arXiv subjects

Stephan Marais

Publications and source records attributed to Stephan Marais.

2 recordsLinked to original sources

arXiv2026

Supervising the Chain Ladder

The chain ladder's volume-weighted pattern minimises an explicit loss function, yet is rarely booked as such. Practitioners adjust the pattern and record the final adjusted ratios. This paper treats the chain ladder's pattern selection as a supervised-learning problem. Judgement on pattern adjustments becomes a framework of defined penalties and hyperparameters on the chain ladder's loss function, treated here as an objective function in machine learning. Data weights are generalised with a decay and a power parameter for recency and volume weighting. Benchmark shaping and smoothness enter through a reference penalty and Whittaker-Henderson smoothing. The assembled objective is strictly convex and minimised by a single linear system. Each hyperparameter becomes an interpretable adjustment in its own right, declarable by judgement and categorised as an experience or a prospective adjustment. Experience adjustments can be set more objectively by a proposed training loop and a reserve validation score on held-out calendar diagonals. Further hyperparameter-based adjustments are written as almost-everywhere differentiable penalties that re-time or reshape the pattern. A worked example carries one real Schedule P triangle through an incurred and then a paid training stage, demonstrating the workflow.

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arXiv2026

A Multiplicative Loss Function for Chain Ladder

Reserving models increasingly rely on loss-based estimation, where the loss function encodes the assumed error structure. Mack demonstrated this for the chain ladder, showing that the volume-weighted average estimator minimises a volume-weighted squared-error loss function that is additive in successive claim developments. This paper instead considers a multiplicative error structure and proposes the corresponding volume-weighted squared log-error loss function. Adapting Mack's distribution-free framework, we show that this loss function is minimised by the volume-weighted geometric average of the individual development ratios. This provides practitioners with an alternative estimator of development ratios for chain-ladder-based models, and a candidate loss function for machine-learning-based reserving models. We further show that the same estimator arises from two independent arguments: a stability requirement on successive ultimate loss projections, and maximum-likelihood estimation under a log-normal model. The estimator thus admits three complementary justifications: loss minimisation, reserve stability, and parametric likelihood. An out-of-sample study of 362 Schedule P company-line datasets supports the use of the proposed estimator in place of the volume-weighted average for the chain ladder, by showing that it improves predictive accuracy and reduces a slight over-prediction bias.

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