Risk Sharing with Distortion Risk Measures Beyond Risk Aversion

Abstract

This thesis studies optimal risk sharing among multiple agents whose preferences are represented by distortion risk measures, equivalently Yaari dual utilities. The central question is how to characterize Pareto-optimal risk allocations when agents may have heterogeneous risk preferences, including risk-averse, risk-seeking, and behavioral attitudes toward risk. Particular attention is paid to the dependence structure of optimal allocations and to the geometry of the Pareto frontier.

The first part of the thesis develops counter-monotonic risk sharing as a counterpart to the classical comonotonic theory. Comonotonicity represents positive dependence and is fundamental in risk sharing among risk-averse agents, while counter-monotonicity represents an extreme form of negative dependence and arises naturally for risk-seeking agents. Chapters 2 and 3 analyze this counter-monotonic structure for distortion risk measures, moving from a homogeneous setting with a common distortion function to a heterogeneous setting with different distortion functions. Inf-convolution is an important tool for studying Pareto optimality. Using this tool, Chapters 2 and 3 compare the usual formulation with variants that restrict allocations to be comonotonic or counter-monotonic, derive explicit formulas for risk-seeking agents, and illustrate the formulas through a portfolio manager’s problem.

Chapter 4 studies markets with mixed risk attitudes, where risk-averse and risk-seeking agents coexist. This setting is more challenging because neither the usual comonotonic arguments for risk-averse agents nor the counter-monotonic arguments for risk-seeking agents apply directly to the whole market. The chapter establishes a reduction theorem showing that the general multi-agent problem can be reduced to a two-agent problem between representative risk-averse and risk-seeking agents. Based on this reduction, the chapter further studies the existence of optimal allocations, identifies cases in which the inf-convolution is unbounded, and derives explicit solutions for piecewise linear distortion functions and Bernoulli-type aggregate risks.

Chapter 5 studies Pareto optimality beyond universal risk aversion, with emphasis on constrained allocation problems. Feasibility constraints, such as nonnegative allocations, are natural in insurance and reinsurance, but they change the geometry of the risk-sharing problem and may limit the applicability of weighted-sum methods. The chapter reduces the two-agent Pareto problem to a one-parameter family of constrained optimization problems. For Bernoulli aggregate risks, the Pareto frontier admits a convex-envelope characterization and can be attained by three-atom allocations. For two-point aggregate risks, finite-atom structural results are developed, showing that efficient allocations can be represented by a bounded number of payment levels. These results provide both theoretical insight into non-risk-averse risk sharing and a tractable framework for numerical computation.