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Ch3cermodel - Introduction to Financial Econometrics...

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IntroductiontoFinancialEconometrics Chapter3 TheConstantExpectedReturnModel EricZivot DepartmentofEconomics UniversityofWashington January6,2000 Thisversion:January23,2001 1 The Constant Expected Return Model of Asset Returns 1.1 Assumptions Let R it denotethe continuously compoundedreturn onanasset i at time t. We makethefollowingassumptionsregardingtheprobabilitydistributionof R it for i = 1 , . . . , N assetsoverthetimehorizon t = 1 , . . . , T. 1. Normalityofreturns: R it N ( μ i , σ 2 i ) for i = 1 , . . . , N and t = 1 , . . . , T. 2. Constantvariancesandcovariances: cov ( R it , R jt ) = σ ij for i = 1 , . . . , N and t = 1 , . . . , T. 3. Noserialcorrelationacrossassetsovertime: cov ( R it , R js ) = 0 for t 6 = s and i, j = 1 , . . . , N. Assumption1statesthatineverytimeperiodassetreturnsarenormallydis- tributedandthatthemeanandthevarianceofeachassetreturnisconstantover time.Inparticular,wehaveforeachasset i E [ R it ] = μ i forallvaluesof t var ( R it ) = σ 2 i forallvaluesof t Thesecondassumptionstatesthatthecontemporaneouscovariancesbetweenassets areconstantovertime. Givenassumption1,assumption2impliesthatthecontem- poraneouscorrelationsbetweenassetsareconstantovertimeaswell.Thatis,forall 1
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assets corr ( R it , R jt ) = ρ ij forallvaluesof t. Thethirdassumptionstipulatesthatalloftheassetreturnsareuncorrelatedover time 1 .Inparticular,foragivenasset i thereturnsontheassetare serially uncorre- lated whichimpliesthat corr ( R it , R is ) = cov ( R it , R is ) = 0 forall t 6 = s. Additionally,thereturnsonallpossiblepairsofassets i and j areseriallyuncorrelated whichimpliesthat corr ( R it , R js ) = cov ( R it , R js ) = 0 forall i 6 = j and t 6 = s. Assumptions1-3indicatethatallassetreturnsatagivenpointintimearejointly (multivariate)normallydistributedandthatthisjointdistributionstaysconstant overtime.Clearlytheseareverystrongassumptions.However,theyallowustode- velopmentastraightforwardprobabilisticmodelforassetreturnsaswellasstatistical toolsforestimatingtheparametersofthemodelandtestinghypothesesaboutthe parametervaluesandassumptions. 1.2 Constant Expected Return Model Representation Aconvenientmathematicalrepresentationor model ofassetreturnscanbegiven basedonassumptions1-3. Thisisthe constant expected return (CER)model. For assets i = 1 , . . . , N andtimeperiods t = 1 , . . . , T theCERmodelisrepresentedas R it = μ i + ε it (1) ε it i.i.d. N (0 , σ 2 i ) cov ( ε it , ε jt ) = σ ij (2) where μ i isaconstantandweassumethat ε it isindependentof ε js foralltimeperiods t 6 = s . Thenotation ε it i.i.d. N (0 , σ 2 i ) stipulatesthattherandomvariable ε it is seriallyindependentandidenticallydistributedasanormalrandomvariablewith meanzeroandvariance σ 2 i . Inparticular,notethat, E [ ε it ] = 0 , var ( ε it ) = σ 2 i and cov ( ε it , ε js ) = 0 for i 6 = j and t 6 = s. Usingthebasicpropertiesofexpectation,varianceandcovariancediscussedin chapter2,wecanderivethefollowingpropertiesofreturns.Forexpectedreturnswe have E [ R it ] = E [ μ i + ε it ] = μ i + E [ ε it ] = μ i , 1 Sinceallassetsareassumedtobenormallydistributed(assumption1),uncorrelatednessimplies thestrongerconditionofindependence.
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