現(xiàn)代資產(chǎn)組合理論和資本資產(chǎn)定價(jià)模型分析

ModernPortfolioTheory

TheFactorModelsand

TheArbitragePricingTheoryChapter8ByDingzhaoyongReturn-generatingProcess

andFactorModelsReturn-generatingprocessIsastatisticalmodelthatdescribehowreturnonasecurityisproduced.ThetaskofidentifyingtheMarkowitzefficientsetcanbegreatlysimplifiedbyintroducingthisprocess.Themarketmodelisakindofthisprocess,andtherearemanyothers.Return-generatingProcess

andFactorModelsFactormodelsThesemodelsassumethatthereturnonasecurityissensitivetothemove-mentsofvariousfactorsorindices.Inattemptingtoaccuratelyestimateexpectedreturns,variances,andcovariancesforsecurities,multiple-factormodelsarepotentiallymoreusefulthanthemarketmodel.Return-generatingProcess

andFactorModelsImplicitintheconstructionofafactormodelistheassumptionthatthereturnsontwosecuritieswillbecorrelatedonlythroughcommonreactionstooneormoreofthespecifiedinthemodel.Anyaspectofasecurity’sreturnunexplainedbythefactormodelisuncorrelatedwiththeuniqueelementsofreturnsonothersecurities.Return-generatingProcess

andFactorModelsAfactormodelisapowerfultoolforportfoliomanagement.Itcansupplytheinformationneededtocalculateexpectedreturns,variances,andcovariancesforeverysecurity,whicharethenecessaryconditionsfordeterminingthecurvedMarkowitzefficientset.Itcanalsobeusedtocharacterizeaportfolio’ssensitivitytomovementinthefactors.Return-generatingProcess

andFactorModelsFactormodelssupplythenecessarylevelofabstractionincalculatingcovariances.Theproblemofcalculatingcovariancesamongsecuritiesrisesexponentiallyasthenumberofsecuritiesanalyzedincrease.Practically,abstractionisanessentialstepinidentifyingtheMarkowitzset.Return-generatingProcess

andFactorModelsFactormodelsprovideinvestmentmanagerswithaframeworktoidentifyimportantfactorsintheeconomyandthemarketplaceandtoassesstheextenttowhichdifferentsecuritiesandportfolioswillrespondtochangesinthesefactors.Aprimarygoalofsecurityanalysisistodeterminethesefactorsandthesensitivitiesofsecurityreturntomovementsinthesefactors.One-FactorModelsTheone-factormodelsrefertothereturn-generatingprocessforsecuritiesinvolvesasinglefactor.Thesefactorsmaybeoneofthefollowings:ThepredictedgrowthrateinGDPTheexpectedreturnonmarketindexThegrowthrateofindustrialproduc-tion,etc.One-FactorModelsAnexamplePage295:Figure11.1One-FactorModelsGeneralizingtheexampleAssumptionsTherandomerrortermandthefactorareuncorrelated.(Why?)Therandomerrortermsofanytwosecuritiesareuncorrelated.(Why?)One-FactorModelsExpectedreturnVarianceCovarianceOne-FactorModelsTwoimportantfeaturesofone-factormodelThetangencyportfolioiseasytoget.Thereturnsonallsecuritiesrespondtoasinglecommonfactorgreatersimplifiesthetaskofidentifyingthetangencyportfolio.Thecommonresponsivenessofsecuritiestothefactoreliminatestheneedtoestimatedirectlythecovariancesbetweenthesecurities.Thenumberofestimates:3N+2One-FactorModelsThefeatureofdiversificationistrueofanyone-factormodel.Factorrisk:Nonfactorrisk:DiversificationleadstoanaveragingoffactorriskDiversificationreducesnonfactorriskOne-FactorModelsMultiple-FactorModelsThehealthoftheeconomyeffectsmostfirms,buttheeconomyisnotasimple,monolithicentity.SeveralcommoninfluenceswithpervasiveeffectsmightbeidentifiedThegrowthrateofGDPThelevelofinterestrateTheinflationrateThelevelofoilpriceMultiple-FactorModelsTwo-FactorModelsAssumethatthereturn-generatingprocesscontainstwofactors.Multiple-FactorModelsThesecondequationprovidesatwo-factormodelofacompany’sstock,whosereturnsareaffectedbyexpectationsconcerningboththegrowthrateinGDPandtherateofinflation.Page301:Figure11.2Tothisscatterofpointsisfitatwo-dimensionalplanebyusingthestatisticaltechniqueofmultiple-regressionanalysis.Multiple-FactorModelsFourparametersneedtobeestimatedforeachsecuritywiththetwo-factormodel:ai,

