Michel Dacorogna, Tail-Dependence an Essential Factor for

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Fitting: Hierarchical ScenarioFitting ScenarioThe Clayton Hierarchical Tree is fit by using the same structure* correlatedwith different copulas for each scenarioThe results are then displayed on a rank-scatter plot and throughdiversification and RAC value calculationCompany Level This is not the usual hierarchical Archimedeancopula of Savu and Trede 2006 but rather adependence on the aggregateAggregate DependenceCopulaMarginal DependenceCopulaMarginal DependenceCopulaRisk Factor 1e.g. Fire, FRRisk Factor 2e.g. Nat Cat, FRRisk Factor 3e.g. Fire, DERisk Factor 4e.g. Nat Cat, DETail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201223Fitting: Flat ScenarioBusiness Scenario:Small company, with a small amount of business in eachbasket baskets are merged.All marginals modeled by one copulaCompany LevelMarginal DependenceCopulaRisk Factor 1e.g. Fire, FRRisk Factor 2e.g. Nat Cat, FRRisk Factor 3e.g. Fire, DERisk Factor 4e.g. Nat Cat, DETail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201224
Estimation of CopulasThe following estimation methods were used:Clayton / Gumbel:Maximum likelihood estimation, i.e. estimates the parameter by maximizing the loglikelihoodfunction LMLF log(c ( x )), where c( xii) is the copula density of point iiRank Correlation:Estimate the Spearman’s correlation RankCorr(X , Y ) 12E[(FX( X ) 0.5)( FY( Y ) 0.5)]for each pair X and Y. The correlation matrix for the Gauss copula can be derived as ij 2sinRankCorr( Xi,Xj) 6Student’s T:2Estimate the Kendall ( X , Y) sign[( Xi X j)( Yi Yj)]N( N 1)ij The correlation matrix can be derived as ij sinij 2 The degree of freedom is estimated with maximum likelihood.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201225Fitting Convergence Plots: MethodologyThe Fitting Convergence Plots are drawn using the following methodology:1. Simulate N observations from the reference scenario2. Fit the corresponding scenario to the N observations3. Resample the fitted scenario with 50’000 observations4. Measure the diversification gainRepeat 10 timesThe mean and standard deviation of the 10 runs per point are calculated.• The fitting convergence plots show the mean ± one standarddeviation for each N.• The fitting error plots show the standard deviation for each N.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201226
Page 1 and 2: Tail-Dependence an Essential Factor
Page 3 and 4: Determining Risk-Adjusted CapitalTh
Page 5 and 6: Agenda1 Risk-Adjusted Capital and d
Page 7 and 8: The Lognormal DistributionLognormal
Page 9 and 10: Elliptical Copula: Rank Correlation
Page 11: Agenda1 Risk-Adjusted Capital and d
Page 15 and 16: Convergence of the Fits for 4 Margi
Page 17 and 18: Results of Fit: Clayton FlatAs expe
Page 19 and 20: Results of Fit: Rank Correlation Fl
Page 21 and 22: Dependence Model mattersmore than D
Page 23 and 24: Strategy for modeling dependencesUs
Page 25 and 26: Agenda1 Risk-Adjusted Capital and d

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