現(xiàn)代資產(chǎn)組合理論和資本資產(chǎn)定價(jià)模型分析第1頁(yè)/共55頁(yè)ModernPortfolioTheory

TheFactorModelsand

TheArbitragePricingTheoryChapter8ByDingzhaoyong第2頁(yè)/共55頁(yè)Return-generatingProcess

andFactorModelsReturn-generatingprocessIsastatisticalmodelthatdescribehowreturnonasecurityisproduced.ThetaskofidentifyingtheMarkowitzefficientsetcanbegreatlysimplifiedbyintroducingthisprocess.Themarketmodelisakindofthisprocess,andtherearemanyothers.第3頁(yè)/共55頁(yè)Return-generatingProcess

andFactorModelsFactormodelsThesemodelsassumethatthereturnonasecurityissensitivetothemove-mentsofvariousfactorsorindices.Inattemptingtoaccuratelyestimateexpectedreturns,variances,andcovariancesforsecurities,multiple-factormodelsarepotentiallymoreusefulthanthemarketmodel.第4頁(yè)/共55頁(yè)Return-generatingProcess

andFactorModelsImplicitintheconstructionofafactormodelistheassumptionthatthereturnsontwosecuritieswillbecorrelatedonlythroughcommonreactionstooneormoreofthespecifiedinthemodel.Anyaspectofasecurity’sreturnunexplainedbythefactormodelisuncorrelatedwiththeuniqueelementsofreturnsonothersecurities.第5頁(yè)/共55頁(yè)Return-generatingProcess

andFactorModelsAfactormodelisapowerfultoolforportfoliomanagement.Itcansupplytheinformationneededtocalculateexpectedreturns,variances,andcovariancesforeverysecurity,whicharethenecessaryconditionsfordeterminingthecurvedMarkowitzefficientset.Itcanalsobeusedtocharacterizeaportfolio’ssensitivitytomovementinthefactors.第6頁(yè)/共55頁(yè)Return-generatingProcess

andFactorModelsFactormodelssupplythenecessarylevelofabstractionincalculatingcovariances.Theproblemofcalculatingcovariancesamongsecuritiesrisesexponentiallyasthenumberofsecuritiesanalyzedincrease.Practically,abstractionisanessentialstepinidentifyingtheMarkowitzset.第7頁(yè)/共55頁(yè)Return-generatingProcess

andFactorModelsFactormodelsprovideinvestmentmanagerswithaframeworktoidentifyimportantfactorsintheeconomyandthemarketplaceandtoassesstheextenttowhichdifferentsecuritiesandportfolioswillrespondtochangesinthesefactors.Aprimarygoalofsecurityanalysisistodeterminethesefactorsandthesensitivitiesofsecurityreturntomovementsinthesefactors.第8頁(yè)/共55頁(yè)One-FactorModelsTheone-factormodelsrefertothereturn-generatingprocessforsecuritiesinvolvesasinglefactor.Thesefactorsmaybeoneofthefollowings:ThepredictedgrowthrateinGDPTheexpectedreturnonmarketindexThegrowthrateofindustrialproduc-tion,etc.第9頁(yè)/共55頁(yè)One-FactorModelsAnexamplePage295:Figure11.1第10頁(yè)/共55頁(yè)One-FactorModelsGeneralizingtheexampleAssumptionsTherandomerrortermandthefactorareuncorrelated.(Why?)Therandomerrortermsofanytwosecuritiesareuncorrelated.(Why?)第11頁(yè)/共55頁(yè)One-FactorModelsExpectedreturnVarianceCovariance第12頁(yè)/共55頁(yè)One-FactorModelsTwoimportantfeaturesofone-factormodelThetangencyportfolioiseasytoget.Thereturnsonallsecuritiesrespondtoasinglecommonfactorgreatersimplifiesthetaskofidentifyingthetangencyportfolio.Thecommonresponsivenessofsecuritiestothefactoreliminatestheneedtoestimatedirectlythecovariancesbetweenthesecurities.Thenumberofestimates:3N+2第13頁(yè)/共55頁(yè)One-FactorModelsThefeatureofdiversificationistrueofanyone-factormodel.Factorrisk:Nonfactorrisk:DiversificationleadstoanaveragingoffactorriskDiversificationreducesnonfactorrisk第14頁(yè)/共55頁(yè)One-FactorModels第15頁(yè)/共55頁(yè)Multiple-FactorModelsThehealthoftheeconomyeffectsmostfirms,buttheeconomyisnotasimple,monolithicentity.SeveralcommoninfluenceswithpervasiveeffectsmightbeidentifiedThegrowthrateofGDPThelevelofinterestrateTheinflationrateThelevelofoilprice第16頁(yè)/共55頁(yè)Multiple-FactorModelsTwo-FactorModelsAssumethatthereturn-generatingprocesscontainstwofactors.第17頁(yè)/共55頁(yè)Multiple-FactorModelsThesecondequationprovidesatwo-factormodelofacompany’sstock,whosereturnsareaffectedbyexpectationsconcerningboththegrowthrateinGDPandtherateofinflation.Page301:Figure11.2Tothisscatterofpointsisfitatwo-dimensionalplanebyusingthestatisticaltechniqueofmultiple-regressionanalysis.第18頁(yè)/共55頁(yè)Multiple-FactorModelsFourparametersneedtobeestimatedforeachsecuritywiththetwo-factormodel:ai,

