Generalized FGM dependence: Geometrical representation and convex bounds on sums

TL;DR

This paper explores the geometric structure of generalized FGM copulas, proving they form a convex polytope, and derives sharp bounds for risk measures, especially in high-dimensional settings with identical risks.

Contribution

It establishes the convex polytope structure of generalized FGM copulas and provides analytical bounds for risk measures in high-dimensional cases with identical risks.

Findings

Generalized FGM copulas form a convex polytope.

Sharp bounds for risk measures are derived from extremal points.

Analytical bounds are provided for identical risks in high dimensions.

Abstract

Building on the one-to-one relationship between generalized FGM copulas and multivariate Bernoulli distributions, we prove that the class of multivariate distributions with generalized FGM copulas is a convex polytope. Therefore, we find sharp bounds in this class for many aggregate risk measures, such as value-at-risk, expected shortfall, and entropic risk measure, by enumerating their values on the extremal points of the convex polytope. This is infeasible in high dimensions. We overcome this limitation by considering the aggregation of identically distributed risks with generalized FGM copula specified by a common parameter $p$. In this case, the analogy with the geometrical structure of the class of Bernoulli distribution allows us to provide sharp analytical bounds for convex risk measures.