Arbitrage-Pricing-Theory-(APT)

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cconstantj2.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esthatallreturncharacteristicscommontodierentassetsareexplainedbytheirlinkwiththemarketreturnIfthiswereempiricallysupported,theCAPMwouldbe\theassetpricingmodelEmpiricalworkpointstotheneedofmorethanonefactorAPTallowsformorefactors.Hopefullyareasonablenumberoffactorswillsuce.UniversityofLeicesterSchoolofManagement5ThemarketmodelLaterwewillgeneralizethemodeltomulti-factorThesingle-factormodelisusefultoestimatethebetasfortheCAPMaslongasweassumestationarityoftherelationbetweenassetreturnandmarketreturnModernPortfolioTheoryrequiresthecomputationofthematrixofvarianceandcovarianceoftheassetreturnsIfasset'sjreturnsfollowaone-factormodelthen2j=2j2M+2jij=ij2MThecomputationoftheecientfrontierforNriskyassetsrequirestocompute:Nexpectedreturns,Nvariancesand(N2N)=2covariancesbutonlyNj's,N2jand2M(2N+1parameters),ifworkingwiththemarketmodelUniversityofLeicesterSchoolofManagement6TheAPT-settingConsideramarketwithalargenumberofassetssuchthesehavedierentcharacteristics.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:UncorrelatedDescribeatimestablerelationshipSummarizeallthatiscommoninindividualassetreturnsUniversityofLeicesterSchoolofManagement10ImplicationsDierentstocksarelikelytohavedierentsensitivitiestotheseveralfactorsbj1;bj2;:::arecalledsensitivitiesorloadingsonthefactorsforstockjAsthefactorsvaryovertime,sodothereturnsonthestocksinquestionThedependencebetweenthedierentstockscomesfromthefactthatdierentstockreturnsdependonthesamecommonfactorsSomeportionsofthereturnonanysecuritycannotbeexplainedbyanyofourfactors.Thisgi