Michel Dacorogna, Tail-Dependence an Essential Factor for

Tail-Dependence an Essential Factorfor Correctly Measuring the Benefitsof DiversificationPresented by Michel M. DacorognaWork done with Roland Bürgi and Roger Iles“New Views on Extreme Events: Coupled Networks, Dragon Kings and ExplosivePercolation”, Workshop of the ETH Risk Center, Zurich, Switzerland October 25-26th, 2012DisclaimerAny views and opinions expressed in this presentation or anymaterial distributed in conjunction with it solely reflect the views ofthe author and nothing herein is intended to, or should be deemed,to reflect the views or opinions of the employer of the presenter.The information, statements, opinions, documents or any othermaterial which is made available to you during this presentation arewithout any warranty, express or implied, including, but not limitedto, warranties of correctness, of completeness, of fitness for anyparticular purpose.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20122

AcknowledgementThis work is based on a team effort by:Roland Bürgi who programmed the whole analysis anddeveloped a method to generate hierarchical multivariatedistributions with copulasRoger IlesAnd Michel DacorognaTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20123Agenda1 Risk-Adjusted Capital and dependence2 Description of the model3 Influence of the number of observations on calibration4 Influence of dependence structure and models5 Ways to model the dependence6 ConclusionTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20124

Determining Risk-Adjusted CapitalThe Risk-Adjusted capital (RAC) of an insurance company isevaluated on the basis of a quantitative model of its differentrisksWe first need to identify the various sources of risk. Oneusually distinguishes four large risk categories:1. Underwriting risk (or liability risk),2. Investment risk (or asset risk),3. Credit risk (or risk of default),4. Operational riskGenerally insurance companies have the know-how tomanage and model their liability risk and are able to modelthe next two categories as well using standard finance modelsTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20125Dependence Between Risks is KeyRisk Diversification reduces a company’s need for Risk-Adjusted capital. This is key to both insurance and investments.However, risks are rarely completely independent:• Stock market crashes are usually not limited to one stock market.• Certain lines of business are affected by economic cycles, likeaviation, credit & surety or life insurances.• Motor insurance is also correlated to motor liability insurance andboth will vary during economic cycles.• Big catastrophes can produce claims in various lines of business.Dependence between risks reduces the benefits ofdiversification.The influence of dependence on the aggregated RAC is thuscrucial and needs to be carefully analyzed.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20126

Influence of Correlation on RACLet us take the same risk twice (lognormally distributed, m=10 ands=1) and bundle them in a portfolio.Let us vary the correlation between the risks from 0 to 0.90.Here are the various diversification benefits, D, in percent:RACPD 1 RACwhere RAC P is theportfolio RAC andRAC i are the RAC’sof the various risks.iiDiversification Benefits40%35%30%25%20%15%10%5%0%0 0.3 0.6 0.9Correlation CoefficientTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20127Dependence is not Always LinearWe have learned to model dependence through linear correlation.The whole modern portfolio theory is based on correlation.Often dependence increases when diversification is most needed:in case of stress. It is thus non-linear.It is possible to use the copulas instead of linear correlation tomodel dependences (copula=“generalized dependence structure”as opposed to “linear dependence”=correlation).The dependence structure will influence greatly the needs for RACand the diversification benefits one can obtain.In the following, we present a statistical study of variousdependence structures and their influence on diversification.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20128

Agenda1 Risk-Adjusted Capital and dependence2 Description of the model3 Influence of the number of observations on calibration4 Influence of dependence structure and models5 Ways to model the dependence6 ConclusionTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 20129Aim & MethodAimTo show the difficulty of statistically estimating the rightdependenceTo illustrate the importance of using the correct Copuladependence when modeling dependent marginal distributionsTo analyze the influence of the dependence structure on thediversification benefitsMethodStochastic simulations and fitting of various dependence modelsTo reproduce the behavior of the hierarchical dependence treethat is correlated through 3 Clayton copulasTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201210

