A Model-Agnostic Bootstrap for Macro-Level Claims Reserving Under the Conditioning Principle

TL;DR

This paper introduces a model-agnostic bootstrap method for claims reserving that adheres to the conditioning principle, ensuring accurate coverage and calibration across various models and data-generating processes.

Contribution

It proposes a new bootstrap approach based on the Dirichlet-Gamma hierarchy that exactly satisfies the conditioning principle and is calibration-robust and model-agnostic.

Findings

The bootstrap achieves $O(I^{-1/2})$ coverage error, independent of development periods.

Under compound Poisson processes, the bootstrap is conservative, overestimating true variability.

The method explains heterogeneity in calibration across portfolios and links classical estimators to credibility models.

Abstract

The correct inferential object in claims reserving is the conditional predictive distribution $p(R \mid \mathcal{D}, \hat\theta)$, where $\mathcal{D}$ is the observed triangle held fixed. We refer to this as the conditioning principle. All existing bootstraps violate it by resampling functions of $\mathcal{D}$ inside the predictive loop, producing an $O(1)$ coverage error that does not vanish as the triangle grows. The Dirichlet-Gamma hierarchy admits a bootstrap that satisfies the principle exactly: $S^{IBNP}_i = X^{obs}_i (1-W_i)/W_i$ with $W_i \sim \mathrm{Beta}(c\hat{F}_{I-i}, c(1-\hat{F}_{I-i}))$ sampled directly from its predictive distribution. Only the allocation proportion $W_i$ is simulated; the observed triangle is held fixed. It thus inherits calibration from any development-proportion method (Chain-Ladder, Bornhuetter-Ferguson, Cape Cod, or other), making it model-agnostic. The coverage deficit is $O(I^{-1/2})$, independent of the number of development periods. Under compound Poisson data-generating processes the bootstrap is conservative for every $F_{I-i} \in (0,1)$: the predictive standard deviation analytically exceeds the true value by the factor $1/\sqrt{F_{I-i}}$. The ODP bootstrap violates the principle through two mechanisms in opposite directions: re-estimation inflates bootstrap variance under the ODP DGP, while missing accident-year frailty deflates it under frailty DGPs. The resulting coverage discrepancy is $\Omega(1)$ regardless of $I$, providing a structural explanation for the cross-portfolio miscalibration heterogeneity documented by Meyers (2015). Chain-Ladder, Bornhuetter-Ferguson and Cape Cod emerge as credibility estimators under diffuse, informative and pooling priors respectively, with identical structure for counts and amounts. The concentration $c$ serves as a diagnostic: $\hat{c} < 30$ signals non-stationary development.