bi1,bi2,andthestandarddeviationoftherandomerrorterm.Foreachofthefactors,twoparametersneedtobeestimated.Theseparametersaretheexpectedvalueofeachfactorandthevarianceofeachfactor.Finally,thecovariancebetweenfactors.第19頁(yè)/共55頁(yè)Multiple-FactorModelsExpectedreturnVarianceCovariance第20頁(yè)/共55頁(yè)Multiple-FactorModelsThetangencyportfolioTheinvestorcanproceedtouseanoptimizertoderivethecurveefficientset.DiversificationDiversificationleadstoanaveragingoffactorrisk.Diversificationcansubstantiallyreducenonfactorrisk.Forawell-diversifiedportfolio,nonfactorriskwillbeinsignificant.第21頁(yè)/共55頁(yè)Multiple-FactorModels第22頁(yè)/共55頁(yè)Multiple-FactorModelsSector-FactorModelsSector-factormodelsarebasedontheacknowledgethatthepricesofsecuritiesinthesameindustryoreconomicsectoroftenmovetogetherinresponsetochangesinprospectsforthatsector.Tocreateasector-factormodel,eachsecuritymustbeassignedtoasector.第23頁(yè)/共55頁(yè)Multiple-FactorModelsAtwo-sector-factormodelTherearetwosectorsandeachsecuritymustbeassignedtooneofthem.Boththenumberofsectorsandwhateachsectorconsistsofisanopenmatterthatislefttotheinvestortodecide.Thereturn-generatingprocessforsecuritiesisofthesamegeneralformasthetwo-factormodel.第24頁(yè)/共55頁(yè)Multiple-FactorModelsDifferingfromthetwo-factormodel,withtwo-sector-factormodel,F1andF2nowdenotesector-factors1and2,respectively.Anyparticularsecuritybelongstoeithersector-factor1orsector-factor2butnotboth.第25頁(yè)/共55頁(yè)Multiple-FactorModelsIngeneral,whereasfourparametersneedtobeestimatedforeachsecuritywithatwo-factormodel(ai1,bi1,bi2

,ei,),onlythreeparametersneedtobeestimatedwithatwo-sector-factormodel.(ai1,ei,andeitherbi1orbi2

).Multiple-factormodels第26頁(yè)/共55頁(yè)EstimatingFactorModelsTherearemanymethodsofestimatingfactormodels.Theremethodscanbegroupedintothreemajorapproaches:Time-seriesapproachesCross-sectionalapproachesFactor-analyticapproaches第27頁(yè)/共55頁(yè)FactorModelsandEquilibriumAfactormodelisnotanequilibriummodelofassetpricing.Bothequationshowthattheexpectedreturnonthestockisrelatedtoacharacteristicofthestock,biori.Thelargerthesizeofthecharacteristic,thelargertheasset’sreturn.第28頁(yè)/共55頁(yè)FactorModelsandEquilibriumThekeydifferenceisaiandrf.TheonlycharacteristicofthestockthatdetermineitsexpectedreturnaccordingtotheCAPMisii,asrffdenotestherisk-freerateandisthesameforallsecurities.Withthefactormodel,thereisasecondcharacteristicofthestockthatneedstobeestimatedtodeterminethestock’sexpectedreturn,aii.第29頁(yè)/共55頁(yè)FactorModelsandEquilibriumAsthesizeofaidiffersfromonestocktoanother,itpresentsthefactormodelfrombeinganequilibriummodel.Twostockswiththesamevalueofbicanhavedramaticallydifferentexpectedreturnsaccordingtoafactormodel.Twostockswiththesamevalueofiwillhavethesameexpectedreturnaccordingtotheequilibrium-basedCAPM.第30頁(yè)/共55頁(yè)FactorModelsandEquilibriumTherelationshipbetweentheparametersaiandbioftheone-factormodelandthesingleparameterioftheCAPM.IftheexpectedreturnsaredeterminedaccordingtotheCAPMandactualreturnsaregeneratedbytheone-factormarketmodel,thentheaboveequationsmustbetrue.第31頁(yè)/共55頁(yè)ArbitragePricingTheoryAPTisatheorywhichdescribeshowasecurityispricedjustlikeCAPM.Movingawayfromconstructionofmean-varianceefficientportfolio,APTinsteadcalculatesrelationsamongexpectedratesofreturnthatwouldruleoutrisklessprofitsbyanyinvestorinwell-functioningcapitalmarkets.第32頁(yè)/共55頁(yè)ArbitragePricingTheoryAPTmakesfewassumptions.Oneprimaryassumptionisthateachinvestor,whengiventheopportunitytoincreasethereturnofhisorherportfoliowithoutincreasingitsrisk,willproceedtodoso.Thereexistsanarbitrageopportunityandtheinvestorcanuseanarbitrageportfolios.第33頁(yè)/共55頁(yè)ArbitrageOpportunitiesArbitrageistheearningofrisklessprofitbytakingadvantageofdifferentialpricingforthesamephysicalassetorsecurity.Ittypicallyentailsthesaleofasecurityatarelativelyhighpriceandthesimultaneouspurchaseofthesamesecurity(oritsfunctionalequivalent)atarelativelylowprice.第34頁(yè)/共55頁(yè)ArbitrageOpportunitiesArbitrageactivityisacriticalelementofmodern,efficientsecuritymarkets.Ittakesrelativelyfewofthisactiveinvestorstoexploitarbitragesituationsand,bytheirbuyingandsellingactions,eliminatetheseprofitopportunities.SomeinvestorshavegreaterresourcesandinclinationtoengageIarbitragethanothers.第35頁(yè)/共55頁(yè)ArbitrageOpportunitiesZero-investmentportfolioAportfolioofzeronetvalue,establishedbybuyingandshortingcomponentsecurities.Arisklessarbitrageopportunityariseswhenaninvestorcanconstructazero-investmentportfoliothatwillyieldasureprofit.第36頁(yè)/共55頁(yè)ArbitrageOpportunitiesToconstructazero-investmentportfolio,onehastobeabletosellshortatleastoneassetandusetheproceedstopurchaseonormoreassets.Evenasmallinvestor,usingborrowedmoneyinthiscase,cantakealargepositioninsuchaportfolio.Therearemanyarbitragetactics.第37頁(yè)/共55頁(yè)ArbitrageOpportunitiesAnexample:Fourstocksandfourpossiblescenariostherateofreturninfourscenarios181inthetextbookTheexpectedreturns,standarddeviationsandcorrelationsdonotrevealanyabnormalitytothenakedeye.第38頁(yè)/共55頁(yè)ArbitrageOpportunitiesThecriticalpropertyofanarbitrageportfolioisthatanyinvestor,regardlessofriskaversionorwealth,willwanttotakeaninfinitepositioninitsothatprofitswillbedriventoaninfinitelevel.Theselargepositionswillforcesomepricesupanddownuntilarbitrageopportunitiesvanishes.第39頁(yè)/共55頁(yè)FactorModelsand

