Loss-Based Risk Measures

TL;DR

This paper introduces loss-based risk measures focusing on portfolio losses, characterizes their properties, and explores robust statistical estimation methods for these measures.

Contribution

It defines and characterizes loss-based risk measures, and develops criteria and examples for statistically robust estimators.

Findings

Loss-based risk measures are characterized by a representation theorem.

A general criterion for the robustness of risk estimators is provided.

Examples of statistically robust estimators for loss-based risk measures are given.

Abstract

Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We characterize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative robustness of risk estimators and compare this criterion with sensitivity analysis of estimators based on influence functions. Finally, we provide examples of statistically robust estimators for loss-based risk measures.