A Dynamic Grouped-T Copula Approachfor High-Dimensional PortfoliosDean FantazziniApril the 21th2007,International Workshop on Computational and FinancialEconometrics, Geneva (Switzerland)Overview of the Presentation1stIntroductionA Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 2Overview of the Presentation1stIntroduction2ndDynamic Grouped-T Copula Modelling: Denition andEstimationA Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 2-aOverview of the Presentation1stIntroduction2ndDynamic Grouped-T Copula Modelling: Denition andEstimation3rdAsymptotic PropertiesA Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 2-bOverview of the Presentation1stIntroduction2ndDynamic Grouped-T Copula Modelling: Denition andEstimation3rdAsymptotic Properties4thA Simulation StudyA Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 2-cOverview of the Presentation1stIntroduction2ndDynamic Grouped-T Copula Modelling: Denition andEstimation3rdAsymptotic Properties4thA Simulation Study5thEmpirical AnalysisA Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 2-dOverview of the Presentation1stIntroduction2ndDynamic Grouped-T Copula Modelling: Denition andEstimation3rdAsymptotic Properties4thA Simulation Study5thEmpirical Analysis6thConclusionsA Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 2-eIntroductionThe increasing complexity of nancial markets has pointed out the needfor advanced dependence modelling in nance. Why? Multivariate models with more exibility than the multivariate normaldistribution are needed; When constructing a model for risk management, the study of bothmarginals and the dependence structure is crucial for the analysis. Awrong choice may lead to severe underestimation of nancial risks.Recent developments in nancial studies have tried to tackle these issuesby using the theory of Copulas: see Cherubini et al. (2004) for a generalreview of copula methods in nance.However, mostly low-dimensional applications have been considered so far,while the rare high-dimensional cases present no dynamics at all.A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 3IntroductionDaul, Giorgi, Lindskog, and McNeil (2003), Demarta and McNeil (2005)and Mc-Neil, Frey, and Embrechts (2005) underlined the ability of thegrouped t-copula to model the dependence present in a large set ofnancial assets into account.We extend their methodology by allowing the copula dependence structureto be time-varying and we show how to estimate its parameters.Furthermore, we prove the consistency and asymptotic normality of thisestimator under some special cases and we examine its nite samplesproperties via simulations.Finally, we apply this methodology for the estimation of the VaR of aportfolio composed of thirty assets.A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 4Dynamic Grouped-T Copula Modelling: Definition andEstimationLet Z|Ft1 Nn(0, Rt), t = 1, . . . T, given the conditioning set Ft1,where Rt is the n n conditional linear correlation matrix which follows aDCC model, and R is the unconditional correlation matrix. Furthermorelet U Uniform(0, 1) be independent of Z .Let G denote the distribution function of

/, where is a chisquare distribution with degrees of freedom, and partition 1, . . . , n intom subsets of sizes s1, . . . , sm. Set Wk = G1k (U) for k = 1, . . . , m and thenY|Ft1 = (W1Z1, . . . , W1Zs1, W2Zs1+1, . . . , W2Zs1+s2, . . . , WmZn), sothat Y has a so-called grouped t distribution. Finally, deneU|Ft1 = (t1(Y1), . . . , t1(Ys1), t2(Ys1+1), . . . , t2(Ys1+s2), . . . , tm(Yn))(1)U has a distribution on [0, 1]nwith components uniformly distributed on[0, 1]. We call its distribution function the dynamic grouped t-copula.A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 5Dynamic Grouped-T Copula Modelling: Definition andEstimationNote that (Y1, . . . , Ys1) has a t distribution with 1 degrees of freedom, andin general for k = 1, . . . , m1, (Ys1+...+sk+1, . . . , Ys1+...+sk+1) has a tdistribution with k+1 degrees of freedom. Similarly, subvectors of U havea t-copula with k+1 degrees of freedom, for k = 0, . . . , m1.In this case no elementary density has been given.However, there is a very useful correlation approximation, obtained byDaul et al. (2003) for the constant correlation case:i,j(zi, zj) sin(ij(ui, uj)/2) (2)where i and j belong to dierent groups and ij is the pairwise Kendallstau. This approximation then allows for Maximum Likelihood estimationfor each subgroup separately.A Dynamic Grouped-T Copula Approach for High-Dimensional Portfolios 6Dynamic Grouped-T Copula Modelling: Definition andEstimationDenition 1 (Dynamic Grouped-T copula estimation).1. Transform the standardized residuals ( 1t, 2t, . . . , nt) obtained from aunivariate GARCH estimation, for example, into uniform variates( u1t, u2t, . . . unt), using either a parametric cumulative distributionfunction (c.d.f.) or an empirical c.d.f..2. Collect all pairwise estimates of the unconditional sample Kendallstau given by i,j( uj, uk) =

T2

11t