Lhasa, China, 2019

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. | |||

WP16 |
Cash-subadditive risk measures without quasi-convexity
(by Xia Han, Qiuqi Wang, Ruodu Wang, Jianming Xia) | ||

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. | |||

WP15 |
One axiom to rule them all: A minimalist axiomatization of quantiles
(by Tolulope Fadina, Peng Liu, Ruodu Wang) | ||

We offer an axiomatic characterization of quantiles through only | |||

WP14 |
A framework for measures of risk under uncertainty
(by Tolulope Fadina, Yang Liu, Ruodu Wang) | ||

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 to appear in | |||

WP13 |
Risk concentration and the mean-Expected Shortfall criterion
(by Xia Han, Bin Wang, Ruodu Wang, Qinyu Wu) | ||

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 to appear in | |||

WP12 |
Star-shaped risk measures
(by Erio Castagnoli, Giacomo Cattelan, Fabio Maccheroni, Claudio Tebaldi, Ruodu Wang) | ||

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 | |||

WP11 |
Bayes risk, elicitability, and the Expected Shortfall
(by Paul Embrechts, Tiantian Mao, Qiuqi Wang, Ruodu Wang) | ||

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 | |||

WP10 |
Distortion riskmetrics on general spaces
(by Qiuqi Wang, Ruodu Wang, Yunran Wei) | ||

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 | |||

WP09 |
Self-consistency, subjective pricing, and a theory of credit rating
(by Nan Guo, Steven Kou, Bin Wang, Ruodu Wang) | ||

With empirical evidence from the post Dodd-Frank period, we develop a theory for credit rating criteria based on a concept of self-consistency. While the expected loss criterion used by Moody's satisfies self-consistency, the probability of default criterion used by S&P does not. A self-consistent rating measure admits a Choquet integral representation, which becomes analytically tractable if one further takes economic scenarios into account (Theorems 4.1 and 5.1). Media coverage: Bloomberg (April 2020). | |||

WP08 |
Distributional transforms, probability distortions, and their applications
(by Peng Liu, Alexander Schied, Ruodu Wang) | ||

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 | |||

WP07 |
An axiomatic foundation for the Expected Shortfall
(by Ruodu Wang, Ricardas Zitikis) | ||

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 | |||

WP06 |
Scenario-based risk evaluation
(by Ruodu Wang, Johanna Ziegel) | ||

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 | |||

WP05 |
Risk functionals with convex level sets
(by Ruodu Wang, Yunran Wei) | ||

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 | |||

WP04 |
Convex risk functionals: representation and applications
(by Fangda Liu, Jun Cai, Christiane Lemieux, Ruodu Wang) | ||

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 | |||

WP03 |
Characterization, robustness and aggregation of signed Choquet integrals
(by Ruodu Wang, Yunran Wei, Gordon Willmot) | ||

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 | |||

WP02 |
A theory for measures of tail risk
(by Fangda Liu, Ruodu Wang) | ||

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 | |||

WP01 | Risk aversion in regulatory capital principles (by Tiantian Mao, Ruodu Wang) | ||

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 |