What Paid Triangles Identify about Reserve Uncertainty: Information Boundaries in Loss Reserving

Abstract

Paid triangles identify some components of reserve uncertainty while others enter through modelling assumptions. We distinguish population identification, recovery by free accident-year-level and two-stage second-moment fits, and precision at practical triangle sizes. In a compound frequency--severity hierarchy, the population law identifies the row dispersion governing non-diversifying reserve variance. With one free level per accident year, its profile likelihood attains its supremum only at homogeneity: the levels absorb realised row heterogeneity on every triangle, in amounts and in counts. Anchoring reopens this channel, but supplies only one observation per accident year. On a $10 \times 10$ triangle the anchored estimate remains uncertain by a factor of several. Within-row allocation dispersion survives conditioning on row totals and hence survives free levels. In the predictive experiments, the treatment of mean-structure estimation error separated the procedures by several times more than the choice among plausible variance laws. A law of the correct form improved coverage only when its dispersion came from a prior instead of the free-level fit. On an outcome-free Schedule~P panel, no amount-only procedure approaches nominal coverage. Strongly one-sided misses reveal a displacement of realised reserves relative to the fitted predictive centres that changing the amount-only variance law cannot repair.