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Counter-monotonic Risk Sharing with Heterogeneous Distortion Risk Measures

Mario Ghossoub, Qinghua Ren, Ruodu Wang

Abstract

We study risk sharing among agents with preferences modeled by heterogeneous distortion risk measures, who are not necessarily risk averse. Pareto optimality for agents using risk measures is often studied through the lens of inf-convolutions, because allocations that attain the inf-convolution are Pareto optimal, and the converse holds true under translation invariance. Our main focus is on groups of agents who exhibit varying levels of risk seeking. Under mild assumptions, we derive explicit solutions for the unconstrained inf-convolution and the counter-monotonic inf-convolution, which can be represented by a generalization of distortion risk measures.

Counter-monotonic Risk Sharing with Heterogeneous Distortion Risk Measures

Abstract

We study risk sharing among agents with preferences modeled by heterogeneous distortion risk measures, who are not necessarily risk averse. Pareto optimality for agents using risk measures is often studied through the lens of inf-convolutions, because allocations that attain the inf-convolution are Pareto optimal, and the converse holds true under translation invariance. Our main focus is on groups of agents who exhibit varying levels of risk seeking. Under mild assumptions, we derive explicit solutions for the unconstrained inf-convolution and the counter-monotonic inf-convolution, which can be represented by a generalization of distortion risk measures.

Paper Structure

This paper contains 11 sections, 10 theorems, 61 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Risk measures and basic terminology
  4. Risk sharing and inf-convolution
  5. Comonotonic and counter-monotonic risk sharing
  6. General relations
  7. Risk-seeking agents
  8. Main results
  9. Three technical lemmas and the proof of Theorem \ref{['thm:convex']}
  10. Sharing a constant payoff
  11. Conclusion

Key Result

Proposition 1

For $X \in \mathcal{X}$ and $n\geqslant 3$, suppose that at least three components of $\left(X_1, \ldots, X_n\right) \in \mathbb{A}_n(X)$ are non-degenerate. Then $\left(X_1, \ldots, X_n\right)$ is counter-monotonic if and only if there exist constants $m_1, \ldots, m_n$ and $\left(A_1, \ldots, A_n\

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Proposition 1: lauzier2023pairwise
  • Theorem 1: lauzier2024negatively
  • Example 1: Inf-convolution of VaRs
  • Example 2: Inf-convolution of ESs
  • Theorem 2
  • proof
  • Corollary 1
  • Example 3
  • ...and 14 more