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ModernPortfolioTheory

TheFactorModelsand

TheArbitragePricingTheoryChapter8ByDingzhaoyongModernPortfolioTheory

TheFa1Return-generatingProcess

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

and2Return-generatingProcess

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

and3Return-generatingProcess

andFactorModelsImplicitintheconstructionofafactormodelistheassumptionthatthereturnsontwosecuritieswillbecorrelatedonlythroughcommonreactionstooneormoreofthespecifiedinthemodel.Anyaspectofasecurity’sreturnunexplainedbythefactormodelisuncorrelatedwiththeuniqueelementsofreturnsonothersecurities.Return-generatingProcess

and4Return-generatingProcess

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

and5Return-generatingProcess

andFactorModelsFactormodelssupplythenecessarylevelofabstractionincalculatingcovariances.Theproblemofcalculatingcovariancesamongsecuritiesrisesexponentiallyasthenumberofsecuritiesanalyzedincrease.Practically,abstractionisanessentialstepinidentifyingtheMarkowitzset.Return-generatingProcess

and6Return-generatingProcess

andFactorModelsFactormodelsprovideinvestmentmanagerswithaframeworktoidentifyimportantfactorsintheeconomyandthemarketplaceandtoassesstheextenttowhichdifferentsecuritiesandportfolioswillrespondtochangesinthesefactors.Aprimarygoalofsecurityanalysisistodeterminethesefactorsandthesensitivitiesofsecurityreturntomovementsinthesefactors.Return-generatingProcess

and7One-FactorModelsTheone-factormodelsrefertothereturn-generatingprocessforsecuritiesinvolvesasinglefactor.Thesefactorsmaybeoneofthefollowings:ThepredictedgrowthrateinGDPTheexpectedreturnonmarketindexThegrowthrateofindustrialproduc-tion,etc.One-FactorModelsTheone-facto8One-FactorModelsAnexamplePage295:Figure11.1One-FactorModelsAnexample9One-FactorModelsGeneralizingtheexampleAssumptionsTherandomerrortermandthefactorareuncorrelated.(Why?)Therandomerrortermsofanytwosecuritiesareuncorrelated.(Why?)One-FactorModelsGeneralizing10One-FactorModelsExpectedreturnVarianceCovarianceOne-FactorModelsExpectedretu11One-FactorModelsTwoimportantfeaturesofone-factormodelThetangencyportfolioiseasytoget.Thereturnsonallsecuritiesrespondtoasinglecommonfactorgreatersimplifiesthetaskofidentifyingthetangencyportfolio.Thecommonresponsivenessofsecuritiestothefactoreliminatestheneedtoestimatedirectlythecovariancesbetweenthesecurities.Thenumberofestimates:3N+2One-FactorModelsTwoimportant12One-FactorModelsThefeatureofdiversificationistrueofanyone-factormodel.Factorrisk:Nonfactorrisk:DiversificationleadstoanaveragingoffactorriskDiversificationreducesnonfactorriskOne-FactorModelsThefeatureo13One-FactorModelsOne-FactorModels14Multiple-FactorModelsThehealthoftheeconomyeffectsmostfirms,buttheeconomyisnotasimple,monolithicentity.SeveralcommoninfluenceswithpervasiveeffectsmightbeidentifiedThegrowthrateofGDPThelevelofinterestrateTheinflationrateThelevelofoilpriceMultiple-FactorModelsTheheal15Multiple-FactorModelsTwo-FactorModelsAssumethatthereturn-generatingprocesscontainstwofactors.Multiple-FactorModelsTwo-Fact16Multiple-FactorModelsThesecondequationprovidesatwo-factormodelofacompany’sstock,whosereturnsareaffectedbyexpectationsconcerningboththegrowthrateinGDPandtherateofinflation.Page301:Figure11.2Tothisscatterofpointsisfitatwo-dimensionalplanebyusingthestatisticaltechniqueofmultiple-regressionanalysis.Multiple-FactorModelsTheseco17Multiple-FactorModelsFourparametersneedtobeestimatedforeachsecuritywiththetwo-factormodel:ai,

