Paper: arXiv 2610.32614

Authors: Wiebke Hansen, Stefan Weber, Mario Wüthrich

Abstract

Insurance tariffs must reconcile risk-based pricing, portfolio coverage, and anti-discrimination requirements. We introduce a framework in which tariffs are constructed through classification, aggregation of benchmark risk prices within classes, and balanced transfers, and are evaluated by several distinct criteria. This yields a multi-objective tariff-design problem whose attainable objective set and Pareto frontier describe the resulting trade-offs. Risk adequacy is measured by the discrepancy between charged premiums and individual benchmark risk prices. Portfolio coverage is assessed through the capital required to make aggregate claims net of premium income acceptable. Anti-discrimination is examined through differences in protected-group composition across tariff classes and through premium effects experienced by protected groups. We analyze the trade-offs among these criteria and prove mathematical results characterizing their relationships: (i) the inferential threshold for perfect risk adequacy with finitely many benchmark-price levels; (ii) lower bounds on risk inadequacy under stricter inferential constraints; (iii) exact changes in risk inadequacy and inferential disparity when classes are merged, and dynamic programming for scalarized optimization over consecutive benchmark-price classes; (iv) attainable protected-group total premium effects and their minimum risk inadequacy costs under balanced transfers; (v) a sharp lower bound linking inferential disparity to the risk inadequacy required under a bound on protected-group premium-effect dispersion; and (vi) conditions for portfolio-coverage improvement and its additional risk inadequacy cost at fixed protected-group total premium effects. A numerical motor-insurance study illustrates the results.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly mathematical framework for a complex multi-objective problem in insurance pricing, demonstrating strong theoretical contributions. While a numerical study is mentioned, the emphasis is on mathematical proofs and characterizations, suggesting a solid but not exhaustive empirical component. The novelty lies in the integrated framework and the specific mathematical results derived for the trade-offs.

Research Flowchart

  flowchart TD
    A[Research Goal: Reconcile Risk-Based Pricing, Portfolio Coverage, and Anti-Discrimination in Insurance Tariffs] --> B{Key Methodology: Framework for Tariff Construction and Evaluation};
    B --> C[Methodology Details: Classification, Aggregation of Benchmark Risk Prices, Balanced Transfers];
    C --> D[Evaluation Criteria: Risk Adequacy, Portfolio Coverage, Anti-Discrimination (Protected Group Composition & Premium Effects)];
    D --> E[Computational Process: Multi-objective Tariff-Design Problem, Attainable Objective Set & Pareto Frontier Analysis];
    E --> F[Data Input: Numerical Motor Insurance Study];
    F --> G[Key Findings/Outcomes: Mathematical Results on Trade-offs & Characterizations of Relationships Between Criteria];