Supermodular and directionally convex comparison results for general factor models

Abstract

This paper provides comparison results for general factor models with respect to the supermodular and directionally convex order. These results extend and strengthen previous ordering results from the literature concerning certain classes of mixture models as mixtures of multivariate normals, multivariate elliptic and exchangeable models to general factor models. For the main results, we first strengthen some known orthant ordering results for the multivariate

-product of the specifications, which represents the copula of the factor model, to the stronger notion of the supermodular ordering. The stronger comparison results are based on classical rearrangement results and in particular are rendered possible by some involved constructions of transfers as arising from mass transfer theory. The ordering results for

-products are then extended to factor models with general conditional dependencies. As a consequence of the ordering results, we derive worst case scenarios in relevant classes of factor models allowing, in particular, interesting applications to deriving sharp bounds in financial and insurance risk models. The results and methods of this paper are a further indication of the particular effectiveness of Sklar‘s copula notion.