Multivariate Fréchet copulas and conditional value‐at‐risk

Abstract

Based on the method of copulas, we construct a parametric family of multivariate distributions using mixtures of independent conditional distributions. The new family of multivariate copulas is a convex combination of products of independent and comonotone subcopulas. It fulfills the four most desirable properties that a multivariate statistical model should satisfy. In particular, the bivariate margins belong to a simple but flexible one-parameter family of bivariate copulas, called linear Spearman copula, which is similar but not identical to the convex family of Fréchet. It is shown that the distribution and stop-loss transform of dependent sums from this multivariate family can be evaluated using explicit integral formulas, and that these dependent sums are bounded in convex order between the corresponding independent and comonotone sums. The model is applied to the evaluation of the economic risk capital for a portfolio of risks using conditional value-at-risk measures. A multivariate conditional value-at-risk vector measure is considered. Its components coincide for the constructed multivariate copula with the conditional value-at-risk measures of the risk components of the portfolio. This yields a “fair” risk allocation in the sense that each risk component becomes allocated to its coherent conditional value-at-risk.

A class of multivariate copulas with bivariate Fréchet marginal copulas

“…, U n ) belong to the family of bivariate Fréchet copulas. Hürlimann (2004) presented a family of copulas with two-dimensional marginal copulas of linear Spearman copulas, that is, for i = m the vector (U i , U m ) has one parameter linear Spearman copulas, given by…”

A Mean-Variance Portfolio Optimal Under Utility Pricing

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Copula based post-processing for improving the NMME precipitation forecasts

“…The copula functions are unit cube functions that relate the multi-dimensional distributions to their one-dimensional marginal (Sklar, 1959). At first, it was commonly used in financial approaches (Cherubini, 2004;de Melo Mendes & de Souza, 2004;Frees et al, 1996;Frees & Valdez, 1998;Hürlimann, 2004). The application of Copula functions in the hydrological domain started in 2003 (De Michele & Salvadori, 2003;Favre et al, 2004;Salvadori & De Michele, 2004a, 2004b.…”