Hansen, Weber and Wüthrich model insurance pricing trade-offs under anti-discrimination rules

In context: this report covers a source published on 23 Oct 2026.

Insurers estimate a risk price for each type of policyholder, but they charge premiums by class: they group customers, average their prices and sometimes shift money between classes. A preprint posted on arXiv on 23 October 2026 by Wiebke Hansen, Stefan Weber and Mario Wüthrich, of Leibniz Universität Hannover and ETH Zürich, asks how far such tariffs can meet anti-discrimination requirements without moving premiums away from each customer's risk. It has not been peer reviewed.

The authors build a framework that judges a tariff on three tests: how close premiums stay to individual risk prices, how much capital the portfolio needs, and how premiums affect protected groups. The last is measured both by how unevenly the groups are spread across classes and by the average premium change each group sees. They prove results on the trade-offs and illustrate them with a stylised, synthetic motor example: eight risk-price levels and three equally sized protected groups with different risk profiles, although the risk price itself ignores group membership. The authors say the model illustrates the mathematics and is not calibrated to data.

The central result is a threshold. Pricing every policyholder exactly at their risk price fixes a minimum level of group imbalance across classes; any rule that demands less imbalance forces premiums away from risk, at a cost the authors bound from below. The threshold is zero only when risk and group membership are independent. In the example, merging eight classes into two cut the imbalance measure from 0.116 to 0.067 but raised the risk-inadequacy measure from zero to about 19. Balanced transfers could reach what the authors call exact compensation for the groups with four or three classes, at extra costs of 0.23 and 1.47, but not with two.

The authors also show that a single score for group imbalance does not settle the premium effects on groups. Tariffs with the same score can need very different adjustments, and every group average can be neutral while individual premiums still stray from risk, in their words 'risk inadequacy remains positive'.

The paper does not test real tariffs or claims data, and its numbers come from a toy model, not estimates of what fairness rules cost insurers. Its main analysis assumes the insurer knows each policyholder's true risk type; the authors call this 'a theoretical benchmark rather than an implementable tariff' and leave pricing on observable data, estimation error, robustness and practical algorithms to future work. No machine learning is used.