Conditional Prediction for Macro-Level Claims Reserving: What a Single Paid Triangle Identifies

Abstract

Macro-level reserve uncertainty is assessed from a single realised run-off triangle. The target is therefore the conditional predictive distribution $p(R\mid D)$ of the outstanding amount given that triangle. We state the requirements for targeting it and supply, in a Dirichlet-Gamma hierarchy, a closed-form conditional predictive that attaches to any method producing a development pattern (Chain-Ladder, Bornhuetter-Ferguson, Cape Cod), the latter two previously lacking a conditional predictive bootstrap. Three results bound what the construction can deliver. First, the concentration $c$ governing allocation uncertainty is estimated from partial-column proportions that form a Dirichlet subcomposition, carrying concentration $c{F}_k$ rather than $c$; the corrected estimator is consistent but slow ($E[\hat{c}]/c=1.31$ at $I=10$), and substituting it costs 6.4 coverage points at that size, of which integrating over its posterior recovers about two. Second, the closed form is exact only under diffuse anchoring, so an oracle supplied with the true parameters remains conditionally miscalibrated by roughly ten points at the ultimate dispersion typical of macro-level portfolios. Third, contrary to the intuition that motivated this work, a residual bootstrap with free accident-year effects does not omit accident-year frailty uncertainty: its resampling of the diagonal reproduces the diffuse-limit posterior of the row level, and under the count hierarchy its coverage is nominal at every frailty variance tested. The under-coverage that Meyers (2015) documents on paid data is therefore not explained by omitted frailty, and the residual bootstrap is a correctly targeted benchmark at practical triangle sizes. The paper's contribution is the accounting: which parameters a single paid triangle determines, what each substitution costs, and where the closed form itself gives way.

57 pages. v3: substantially revised and retitled. Withdrawn: the claim that residual bootstraps miscalibrate through re-estimation inflation and omitted accident-year frailty, and the explanation of the Meyers (2015) under-coverage built on it. Both are components of the conditional predictive variance; in a count-hierarchy simulation ODP coverage stays nominal across frailty variances