第十一章套利定价理论

27五月2023第十一章套利定价理论ArbitragePricingTheoryArbitrage-arisesifaninvestorcanconstructazeroinvestmentportfoliowithasureprofit.Sincenoinvestmentisrequired,aninvestorcancreatelargepositionstosecurelargelevelsofprofit.Inefficientmarkets,profitablearbitrageopportunitieswillquicklydisappear.11.1FactorModels:Announcements,Surprises,andExpectedReturnsThereturnonanysecurityconsistsoftwoparts.FirsttheexpectedreturnsSecondistheunexpectedorriskyreturns.Awaytowritethereturnonastockinthecomingmonthis:11.1FactorModels:Announcements,Surprises,andExpectedReturnsAnyannouncementcanbebrokendownintotwoparts,theanticipatedorexpectedpartandthesurpriseorinnovation:Announcement=Expectedpart+Surprise.Theexpectedpartofanyannouncementispartoftheinformationthemarketusestoformtheexpectation,Rofthereturnonthestock.Thesurpriseisthenewsthatinfluencestheunanticipatedreturnonthestock,U.11.2Risk:SystematicandUnsystematicAsystematicriskisanyriskthataffectsalargenumberofassets,eachtoagreaterorlesserdegree.Anunsystematicriskisariskthatspecificallyaffectsasingleassetorsmallgroupofassets.Unsystematicriskcanbediversifiedaway.Examplesofsystematicriskincludeuncertaintyaboutgeneraleconomicconditions,suchasGNP,interestratesorinflation.Ontheotherhand,announcementsspecifictoacompany,suchasagoldminingcompanystrikinggold,areexamplesofunsystematicrisk.11.2Risk:SystematicandUnsystematicSystematicRisk;m

NonsystematicRisk;nTotalrisk;UWecanbreakdowntherisk,U,ofholdingastockintotwocomponents:systematicriskandunsystematicrisk:11.3SystematicRiskandBetasThebetacoefficient,b,tellsustheresponseofthestock’sreturntoasystematicrisk.IntheCAPM,bmeasuredtheresponsivenessofasecurity’sreturntoaspecificriskfactor,thereturnonthemarketportfolio.Weshallnowconsidermanytypesofsystematicrisk.11.3SystematicRiskandBetasForexample,supposewehaveidentifiedthreesystematicrisksonwhichwewanttofocus:Inflation

GDPgrowthThedollar-eurospotexchangerate,S($,€)Ourmodelis:SystematicRiskandBetas:ExampleSupposewehavemadethefollowingestimates:bI=-2.30bGDP=1.50bS=0.50.Finally,thefirmwasabletoattracta“superstar”CEOandthisunanticipateddevelopmentcontributes1%tothereturn.SystematicRiskandBetas:ExampleWemustdecidewhatsurprisestookplaceinthesystematicfactors.Ifitwasthecasethattheinflationratewasexpectedtobeby3%,butinfactwas8%duringthetimeperiod,thenFI=Surpriseintheinflationrate=actual–expected=8%-3%=5%SystematicRiskandBetas:ExampleIfitwasthecasethattherateofGDPgrowthwasexpectedtobe4%,butinfactwas1%,thenFGDP=SurpriseintherateofGDPgrowth =actual–expected =1%-4% =-3%SystematicRiskandBetas:ExampleIfitwasthecasethatdollar-eurospotexchangerate,S($,€),wasexpectedtoincreaseby10%,butinfactremainedstableduringthetimeperiod,thenFS=Surpriseintheexchangerate =actual–expected=0%-10%=-10%SystematicRiskandBetas:ExampleFinally,ifitwasthecasethattheexpectedreturnonthestockwas8%,then11.4PortfoliosandFactorModelsNowletusconsiderwhathappenstoportfoliosofstockswheneachofthestocksfollowsaone-factormodel.WewillcreateportfoliosfromalistofNstocksandwillcapturethesystematicriskwitha1-factormodel.Theithstockinthelisthavereturns:RelationshipBetweentheReturnontheCommonFactor&ExcessReturnExcessreturnThereturnonthefactorFIfweassumethatthereisnounsystematicrisk,thenei=0RelationshipBetweentheReturnontheCommonFactor&ExcessReturnExcessreturnThereturnonthefactorFIfweassumethatthereisnounsystematicrisk,thenei=0RelationshipBetweentheReturnontheCommonFactor&ExcessReturnExcessreturnThereturnonthefactorFDifferentsecuritieswillhavedifferentbetasPortfoliosandDiversificationWeknowthattheportfolioreturnistheweightedaverageofthereturnsontheindividualassetsintheportfolio:PortfoliosandDiversificationThereturnonanyportfolioisdeterminedbythreesetsofparameters:Inalargeportfolio,thethirdrowofthisequationdisappearsastheunsystematicriskisdiversifiedaway.Theweighedaverageofexpectedreturns.Theweightedaverageofthebetastimesthefactor.Theweightedaverageoftheunsystematicrisks.PortfoliosandDiversificationSothereturnonadiversifiedportfolioisdeterminedbytwosetsofparameters:Theweighedaverageofexpectedreturns.TheweightedaverageofthebetastimesthefactorF.Inalargeportfolio,theonlysourceofuncertaintyistheportfolio’ssensitivitytothefactor.11.5BetasandExpectedReturnsThereturnonadiversifiedportfolioisthesumoftheexpectedreturnplusthesensitivityoftheportfoliotothefactor.RelationshipBetweenb&ExpectedReturnIfshareholdersareignoringunsystematicrisk,onlythesystematicriskofastockcanberelatedtoitsexpectedreturn.RelationshipBetweenb&ExpectedReturnExpectedreturnbABCDSML11.6TheCapitalAssetPricingModelandtheArbitragePricingTheoryAPTappliestowelldiversifiedportfoliosandnotnecessarilytoindividualstocks.WithAPTitispossiblefor