Self-protection and insurance demand with convex premium principles

Paper: arXiv 2411.19436

Authors: Qiqi Li, Wei Wang, Yiying Zhang

Abstract

In economic analysis, rational decision-makers often take actions to reduce their risk exposure. These actions include purchasing market insurance and implementing prevention measures to modify the shape of the loss distribution. Under the assumption that the insureds’ actions are fully observed by the insurer, this paper investigates the interaction between self-protection and insurance demand when insurance premiums are determined by convex premium principles within the framework of distortion risk measures. Specifically, the insured selects an optimal proportional insurance share and prevention effort to minimize the risk measure of their end-of-period exposure. We explicitly characterize the optimal combination of prevention effort and insurance demand in a self-protection model when the insured adopts tail value-at-risk or a subclass with strictly concave distortion functions. Additionally, we conduct comparative static analyses to illustrate our main findings under various premium structures, risk aversion levels, and loss distributions. Our results indicate that market insurance and self-protection are complementary, supporting classical insights from the literature regarding corner insurance policies (i.e., null and full insurance) in the absence of ex ante moral hazard. Finally, we consider the effects of moral hazard on the interaction between self-protection and insurance demand. Our findings show that ex ante moral hazard shifts the complementary effect into substitution effect.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of self-protection and optimal insurance demand using distortion risk measures and convex premium principles. It is heavily mathematical with formal derivations and proofs, but lacks empirical backtesting or real-world dataset validation, relying primarily on theoretical and numerical analyses.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze interaction between self-protection effort & insurance demand under convex premium principles] --> B[Model Setup: Define end-of-period risk exposure using Distortion Risk Measures & TVaR]
    B --> C[Optimization Process: Jointly minimize risk measure for optimal prevention effort & proportional insurance share]
    C --> D{Evaluate Moral Hazard Conditions}
    D -- Observable Effort No Moral Hazard --> E[Analytical Characterization: Derivation of optimal coverage & comparative statics]
    D -- Unobservable Effort Ex Ante Moral Hazard --> F[Moral Hazard Analysis: Re-evaluate optimal choices under asymmetric information]
    E --> G[Finding 1: Insurance and self-protection act as Complements under full observability]
    F --> H[Finding 2: Ex ante moral hazard shifts interaction from Complements to Substitutes]

Cited References (76)

1. Comparison of risks based on the expected proportional shortfall (2012)
2. Optimal insurance contracts for a shot-noise Cox claim process and persistent insured’s actions (2023)
3. An extension of Alexandrov’s theorem on second derivatives of convex functions (2011)
4. A systematic review of strategies used for controlling consumer moral hazard in health systems (2022)
5. The impact of prudence on optimal prevention revisited (2011)
6. The dual theory of choice under risk (1987)
7. Estimating the effects of wind loss mitigation on home value (2023)
8. Does regulation of built-in security reduce crime? Evidence from a natural experiment (2011)

2 of 76 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.