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

Wiebke Hansen; Stefan Weber; Mario Wüthrich

Home/Archive/arXiv

arXiv · 2610.32614

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

Wiebke Hansen·Stefan Weber·Mario Wüthrich

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

q-fin.RM q-fin.MF

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Wiebke Hansen, Stefan Weber, Mario Wüthrich. 2026-10-23. Risk Adequacy and Anti-Discrimination in Insurance Pricing: Classification, Aggregation, and Transfers. https://arxiv.org/abs/2610.32614

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Distributions or Quantiles? A Convex Variational Theory of Worst-Case Risk

Worst-case risk evaluation is central to finance and operations research, yet existing analytical results remain fragmented and largely confined to ambiguity sets with specific structures. Using spectral risk measures as the underlying class, we develop a convex variational theory that unifies these analyses and extends them to richer ambiguity models. The theory rests on a representation principle: use the quantile or distribution formulation that preserves convexity of the full ambiguity set. In both representations, the resulting optimality conditions provide a systematic route to identify the worst-case law and determine its risk value. This approach recovers classical central-moment bounds and yields closed-form or semi-closed-form solutions for practically motivated models involving Wasserstein ambiguity with tail stress, two-sided expected shortfall (ES) bands, asymmetric partial moments, geometric-moment bounds, and exponential- or Gaussian-reference Kullback--Leibler (KL) neighborhoods. Beyond these analytical results, the same theory yields projection-based algorithms with fixed-grid convergence under the stated assumptions. In a high-dimensional portfolio application, we establish an exact projection identity for mean-anchored joint KL ambiguity under a Gaussian reference. The resulting standardized worst-case ES coefficient applies to every portfolio, reducing robust portfolio selection to an exact second-order cone program after a single calibration.

q-fin.RM↗

arXiv2026

General bounds on functionals of the lifetime under life table constraints in a joint actuarial-financial framework

In life insurance, life tables provide survival probabilities at integer ages but leave the distribution of deaths within each year unspecified. This incompleteness is particularly relevant for modern insurance products such as variable annuities, whose payoffs depend jointly on lifetime uncertainty and financial market performance. The valuation of such contracts must therefore be carried out in a joint actuarial-financial framework, as their values depend not only on the full information about mortality rates but also on the interaction between mortality risk, asset dynamics and embedded guarantees. Standard approaches impose fractional-age assumptions or mortality rate models, but the resulting valuations strongly depend on these assumptions, which makes it difficult to generalize them. We hence derive upper and lower bounds of hybrid functionals of the lifetime with respect to mortality rates, which are compatible with the observed life table at integer ages and the given financial market. We derive two sets of results under distinct assumptions. In the first, we assume that each mortality trajectory is almost surely consistent with all the tabulated one-year survival probabilities. In the second, we consider a relaxed formulation that allows for deviations of the mortality rates while still being consistent in expectation with the given reference life table. These distinct yet complementary formulations characterize the worst- and best-case contract values over all mortality processes that remain compatible with the observed life-table information and the financial market. The resulting bounds provide insurers with a practical and robust risk-management tool for assessing the impact of adverse mortality dynamics on variable annuities, quantifying the uncertainty arising from incomplete life-table information, and incorporating mortality model risk into pricing decisions.

q-fin.RM↗

arXiv2026

Coherent risk from finite models

Coherent estimators at different sample sizes need not describe the same population risk. We prove that dyadic estimators do so exactly when duplicating every scenario leaves the estimate unchanged, and recover the population dual set from the finite models. The risk of a Bernoulli loss determines the exact Wasserstein, Kolmogorov and total-variation moduli on a common bounded support. It also characterises uniform estimability, with an impossibility bound for arbitrary randomised estimators. An intrinsic floor on tail levels gives an explicit optimal Lipschitz constant and a uniform approximation defect; every deterministic vanishing floor preserves pointwise strong consistency. A nonstandard construction supplies compatible estimators at all finite sample sizes, explicit block-sum weights, and a formula for the worst-case mass created by infinitesimal tail levels. Finally, we derive uniform regret bounds for data-driven risk-averse optimisation and specialise them to stop-loss retention. Exact binomial calculations compare regularised and original estimators against the same population target.

q-fin.RM↗