coherent representations of subjective risk aversion

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1、CoherentRepresentationsofSubjectiveRiskAversionCarloAcerbi∗July25,2003∗Abaxbank,CorsoMonforte34,20122Milano,Italy1CONTENTS2Contents1ForewordsandMotivations31.1Indefenseofaxiomatics............................31.2Scopeandobjectives...........................

2、...71.3Outlineofthework...............................92BuildingaRiskMeasure:theExpectedShortfall112.1AcloselookintoVaR’sdefinition.......................112.2Anaturalremedytoprobethetail:theExpectedShortfall.........142.3CoherencyofES.......................

3、.........192.4EstimationofES................................263SpectralMeasuresofRisk303.1EstimationofSpectralMeasuresofRisk...................393.2CharacterizationofSpectralMeasuresviaadditionalconditions......433.2.1ComonotonicAdditivity..................

4、......443.2.2Law–invariance.............................453.2.3Firststochasticdominance.......................473.3Spectralmeasuresandcapitaladequacy....................504OptimizationofSpectralMeasuresofRisk524.1Coherentmeasuresandconvexrisksurfaces.......

5、..........524.2MinimizationofExpectedShortfall......................594.3MinimizationofgeneralSpectralMeasures..................624.4Risk–rewardoptimization...........................665StatisticalerrorsofSpectralMeasuresofRisk705.1Varianceoftheestimator...

6、.........................705.2Somemeaningfulexamples...........................721FOREWORDSANDMOTIVATIONS31ForewordsandMotivations1.1Indefenseofaxiomatics
 
 Euclid1TheappearanceoftheseminalworksofArtzneretal.[7,8,11]represen

7、tedmorethanasignificativebreakthroughinthefinancialmathematicsliterature.Itprobablymarkedthefirstseriousattempttothefoundationofaquantitativetheoryoffinancialrisk.Thedeclaredaimoftheauthorstosingleoutaminimalsetofpropertiesthataportfoliostatisticshastosatisfyin

8、ordertobea“coherentmeasureofrisk”ledthemtodefinewithremarkablefinancialinsightthecelebrated“axiomsofcoherency”andtofindtheirmostgeneralsolution,i.e.theexplicitrepresentationofanycoherentriskmeasure[11].Th