bi1,bi2,andthestandarddeviationoftherandomerrorterm.Foreachofthefactors,twoparametersneedtobeestimated.Theseparametersaretheexpectedvalueofeachfactorandthevarianceofeachfactor.Finally,thecovariancebetweenfactors.Multiple-FactorModelsExpectedreturnVarianceCovarianceMultiple-FactorModelsThetangencyportfolioTheinvestorcanproceedtouseanoptimizertoderivethecurveefficientset.DiversificationDiversificationleadstoanaveragingoffactorrisk.Diversificationcansubstantiallyreducenonfactorrisk.Forawell-diversifiedportfolio,nonfactorriskwillbeinsignificant.Multiple-FactorModelsMultiple-FactorModelsSector-FactorModelsSector-factormodelsarebasedontheacknowledgethatthepricesofsecuritiesinthesameindustryoreconomicsectoroftenmovetogetherinresponsetochangesinprospectsforthatsector.Tocreateasector-factormodel,eachsecuritymustbeassignedtoasector.Multiple-FactorModelsAtwo-sector-factormodelTherearetwosectorsandeachsecuritymustbeassignedtooneofthem.Boththenumberofsectorsandwhateachsectorconsistsofisanopenmatterthatislefttotheinvestortodecide.Thereturn-generatingprocessforsecuritiesisofthesamegeneralformasthetwo-factormodel.Multiple-FactorModelsDifferingfromthetwo-factormodel,withtwo-sector-factormodel,F1andF2nowdenotesector-factors1and2,respectively.Anyparticularsecuritybelongstoeithersector-factor1orsector-factor2butnotboth.Multiple-FactorModelsIngeneral,whereasfourparametersneedtobeestimatedforeachsecuritywithatwo-factormodel(ai1,bi1,bi2

,

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

ei,andeitherbi1orbi2

).Multiple-factormodelsEstimatingFactorModelsTherearemanymethodsofestimatingfactormodels.Theremethodscanbegroupedintothreemajorapproaches:Time-seriesapproachesCross-sectionalapproachesFactor-analyticapproachesFactorModelsandEquilibriumAfactormodelisnotanequilibriummodelofassetpricing.Bothequationshowthattheexpectedreturnonthestockisrelatedtoacharacteristicofthestock,bior

i.Thelargerthesizeofthecharacteristic,thelargertheasset’sreturn.FactorModelsandEquilibriumThekeydifferenceisaiandrf.TheonlycharacteristicofthestockthatdetermineitsexpectedreturnaccordingtotheCAPMis

ii,asrffdenotestherisk-freerateandisthesameforallsecurities.Withthefactormodel,thereisasecondcharacteristicofthestockthatneedstobeestimatedtodeterminethestock’sexpectedreturn,aii.FactorModelsandEquilibriumAsthesizeofaidiffersfromonestocktoanother,itpresentsthefactormodelfrombeinganequilibriummodel.Twostockswiththesamevalueofbicanhavedramaticallydifferentexpectedreturnsaccordingtoafactormodel.Twostockswiththesamevalueof

iwillhavethesameexpectedreturnaccordingtotheequilibrium-basedCAPM.FactorModelsandEquilibriumTherelationshipbetweentheparametersaiandbioftheone-factormodelandthesingleparameter

