Ordering of multivariate probability distributions with respect to extreme portfolio losses

TL;DR

This paper introduces a new stochastic ordering concept for multivariate risk models focusing on extreme portfolio losses, with criteria based on spectral measures, applicable to various distribution classes.

Contribution

It presents a novel stochastic ordering framework for multivariate risks, linking spectral measures to extreme loss comparisons, and provides practical verification methods.

Findings

Comparison criteria for spectral measures are established.

Application examples include elliptical and copula-based models.

Framework enables analytical and numerical risk assessment.

Abstract

A new notion of stochastic ordering is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of ordering conditions on the spectral measures, which allows for analytical or numerical verification in practical applications. Additional comparison criteria in terms of further stochastic orderings are derived. The application examples include worst case and best case scenarios, elliptically contoured distributions, and multivariate regularly varying models with Gumbel, Archimedean, and Galambos copulas.