International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 7, Pages 345-364
Multivariate Fréchet copulas and conditional value-at-risk
Aon Re and IRMG (Switzerland) Ltd., Sternengasse 21, Basel CH-4010, Swaziland
Received 18 October 2002
Copyright © 2004 Werner Hürlimann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.