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 × 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.
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BibTeX
@misc{oirbeek2026what,
title = {{What Paid Triangles Identify about Reserve Uncertainty: Information Boundaries in Loss Reserving}},
author = {Robin Van Oirbeek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41087},
eprint = {2610.41087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41087},
}
RIS
TY - UNPB
TI - What Paid Triangles Identify about Reserve Uncertainty: Information Boundaries in Loss Reserving
AU - Robin Van Oirbeek
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41087
UR - https://arxiv.org/abs/2610.41087
AB - 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 × 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.
ER -