{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16561",
  "author_total": 2,
  "authors": [
    {
      "name": "James Grove"
    },
    {
      "name": "Stephan Marais"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frjjpez3a2e0w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16561",
  "fulltext_url": null,
  "id": "frjjpez3a2e0w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16561",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "1803|Probability and Risk Models"
    ]
  ],
  "openalex": "W7213392083",
  "pdf_url": "https://arxiv.org/pdf/2609.16561",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frjjpez3a2e0w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:51Z",
      "native_id": "2609.16561",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16561"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "A Multiplicative Loss Function for Chain Ladder",
  "title_text": "A Multiplicative Loss Function for Chain Ladder",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/frjjpez3a2e0w",
  "venue": "arXiv",
  "versions": []
}