A Framework for Measures of Risk under Uncertainty

TL;DR

This paper introduces a unified axiomatic framework for risk evaluation that jointly considers loss variables and multiple economic scenarios, advancing the theoretical understanding of risk measures under uncertainty.

Contribution

It develops a generalized risk measure framework that unifies various risk evaluation principles and characterizes their properties through axioms, highlighting differences from traditional models.

Findings

Characterization of worst-case, coherent, and robust risk measures.

Establishment of equivalence between forms of law invariance.

Support for coherent risk measures like Expected Shortfall over Value-at-Risk.

Abstract

A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Value-at-Risk, a relevant issue for risk management practice.