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Ch4Portfolio - Introduction to Financial Econometrics...

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IntroductiontoFinancialEconometrics Chapter4 IntroductiontoPortfolioTheory EricZivot DepartmentofEconomics UniversityofWashington January26,2000 Thisversion:February20,2001 1 Introduction to Portfolio Theory Considerthefollowinginvestmentproblem.Wecaninvestintwonon-dividendpaying stocksAandBoverthenextmonth.Let R A denotemonthlyreturnonstockAand R B denotethemonthlyreturnonstockB.Thesereturnsaretobetreatedasrandom variablessincethereturnswillnotberealizeduntiltheendofthemonth.Weassume thatthereturns R A and R B arejointlynormallydistributedandthatwehavethe followinginformationaboutthemeans,variancesandcovariancesoftheprobability distributionofthetworeturns: μ A = E [ R A ] , σ 2 A = V ar ( R A ) , μ B = E [ R B ] , σ 2 B = V ar ( R B ) , σ AB = Cov ( R A , R B ) . Weassumethatthesevaluesaretakenasgiven.Wemightwonderwheresuchvalues comefrom.Onepossibilityisthattheyareestimatedfromhistoricalreturndatafor thetwostocks.Anotherpossibilityisthattheyaresubjectiveguesses. Theexpectedreturns, μ A and μ B ,areourbestguessesforthemonthlyreturnson eachofthestocks.However,sincetheinvestmentsarerandomwemustrecognizethat therealizedreturnsmaybedi ff erentfromourexpectations. Thevariances, σ 2 A and σ 2 B ,providemeasuresoftheuncertaintyassociatedwiththesemonthlyreturns. We canalsothinkofthevariancesasmeasuringtheriskassociatedwiththeinvestments. Assetsthathavereturnswithhighvariability(orvolatility)areoftenthoughtto be risky andassets withlow returnvolatility are often thought to be safe. The covariance σ AB givesusinformationaboutthe direction ofanylineardependence betweenreturns.If σ AB > 0 thenthereturnsonassetsAandBtendtomoveinthe 1
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samedirection;if σ AB < 0 thereturnstendtomoveinoppositedirections;if σ AB = 0 thenthereturnstendtomoveindependently.Thestrengthofthedependencebetween thereturnsismeasuredbythecorrelationcoe cient ρ AB = σ AB σ A σ B . If ρ AB iscloseto oneinabsolutevaluethenreturnsmimiceachotherextremelycloselywhereasif ρ AB isclosetozerothenthereturnsmayshowverylittlerelationship. Theportfolioproblemisset-upasfollows.Wehaveagivenamountofwealthand itisassumedthatwewillexhaustallofourwealthbetweeninvestmentsinthetwo stocks. Theinvestor° sproblemistodecidehowmuchwealthtoputinassetAand howmuchtoputinassetB.Let x A denotetheshareofwealthinvestedinstockA and x B denotetheshareofwealthinvestedinstockB.Sinceallwealthisputinto thetwoinvestmentsitfollowsthat x A + x B = 1 . (Aside: Whatdoesitmeanfor x A or x B tobenegativenumbers?)Theinvestormustchoosethevaluesof x A and x B . Ourinvestmentinthetwostocksformsa portfolio andtheshares x A and x B are referredtoas portfolioshares orweights. Thereturnontheportfoliooverthenext monthisarandomvariableandisgivenby R p = x A R A + x B R B , (1) whichisjustasimplelinearcombinationorweightedaverageoftherandomreturn variables R A and R B .Since R A and R B areassumedtobenormallydistributed, R p isalsonormallydistributed.
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