mathematical finance coherent measures of risk

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1、CoherentMeasuresofRiskArtznerP.,F.Delbaen,J.EberandD.Heath,1999.MathematicalFinance,9(3),203{228.PeterMAddoUniversityofCaFoscaripkaddo2000@yahoo.comNovember11,2010PeterMAddo(UniversitaCaFoscari-Venezia)CoherentMeasuresofRiskNovember11,20101/22Outline1Introduction2RiskMeasureDenitio

2、n3ValueatRiskProblemswithVaR4CoherentRiskMeasuresExamplesofCoherentRiskMeasuresIllustrationofnon-coherenceofvalue-at-riskEmpiricalandGraphicalDisplayAxiomofRelevance5QuestionsandRemarksSomeIssuesReferencesPeterMAddo(UniversitaCaFoscari-Venezia)CoherentMeasuresofRiskNovember11,20102/

3、22IntroductionObjectivesIntheeldofnancialeconomicsthereareanumberofwaysthatriskcanbedened.Thereislittlewonderthatpeopleareconfusedbywhatthetermriskoughttomean.Forinstance,thewebsitebusinessdictionary.comdenesriskinseventeendierentgeneralcategories.Toclarifytheconcepttheoreticia

4、nshavedescribedanumberofpropertiesthatariskmeasuremightormightnothaveSomeoftheseriskmeasuresarecalledCoherentmeasuresofrisk.Ofcoursethiswasanintelligentchoiceofnamesfortheirclassofmeasures,asnobodywouldwanttobeaccusedofusingameasurethatisincoherent."PeterMAddo(UniversitaCaFoscari-V

5、enezia)CoherentMeasuresofRiskNovember11,20103/22IntroductionDenition(Artzneretal.,1997)LetXbearandomvariable.Thenisariskmeasureifitsatisesthefollowingproperties:MonotonicityifX0,then(X)0PositiveHomogeneity(X)=(X)80TranslationInvariance(X+a)=(X)a8a2RPeterMAddo(Universita

6、CaFoscari-Venezia)CoherentMeasuresofRiskNovember11,20104/22IntroductionWhatistheIdeabehindthisdenition?ApositivenumberimpliesthatoneisatriskforlosingcapitalandshouldhavethatpositivenumberofacashbalanceonhandtoosetthispotentiallossAnegativenumberwouldsaythatthecompanyhasenoughcapit

7、altotakeonmoreriskortoreturnsomeofitscashtootheroperationsortoitsshareholdersThemonotonicitypropertystatesthataninvestmentthatalwayshaspositivepayogivesthecompanytheabilitytotakeonmorerisk.Positivehomogeneityimpliesthatmultiplyingyourinvestmentbyktimesgivesyouariskofalossthatisktim

8、eslargerTranslation