Additive Tree Latent Variable Models for Insurance Loss Prediction

Abstract

Accurate prediction of insurance losses is essential for premium pricing, reserve estimation, and risk management in the property and casualty insurance industry. Traditional actuarial models, such as generalized linear models, often rely on predefined distributional assumptions and struggle to capture complex nonlinear interactions among risk factors. Meanwhile, tree based ensemble methods have demonstrated strong predictive power but typically lack the ability to model latent heterogeneity in policyholder risk profiles. This paper proposes an Additive Tree Latent Variable (ATLV) model that integrates gradient boosted additive tree structures with a latent variable framework to improve insurance loss prediction. A Latent Risk Index (LRI) is developed to capture unobserved policyholder risk heterogeneity through variational inference. A Tree Ensemble Loss Predictor (TELP) is formulated to combine the latent representations with observed covariates in a gradient boosting framework. A Distributional Calibration Module (DCM) is introduced to ensure that predicted loss distributions are well calibrated under the Tweedie compound Poisson family. Empirical analysis is conducted on the French Motor Third Party Liability (MTPL) insurance dataset comprising 678,013 policies. The results demonstrate that the ATLV model significantly outperforms both traditional generalized linear models and standard gradient boosting approaches across multiple evaluation metrics. The ATLV model achieves a mean Deviance of 1.247, a Gini coefficient of 0.312, and a calibration error of 0.034, representing improvements of 8.6%, 14.3%, and 41.4% respectively over the best performing baseline. An ablation study further reveals that the LRI module contributes most significantly to overall performance, confirming the importance of modeling latent risk heterogeneity in insurance loss prediction. These findings provide both theoretical insights and practical tools for actuaries seeking to enhance loss prediction accuracy through the integration of machine learning and latent variable modeling.