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1、ATwoStateCapitalAssetPricingModelMosheFridmanInstituteforMathematicsanditsApplicationsUniversityofMinnesota514VincentHallMinneapolis,Minnesota55455March16,1994AbstractAfamousmodelin nancialtheoryistheCapitalAssetPricingModel(CAPM).InthispaperweproposeatwostateCAPMinwhichweassumethatexcessreturnsfo
2、rthemarketandforaparticularsecurityarebivariatenormallydistributed.TheparametersofthedistributionaredeterminedbythestateofanunobservedstationaryMarkovchain.Twostatesrepresenttwobusinessregimesthatarecharacterizedbylowandhighvolatility.Maximumlikelihoodestimatesfortheparametersofthemodelareobtainedv
3、iatheBaum-WelchalgorithmforlocalmaximizationofthelikelihoodfunctionforMarkovRegimeModels(MRM),alsoknownasHiddenMarkovModels(HMM).Weapplythemodeltomonthlyreturndataforthreeoilindustrycorporationsecurities.Acomparisonoftheresultswithtwosimplermodels,theindependentswitchingregressionmodelandthestandar
4、dCAPMwithoneregime,showsasigni cantimprovementingoodnessof tobtainedbytheproposedtwostateMRM.Estimatesofthepe-riodsofhighvolatilityforeachcorporationdepictagenerale ectofthebusinesscyclesonallthreeprocessescombinedwithuniquee ectssteamingfromcorporationrelatedevents.Keywords:CapitalAssetPricingMode
5、l,MarkovRegimeModel,HiddenMarkovModel,Baum-Welchalgorithm.AMS1991subjectclassi cations.Primary60J10;Secondary90A12.ResearchpartiallysupportedbyAROGrantDAAL03-91-G-01101IntroductionAfamousandwidelyusedmodelin nancialtheoryistheCapitalAssetPricingModel(CAPM)introducedinSharpe(1964)andLintner(1965).T
6、hemodelrelatesexpectedreturnsoncapitalassetsinvestmentswithex-pectedmarketreturns.Underthemodel,thereturnperperiodtofanassetRishypothesizedtobelinearlyrelatedtothemarketreturnRforthattmtperiodinthefollowingway:(R?R)=(R?R)+;tfmftttt2whereareindependentandidenticallydistributedasN(0;)andRistftther
7、eturnforperiodtonarisk-freeasset.Thequantities(R?R)andtft(R?R)arereferredtoastheexcessreturns.Theparameterofinterestmfttforinvestorsisthesystematicriskassociatedwiththeasset,measuredbytheregressionslope.The
