链梯法的一种乘法损失函数

中文摘要 AI 辅助

准备金模型越来越依赖于基于损失的估计，其中损失函数编码了假定的误差结构。Mack 证明了这一点对于链梯法成立，表明体积加权平均估计量最小化了一个在连续索赔发展中可加的体积加权平方误差损失函数。本文转而考虑乘法误差结构，并提出了相应的体积加权平方对数误差损失函数。通过调整 Mack 的无分布框架，我们证明该损失函数由个体发展比率的体积加权几何平均所最小化。这为从业者提供了基于链梯法的模型中发展比率的一种替代估计量，以及用于基于机器学习的准备金模型的候选损失函数。我们进一步表明，相同的估计量源于两个独立的论证：对连续最终损失预测的稳定性要求，以及对数正态模型下的最大似然估计。因此，该估计量具有三个互补的合理性：损失最小化、准备金稳定性和参数似然性。一项对 362 个 Schedule P 公司-业务线数据集的样本外研究支持在链梯法中使用所提出的估计量替代体积加权平均，表明它提高了预测准确性并减少了轻微的过度预测偏差。

英文摘要

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.

发表机构

• Dynamo Analytics

机构由 AI 辅助整理，请以论文原文为准。