PrincipleofArbitrageAlmostarbitrageopportunitiescaninvolvesimilarsecuritiesorportfolios.Thatsimilaritycanbedefinedinmanyways.Onewayistheexposuretopervasivefactorsthataffectsecurityprices.AnexamplePage324第40頁(yè)/共55頁(yè)FactorModelsand

PrincipleofArbitrageAfactormodelimpliesthatsecuritiesorportfolioswithequal-factorsensitivitieswillbehaveinthesamewayexceptfornonfactorrisk.APTstartsoutbymakingtheassumptionthatsecurityreturnsarerelatedtoanunknownnumberofunknownfactors.Securitieswiththesamefactorsensitivitiesshouldofferthesameexpectedreturns.第41頁(yè)/共55頁(yè)ArbitragePortfoliosAnarbitrageportfoliomustsatisfy:AnetmarketvalueofzeroNosensitivitytoanyfactorApositiveexpectedreturn第42頁(yè)/共55頁(yè)ArbitragePortfoliosThearbitrageportfolioisattractivetoanyinvestorwhodesiresahigherreturnandisnotconcernedwithnonfactorrisk.Itrequiresnoadditionaldollarinvestment,ithasnofactorrisk,andithasapositiveexpectedreturn.第43頁(yè)/共55頁(yè)One-FactorModelandAPTPricingeffectsonarbitrageportfolioThebuying-and-sellingactivitywillcontinueuntilallarbitragepossibilitiesaresignificantreducedoreliminatedTherewillexistanapproximatelylinearrelationshipbetweenexpectedreturnsandsensitivitiesofthefollowingsort:第44頁(yè)/共55頁(yè)One-FactorModelandAPTTheequationistheassetpricingequationoftheAPTwhenreturnsaregeneratedbyonefactorThelinearequationmeansthatinequili-briumtherewillbealinearrelationshipbetweenexpectedreturnsandsensitivities.Theexpectedreturnonanysecurityis,inequilibrium,alinearfunctionofthesecurity’ssensitivitytothefactor,bi第45頁(yè)/共55頁(yè)One-FactorModelandAPTAnysecuritythathasafactorsensitivityandexpectedreturnsuchthatitliesoffthelinewillbemispricedaccordingtotheAPTandwillpresentinvestorswiththeopportunityofformingarbitrageportfolios.Page327:Figure12.1第46頁(yè)/共55頁(yè)One-FactorModelandAPTInterpretingtheAPTpricingequationRiskfreeasset,rfPurefactorportfolio,p*第47頁(yè)/共55頁(yè)Two-FactorModelAndAPTThetwo-factormodelArbitrageportfoliosAnetmarketvalueofzeroNosensitivitytoanyfactorApositiveexpectedreturn第48頁(yè)/共55