bi1,bi2,andthestandarddeviationoftherandomerrorterm.Foreachofthefactors,twoparametersneedtobeestimated.Theseparametersaretheexpectedvalueofeachfactorandthevarianceofeachfactor.Finally,thecovariancebetweenfactors.Multiple-FactorModelsFourpar18Multiple-FactorModelsExpectedreturnVarianceCovarianceMultiple-FactorModelsExpected19Multiple-FactorModelsThetangencyportfolioTheinvestorcanproceedtouseanoptimizertoderivethecurveefficientset.DiversificationDiversificationleadstoanaveragingoffactorrisk.Diversificationcansubstantiallyreducenonfactorrisk.Forawell-diversifiedportfolio,nonfactorriskwillbeinsignificant.Multiple-FactorModelsThetang20Multiple-FactorModelsMultiple-FactorModels21Multiple-FactorModelsSector-FactorModelsSector-factormodelsarebasedontheacknowledgethatthepricesofsecuritiesinthesameindustryoreconomicsectoroftenmovetogetherinresponsetochangesinprospectsforthatsector.Tocreateasector-factormodel,eachsecuritymustbeassignedtoasector.Multiple-FactorModelsSector-F22Multiple-FactorModelsAtwo-sector-factormodelTherearetwosectorsandeachsecuritymustbeassignedtooneofthem.Boththenumberofsectorsandwhateachsectorconsistsofisanopenmatterthatislefttotheinvestortodecide.Thereturn-generatingprocessforsecuritiesisofthesamegeneralformasthetwo-factormodel.Multiple-FactorModelsAtwo-se23Multiple-FactorModelsDifferingfromthetwo-factormodel,withtwo-sector-factormodel,F1andF2nowdenotesector-factors1and2,respectively.Anyparticularsecuritybelongstoeithersector-factor1orsector-factor2butnotboth.Multiple-FactorModelsDifferin24Multiple-FactorModelsIngeneral,whereasfourparametersneedtobeestimatedforeachsecuritywithatwo-factormodel(ai1,bi1,bi2

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

).Multiple-factormodelsMultiple-FactorModelsIngener25EstimatingFactorModelsTherearemanymethodsofestimatingfactormodels.Theremethodscanbegroupedintothreemajorapproaches:Time-seriesapproachesCross-sectionalapproachesFactor-analyticapproachesEstimatingFactorModelsThere26FactorModelsandEquilibriumAfactormodelisnotanequilibriummodelofassetpricing.Bothequationshowthattheexpectedreturnonthestockisrelatedtoacharacteristicofthestock,biori.Thelargerthesizeofthecharacteristic,thelargertheasset’sreturn.FactorModelsandEquilibrium27FactorModelsandEquilibriumThekeydifferenceisaiandrf.TheonlycharacteristicofthestockthatdetermineitsexpectedreturnaccordingtotheCAPMisii,asrffdenotestherisk-freerateandisthesameforallsecurities.Withthefactormodel,thereisasecondcharacteristicofthestockthatneedstobeestimatedtodeterminethestock’sexpectedreturn,aii.FactorModelsandEquilibriumT28FactorModelsandEquilibriumAsthesizeofaidiffersfromonestocktoanother,itpresentsthefactormodelfrombeinganequilibriummodel.Twostockswiththesamevalueofbicanhavedramaticallydifferentexpectedreturnsaccordingtoafactormodel.Twostockswiththesamevalueofiwillhavethesameexpectedreturnaccordingtotheequilibrium-basedCAPM.FactorModelsandEquilibriumA29FactorModelsandEquilibriumTherelationshipbetweentheparametersaiandbioftheone-factormodelandthesingleparameterioftheCAPM.IftheexpectedreturnsaredeterminedaccordingtotheCAPMandactualreturnsaregeneratedbytheone-factormarketmodel,thentheaboveequationsmustbetrue.FactorModelsandEquilibriumT30ArbitragePricingTheoryAPTisatheorywhichdescribeshowasecurityispricedjustlikeCAPM.Movingawayfromconstructionofmean-varianceefficientportfolio,APTinsteadcalculatesrelationsamongexpectedratesofreturnthatwouldruleoutrisklessprofitsbyanyinvestorinwell-functioningcapitalmarkets.ArbitragePricingTheoryAPTi31ArbitragePricingTheoryAPTmakesfewassumptions.Oneprimaryassumptionisthateachinvestor,whengiventheopportunitytoincreasethereturnofhisorherportfoliowithoutincreasingitsrisk,willproceedtodoso.Thereexistsanarbitrageopportunityandtheinvestorcanuseanarbitrageportfolios.ArbitragePricingTheoryAPTma32ArbitrageOpportunitiesArbitrageistheearningofrisklessprofitbytakingadvantageofdifferentialpricingforthesamephysicalassetorsecurity.Ittypicallyentailsthesaleofasecurityatarelativelyhighpriceandthesimultaneouspurchaseofthesamesecurity(oritsfunctionalequivalent)atarelativelylowprice.ArbitrageOpportunitiesArbitra33ArbitrageOpportunitiesArbitrageactivityisacriticalelementofmodern,efficientsecuritymarkets.Ittakesrelativelyfewofthisactiveinvestorstoexploitarbitragesituationsand,bytheirbuyingandsellingactions,eliminatetheseprofitopportunities.SomeinvestorshavegreaterresourcesandinclinationtoengageIarbitragethanothers.ArbitrageOpportunitiesArbitra34ArbitrageOpportunitiesZero-investmentportfolioAportfolioofzeronetvalue,establishedbybuyingandshortingcomponentsecurities.Arisklessarbitrageopportunityariseswhenaninvestorcanconstructazero-investmentportfoliothatwillyieldasureprofit.ArbitrageOpportunitiesZero-in35ArbitrageOpportunitiesToconstructazero-investmentportfolio,onehastobeabletosellshortatleastoneassetandusetheproceedstopurchaseonormoreassets.Evenasmallinvestor,usingborrowedmoneyinthiscase,cantakealargepositioninsuchaportfolio.Therearemanyarbitragetactics.ArbitrageOpportunitiesTocons36ArbitrageOpportunitiesAnexample:Fourstocksandfourpossiblescenariostherateofreturninfourscenarios181inthetextbookTheexpectedreturns,standarddeviationsandcorrelationsdonotrevealanyabnormalitytothenakedeye.ArbitrageOpportunitiesAnexam37ArbitrageOpportunitiesThecriticalpropertyofanarbitrageportfolioisthatanyinvestor,regardlessofriskaversionorwealth,willwanttotakeaninfinitepositioninitsothatprofitswillbedriventoaninfinitelevel.Theselargepositionswillforcesomepricesupanddownuntilarbitrageopportunitiesvanishes.ArbitrageOpportunitiesThecri38FactorModelsand

