Coherent estimation of risk measures

2025-10-07

AI Summary

This paper addresses the disconnect between axiomatic risk measure theory, regulatory capital requirements, and statistical estimation practice in market risk measurement. Methodologically, it introduces the concept of “compatible risk estimators,” leveraging the robust representation of L-estimators and integrating convex analysis with asymptotic statistics to achieve consistent estimation and rigorous error characterization for coherent risk measures—particularly Expected Shortfall (ES)—under both i.i.d. and overlapping sampling schemes. The main contributions are threefold: (i) it establishes, for the first time, a canonical linkage between the axiomatic foundation of risk measures and statistical estimation; (ii) it ensures that the resulting estimators possess economic interpretability, statistical robustness, and regulatory compatibility; and (iii) numerical experiments demonstrate high accuracy and strong robustness in regulatory capital frameworks such as the Fundamental Review of the Trading Book (FRTB).

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

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.

Problem

• Developing coherent risk estimators with robust statistical properties
• Unifying risk measure theory with practical capital adequacy requirements
• Addressing expected shortfall estimation challenges for regulatory FRTB models

Innovation

• Coherent risk estimators derived from axiomatic theory
• Robust representations linked to L-estimators methodology
• Unified framework for financial and statistical properties