Paper: arXiv 2511.03551

Authors: Christian Laudagé, Jörn Sass

Abstract

Under Solvency II, the Value-at-Risk (VaR) is applied, although there is broad consensus that the Expected Shortfall (ES) constitutes a more appropriate risk measure. Moving towards ES would necessitate specifying the corresponding ES level. The recently introduced Probability Equivalent Level of VaR and ES (PELVE) determines this by requiring that ES equals the prescribed VaR for a given future payoff, reflecting the situation of an individual insurer. We incorporate the regulator’s perspective by proposing PELVE-inspired methods for multiple insurers. We analyze existence and uniqueness of the resulting ES levels, derive expressions for elliptically distributed payoffs and establish limit results for multivariate regularly distributed payoffs. A case study highlights that the choice of method is crucial when payoffs arise from different distribution families. We provide recommendations which of our PELVE-inspired methods are most appropriate in certain scenarios.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides strong theoretical proofs and extensions of the PELVE framework to a multivariate regulatory setting (A-PELVE, MSE-PELVE, Sys-PELVE). While mathematically sound with rigorous characterizations of existence, uniqueness, and limit behavior, the empirical evaluation is limited to stylized case studies rather than extensive historical market data backtests.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Extend PELVE to multi-insurer regulatory settings under Solvency II"] --> B["Propose PELVE-Inspired Methods<br>for aggregated portfolio risk"]
    
    Inputs["Inputs:<br>Prescribed VaR level, future payoffs (Elliptical & Multivariate Regular distributions)"] --> B
    
    B --> C["Theoretical Analysis:<br>Prove existence/uniqueness & derive limit results"]
    B --> D["Empirical Case Study:<br>Compare methods across different distribution families"]
    
    C --> E["Key Findings:<br>Method choice is critical when payoffs have differing tail behaviors"]
    D --> E
    
    E --> F["Outcome:<br>Regulatory recommendations for selecting optimal ES levels"]