PrincipleofArbitrageAlmostarbitrageopportunitiescaninvolvesimilarsecuritiesorportfolios.Thatsimilaritycanbedefinedinmanyways.Onewayistheexposuretopervasivefactorsthataffectsecurityprices.AnexamplePage324FactorModelsand

Principleo39FactorModelsand

PrincipleofArbitrageAfactormodelimpliesthatsecuritiesorportfolioswithequal-factorsensitivitieswillbehaveinthesamewayexceptfornonfactorrisk.APTstartsoutbymakingtheassumptionthatsecurityreturnsarerelatedtoanunknownnumberofunknownfactors.Securitieswiththesamefactorsensitivitiesshouldofferthesameexpectedreturns.FactorModelsand

Principleo40ArbitragePortfoliosAnarbitrageportfoliomustsatisfy:AnetmarketvalueofzeroNosensitivitytoanyfactorApositiveexpectedreturnArbitragePortfoliosAnarbitra41ArbitragePortfoliosThearbitrageportfolioisattractivetoanyinvestorwhodesiresahigherreturnandisnotconcernedwithnonfactorrisk.Itrequiresnoadditionaldollarinvestment,ithasnofactorrisk,andithasapositiveexpectedreturn.ArbitragePortfoliosThearbitr42One-FactorModelandAPTPricingeffectsonarbitrageportfolioThebuying-and-sellingactivitywillcontinueuntilallarbitragepossibilitiesaresignificantreducedoreliminatedTherewillexistanapproximatelylinearrelationshipbetweenexpectedreturnsandsensitivitiesofthefollowingsort:One-FactorModelandAPTPricin43One-FactorModelandAPTTheequationistheassetpricingequationoftheAPTwhenreturnsaregeneratedbyonefactorThelinearequationmeansthatinequili-briumtherewillbealinearrelationshipbetweenexpectedreturnsandsensitivities.Theexpectedreturnonanysecurityis,inequilibrium,alinearfunctionofthesecurity’ssensitivitytothefactor,biOne-FactorModelandAPTTheeq44One-FactorModelandAPTAnysecuritythathasafactorsensitivityandexpectedreturnsuchthatitliesoffthelinewillbemispricedaccordingtotheAPTandwillpresentinvestorswiththeopportunityofformingarbitrageportfolios.Page327:Figure12.1One-FactorModelandAPTAnyse45One-FactorModelandAPTInterpretingtheAPTpricingequationRiskfreeasset,rfPurefactorportfolio,p*One-FactorModelandAPTInterp46Two-FactorModelAndAPTThetwo-factormodelArbitrageportfoliosAnetmarketvalueofzeroNosensitivitytoanyfactorApositiveexpectedreturnTwo-FactorModelAndAPTThetw47Two-FactorModelAndAPTPricingeffectsTwo-FactorModelAndAPTPricin48Two-FactorModelAndAPT1istheexpectedreturnontheportfoliowhichisknownasapurefactorportfolioorpurefactorplay,becauseithas:Unitsensitivitytoonefactor(F1,b1=1)Nosensitivitytoanyotherfactor(F2,b2=0)ZerononfactorriskThisportfolioisawell-diversificationportfoliothathasunitsensitivitytothefirstfactorandzerosensitivitytothesecondfactor.Two-FactorModelAndAPT1is49Two-FactorModelAndAPTItisthesamewith2.Itisthewell-diversificationportfoliothathaszerosensitivitytothefirstfactorandunitsensitivitytothesecondfactor,meaningthatithasb1=0andb2=1.Suchasaportfoliothathaszerosensitivitytopredictedindustrialproductionandunitsensitivitytopredictedinflationwouldhaveanexpectedreturnof6%.Two-FactorModelAndAPTItis50Multiple-factormodelTheAPTpricingequationMultiple-FactorModel

AndAPTMultiple-factormodelMultiple-51TheAPTAndTheCAPMCommonpointBothrequireequilibriumBothhavealmostsimilarequationDistinctionsDifferentequilibriummechanismManyinvestorsv.s.FewinvestorsDifferentPortfolioMarketportfoliov.s.Well-diversifyedP.TheAPTAndTheCAPMCommonpoi52SummaryTheFactorModelsOne-factormodelsMulti-factormodelsFactormodelsandequilibriumArbitrageopportunityandportfolioThearbitragepricingequationOne-factorequationMulti-factorequationSummaryTheFactorModels53AssignmentsForchapter8ReadingsPage282through301Page308through321ExercisesPage304:14,15;Page323:4,13Q/A:Page302:3Page324:8AssignmentsForchapter8Readi54ModernPortfolioTheory

TheFactorModelsand

TheArbitragePricingTheoryChapter8ByDingzhaoyongModernPortfolioTheory

TheFa55Return-generatingProcess

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

and56Return-generatingProcess

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

and57Return-generatingProcess

andFactorModelsImplicitintheconstructionofafactormodelistheassumptionthatthereturnsontwosecuritieswillbecorrelatedonlythroughcommonreactionstooneormoreofthespecifiedinthemodel.Anyaspectofasecurity’sreturnunexplainedbythefactormodelisuncorrelatedwiththeuniqueelementsofreturnsonothersecurities.Return-generatingProcess

and58Return-generatingProcess

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

and59Return-generatingProcess

andFactorModelsFactormodelssupplythenecessarylevelofabstractionincalculatingcovariances.Theproblemofcalculatingcovariancesamongsecuritiesrisesexponentiallyasthenumberofsecuritiesanalyzedincrease.Practically,abstractionisanessentialstepinidentifyingtheMarkowitzset.Return-generatingProcess

and60Return-generatingProcess

andFactorModelsFactormodelsprovideinvestmentmanagerswithaframeworktoidentifyimportantfactorsintheeconomyandthemarketplaceandtoassesstheextenttowhichdifferentsecuritiesandportfolioswillrespondtochangesinthesefactors.Aprimarygoalofsecurityanalysisistodeterminethesefactorsandthesensitivitiesofsecurityreturntomovementsinthesefactors.Return-generatingProcess

