Paper: arXiv 2408.09799

Authors: Tim J. Boonen, Yuyu Chen, Xia Han, Qiuqi Wang

Abstract

This paper explores optimal insurance solutions based on the Lambda-Value-at-Risk ($Λ\VaR$). If the expected value premium principle is used, our findings confirm that, similar to the VaR model, a truncated stop-loss indemnity is optimal in the $Λ\VaR$ model. We further provide a closed-form expression of the deductible parameter under certain conditions. Moreover, we study the use of a $Λ’\VaR$ as premium principle as well, and show that full or no insurance is optimal. Dual stop-loss is shown to be optimal if we use a $Λ’\VaR$ only to determine the risk-loading in the premium principle. Moreover, we study the impact of model uncertainty, considering situations where the loss distribution is unknown but falls within a defined uncertainty set. Our findings indicate that a truncated stop-loss indemnity is optimal when the uncertainty set is based on a likelihood ratio. However, when uncertainty arises from the first two moments of the loss variable, we provide the closed-form optimal deductible in a stop-loss indemnity.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of optimal insurance design using Lambda-Value-at-Risk (Lambda-VaR) across various premium principles and model uncertainty scenarios. It is purely theoretical, featuring advanced mathematical proofs and closed-form expressions, but lacks real-world data application and backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimal Insurance Design with Lambda-VaR (ΛVaR)"] --> B["Define Inputs & Constraints: Loss Distributions & Model Uncertainty"]
    
    B --> C["Analyze Premium Principles & Risk Loadings"]
    
    C --> D1["Expected Value Premium Principle"]
    C --> D2["Λ'VaR Premium Principle"]
    C --> D3["Model Uncertainty Sets (Likelihood Ratio vs. First Two Moments)"]
    
    D1 --> E1["Outcome: Truncated Stop-Loss Indemnity Optimal (Closed-form deductible)"]
    D2 --> E2["Outcome: Full/No Insurance OR Dual Stop-Loss Optimal"]
    D3 --> E3["Outcome: Truncated Stop-Loss (Likelihood) OR Closed-Form Deductible (Moments)"]