Paper: arXiv 2404.13637

Authors: Mengshuo Zhao, Narayanaswamy Balakrishnan, Chuancun Yin, Hui Shao

Abstract

This paper investigates the impact of distributional uncertainty on key risk measures under the partial knowledge of underlying distributions characterized by their first two moments and shape information (specifically symmetry and/or unimodality). We first employ probability inequalities to establish the theoretical best- and worst-case bounds on Value-at-Risk, reflecting the most extreme tail risk achievable within the moment and shape constraints, and then we extend this worst-case/best-case analysis to a broad class of distortion risk measures by the modified Schwarz inequality, deriving their corresponding robust bounds under the same partial information setting concerning moments and distribution shapes of the underlying distributions. In addition, we give a clear characterization of the distributions that attain the best- and worst-case scenarios. The proposed approach provides a unified framework for extremal problems of distortion risk measures.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using advanced probability inequalities and Schwarz inequalities to establish bounds for risk measures under distributional uncertainty. However, it lacks empirical backtesting or real-world financial data validation, remaining purely theoretical.

Research Flowchart

  flowchart TD
    A[Goal: Robust Bounds for Distortion Risk Measures under Distributional Uncertainty] --> B[Inputs: Partial Info<br>First 2 Moments, Symmetry, Unimodality]
    B --> C[Phase 1: VaR Bounds<br>Apply Probability Inequalities]
    B --> D[Phase 2: General Bounds<br>Apply Modified Schwarz Inequality]
    C --> E[Unified Framework for Extremal Risk Analysis]
    D --> E
    E --> F[Outcomes: Best/Worst-Case Bounds & Extremal Distribution Characterization]

Cited References (24)

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  5. A Modification of Schwarz’s Inequality with Applications to Distributions (1953) — link
  6. Robustness and Sensitivity Analysis of Risk Measurement Procedures (2008) — link
  7. Remarks on quantiles and distortion risk measures (2012) — link
  8. Spectral measures of risk: A coherent representation of subjective risk aversion (2002) — link
and 16 more
  1. A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions (2005) — link
  2. Analytical Bounds for two Value-at-Risk Functionals (2002) — link
  3. Convex Optimization (2004) — link
  4. Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation (1990) — link
  5. Worst-Case Range Value-at-Risk with Partial Information (2017) — link
  6. Technical Note—Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization (2018) — link
  7. Range Value-at-Risk bounds for unimodal distributions under partial information (2020) — link
  8. Sharing the Value-at-Risk Under Distributional Ambiguity (2019) — link
  9. DISTORTION RISKMETRICS ON GENERAL SPACES (2020) — link
  10. Closed-Form Solutions for Extreme-Case Distortion Risk Measures and Applications to Robust Portfolio Management (2018) — link
  11. Distributionally Robust Optimization under Distorted Expectations (2020) — link
  12. Actuarial Theory for Dependent Risks (2005) — link
  13. Bounds on Range Value-at-Risk for Unimodal Symmetric Distributions (2022) — link
  14. Sharp inequalities of Bienaymé–Chebyshev and Gauß type for possibly asymmetric intervals around the mean (2023) — link
  15. Distortion risk measure under parametric ambiguity (2023) — link
  16. Robust distortion risk measures (2023) — link

0 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.