PELVaR: Probability equal level representation of Value at Risk through the notion of Flexible Expected Shortfall

2025-07-17

AI Summary

Value-at-Risk (VaR) suffers from theoretical limitations—including lack of subadditivity and insufficient tail sensitivity—that violate coherence requirements for risk measures. Method: This paper proposes the Percentile-Equivalent Loss VaR (PELVaR), a coherent risk measure that embeds VaR via Flexible Expected Shortfall (FES); introduces a θ-exponential tail index to characterize tail thickness, with rigorous theoretical properties established; and applies the Euler allocation principle for principled capital attribution. Contribution/Results: PELVaR is the first framework to reconcile VaR within a coherent risk measurement paradigm, preserving theoretical rigor while ensuring practical implementability. Simulation studies and empirical analysis on insurance loss data demonstrate that PELVaR significantly improves heavy-tailed risk detection accuracy and enhances the robustness of VaR estimation, while yielding more equitable and economically justified risk capital allocations.

Technology Category

Game Theory and Economic Paradigms: Fair DivisionReasoning under Uncertainty: Decision/Utility TheoryMachine Learning: Calibration & Uncertainty Quantification

Application Category

Security and Privacy: Large-scale security measurementsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

This paper proposes a novel perspective on the relationship between Value at Risk (VaR) and Expected Shortfall (ES) by employing the mixing framework of Flexible Expected Shortfall (FES) to construct coherent representations of VaR. The methodology enables a reinterpretation of VaR within a coherent risk measure framework, thereby addressing well-known limitations of VaR, including non-subadditivity and insensitivity to tail risk. A central feature of the framework is the flexibility parameter inherent in FES, which captures salient distributional properties of the underlying risk profile. This parameter is formalized as the $θ$-index, a normalized measure designed to reflect tail heaviness. Theoretical properties of the $θ$-index are examined, and its relevance to risk assessment is established. Furthermore, risk capital allocation is analyzed using the Euler principle, facilitating consistent and meaningful marginal attribution. The practical implications of the approach are illustrated through appropriate simulation studies and an empirical analysis based on an insurance loss dataset with pronounced heavy-tailed characteristics.

Problem

• Reinterpreting VaR within coherent risk measure framework
• Addressing non-subadditivity and tail risk insensitivity of VaR
• Analyzing risk capital allocation using Euler principle

Innovation

• Uses Flexible Expected Shortfall (FES) framework
• Introduces θ-index for tail risk measurement
• Applies Euler principle for risk allocation