Pareto-optimal reinsurance policies in the presence of individual risk constraints

Abstract

The notion of Pareto optimality is commonly employed to formulate decisions that reconcile the conflicting interests of multiple agents with possibly different risk preferences. In the context of a one-period reinsurance market comprising an insurer and a reinsurer, both of which perceive risk via distortion risk measures, also known as dual utilities, this article characterizes the set of Pareto-optimal reinsurance policies analytically and visualizes the insurer–reinsurer trade-off structure geometrically. The search of these policies is tackled by translating it mathematically into a functional minimization problem involving a weighted average of the insurer’s risk and the reinsurer’s risk. The resulting solutions not only cast light on the structure of the Pareto-optimal contracts, but also allow us to portray the resulting insurer–reinsurer Pareto frontier graphically. In addition to providing a pictorial manifestation of the compromise reached between the insurer and reinsurer, an enormous merit of developing the Pareto frontier is the considerable ease with which Pareto-optimal reinsurance policies can be constructed even in the presence of the insurer’s and reinsurer’s individual risk constraints. A strikingly simple graphical search of these constrained policies is performed in the special cases of Value-at-Risk and Tail Value-at-Risk.

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Notes

  1. 1.

    This condition, which is stronger than the usual \(g(0)=0\), is a necessary condition for the finiteness of the DRM of unbounded random variables, which are of particular relevance to reinsurance.

  2. 2.

    To ensure that the Lebesgue–Stieltjes integral with respect to \(g(S_{Y}(\cdot ))\) is well-defined, the left- or right-continuity of g is required. See the proof of Lemma 2.1 of Cheung and Lo (2017) about how general distortion functions (not necessarily left-continuous or right-continuous) can be dealt with.

  3. 3.

    TVaR is also known variously as Average Value-at-Risk (AVaR), Conditional Value-at-Risk (CVaR), and Expected Shortfall (ES), although there are subtle differences between these terms.

  4. 4.

    Note that \([F_{X}^{-1}(\beta ),F_{X}^{-1}(\alpha ))\) is the empty set when \(\alpha \le \beta \).

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Acknowledgements

This work was supported by a start-up fund provided by the College of Liberal Arts and Sciences, The University of Iowa, and a Centers of Actuarial Excellence (CAE) Research Grant (2018-2021) from the Society of Actuaries (SOA). Any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA. The authors are also grateful to a Stanley International Travel Award from International Programs, The University of Iowa, and the anonymous reviewers for their careful reading and insightful comments.

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Correspondence to Ambrose Lo.

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Lo, A., Tang, Z. Pareto-optimal reinsurance policies in the presence of individual risk constraints. Ann Oper Res 274, 395–423 (2019). https://doi.org/10.1007/s10479-018-2820-4

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Keywords

  • Distortion
  • 1-Lipschitz
  • Value-at-Risk
  • Pareto frontier
  • Multi-criteria optimization
  • Risk sharing