Basics of the Model (1/2)We use lognormal distributions as the basic risk of our portfolio:1f x e x2x2(ln x)22( ) 0; 0We choose m=10 and s=1 for all the risks*We want a simple risk model to study the influence of thedependence structure and functionThe basic risk is then used in various dependenceconfigurations and with different dependence functionsWe choose a configuration that we assume to be the real one,which we fit with various other models* Close to the parameters proposed by M. Bagarry For modeling insurance risks.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201211Basics of the Model (2/2)RAC is calculated with Expected Shortfall (tVaR) for variousrisk tolerance levels. We summarize the results for the 1/100tVaR.This is the risk measure used in the Swiss Solvency Test andat the basis of our own capital allocation modelWe also compute the VaR at 1/200 as it is the risk measurerecommended by Solvency II and we compare both resultsTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201212

The Lognormal DistributionLognormal distribution is the singletailedprobability distribution of anyrandom variable whose logarithm isnormally distributed.Lognormal Probability DensityFunction2(ln x)122f( x) e x0; 02xLognormal Cumulative DistributionFunctionlnx F( x) x0; 0 Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201213Hierarchical Dependence TreeBase scenario: Hierarchical Dependence Tree*• Hierarchy of 4 related marginal distributions• Using significantly different dependence parameters QCompany LevelClayton CopulaΘ=1 This is not the usual hierarchical Archimedeancopula of Savu and Trede 2006 but rather adependence on the aggregateProperty FRClayton CopulaΘ=2Property DEClayton CopulaΘ=3Risk 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, 201214

Archimedean Copula: Clayton Copulaθ = 0.1 θ = 0.5 θ = 1.0 θ = 2.0The Clayton Copula CDF is definedby:With a Generator of the Copula:Diversification Benefits35%30%25%20%15%10%5%The Clayton copula is Archimedean0%0.1 0.5 1.0 2.0Correlation CoefficientTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201215Archimedean Copula: Gumbel Copulaθ = 1.0 θ = 1.5 θ = 2.0 θ = 3.0The Gumbel Copula CDF is definedby:where the Generator of the Copula isgiven by:The Gumbel copula is ArchimedeanDiversification Benefits40%35%30%25%20%15%10%5%0%1.0 1.5 2.0 3.0Correlation CoefficientTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201216

Elliptical Copula: Rank Correlationm1 m2m1 1 0m2 0 1m1 m2m1 1 0.3m2 0.3 1m1 m2m1 1 0.6m2 0.6 1m1 m2m1 1 0.9m2 0.9 1The multivariate Normal distribution copula has a matrix as a parameter.The PDF of a Normal copula is:where,is the inverse of the CDF N(0,1) and I is theidentity matrix of size n.The rank correlation is an elliptical copula.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201217Elliptical Copula: Student’s TThe multivariate Student’s T distribution copula also has a matrix as a parameter. ThePDF of a Student’s T copula is:Where,is the inverse of the CDF of the univariate T distribution withn degrees of freedomThe Student’s T copula is an elliptical copulaEffects of matrix parameter changes:n =3m1 m2m1 1 0m2 0 1m1 m2m1 1 0.3m2 0.3 1m1 m2m1 1 0.6m2 0.6 1m1 m2m1 1 0.9m2 0.9 1Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201218

Student’s TEffect of changing degrees of freedom:m1 m2m1 1 0.6m2 0.6 1n = 1n = 3 n = 6 n = 9Diversification benefits from varying the Diversification benefits from varying thematrix parameter (3 degrees of freedom): degrees of freedom:35%16%30%14%Diversification Benefits25%20%15%10%Diversification Benefits12%10%8%6%4%5%2%0%0 0.3 0.6 0.9Matrix Parameter0%1 3 6 9Degrees of FreedomTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201219Value versus Rank ScatterSignificance of Value versusRank ScatterRank ScatterValue ScatterValue scatter is used tocharacterize the spread ofthe marginal distributionsΘ=2Rank scatter shows theunderlying dependencebetween the marginalsΘ=1For the purpose of analysingdependence we shall displayonly rank scatter from hereonΘ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201220

Agenda1 Risk-Adjusted Capital and dependence2 Description of the model3 Influence of the number of observations on calibration4 Influence of dependence structure and models5 Ways to model the dependence6 ConclusionTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201221Fitting ScenariosScenarios: We fit the original dependence using the followingcopula scenarios and calculate the Diversification and RACHierarchical Dependence• Gumbel Hierarchy• Rank Correlation Hierarchy• Student-T HierarchyFlat Dependence• Clayton Flat• Gumbel Flat• Rank Correlation Flat• Student-T FlatThe base scenario is the hierarchical Clayton scenario presentedbeforeTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201222