ioftheCAPM.IftheexpectedreturnsaredeterminedaccordingtotheCAPMandactualreturnsaregeneratedbytheone-factormarketmodel,thentheaboveequationsmustbetrue.ArbitragePricingTheoryAPTisatheorywhichdescribeshowasecurityispricedjustlikeCAPM.Movingawayfromconstructionofmean-varianceefficientportfolio,APTinsteadcalculatesrelationsamongexpectedratesofreturnthatwouldruleoutrisklessprofitsbyanyinvestorinwell-functioningcapitalmarkets.ArbitragePricingTheoryAPTmakesfewassumptions.Oneprimaryassumptionisthateachinvestor,whengiventheopportunitytoincreasethereturnofhisorherportfoliowithoutincreasingitsrisk,willproceedtodoso.Thereexistsanarbitrageopportunityandtheinvestorcanuseanarbitrageportfolios.ArbitrageOpportunitiesArbitrageistheearningofrisklessprofitbytakingadvantageofdifferentialpricingforthesamephysicalassetorsecurity.Ittypicallyentailsthesaleofasecurityatarelativelyhighpriceandthesimultaneouspurchaseofthesamesecurity(oritsfunctionalequivalent)atarelativelylowprice.ArbitrageOpportunitiesArbitrageactivityisacriticalelementofmodern,efficientsecuritymarkets.Ittakesrelativelyfewofthisactiveinvestorstoexploitarbitragesituationsand,bytheirbuyingandsellingactions,eliminatetheseprofitopportunities.SomeinvestorshavegreaterresourcesandinclinationtoengageIarbitragethanothers.ArbitrageOpportunitiesZero-investmentportfolioAportfolioofzeronetvalue,establishedbybuyingandshortingcomponentsecurities.Arisklessarbitrageopportunityariseswhenaninvestorcanconstructazero-investmentportfoliothatwillyieldasureprofit.ArbitrageOpportunitiesToconstructazero-investmentportfolio,onehastobeabletosellshortatleastoneassetandusetheproceedstopurchaseonormoreassets.Evenasmallinvestor,usingborrowedmoneyinthiscase,cantakealargepositioninsuchaportfolio.Therearemanyarbitragetactics.ArbitrageOpportunitiesAnexample:Fourstocksandfourpossiblescenariostherateofreturninfourscenarios181inthetextbookTheexpectedreturns,standarddeviationsandcorrelationsdonotrevealanyabnormalitytothenakedeye.ArbitrageOpportunitiesThecriticalpropertyofanarbitrageportfolioisthatanyinvestor,regardlessofriskaversionorwealth,willwanttotakeaninfinitepositioninitsothatprofitswillbedriventoaninfinitelevel.Theselargepositionswillforcesomepricesupanddownuntilarbitrageopportunitiesvanishes.FactorModelsand

PrincipleofArbitrageAlmostarbitrageopportunitiescaninvolvesimilarsecuritiesorportfolios.Thatsimilaritycanbedefinedinmanyways.Onewayistheexposuretopervasivefactorsthataffectsecurityprices.AnexamplePage324FactorModelsand

PrincipleofArbitrageAfactormodelimpliesthatsecuritiesorportfolioswithequal-factorsensitivitieswillbehaveinthesamewayexceptfornonfactorrisk.APTstartsoutbymakingtheassumptionthatsecurityreturnsarerelatedtoanunknownnumberofunknownfactors.Securitieswiththesamefactorsensitivitiesshouldofferthesameexpectedreturns.ArbitragePortfoliosAnarbitrageportfoliomustsatisfy:AnetmarketvalueofzeroNosensitivitytoanyfactorApositiveexpectedreturnArbitragePortfoliosThearbitrageportfolioisattractivetoanyinvestorwhodesiresahigherreturnandisnotconcernedwithnonfactorrisk.Itrequiresnoadditionaldollarinvestment,ithasnofactorrisk,andithasapositiveexpectedreturn.One-FactorModelandAPTPricingeffectsonarbitrageportfolioThebuying-and-sellingactivitywillcontinueuntilallarbitragepossibilitiesaresignificantreducedoreliminatedTherewillexistanapproximatelylinearrelationshipbetweenexpectedreturnsandsensitivitiesofthefollowingsort:One-FactorModelandAPTTheequationistheassetpricingequationoftheAPTwhenreturnsaregeneratedbyonefactorThelinearequationmeansthatinequili-briumtherewillbealinearrelationshipbetweenexpectedreturnsandsensitivities.Theexpectedreturnonanysecurityis,inequilibrium,alinearfunctionofthesecurity’ssensitivitytothefactor,biOne-FactorModelandAPTAnysecuritythathasafactorsensitivityandexpectedreturnsuchthatitliesoffthelinewillbemispricedaccordingtotheAPTandwillpresentinvestorswiththeopportunityofformingarbitrageportfolios.Page327:Figure12.1One-FactorModelandAPTInterpretingtheAPTpricingequationRiskfreeasset,rfPurefactorportfolio,p*Two-FactorModelAndAPTThetwo-factormodelArbitrageportfoliosAnetmarketvalueofzeroNosensitivitytoanyfactorApositiveexpectedreturnTwo-FactorModelAndAPTPricingeffectsTwo-FactorModelAndAPT

1istheexpectedreturnonthepo