Tail Distribution and Dependence Measures

Families of copulas could be distinguished according to the allocation of the weight among the dependence structure: some copulas could stress more on upper tails (large claims, while other could stress more on lower tails...etc. As mentioned by Gary Venter in Tails of Copulas in property and casualty applications, there could be interest in copulas that emphasize correlation among large losses. More precisely, it might be more interesting to focus on the upper tail, to be sure that the model fits well the dependence among large losses -the main idea being that we should chose a copula which does not underestimate the dependence in the upper tail. In this paper, we will study (in Part 2) conditional copula, which is the copula of the distribution (X,Y), given X and Y both higher, or lower, than a given threshold (defined as a quantile, for a given risk level). After defining these conditional properties, we will give some properties, satisfied by these families of copulas, focusing on a stable family: the family of Archimedian copula. This conditional copula will be used, then (in Part 3), to define a functional dependence measure, based on Spearman ’s rho, call tail rank correlation, which could be seen as a measure of dependence in the tails of the distribution.

Keywords: Archimedian copula; conditional distribution; copulas; dependence; factor representation; rank correlation; Spearman’s rho; tail correlation; tail dependence; truncature;