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

2026-05-15

AI Summary

This study addresses a fundamental limitation of existing actuarial bootstrap methods, which violate the conditioning principle in predictive distribution inference, leading to coverage errors that do not vanish asymptotically with increasing sample size. The paper proposes the first model-agnostic bootstrap approach that strictly adheres to the conditioning principle: by fixing the observed loss triangle, it directly samples development factors from their predictive distribution, making it applicable to Chain-Ladder, Bornhuetter–Ferguson, and Cape Cod models. Built upon a Dirichlet–Gamma hierarchical structure, the method employs Beta-distributed sampling for allocation ratios and introduces a concentration parameter to diagnose non-stationarity. Theoretical and empirical results demonstrate that the proposed method reduces coverage error to $O(I^{-1/2})$ and achieves good calibration under compound Poisson data; furthermore, a concentration estimate $\hat{c} < 30$ effectively flags non-stationary patterns, resolving the structural biases and cross-portfolio calibration issues inherent in the ODP bootstrap arising from re-estimation and accident-year heterogeneity.

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.

Problem

• claims reserving
• conditioning principle
• bootstrap
• coverage error
• model-agnostic

Innovation

• model-agnostic bootstrap
• conditioning principle
• claims reserving
• Dirichlet-Gamma hierarchy
• coverage error