarianceoftheassetreturnsIfasset'sjreturnsfollowaone-factormodelthen2j= 2j2M+2jij= i j2MThecomputationoftheecientfrontierforNriskyassetsrequirestocompute:Nexpectedreturns,Nvariancesand(N2N)=2covariancesbutonlyN j's,N2jand2M(2N+1parameters),ifworkingwiththemarketmodelUniversityofLeicesterSchoolofManagement6TheAPT-settingConsideramarketwithalargenumberofassetssuchthesehavedi erentcharacteristics.ConsideraportfolioPsuchthat:1.Denotebyxithevalue`ofthepositioninassetiinportfolioP.Pisthende nedbyx0=[x1;x2;:::;xN].Phaszerocost:NXi=1xi=0=x012.Phaszerosensitivity(zerobeta)tothecommonfactor:NXi=1xi i=0=x0 3.Pisawell-diversi edportfolio.Thespeci criskofPis(almost)totallyeliminatedNXi=1x2i2i=0UniversityofLeicesterSchoolofManagement7TheAPT-no-arbitragePoint1inpreviousslidestatesthatxisorthogonalto1(seegraphforN=3)Point2inpreviousslidestatesthatxisorthogonalto .Points2and3inpreviousslideimplythatPisriskless.IfPhasazerocostandisrisklessthenanarbitrageopportunitywillexistunless:rP=0=x0rIfx?1,x? andx?rThentheArbitragePricingTheoremstatesthatrmustbealinearcombinationof1and ,i.e.,thereexistscalars0and1,suchthat:ri=0+1 i;(1)UniversityofLeicesterSchoolofManagement8Themeaningof0and1Supposethatthereexistsarisk-freeassetwhichhasazerosensitivitytothecommonfactor.Then,rf=rf=0NowconsideraportfolioQsuchthatits =1.ApplyingAPTwehavethatrQ=rf+11Thus,1=rQrfistheexcessreturnonthepure-factorportfolioQ.Substitutingin(1):ri=rf+ i(rQrf)IfweassumethattheuniquefactoristhemarketportfoliowereencountertheCAPMequation:ri=rf+ i(rMrf):UniversityofLeicesterSchoolofManagement9Multi-factormodelsandtheAPTTheAPTmodelcanhaveanynumberoffactors.Advantage:ItisveryexibleDisadvantage:ItgivesnoclueaboutwhichfactorstouseConsiderthetwo-factormodel:~rj= j+systematicz}|{bj1~F1+bj2~F2+speci cz}|{~ejwithE(~ej)=0,cov(~F1;~ej)=cov(~F2;~ej)=0,8j,andcov(~ek;~ej)=0,8j6=k.Factorsareassumed:UncorrelatedDescribeatimestablerelationshipSummarizeallthatiscommoninindividualassetreturnsUniversityofLeicesterSchoolofManagement10ImplicationsDi erentstocksarelikelytohavedi erentsensitivitiestotheseveralfactorsbj1;bj2;:::arecalledsensitivitiesorloadingsonthefactorsforstockjAsthefactorsvaryovertime,sodothereturnsonthestocksinquestionThedependencebetweenthedi erentstockscomesfromthefactthatdi erentstockreturnsdependonthesamecommonfactorsSomeportionsofthereturnonanysecuritycannotbeexplainedbyanyofourfactors.Thisgi

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