Risk Sharing and Reinsurance (CREST)

Objective

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.

Planning

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

References

[1] K.J. Arrow. Essays in the Theory of Risk-Bearing. Chicago: Markham Publishing Company, 1971.
[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.
[3] H. Assa. On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums. Insurance: Mathematics and Economics, 61:70{75, 2015.
[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.
[8] G. Carlier and R.A. Dana. Pareto Ecient Insurance Contracts when the Insurer's Cost Function is Discontinuous. Economic Theory, 21(4):871{893, 2003.
[9] G. Carlier and R.A. Dana. Rearrangement Inequalities in Non-convex Insurance Models.Journal of Mathematical Economics, 41(4-5):483{503, 2005.
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[19] M. Ghossoub. Budget-Constrained Optimal Insurance With Belief Heterogeneity. Insurance Mathematics and Economics, 89:79{91, 2019.
[20] M. Ghossoub. Optimal Insurance under Rank-Dependent Expected Utility. Insurance Mathematics and Economics, 87:51{66, 2019.
[21] I. Gilboa. Theory of Decision under Uncertainty. Cambridge University Press, 2009.
[22] I. Gilboa and M. Marinacci. Ambiguity and the Bayesian Paradigm. In D. Acemoglu, M. Arellano, and E. Dekel (eds.), Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press, 2013.
[23] M. Marinacci and L. Montrucchio. Introduction to the Mathematics of Ambiguity. In I. Gilboa (ed.), Uncertainty in Economic Theory: Essays in Honor of David Schmeidler's 65th Birthday, pages 46{107. Routledge, London, 2004.
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[25] Z.Q. Xu, X.Y. Zhou, and S.C. Zhuang. Optimal Insurance under Rank-Dependent Utility and Incentive Compatibility. Mathematical Finance, 21(4):503{34, 2018.4