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

Convergence of Fits for 2 MarginalsStarting from Clayton = 1Theoretical Diversification GainTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201227Standard Deviations of the Fits for 2 MarginalsStarting from Clayton = 1Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201228

Convergence of the Fits for 4 MarginalsTheoretical Diversification GainTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201229Standard Deviation of the Fits for 4 MarginalsTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201230

Number of Observations mattersless than Dependence ModelsWe see that the elliptical copulas keep a systematic biaswhatever the number of observationsThe Archimedean copulas fit much better the theoretical valuewith Clayton doing it the best, as expectedThe error of the estimation decreases with the number ofobservations and remains at a certain level even with 50’000observationsThe structure of the dependence (hierarchical or flat) does notaffect really the diversification benefit with hierarchical beingslightly better for Archimedean copulasWhen the dependence is asymmetric (as it is usually the casefor insurance liabilities), it is difficult to model it with symmetricdependence models (use asymmetric copulas)Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201231Agenda1 Risk-Adjusted Capital and dependence2 Description of the model3 Influence of the number of observations on calibration4 Influence of dependence structure and models5 Ways to model the dependence6 ConclusionTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201232

Results of Fit: Clayton FlatAs expected, the two strongerdependences are reduced and the weakerdependence is strengthened.Fit: = 1.2The RAC and diversification gain are evenslightly more conservative. The error isalmost negligible.Clayton FlatClayton HierarchyM3,M4 Θ=2The Q/Q plot shows good agreement ofthe models.Clayton ClaytonHierarchy FlatExpected 145‘550 145‘407Std Dev 172‘587 172‘948LH Div. Gain 9.56% 9.14%RH VaR 1‘070‘462 1‘078‘190RH Shortfall 1‘248‘146 1‘251‘138RH RAC 1‘102‘595 1‘105‘731RH Div. Gain 8.17% 7.73%Aggregate Θ=1M5, M6 Θ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201233Results of Fit: Gumbel HierarchyAs the Clayton, the Gumbel copula is anArchimedean Copula.Gumbel HierarchyClayton HierarchyGumbel represents the Clayton well in the tailregion for all 3 distributionsIn contrast to the Clayton, the Gumbelintroduces a dependence also in the lowertail.RAC is slightly underestimatedThe Q/Q shows fair similarity of the twoCopula types.ClaytonHierarchyGumbelHierarchyExpected 145‘550 144‘964Std Dev 172‘587 164‘507LH Div. Gain 9.56% 5.70%RH VaR 1‘070‘462 1‘021‘144RH Shortfall 1‘248‘146 1‘195‘423RH RAC 1‘102‘595 1‘050‘458RH Div. Gain 8.17% 11.73%Θ=2.07Θ=1.54Θ=2.61M3,M4 Θ=2Aggregate Θ=1M5, M6 Θ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201234

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

Results of Fit: Rank Correlation FlatThe flat rank correlation produces alsostthe same results as the hierarchical one.Rank Corr. FlatClayton HierarchyThis can be seen in all graphics as well asthe RAC calculations. i, j1.000.700.450.460.701.000.450.460.450.451.000.800.460.460.801.00ClaytonHierarchyRankCorr.Expected 145‘550 145‘293Std Dev 172‘587 144‘448LH Div. Gain 9.56% 3.98%RH VaR 1‘070‘462 870‘389RH Shortfall 1‘248‘146 977‘617RH RAC 1‘102‘595 832‘324RH Div. Gain 8.17% 30.70%M3,M4 Θ=2Aggregate Θ=1M5, M6 Θ=3Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201237Results of Fit: Student’s T HierarchyAs the Rank Correlation, the Student’s Tcopula is an elliptical copula.The Student‘s T has one parameter moreper dependence than the Rank CorrelationDependence is symmetric, i.e. alsointroduced in the lower tail.RAC is substantially underestimated.Unrealistically high diversification gainThe Q/Q plot looks similar as for the RankCorrelation.ClaytonHierarchyStudent THierarchyExpected 145‘550 145‘088Std Dev 172‘587 150‘290LH Div. Gain 9.56% 3.05%RH VaR 1‘070‘462 909‘335RH Shortfall 1‘248‘146 1‘045‘099RH RAC 1‘102‘595 900‘012RH Div. Gain 8.17% 25%= 0.81; = 4 = 0.51; = 9 = 0.71; = 6Student-T HierarchyClayton HierarchyTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201238M3,M4 Θ=2Aggregate Θ=1M5, M6 Θ=3

