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Distortion risk measures of sums of two counter-monotonic risks

Chunle Huang

TL;DR

The paper addresses how distortion risk measures of the sum of two counter-monotonic risks can be decomposed into marginal components. By assuming symmetric, continuously distributed marginals and a dispersive order between them, the authors prove that for any distortion function $g$, $\rho_g[S^-]$ equals the sum of the marginal distortion measures with $X_2$ evaluated under the dual distortion $\overline{g}$, i.e., $\rho_g[S^-] = \rho_g[X_1] + \rho_{\overline{g}}[X_2]$, with the proof leveraging Lebesgue-Stieltjes representations and a decomposition of $g$ into left- and right-continuous parts. The results extend known decompositions from VaR/TVaR to the full class of distortion risk measures and provide explicit formulas for common distributions (Gaussian, Student, log-normal), including identically and non-identically distributed marginals, and even outline extensions to sums of more than two risks via comonotone decompositions. These findings facilitate analytic risk aggregation under extreme negative dependence, aiding reserving, capital requirements, and risk management decisions across actuarial and financial contexts.

Abstract

In this paper, we will show that under certain conditions, associated to any fixed distortion function $g$, the distortion risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals involved, one associated to the original distortion function $g$ and the other associated to the dual distortion function of $g$. This result extends some of the work in \cite{Chaoubi et al. (2020)} and \cite{HLD} since the class of distortion risk measures includes the risk measure of VaR and TVaR as special cases.

Distortion risk measures of sums of two counter-monotonic risks

TL;DR

The paper addresses how distortion risk measures of the sum of two counter-monotonic risks can be decomposed into marginal components. By assuming symmetric, continuously distributed marginals and a dispersive order between them, the authors prove that for any distortion function , equals the sum of the marginal distortion measures with evaluated under the dual distortion , i.e., , with the proof leveraging Lebesgue-Stieltjes representations and a decomposition of into left- and right-continuous parts. The results extend known decompositions from VaR/TVaR to the full class of distortion risk measures and provide explicit formulas for common distributions (Gaussian, Student, log-normal), including identically and non-identically distributed marginals, and even outline extensions to sums of more than two risks via comonotone decompositions. These findings facilitate analytic risk aggregation under extreme negative dependence, aiding reserving, capital requirements, and risk management decisions across actuarial and financial contexts.

Abstract

In this paper, we will show that under certain conditions, associated to any fixed distortion function , the distortion risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals involved, one associated to the original distortion function and the other associated to the dual distortion function of . This result extends some of the work in \cite{Chaoubi et al. (2020)} and \cite{HLD} since the class of distortion risk measures includes the risk measure of VaR and TVaR as special cases.

Paper Structure

This paper contains 10 sections, 12 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantile functions
  4. Distortion risk measures
  5. Fréchet spaces and Fréchet bounds
  6. Comomontonic and counter-monotonic risks
  7. Dispersive order
  8. Main result
  9. Applications
  10. Further discussions

Key Result

Theorem 3.1

Assume that $(X^{-}_1, X^{-}_2)$ is a pair of symmetric random variables such that their distribution functions $F_{X_i}(x), i = 1, 2$, are continuous and strictly increasing on $(F^{-1+}_{X_{i}}(0), F^{-1}_{X_i}(1))$. If $X_2 \leq_{disp} X_1$, then for any distortion function $g$ we have where $\overline{g}$ is the dual distortion function of $g$.

Theorems & Definitions (29)

  • Example 2.1: VaR is a distortion risk measure
  • Example 2.2: TVaR is a distortion risk measure
  • Example 2.3: The Wang Transform Risk Measure
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8: See Example 1 in Chaoubi et al. (2020) and Example 3 in Jeon et al. (2006)
  • Theorem 3.1
  • proof
  • ...and 19 more