and61One-FactorModelsTheone-factormodelsrefertothereturn-generatingprocessforsecuritiesinvolvesasinglefactor.Thesefactorsmaybeoneofthefollowings:ThepredictedgrowthrateinGDPTheexpectedreturnonmarketindexThegrowthrateofindustrialproduc-tion,etc.One-FactorModelsTheone-facto62One-FactorModelsAnexamplePage295:Figure11.1One-FactorModelsAnexample63One-FactorModelsGeneralizingtheexampleAssumptionsTherandomerrortermandthefactorareuncorrelated.(Why?)Therandomerrortermsofanytwosecuritiesareuncorrelated.(Why?)One-FactorModelsGeneralizing64One-FactorModelsExpectedreturnVarianceCovarianceOne-FactorModelsExpectedretu65One-FactorModelsTwoimportantfeaturesofone-factormodelThetangencyportfolioiseasytoget.Thereturnsonallsecuritiesrespondtoasinglecommonfactorgreatersimplifiesthetaskofidentifyingthetangencyportfolio.Thecommonresponsivenessofsecuritiestothefactoreliminatestheneedtoestimatedirectlythecovariancesbetweenthesecurities.Thenumberofestimates:3N+2One-FactorModelsTwoimportant66One-FactorModelsThefeatureofdiversificationistrueofanyone-factormodel.Factorrisk:Nonfactorrisk:DiversificationleadstoanaveragingoffactorriskDiversificationreducesnonfactorriskOne-FactorModelsThefeatureo67One-FactorModelsOne-FactorModels68Multiple-FactorModelsThehealthoftheeconomyeffectsmostfirms,buttheeconomyisnotasimple,monolithicentity.SeveralcommoninfluenceswithpervasiveeffectsmightbeidentifiedThegrowthrateofGDPThelevelofinterestrateTheinflationrateThelevelofoilpriceMultiple-FactorModelsTheheal69Multiple-FactorModelsTwo-FactorModelsAssumethatthereturn-generatingprocesscontainstwofactors.Multiple-FactorModelsTwo-Fact70Multiple-FactorModelsThesecondequationprovidesatwo-factormodelofacompany’sstock,whosereturnsareaffectedbyexpectationsconcerningboththegrowthrateinGDPandtherateofinflation.Page301:Figure11.2Tothisscatterofpointsisfitatwo-dimensionalplanebyusingthestatisticaltechniqueofmultiple-regressionanalysis.Multiple-FactorModelsTheseco71Multiple-FactorModelsFourparametersneedtobeestimatedforeachsecuritywiththetwo-factormodel:ai,

bi1,bi2,andthestandarddeviationoftherandomerrorterm.Foreachofthefactors,twoparametersneedtobeestimated.Theseparametersaretheexpectedvalueofeachfactorandthevarianceofeachfactor.Finally,thecovariancebetweenfactors.Multiple-FactorModelsFourpar72Multiple-FactorModelsExpectedreturnVarianceCovarianceMultiple-FactorModelsExpected73Multiple-FactorModelsThetangencyportfolioTheinvestorcanproceedtouseanoptimizertoderivethecurveefficientset.DiversificationDiversificationleadstoanaveragingoffactorrisk.Diversificationcansubstantiallyreducenonfactorrisk.Forawell-diversifiedportfolio,nonfactorriskwillbeinsignificant.Multiple-FactorModelsThetang74Multiple-FactorModelsMultiple-FactorModels75Multiple-FactorModelsSector-FactorModelsSector-factormodelsarebasedontheacknowledgethatthepricesofsecuritiesinthesameindustryoreconomicsectoroftenmovetogetherinresponsetochangesinprospectsforthatsector.Tocreateasector-factormodel,eachsecuritymustbeassignedtoasector.Multiple-FactorModelsSector-F76Multiple-FactorModelsAtwo-sector-factormodelTherearetwosectorsandeachsecuritymustbeassignedtooneofthem.Boththenumberofsectorsandwhateachsectorconsistsofisanopenmatterthatislefttotheinvestortodecide.Thereturn-generatingprocessforsecuritiesisofthesamegeneralformasthetwo-factormodel.Multiple-FactorModelsAtwo-se77Multiple-FactorModelsDifferingfromthetwo-factormodel,withtwo-sector-factormodel,F1andF2nowdenotesector-factors1and2,respectively.Anyparticularsecuritybelongstoeithersector-factor1orsector-factor2butnotboth.Multiple-FactorModelsDifferin78Multiple-FactorModelsIngeneral,whereasfourparametersneedtobeestimatedforeachsecuritywithatwo-factormodel(ai1,bi1,bi2

