Ordering risk bounds in factor models

Abstract

Conditionally comonotonic risk vectors have been proved in [4] to yield worst case dependence structures maximizing the risk of the portfolio sum in partially specified risk factor models. In this paper we investigate the question how risk bounds depend on the specification of the pairwise copulas of the risk components Xiwith the systemic risk factor. As basic toolwe introduce a new ordering based on sign changes of the derivatives of copulas. This together with discretization by n-grids and the theory of supermodular transfers allows us to derive concrete ordering criteria for the maximal risks.

Keywords

• products of copulas
• supermodular ordering
• risk bounds
• conditionally comonotonic distributions
• mass transfer theory
• elliptical distributions
• archimedean copulas