Michel Dacorogna, Tail-Dependence an Essential Factor for

Results of Fit: Gumbel FlatThe coupling on both ends is again visibleGumbel FlatClayton HierarchyOverall, the fit of the tail region is stillreasonableSince the dependence of the lower tailreduces the tail dependence in the uppertail in the fit, the RAC is slightlyunderestimated.The Q/Q plot still shows a reasonablygood agreement between the copulas.Clayton GumbelHierarchy FlatExpected 145‘550 145‘063Std Dev 172‘587 162‘936LH Div. Gain 9.56% 7.20%RH VaR 1‘070‘462 1‘028‘533RH Shortfall 1‘248‘146 1‘188‘803RH RAC 1‘102‘595 1‘043‘740RH Div. Gain 8.17% 12.69%M3,M4 Θ=2Aggregate Θ=1M5, M6 Θ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201235Results of Fit: Rank Correlation HierarchyRank correlation is symmetric strongcorrelation also for the lower tail.Rank Corr. HierarchyClayton HierarchyThe upper tail is much less pointed thanfor the Clayton.The RAC is substantially underestimated.The diversification gain is unrealisticallyhigh.The Q/Q plot shows the deviation in thetails.ClaytonHierarchyRankCorr.HierarchyExpected 145‘550 145‘464Std Dev 172‘587 146‘133LH Div. Gain 9.56% 3.90%RH VaR 1‘070‘462 877‘052RH Shortfall 1‘248‘146 990‘103RH RAC 1‘102‘595 844‘639RH Div. Gain 8.17% 30.33%= 0.7= 0.5 = 0.8M3,M4 Θ=2Aggregate Θ=1M5, M6 Θ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201236