Risk sharing, allocation, and equilibria

The series of working papers contains recent advances on risk sharing problems and equilibria. Objectives, preferences, and constraints in most papers are modelled by risk measures, with a special focus on the industry standards such as the Value-at-Risk (VaR), the Expected Shortfall (ES, also known as TVaR, CVaR and AVaR) and the Range-Value-at-Risk (RVaR, WP02). In particular, the framework of quantile-based risk sharing is initiated in WP02, which is further explored in WP04, WP06 and WP08. In addition to risk allocation and financial market equilibria, insurance and reinsurance arrangements are also common forms of risk sharing in an incomplete market (WP03, WP05).

The inf-convolution is a common mathematical tool to study Pareto and competitive equilibria. Technical challenges often arise from the non-convexity of most of the interesting objectives (such as VaR, RVaR and other distortion risk measures), whereas inf-convolution and risk sharing are extensively studied in the literature with convexity.

Most papers provide theoretical results with the hope to offer insights for regulation and risk management practice. A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

We study risk sharing problems for which the optimal allocation is counter-monotonic. We call such allocations jackpot allocations and scapegoat allocations, and they appear in many natural contexts, such as gambling, lottery tickets, and technological innovation.

We systemically study the problem of sharing risk among agents with preferences modelled by distortion riskmetrics (WP10 of Axiomatic theory of risk measures). The setting of inter-quantile differences is particularly interesting, for which explicit optimal allocations are obtained (Theorem 2).

We show the perhaps surprising inequality (Theorem 1) that the weighted average of iid ultra heavy-tailed (i.e., infinite mean) Pareto losses is larger than a standalone loss in the sense of first-order stochastic dominance. The main result applies to super-Pareto distributions and negative dependence. Our finding has some implications for a risk exchange market, but they are omitted from the final version of the paper for brevity. The journal version is to appear in Operations Research (2024).

We study an axiomatic framework for anonymized risk sharing. Four natural axioms, actuarial fairness, risk fairness, risk anonymity, and operational anonymity, are proposed and discussed. We establish in Theorem 1 that the four axioms characterizes the conditional mean risk sharing rule, revealing the unique and prominent role of this popular risk sharing rule among all others in relevant applications of anonymized risk sharing. A characterization of the conditional expectation in Theorem 2 may be of independent interest.

This paper is the fourth in a series of papers on quantile-based risk sharing. We extend the setting of WP02 to general tail risk measures (coined in WP04 of Axiomatic theory of risk measures) based on results from WP07, with a few technical results distinguishing the left-VaR and the right-VaR. Two types of model uncertainty are studied in the setting of risk sharing, and useful formulas are obtained on worst-case values of risk measures under uncertainty modeled by the Wasserstein metric. The journal version is published in Mathematics of Operations Research (2022).

To the question reflected in the paper's title, an answer is generally negative even in an atomless probability space, but it becomes positive with minor assumptions. The paper contains many subtle examples. The journal version is published in Insurance: Mathematics and Economics (2020).

This paper is the third in a series of papers on quantile-based risk sharing. Via several technical results, the paper offers results on existence, uniqueness, and characterization of Pareto-optimal allocations considered in WP02. In addition, it contains some auxiliary results on VaR and ES optimization. The journal version is published in Insurance: Mathematics and Economics (2020).

This paper is a study of competitive equilibria in a comonotone market and their relation to equilibria in a complete financial market. Many of our results are in sharp contrast to the ones on complete markets in terms of existence, uniqueness, and closed-form solutions of the equilibria, and monotonicity of the pricing kernel. The journal version is to appear in Economic Theory (2020).

This paper is the second in a series of papers on quantile-based risk sharing, by generalizing WP02 to the setting of heterogeneous beliefs. Pareto-optimal and equilibrium allocations for ES agents are obtained in explicit formulas (Theorems 1-3), whereas they are inexplicit for VaR agents. The journal version is published in Mathematical Programming (2020).

In this paper is a treatment of Pareto optimality in the context of reinsurance arrangements. Main results are necessary and sufficient conditions for Pareto-optimality and its existence, which are based on classic results in game theory. The journal version is published in Insurance: Mathematics and Economics (2017).

This paper is the first in a series of papers on quantile-based risk sharing. The main focus is Pareto and Arrow-Debreu equilibria for VaR, ES and RVaR. Main results include a fundamental quantile inequality (Theorem 1) and explicit formulas for optimal allocation (Theorem 2) and for equilibria (Theorem 3). The journal version is published in Operations Research (2018).

Regulatory arbitrage is the amount of possible capital requirement reduction through splitting a financial risk into several fragments, and in this paper it is formulated via self-convolution of a risk measure. In particular, VaR has infinite regulatory arbitrage. The journal version is published in Quantitative Finance (2016).