投资组合与方差分析分析解析.docx

投资组合与方差分析分析解析.docx

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投资组合与方差分析分析解析

Portfoliooptimizationmodelsandmean-variancespanningtests

Wei-PengChen*

DepartmentofFinance,Hsih-ShinUniversity,Taiwan

c8145666@

HuiminChung

GraduateInstituteofFinance,NationalChiaoTungUniversity,Taiwan

Keng-YuHo

DepartmentofFinance,NationalCentralUniversity,Taiwan

kengyuho@cc.ncu.edu.tw

Tsui-LingHsu

GraduateInstituteofFinance,NationalChiaoTungUniversity,Taiwan

tracy.shu@

PreparedforHandbookofQuantitativeFinanceandRiskManagement

Inthischapterweintroducethetheoryandtheapplicationofcomputerprogramofmodernportfoliotheory.Thenotionofdiversificationisage-old“don'tputyoureggsinonebasket”,obviouslypredateseconomictheory.Howeveraformalmodelshowinghowtomakethemostofthepowerofdiversificationwasnotdeviseduntil1952,afeatforwhichHarryMarkowitzeventuallywonNobelPrizeineconomics.

Markowitzportfolioshowsthatasyouaddassetstoaninvestmentportfoliothetotalriskofthatportfolio-asmeasuredbythevariance（orstandarddeviation）oftotalreturn-declinescontinuously,buttheexpectedreturnoftheportfolioisaweightedaverageoftheexpectedreturnsoftheindividualassets.Inotherwords,byinvestinginportfoliosratherthaninindividualassets,investorscouldlowerthetotalriskofinvestingwithoutsacrificingreturn.

Inthesecondpartweintroducethemean-variancespanningtestwhichfollowsdirectlyfromtheportfoliooptimizationproblem.

INTRODUCTIONOFMARKOWITZPORTFOLIO-SELECTIONMODEL

HarryMarkowitz（1952,1959）developedhisportfolio-selectiontechnique,whichcametobecalledmodernportfoliotheory（MPT）.PriortoMarkowitz'swork,security-selectionmodelsfocusedprimarilyonthereturnsgeneratedbyinvestmentopportunities.Standardinvestmentadvicewastoidentifythosesecuritiesthatofferedthebestopportunitiesforgainwiththeleastriskandthenconstructaportfoliofromthese.Followingthisadvice,aninvestormightconcludethatrailroadstocksallofferedgoodrisk-rewardcharacteristicsandcompileaportfolioentirelyfromthese.TheMarkowitztheoryretainedtheemphasisonreturn;butitelevatedrisktoacoequallevelofimportance,andtheconceptofportfolioriskwasborn.Whereasriskhasbeenconsideredanimportantfactorandvarianceanacceptedwayofmeasuringrisk,Markowitzwasthefirsttoclearlyandrigorouslyshowhowthevarianceofaportfoliocanbereducedthroughtheimpactofdiversification,heproposedthatinvestorsfocusonselectingportfoliosbasedontheiroverallrisk-rewardcharacteristicsinsteadofmerelycompilingportfoliosfromsecuritiesthateachindividuallyhaveattractiverisk-rewardcharacteristics.

AMarkowitzportfoliomodelisonewherenoaddeddiversificationcanlowertheportfolio'sriskforagivenreturnexpectation（alternately,noadditionalexpectedreturncanbegainedwithoutincreasingtheriskoftheportfolio）.TheMarkowitzEfficientFrontieristhesetofallportfoliosofwhichexpectedreturnsreachthemaximumgivenacertainlevelofrisk.

TheMarkowitzmodelisbasedonseveralassumptionsconcerningthebehaviorofinvestorsandfinancialmarkets:

1.Aprobabilitydistributionofpossiblereturnsoversomeholdingperiodcanbeestimatedbyinvestors.

2.Investorshavesingle-periodutilityfunctionsinwhichtheymaximizeutilitywithintheframeworkofdiminishingmarginalutilityofwealth.

3.Variabilityaboutthepossiblevaluesofreturnisusedbyinvestorstomeasurerisk.

4.Investorscareonlyaboutthemeansandvarianceofthereturnsoftheirportfoliosoveraparticularperiod.

5.Expectedreturnandriskasusedbyinvestorsaremeasuredbythefirsttwomomentsoftheprobabilitydistributionofreturns-expectedvalueandvariance.

6.Returnisdesirable;riskistobeavoided.

7.Financialmarketsarefrictionless.

MEASURMENTOFRETURNANDRISK

Throughoutthischapter,investorsareassumedtomeasurethelevelofreturnbycomputingtheexpectedvalueofthedistribution,usingtheprobabilitydistributionofexpectedreturnsforaportfolio.Riskisassumedtobemeasurablebythevariabilityaroundtheexpectedvalueoftheprobabilitydistributionofreturns.Themostacceptedmeasuresofthisvariabilityarethevarianceandstandarddeviation.

Return

Givenanysetofriskyassetsandasetofweightsthatdescribehowtheportfolioinvestmentissplit,thegeneralformulasofexpectedreturnfornassetsis:

（X.1）

where:

=

1.0;

n

=

thenumberofsecurities;

=

theproportionofthefundsinvestedinsecurityi;

=

thereturnonithsecurityandportfoliop;and

=

theexpectationofthevariableintheparentheses.

Thereturncomputationisnothingmorethanfindingtheweightedaveragereturnofthesecuritiesincludedintheportfolio.