AnAlternativeViewofRiskandReturn:TheArbitragePricingTheoryChapter120KeyConceptsandSkillsDiscusstherelativeimportanceofsystematicandunsystematicriskindeterminingaportfolio’sreturnCompareandcontrasttheCAPMandArbitragePricingTheory1ChapterOutline12.1Introduction12.2 SystematicRiskandBetas12.3 PortfoliosandFactorModels12.4 BetasandExpectedReturns12.5 TheCapitalAssetPricingModelandtheArbitragePricingTheory12.6 EmpiricalApproachestoAssetPricing212.1Introduction

ArbitragePricingTheoryArbitragearisesifaninvestorcanconstructazeroinvestmentportfoliowithasureprofit.Sincenoinvestmentisrequired,aninvestorcancreatelargepositionstosecurelargelevelsofprofit.Inefficientmarkets,profitablearbitrageopportunitieswillquicklydisappear.3TotalRiskTotalrisk=systematicrisk+unsystematicriskThestandarddeviationofreturnsisameasureoftotalrisk.Forwell-diversifiedportfolios,unsystematicriskisverysmall.Consequently,thetotalriskforadiversifiedportfolioisessentiallyequivalenttothesystematicrisk.4Risk:SystematicandUnsystematicSystematicRisk:m

NonsystematicRisk:

n

2Totalrisk

Wecanbreakdownthetotalriskofholdingastockintotwocomponents:systematicriskandunsystematicrisk:

512.2SystematicRiskandBetasThebetacoefficient,b,tellsustheresponseofthestock’sreturntoasystematicrisk.IntheCAPM,bmeasurestheresponsivenessofasecurity’sreturntoaspecificriskfactor,thereturnonthemarketportfolio.Weshallnowconsiderothertypesofsystematicrisk.6SystematicRiskandBetasForexample,supposewehaveidentifiedthreesystematicrisks:inflation,GNPgrowth,andthedollar-eurospotexchangerate,S($,€).Ourmodelis:7SystematicRiskandBetas:ExampleSupposewehavemadethefollowingestimates:bI=-2.30bGNP=1.50bS=0.50Finally,thefirmwasabletoattracta“superstar”CEO,andthisunanticipateddevelopmentcontributes1%tothereturn.8SystematicRiskandBetas:ExampleWemustdecidewhatsurprisestookplaceinthesystematicfactors.Ifitwerethecasethattheinflationratewasexpectedtobe3%,butinfactwas8%duringthetimeperiod,then:FI=Surpriseintheinflationrate=actual–expected=8%–3%=5%9SystematicRiskandBetas:ExampleIfitwerethecasethattherateofGNPgrowthwasexpectedtobe4%,butinfactwas1%,then:FGNP=SurpriseintherateofGNPgrowth =actual–expected=1%–4%=–3%10SystematicRiskandBetas:ExampleIfitwerethecasethatthedollar-eurospotexchangerate,S($,€),wasexpectedtoincreaseby10%,butinfactremainedstableduringthetimeperiod,then:FS=Surpriseintheexchangerate =actual–expected=0%–10%=–10%11SystematicRiskandBetas:ExampleFinally,ifitwerethecasethattheexpectedreturnonthestockwas8%,then:1212.3PortfoliosandFactorModelsNowletusconsiderwhathappenstoportfoliosofstockswheneachofthestocksfollowsaone-factormodel.WewillcreateportfoliosfromalistofNstocksandwillcapturethesystematicriskwitha1-factormodel.Theithstockinthelisthasreturn:13RelationshipBetweentheReturnontheCommonFactor&ExcessReturnExcessreturnThereturnonthefactorFIfweassumethatthereisnounsystematicrisk,thenei=0.14RelationshipBetweentheReturnontheCommonFactor&ExcessReturnExcessreturnThereturnonthefactorFIfweassumethatthereisnounsystematicrisk,thenei=0.15RelationshipBetweentheReturnontheCommonFactor&ExcessReturnExcessreturnThereturnonthefactorFDifferentsecuritieswillhavedifferentbetas.16PortfoliosandDiversificationWeknowthattheportfolioreturnistheweightedaverageofthereturnsontheindividualassetsintheportfolio:17PortfoliosandDiversificationThereturnonanyportfolioisdeterminedbythreesetsofparameters:Inalargeportfolio,thethirdrowofthisequationdisappearsastheunsystematicriskisdiversifiedaway.Theweightedaverageofexpectedreturns.Theweightedaverageofthebetastimesthefactor.Theweightedaverageoftheunsystematicrisks.18PortfoliosandDiversificationSothereturnonadiversifiedportfolioisdeterminedbytwosetsofparameters:Theweightedaverageofexpectedreturns.TheweightedaverageofthebetastimesthefactorF.Inalargeportfolio,theonlysourceofuncertaintyistheportfolio’ssensitivitytothefactor.1912.4BetasandExpectedReturnsThereturnonadiversifiedportfolioisthesumoftheexpectedreturnplusthesensitivityoftheportfoliotothefactor.20RelationshipBetweenb&ExpectedReturnIfshareholdersareignoringunsystematicrisk,onlythesystematicriskofastockcanberelatedtoitsexpectedreturn.21RelationshipBetweenb&ExpectedReturnExpectedreturnbABCDSML2212.5TheCapitalAssetPricingModelandtheArbitragePricingTheoryAPTappliestowelldiversifiedportfoliosandnotnecessarilytoindividualstocks.WithAPTitispossibleforsomeindividualstockstobemispriced-notlieontheSML.APTismoregeneralinthatitgetstoanexpectedreturnandbetarelationshipwithouttheassumptionofthemarketportfolio.APTcanbeextendedtomultifactormodels.2312.6EmpiricalApproachestoAssetPricingBoththeCAPMandAPTarerisk-basedmodels.Empiricalmethodsarebasedlessontheoryandmoreonlookingforsomeregularitiesinthehistoricalrecord.Beawarethatcorrelationdoesnotimplycausality.Relatedtoempiricalmethodsisthepracticeofclassifyingportfoliosbystyle,e.g.,ValueportfolioGrowthportfolio24QuickQuizDifferentiatesystematicriskfromunsystematicrisk.Whichtypeisessentiallyeliminatedwithwelldiversifiedportfolios?Definearbitrage.ExplainhowtheCAPMbeconsideredaspecialcaseofArbitragePricingTheory?251.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.Supposeafactormodelisappropriatetodescribethereturnsonastock.Informationaboutthosefactorsis1.S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