Coherent estimation of risk measures

ArXiv ID: 2510.05809 “View on arXiv”

Authors: Martin Aichele, Igor Cialenco, Damian Jelito, Marcin Pitera

Abstract

We develop a statistical framework for risk estimation, inspired by the axiomatic theory of risk measures. Coherent risk estimators – functionals of P&L samples inheriting the economic properties of risk measures – are defined and characterized through robust representations linked to $L$-estimators. The framework provides a canonical methodology for constructing estimators with sound financial and statistical properties, unifying risk measure theory, principles for capital adequacy, and practical statistical challenges in market risk. A numerical study illustrates the approach, focusing on expected shortfall estimation under both i.i.d. and overlapping samples relevant for regulatory FRTB model applications.

Keywords: Coherent risk estimators, Risk measures, Expected shortfall, L-estimators, FRTB, Market Risk

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced robust representation theory, duality arguments, and characterizations via L-estimators. While it includes a numerical study, it lacks code, detailed backtesting frameworks, or implementation-heavy data analysis, focusing primarily on theoretical construction and illustration rather than empirical validation.

  flowchart TD
    A["Research Goal:<br>Develop a statistical framework for coherent risk estimation"] --> B["Methodology:<br>Axiomatic risk theory & robust representations"]
    B --> C["Inputs:<br>Price/return samples P&L"]
    C --> D["Computation:<br>Construct & apply coherent risk estimators<br>e.g., L-estimator for Expected Shortfall"]
    D --> E["Analysis:<br>Backtesting & simulation under i.i.d.<br>and overlapping samples"]
    E --> F["Outcomes:<br>Unified methodology for market risk;<br>Improved capital adequacy for FRTB"]