Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coherent estimation of risk measures

Martin Aichele, Igor Cialenco, Damian Jelito, Marcin Pitera

TL;DR

The paper develops a coherent risk estimator (CRE) framework that mirrors the axioms of coherent risk measures, links CREs to robust representations via $L$-estimators, and shows that law-invariant CREs correspond to suprema over weighted order-statistic estimators. It proves that comonotonic, law-invariant CREs reduce to a single $L$-estimator and analyzes consistency results for CREs in i.i.d. settings, with a focus on spectral risk measures and ES plug-in estimators. A detailed numerical study compares six ES estimators based on $L$-statistics under i.i.d. and overlapping data, illustrating how weighting schemes affect accuracy, bias, and regulatory applicability (FRTB). The results provide guidance on constructing CREs with sound financial and statistical properties and on selecting ES estimators in practice, especially in regulatory contexts.

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.

Coherent estimation of risk measures

TL;DR

The paper develops a coherent risk estimator (CRE) framework that mirrors the axioms of coherent risk measures, links CREs to robust representations via -estimators, and shows that law-invariant CREs correspond to suprema over weighted order-statistic estimators. It proves that comonotonic, law-invariant CREs reduce to a single -estimator and analyzes consistency results for CREs in i.i.d. settings, with a focus on spectral risk measures and ES plug-in estimators. A detailed numerical study compares six ES estimators based on -statistics under i.i.d. and overlapping data, illustrating how weighting schemes affect accuracy, bias, and regulatory applicability (FRTB). The results provide guidance on constructing CREs with sound financial and statistical properties and on selecting ES estimators in practice, especially in regulatory contexts.

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 -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.

Paper Structure

This paper contains 7 sections, 8 theorems, 67 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coherent risk estimators
  4. Robust representations of a CRE
  5. Comonotonic CREs and their representation as L-estimators
  6. Consistency of CREs for i.i.d. samples
  7. Numerical study: comparison of ES estimators based on L-statistics

Key Result

Theorem 2.1

Let $\mathcal{X}=L^\infty(\Omega,\mathscr{G},\mathbb{P})$. A functional $\rho : \mathcal{X} \to \mathbb{R}$ is a coherent risk measure if and only if there exists $M_\rho \subset \mathcal{M}^{f}$ such that Moreover, $M_\rho$ can be chosen as a convex set for which the supremum is attained. That is, for any $X\in\mathcal{X}$, there exists $Q_X^*\in M_\rho$ such that $\rho(X) = \mathbb{E}_{Q_X^*}[-

Theorems & Definitions (33)

  • Theorem 2.1: Robust representation of CRMs
  • Definition 3.1: Coherent risk estimator
  • Definition 3.2: Law-invariant estimator
  • Proposition 3.3: Empirical plug-in risk estimator for CRM is coherent
  • proof
  • Example 3.4: Gaussian parameteric plug-in ES estimator is not coherent
  • Example 3.5: Average tail loss ES estimator is coherent
  • Example 3.6: Empirical quantile $\mathop{\mathrm{\mathrm{VaR}}}\nolimits$ estimator is not coherent
  • Example 3.7: Non-parametric plug-in ES estimator is coherent
  • Theorem 4.1: Robust representation of CREs
  • ...and 23 more