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

).Multiple-factormodelsMultiple-FactorModelsIngener79EstimatingFactorModelsTherearemanymethodsofestimatingfactormodels.Theremethodscanbegroupedintothreemajorapproaches:Time-seriesapproachesCross-sectionalapproachesFactor-analyticapproachesEstimatingFactorModelsThere80FactorModelsandEquilibriumAfactormodelisnotanequilibriummodelofassetpricing.Bothequationshowthattheexpectedreturnonthestockisrelatedtoacharacteristicofthestock,biori.Thelargerthesizeofthecharacteristic,thelargertheasset’sreturn.FactorModelsandEquilibrium81FactorModelsandEquilibriumThekeydifferenceisaiandrf.TheonlycharacteristicofthestockthatdetermineitsexpectedreturnaccordingtotheCAPMisii,asrffdenotestherisk-freerateandisthesameforallsecurities.Withthefactormodel,thereisasecondcharacteristicofthestockthatneedstobeestimatedtodeterminethestock’sexpectedreturn,aii.FactorModelsandEquilibriumT82FactorModelsandEquilibriumAsthesizeofaidiffersfromonestocktoanother,itpresentsthefactormodelfrombeinganequilibriummodel.Twostockswiththesamevalueofbicanhavedramaticallydifferentexpectedreturnsaccordingtoafactormodel.Twostockswiththesamevalueofiwillhavethesameexpectedreturnaccordingtotheequilibrium-basedCAPM.FactorModelsandEquilibriumA83FactorModelsandEquilibriumTherelationshipbetweentheparametersaiandbioftheone-factormodelandthesingleparameterioftheCAPM.IftheexpectedreturnsaredeterminedaccordingtotheCAPMandactualreturnsaregeneratedbytheone-factormarketmodel,thentheaboveequationsmustbetrue.FactorModelsandEquilibriumT84ArbitragePricingTheoryAPTisatheorywhichdescribeshowasecurityispricedjustlikeCAPM.Movingawayfromconstructionofmean-varianceefficientportfolio,APTinsteadcalculatesrelationsamongexpectedratesofreturnthatwouldruleoutrisklessprofitsbyanyinvestorinwell-functioningcapitalmarkets.ArbitragePricingTheoryAPTi85ArbitragePricingTheoryAPTmakesfewassumptions.Oneprimaryassumptionisthateachinvestor,whengiventheopportunitytoincreasethereturnofhisorherportfoliowithoutincreasingitsrisk,willproceedtodoso.Thereexistsanarbitrageopportunityandtheinvestorcanuseanarbitrageportfolios.ArbitragePricingTheoryAPTma86ArbitrageOpportunitiesArbitrageistheearningofrisklessprofitbytakingadvantageofdifferentialpricingforthesamephysicalassetorsecurity.Ittypicallyentailsthesaleofasecurityatarelativelyhighpriceandthesimultaneouspurchaseofthesamesecurity(oritsfunctionalequivalent)atarelativelylowprice.ArbitrageOpportunitiesArbitra87ArbitrageOpportunitiesArbitrageactivityisacriticalelementofmodern,efficientsecuritymarkets.Ittakesrelativelyfewofthisactiveinvestorstoexploitarbitragesituationsand,bytheirbuyingandsellingactions,eliminatetheseprofitopportunities.SomeinvestorshavegreaterresourcesandinclinationtoengageIarbitragethanothers.ArbitrageOpportunitiesArbitra88ArbitrageOpportunitiesZero-investmentportfolioAportfolioofzeronetvalue,establishedbybuyingandshortingcomponentsecurities.Arisklessarbitrageopportunityariseswhenaninvestorcanconstructazero-investmentportfoliothatwillyieldasureprofit.ArbitrageOpportunitiesZero-in89ArbitrageOpportunitiesToconstructazero-investmentportfolio,onehastobeabletosellshortatleastoneassetandusetheproceedstopurchaseonormoreassets.Evenasmallinvestor,usingborrowedmoneyinthiscase,cantakealargepositioninsuchaportfolio.Therearemanyarbitragetactics.ArbitrageOpportunitiesTocons90ArbitrageOpportunitiesAnexample:Fourstocksandfourpossiblescenariostherateofreturninfourscenarios181inthetextbookTheexpectedreturns,standarddeviationsandcorrelationsdonotrevealanyabnormalitytothenakedeye.ArbitrageOpportunitiesAnexam91ArbitrageOpportunitiesThecriticalprope

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