Michel Dacorogna, Tail-Dependence an Essential Factor for

Results of Fit: Student-T FlatSimilar to Student-T HierarchyStudent-T FlatClayton HierarchyHas one parameter more than the RankCorrelation Slightly betterHas two parameters less than thehierarchical Student‘s T Slightly worseM3,M4 Θ=21.000.71 0.45 0.460.711.00 0.45 0.46i, j0.45 0.45 1.00 0.81 = 100.460.46 0.81 1.00 Clayton Student's THierarchyExpected 145‘550 144‘980Std Dev 172‘587 150‘838LH Div. Gain 9.56% 3.20%RH VaR 1‘070‘462 901‘757RH Shortfall 1‘248‘146 1‘043‘301RH RAC 1‘102‘595 898‘322RH Div. Gain 8.17% 25.42%Aggregate Θ=1M5, M6 Θ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201239Summary of the Statistical ResultsStatisticalResults250,000 simulationsClaytonHierarchyClaytonFlatGumbelHierarchyGumbelFlatRankCorr.Rank Corr. Student's T Student'sHierarchyTHierarchyExpected 145‘550 145‘407 144‘965 145‘063 145‘293 145‘464 144‘980 145‘088Std Dev 172‘587 172‘948 164‘507 162‘936 144‘448 146‘133 150‘838 150‘290LH Div. Gain 9.56% 9.14% 5.70% 7.20% 3.98% 3.90% 3.20% 3.05%RH VaR 1‘070‘462 1‘078‘190 1‘021‘144 1‘028‘533 870‘389 877‘052 901‘757 909‘335RH Shortfall 1‘248‘146 1‘251‘138 1‘195‘423 1‘188‘803 977‘617 990‘103 1‘043‘301 1‘045‘099RH RAC 1‘102‘595 1‘105‘731 1‘050‘458 1‘043‘740 832‘324 844‘639 898‘322 900‘012RH Div. Gain 8.17% 7.73% 11.73% 12.69% 30.70% 30.33% 25.42% 25%Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201240