On evaluation of joint risk for non-negative multivariate risks under dependence uncertainty

TL;DR

This paper introduces a new axiomatic framework for evaluating joint risk of non-negative multivariate risks under dependence uncertainty, including scalar and vector-valued distortion measures, with theoretical characterizations and comparisons to existing measures.

Contribution

It proposes a novel axiomatic approach and new classes of distortion joint risk measures for multivariate risks, expanding the theoretical understanding of risk evaluation under dependence uncertainty.

Findings

Established fundamental properties of the scalar distortion joint risk measure.

Provided an axiomatic characterization including a new positive homogeneity axiom.

Compared new measures with existing multivariate risk measures, showing their relationships.

Abstract

In this paper, we propose a novel axiomatic approach to evaluating the joint risk of multiple insurance risks under dependence uncertainty. Motivated by both the theory of expected utility and the Cobb-Dauglas utility function, we establish a joint risk measure for non-negative multivariate risks, which we refer to as a scalar distortion joint risk measure. After having studied its fundamental properties, we provide an axiomatic characterization of it by proposing a set of new axioms. The most novel axiom is the component-wise positive homogeneity. Then, based on the resulting distortion joint risk measures, we also propose a new class of vector-valued distortion joint risk measures for non-negative multivariate risks. Finally, we make comparisons with some vector-valued multivariate risk measures known in the literature, such as multivariate lower-orthant value at risk, multivariate upper-orthant conditional-tail-expectation, multivariate tail conditional expectation and multivariate tail distortion risk measures. It turns out that those vector-valued multivariate risk measures have forms of vector-valued distortion joint risk measures, respectively. This paper mainly gives some theoretical results about the evaluation of joint risk under dependence uncertainty, and it is expected to be helpful for measuring joint risk.