Risk Sharing and Reinsurance (CREST)


This course will deal with the state-of-the-art in the theory of Pareto-optimal (re)insurance design under model uncertainty and/or non-Expected-Utility preferences, as well as provide an introduction to Pareto optimality in problems of peer-to-peer collaborative insurance. Specifically:

  •  We will start with some background material on the theory of decision-making under uncertainty, from the classical work on Expected-Utility Theory (EUT) of von Neumann and Morgenstern, Savage, and De Finetti, to the more recent work on ambiguity and probability distortions (Quiggin, Yaari, Schmeidler, Gilboa, Amarante, Maccheroni-Marinacci).
  • We will cover some required mathematical tools, such as non-additive measure theory, probability distortions, Choquet integration, the theory of equimeasurable rearrangements, as well as risk measures, their properties, and their representations. We will pay special attention to distortion risk measures and spectral risk measures.
  • We will then formally introduce a general model of the insurance market, following the work of Carlier and Dana [8], and study the existence and characterization of Pareto optima in this general setup. As a special case, we will consider the classical formulation of the optimal (re)insurance problem due to Arrow, in the framework of EUT, as well as more recent work extending Arrow's setting to situations of belief heterogeneity between the two agents.
  • We will then proceed to formulating several problems that extend the insurance model above to more general setting with non-EUT preferences, distortion risk measures, or situations of model uncertainty.
  •  Finally, we will go over a brief introduction to conditional mean risk sharing and peer-to-peer collaborative insurance.


  1. Introduction to Choice under Uncertainty 
    General overview of decision theory
    Models of decision-making under uncertainty
  2. Pareto-Optimal Insurance Contracts under EUT 
    Some mathematical tools 
    General problem formulation
    Pareto optimality
    The special case of Arrow's setting
    Pareto optima and comonotonicity
    Pareto optima under belief heterogeneity
  3. Non-Additive Measures and Choquet Integration 
    Capacities and their properties
    The Choquet integral and its properties
    Mathematics of the RDEU model
    Distortion risk measures and spectral risk measures
  4. Pareto-Optimal Insurance Contracts under non-EUT Models 
    Problem formulation
    The case of RDEU
    The case of Choquet EU
    The case of MEU
    A note on belief heterogeneity under non-EUT
  5. Pareto-Optimality in Reinsurance Markets
    General problem formulation
    Pareto optimality with translation invariant preferences
    Distortion risk measures and the marginal indemnification approach
  6. Peer-to-Peer Risk Sharing Mechanisms: An Introduction 
    General problem formulation
    Pareto-optimality, convex ordering, and conditional mean risk sharing


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[2] V.A. Asimit and T.J. Boonen. Insurance with Multiple Insurers: a Game-Theoretic Approach. European Journal of Operational Research, 267(2):778{790, 2018.
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[4] C. Bernard, X. He, J.A. Yan, and X.Y. Zhou. Oprimal Insurance Design under Rank-Dependent Expected Utility. Mathematical Finance, 25(1):154{186, 2015.
[5] C. Birghila, M. Ghossoub, and T. Boonen. Optimal Insurance under Maxmin Expected Utility. Working Paper, 2020.
[6] T.J. Boonen and M. Ghossoub. On the Existence of a Representative Reinsurer under Heterogeneous Beliefs. Insurance Mathematics and Economics, 88:209{225, 2019.
[7] T.J. Boonen and M. Ghossoub. Optimal Reinsurance with Multiple Reinsurers: Distortion Risk Measures, Distortion Premium Principles, and Heterogeneous Beliefs. Insurance Mathematics and Economics, 2021. Forthcoming.
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