Distortion risk measures of sums of two counter-monotonic risks

Paper: arXiv 2503.05256

Authors: Chunle Huang

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous theoretical mathematical derivations extending distortion risk measure decompositions for counter-monotonic sums from VaR/TVaR to general distortion functions. However, it is purely theoretical and mathematical, lacking any empirical data, backtesting, or real-world asset applications.

Research Flowchart

  flowchart TD
    A[Research Goal: Express distortion risk measure of two counter-monotonic risks] --> B[Input: Marginals X, Y & Distortion Function g]
    B --> C[Methodology: Apply dual distortion function g* and extend previous frameworks]
    C --> D[Compute: Calculate distortion risk measures for individual marginals]
    D --> E[Key Finding: Sum of counter-monotonic risks = Measure with g + Measure with g*]
    E --> F[Outcome: Generalizes VaR & TVaR for cross-asset risk aggregation]