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

Dependence Model mattersmore than Dependence StructureWe see that the elliptical copulas do not improve by movingfrom a flat structure to a hierarchical one.The Gumbel copula slightly improves if used in the appropriatedependence structure.The elliptical copulas grossly underestimate the risk and showundue diversification benefits.Gumbel copula is able to produce reasonable results on the lefttail but also emphasizes a dependence on the right tail thatdoes not exist in the benchmark model.Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201241Underestimation starts alreadywith weak dependenceTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201242

Agenda1 Risk-Adjusted Capital and dependence2 Description of the model3 Influence of the number of observations on calibration4 Influence of dependence structure and models5 Ways to model the dependence6 ConclusionTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201243How to Estimate Dependences?Dependences can hardly be described by one number such as a linearcorrelation coefficient.We just saw that it is possible to use the copulas to model dependences.In insurance, there is often not enough liability data to estimate thecopulas.Nevertheless, copulas can be used to translate an opinion aboutdependences in the portfolio into a model:Part III, SCOR Switzerland’s internal modelSelect a copula with an appropriate shape• increased dependences in the tail– this feature is observable in historic insurance loss dataTry to estimate conditional probabilities by asking questions such as“What if a particular risk turned very bad?”• Think about adverse scenarios in the portfolio• Look at causal relations between risksTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201244

Strategy for modeling dependencesUsing the knowledge of the underlying business, develop ahierarchical model for dependences in order to reduce theparameter space and describe more accurately the mainsources of dependent behaviorWherever we know a causal dependence, we model it explicitlySystematically usage of non-symmetric copulas: ClaytoncopulaWherever there is enough data, we calibrate statistically theparametersIn absence of data, we use stress scenarios and expertopinions to estimate conditional probabilities (PrObEx)Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201245Combining three sources of information(Up to) three sources of information can be combined (PrObEx*):PriorA prior density, e.g. from previous years or from regulators.ObservationExpertsN independent observations of joint realizations from .The set of observation is denoted byK experts, each providing one point estimate of .The set of expert assessments is denoted by We replace the prior density by a posterior density of given andBayes‘ Theorem leads to the relation:*) See SCOR Paper no 10, on http://scor.com/images/stories/pdf/scorpapers/scorpapers10_en.pdfTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201246

Our modelWe make the following assumptions:The expert assessments and the observations are independentThe observations are independentThe experts form their opinion independently of each otherUnder these assumptions, the posterior distribution of the value of the dependence measure reads as:PriorObservationExpertsThrough this posterior distribution we can:Estimate , e.g. via .Assess the uncertainty of our estimate, e.g. via .Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201247An illustrative exampleThe best estimate using all information is then:Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201248

Agenda1 Risk-Adjusted Capital and dependence2 Description of the model3 Influence of the number of observations on calibration4 Influence of dependence structure and models5 Ways to model the dependence6 ConclusionTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201249SummaryNeglecting dependences leads to a gross underestimation of the Risk-Adjusted capitalNeglecting the non-linearity of dependences leads also to anoverestimation of the diversification benefitsDependences in insurance risks are usually asymmetric: stronger onthe negative side than on the positive oneA suitable copula to model those type of dependences is the ClaytoncopulaTo get the right diversification benefit the choice of the rightdependence model matters mostWith a relatively modest number of data it is possible to obtain areasonable estimate of the diversification benefit with Clayton copulaEspecially when combined with expert opinions on stress tests(PrObEx)Tail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201250

Further researchDevelop the empirical exploration of dependencesAnalyze retarded dependences: causal relationsPursue the study of the influence of the hierarchical tree ontotal RAC with more branches and depthApply the PrObEx methodology for all the modeled insurancerisksExplore analytical solutions for risk aggregation with copuladependenceTail DependenceMichel M. DacorognaETH Risk Center Workshop, Oct. 25-26, 201251