Axiomatic theory of risk measures

The series of working papers contains advances on the axiomatic theory of risk measures and other decision criteria used in economics, banking, and insurance. An axiomatic theory builds on a set of economically or statistically motivated properties, termed axioms, which leads to characterizations of risk measures or decision criteria. The axiomatic study can help to design new risk and decision procedures, to justify or to warn about the use and misuses of current risk measures and their implementation, and to enhance our understanding of risk, uncertainty and their management. Many papers focus on and relate to the Value-at-Risk (VaR, quantiles) and the Expected Shortfall (ES, also known as TVaR, CVaR and AVaR), which are widely used in global financial regulation and internal risk management. Some particular theoretical contributions are:

A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

We introduce partial law invariance for risk measures. This property is motivated by model uncertainty and is weaker than the usual law invariance. We give examples and characterization results.

This paper is the first systemic study on cash-subadditive risk measure that are not assumed quasi-convex. A general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex ones, generalizing several results in WP01 and WP12 on cash-additive risk measures. We also introduce quasi-star-shapedness.

We offer an axiomatic characterization of quantiles through only one axiom, that is, commutativity with respect to increasing and continuous transforms, also called ordinality. Based on this result, we obtain axiomatizations of the median, the median interval, and quantile maximization. The journal version is published in SIAM Journal on Financial Mathematics (2023).

In the traditional setting, all risk measures are mappings from random lesses to real numbers. We introduce a unified axiomatic framework of generalized risk measures using random losses and scenario sets representing model uncertainty as the joint input. Through various results, we observe a sharp technical contrast between our framework and the traditional one. The journal version is published in Finance and Stochastics (2024).

This paper is a follow-up of WP07 and it contains a new axiomatic characterization of ES. The main idea is to introduce a new axiom of concentration aversion, which plays a similar role as no reward for concentration in WP07, but without a strong additive form. Our results lead to an axiomatization of the mean-ES portfolio selection criterion and new analytical examples of consistent risk measures proposed in WP01. The journal version is published in Mathematical Finance (2024).

Star-shaped risk measures are introduced and characterized via an "inf-convex" representation. The new class includes all practically used risk measures, in particular, VaR and convex risk measures, as well as their robustifications, and it has manifold motivations, including liquidity risk, competitive delegation, robust aggregation, ambiguity aversion, and non-concave utility functions. The journal version is published in Operations Research (2022).

We introduce the class of Bayes risk measures, which are the counterpart of elicitable risk measures. We obtain a new characterization of ES as unique coherent Bayes risk measures (Theorem 3.1). A second main result (Theorem 6.1) is that entropic risk measures are the only risk measures which are both elicitable and Bayes. The journal version is published in Mathematical Finance (2021).

Distortion riskmetrics, which are defined through signed Choquet integrals, are the unifying term for all distortion risk measures, many variability measures, and other functionals. This paper is a follow-up of WP03 and it serves as a comprehensive collection of results on distortion riskmetrics on general spaces. The journal version is published in ASTIN Bulletin (2020).
- Erratum in Table 1 of the journal version: the domain of RVaR should be L⁰ instead of L¹.

With empirical evidence from the post Dodd-Frank period, we develop a theory for credit rating criteria. We consider market models with two types of investors, simple investors and model-based investors. Concepts of self-consistency and information gap are proposed to study different rating criteria. The expected loss criterion used by Moody’s satisfies self-consistency but the probability of default criterion used by S&P does not. Moreover, the probability of default criterion typically has a higher information gap than the expected loss criterion. A self-consistent rating measure admits an axiomatic characterization via Choquet integrals. The journal version is to appear in Management Science (2024). Media coverage: Bloomberg (April 2020).

This paper provides a general mathematical framework for distributional transforms. As a particular result, a probability distortion is characterized by commutativity with shape transforms (Theorem 1). Probability distortions can be used to generate many risk measures such as VaR and ES. The journal version is published in Mathematics of Operations Research (2021).

This paper contains the first axiomatic characterization of ES and related results. A risk measure is an ES if and only if it satisfies monotonicity, law-invariance, lower semi-continuity, and no reward for concentration (Theorem 1). As the main message, ES rewards portfolio diversification and penalizes risk concentration in a unique and intuitive way. The journal version is published in Management Science (2021).

This paper is the first attempt to bring the practice of scenario analysis as an axiom for scenario-based risk measures, which bridge between generic risk measures and law-invariant ones. Several technical results are obtained including a mixture-ES based representation (Theorem 3.8 and Proposition 4.11). The journal version is published in Finance and Stochastics (2021).

This paper is a comprehensive study of the "convex level sets" (CxLS) property, a necessary condition for elicitability. The paper contains full characterizations of the CxLS property for one-dimensional signed Choquet integrals (Theorem 3.2) and for two-dimensional signed Choquet integrals with a VaR component (Theorem 5.3). Further, under a continuity condition, a spectral risk measure is co-elicitable with VaR if and only if it is the corresponding ES (Theorem 6.9). The journal version is published in Mathematical Finance (2020).

This paper is a theoretical treatment of law-invariant convex risk functionals, including a wide majority of practically used convex risk measures and deviation measures. A unified representation theorem (Theorem 2.2) is obtained, and some related optimization problems are solved. The journal version is published in Insurance: Mathematics and Economics (2020).

This paper is dedicated to a comprehensive study of signed Choquet integrals, a class of non-monotone law-invariant functionals, which can be characterized via comonotonic additivity (Theorem 1). Technical results include, among others, six equivalent conditions for convexity (Theorem 3) and equivalence in risk aggregation (Theorem 5). The journal version is published in Mathematics of Operations Research (2020).

In this paper, the notion of tail risk, popular in risk management, is rigorously formulated in an axiomatic framework. A tail risk measure, such as VaR, ES and GS (WP03), relies only on the tail distribution of a risk. Relevant properties and representations are provided. A new characterization of VaR is obtained: an elicitable, positively homogeneous, and monetary tail risk measure has to be a VaR (Theorems 5 and 6). A general result on tail risk measures in robust risk aggregation is established (Theorem 3). The journal version is published in Mathematics of Operations Research (2021).

It is well known that any law-invariant convex risk measure is consistent with respect to convex order. With convexity (or quasi-convexity) relaxed, we show that any consistent monetary risk measure admits a representation as the minimum of some law-invariant convex risk measures (Theorems 3.1 and 3.3). The result leads to some analytical solutions to risk sharing and optimal investment problems with such risk measures. The journal version is published in SIAM Journal on Financial Mathematics (2020).