Paper: arXiv 2312.01813

Authors: Tim J. Boonen, Xia Han

Abstract

This paper studies an optimal insurance contracting problem in which the preferences of the decision maker given by the sum of the expected loss and a convex, increasing function of a deviation measure. As for the deviation measure, our focus is on convex signed Choquet integrals (such as the Gini coefficient and a convex distortion risk measure minus the expected value) and on the standard deviation. We find that if the expected value premium principle is used, then stop-loss indemnities are optimal, and we provide a precise characterization of the corresponding deductible. Moreover, if the premium principle is based on Value-at-Risk or Expected Shortfall, then a particular layer-type indemnity is optimal, in which there is coverage for small losses up to a limit, and additionally for losses beyond another deductible. The structure of these optimal indemnities remains unchanged if there is a limit on the insurance premium budget. If the unconstrained solution is not feasible, then the deductible is increased to make the budget constraint binding. We provide several examples of these results based on the Gini coefficient and the standard deviation.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of optimal insurance contracting using non-linear deviation measures such as Choquet integrals and standard deviation. While purely theoretical with stylized numerical examples rather than empirical backtests on real historical insurance claims data, the analytical proofs and characterizations of optimal indemnity structures under premium constraints are highly thorough.

Research Flowchart

  flowchart TD
    A[Goal: Find optimal insurance contracts using mean-deviation measures] --> B[Model DM Preferences: Expected Loss + Convex Deviation Measure]
    B --> C[Inputs: Choquet Integrals, Standard Deviation, & Premium Principles]
    C --> D[Solve Optimization Problem under Budget Constraints]
    D --> E[Expected Value Principle: Stop-Loss Indemnity Optimal]
    D --> F[VaR / Expected Shortfall Principle: Layer-Type Indemnity Optimal]
    E --> G[Outcomes: Precise deductible characterizations & budget impact analysis]
    F --> G

Cited References (38)

  1. Monotonic mean–deviation risk measures — → 2312.01034
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  3. Optimal Insurance to Maximize RDEU Under a Distortion-Deviation Premium Principle (2021) — link
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  6. Optimal insurance design under narrow framing (2020) — link
  7. DISTORTION RISKMETRICS ON GENERAL SPACES (2020) — link
  8. Characterization, Robustness, and Aggregation of Signed Choquet Integrals (2020) — link
and 30 more
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  3. Optimal insurance under rank-dependent expected utility (2019) — link
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  5. A NEYMAN-PEARSON PERSPECTIVE ON OPTIMAL REINSURANCE WITH CONSTRAINTS (2017) — link
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  7. Optimal reinsurance with expectile (2016) — link
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  30. (ii) For t > 1 − p , or equivalently, x 6 x p , we have H ( x ) = 1 − S X ( x ) − λh 1 ( S X ( x ))

1 of 38 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.