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1、Worst-CaseConditionalValue-at-RiskwithApplicationtoRobustPortfolioManagementShu-ShangZhuDepartmentofManagementScience,SchoolofManagement,FudanUniversity,Shanghai200433,China,sszhu@fudan.edu.cnMasaoFukushimaDepartmentofAppliedMathematicsandPhysics,GraduateSchoolofInformatic
2、s,KyotoUniversity,Kyoto606-8501,Japan,fuku@i.kyoto-u.ac.jp(Originalversion:July5,2005;Revisedversion:June23,2006)Thispaperconsiderstheworst-caseCVaRinsituationwhereonlypartialinformationontheunderlyingprobabilitydistributionisgiven.Itisshownthat,likeCVaR,worst-caseCVaRrema
3、insacoherentriskmeasure.Theminimizationofworst-caseCVaRundermix-turedistributionuncertainty,boxuncertaintyandellipsoidaluncertaintyareinvestigated.Theapplicationofworst-caseCVaRtorobustportfoliooptimizationisproposed,andthecorrespondingproblemsarecastaslinearprogramsandsec
4、ond-orderconeprogramswhichcanbee±cientlysolved.MarketdatasimulationandMonteCarlosimulationexamplesarepresentedtoillustratethemethods.Ourapproachescanbeappliedinmanysituations,includingthoseoutsideof¯nancialriskmanagement.SubjectClassi¯cations:Finance,portfolio:conditionalV
5、aR,portfoliooptimization;Pro-gramming,linear,nonlinear:robustoptimization,second-orderconeprogramming.1IntroductionGenerally,twotypesofdecisionmakingframeworksareadoptedin¯nancialoptimization:theutilitymaximizationandthereturn-risktrade-o®analysis.Inthereturn-risktrade-o®a
6、nalysis,theriskisexplicitlyquanti¯edbyariskmeasurethatmapsthelosstoarealnumber.Sinceitfacilitatestheunderstandingofrisk,thisapproachiswidelyadoptedinbothpracticalapplicationsandtheoreticalstudy.Althoughtheutilitymaximizationexhibitssometheoreticalelegance,itdescribestheris
7、kinanindirectway,andconsequentlyisrarelyadoptedinpractice.Markowitz(1952)pavedthefoundationformodernportfoliotheory.Hismean-varianceanalysisisarepresentativemethodologyintheframeworkofreturn-risktrade-o®analysis,wherevarianceisadoptedasthemeasureofrisk.Sincethemiddleof1990
8、s,Value-at-Risk1(VaR,seeRiskMetricsTM1996),anewmeasureofdownsiderisk,hasbecomepopularin¯n
