Our article, Recoverability of market-wide fair insurance premiums under selection bias, with Marie-Pier Côté and Olivier Côté, was published in Insurance: Mathematics & Economics.
Fairness adjustments in insurance pricing are defined relative to a reference population, i.e., to a joint distribution of (X, D, Y) where X are rating factors, D protected attributes, and Y is claim amount. Because an insurer’s portfolio is generally a selected subpopulation, portfolio and population reference distributions typically differ, so portfolio-calibrated and population-calibrated fairness adjustments need not coincide. In what follows, we use selection bias as an umbrella term for any discrepancy between an observed sample and its target population. We call portfolio composition bias the insurance-specific form of selection bias induced by the portfolio inclusion mechanism (underwriting/marketing), which makes each insurer’s portfolio a selected subpopulation. Relying on causal inference and a portfolio composition indicator, we characterize how portfolio composition bias affects common premium adjustments (unawareness, discrimination-free pricing, and transport-based corrective pricing), and we provide restrictive conditions under which portfolio and population adjustments coincide. We propose estimators to recover the fairness-adjusted premiums on the regulator-intended target population from selection-biased data, by using externally available information on the population marginal distribution of the prohibited attribute D. We study this scope mismatch from the policyholder’s perspective: we model the market premium faced by a newly entering policyholder (not yet assigned to any portfolio) as a mixture of insurer-specific premiums, weighted by the probability of being assigned to each insurer. Under this view, a pricing rule can satisfy a fairness criterion within each insurer’s portfolio yet produce direct or proxy discrimination in the market when portfolio inclusion depends on X and/or D. Finally, we show that enforcing portfolio-level balance on population-intended fair premiums can reintroduce portfolio composition bias, highlighting a regulatory trade-off between portfolio balancing and market-wide fairness. We focus on recoverability: which population-level fairness targets are identifiable from portfolio data, and what minimal external information is required to recover them.
Depuis le début de la semaine, après avoir déposé les enfants au camp de jour du Musée des Beaux Arts, je viens à pieds à l’université. Chaque fois, je me dis que je pourrais prendre le bus, mais comme aucun ne vient, je commence en marchant, en me disant que si un bus passe, je le prendrais. Et tous les matins, j’arrive au bureau sans m’être fait dépassé par le moindre bus. Et bien sur, j’en ai croisé un paquet qui passaient en sens inverse…
à parcourir, on suppose que les bus sont espacés d’une distance 
, et qu’ils avancent à une vitesse 
. Autrement dit, les bus passent tous les 
secondes (si ma vitesse est exprimée en secondes).
avec 
(oui, on va supposer que je vais moins vite que le bus… ce qui n’est pas forcément une hypothèse faible aux heures de pointes, mais disons que le problème n’a de sens que si aller en bus me permet d’aller plus vite). Le temps que je vais mettre si je fais tous le trajet à pied est 
. Maintenant, comptons les bus qui passent en face. Je vais croiser tous ceux qui sont déjà sur ma portion de trajet, et il y en a 
. En plus, je croiserais tous ceux qui vont arriver à l’université, et qui n’y sont pas encore, soit 
, i.e. le temps qu’il me reste à marcher divisé par le temps qui s’écoule entre deux bus. On a alors un total de 
bus à croiser, en face.
