The Two-Parameter Portfolio__ Model

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1、TheTwo-ParameterPortfolioModel”FoundationsofFinance”Chapter7October12,20101/22OutlineTheFrameworkTheTwo-ParameterPortfolioModelTheGeometryoftheEfficientSetTheEffectofDiversificationAnExtendedModelSomeEmpiricalPuzzles2/22TheFrameworkConsideroneinvestor’sdecision,whi

2、chisanatomisticcompetitorinfrictionlessmarketsTwoperiod:t=1,hehaswealthw1,whichhemustallocatetocurrentconsumptionc1andtoaninvestmentw1c1insomeportfolioofsecuritiest=2,thevalueofhisportfolioprovideshisconsumptionc2ThenhisproblemisMaxU(c1;c˜2)s:t:c˜2=(w1c1)(1+R˜p

3、)whereR˜pisthereturnontheportfoliopfromtime1totime23/22TheTwo-ParameterPortfolioModelSupposeU(c1;c˜2)=u(c1)+u(˜c2),thenwhatisu(˜c2)?ThroughTaylorexpansion01002u(˜c2)=u(E(˜c2))+u(E(˜c2))(˜c2E(˜c2))+u(E(˜c2))(˜c2E(˜c2))+R32thenhisexpectedutility:1002E(u(˜c2))=u(E

4、(˜c2))+u(E(˜c2))(˜c2))+E(R3)2Howitwillbecometwo-parameterportfoliomodel,ormean-variancemodel?4/22TheTwo-ParameterPortfolioModelForarbitrarydistributions,themean-variancemodelcanbemotivatedbyassumingquadraticutilityquadraticutility:u(c)=10c2c2,thenR=03butithasun

5、desirablepropertiesofsatiationandincreasingabsoluteriskaversionForarbitrarypreferences,themean-variancemodelcanbemotivatedbyassumingtheratesofreturnaremultivariatenormallydistributedthenE(R3)canbeexpressedasfunctionsofthemeanandvariancenormaldistributionarealsost

6、ableunderaddition5/22TheTwo-ParameterPortfolioModelAssumenormaldistribution1002E(u(˜c2))=u(E(˜c2))+u(E(˜c2))(˜c2)+E(R3)2E(˜c2)=(w1c1)(1+E(R˜p))(˜c2)=(w1c1)(R˜p)Wheninvestorisrisk-averse,u0(c)>0u00(c)<0,so,givenE(R˜p),investorprefersless(R˜p)ofportfoliotomor

7、e;given(R˜p),heprefersmoreE(R˜p)toless.Efficientportfolioset:Nootherportfoliowiththesameorhigherexpectedreturnhaslowerstandarddeviationofreturn.6/22TheTwo-ParameterPortfolioModelGeometricinterpretation:7/22TheGeometryoftheEfficientSetConsiderthecombinationsoftwose

8、curitiesR˜p=xR˜q+(1x)R˜sE(R˜p)=xE(R˜q)+(1x)E(R˜s)(R˜p)=(x22(R˜q)+(1x)22(R˜s)+2x(1x)corr(R˜q;R˜s)(R˜q)(R˜s))1=2wherecov(R˜q;R˜s)corr(R˜q;R˜s)=(R˜q)(R