Risk Adequacy and Anti-Discrimination in Insurance Pricing: Classification, Aggregation, and Transfers

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.