Marked Cox Models for IBNR Claims Count: Continuous and Discretized Approaches with Dirichlet-Driven Reporting Delays

TL;DR

This paper introduces novel discrete and continuous Cox models based on hidden Markov models for improved IBNR claims count estimation, addressing challenges in likelihood maximization and delay modeling.

Contribution

It develops a Dirichlet-driven discrete-time Cox model for IBNR claims, enhancing delay variability modeling and estimation accuracy over existing methods.

Findings

Discrete models outperform continuous in delay and frequency joint modeling

Dirichlet-based model captures additional reporting delay variability

All models perform well, with discrete versions showing superior results

Abstract

Accurate loss reserving is crucial in Property and Casualty (P&C) insurance for financial stability, regulatory compliance, and effective risk management. We propose a novel micro-level Cox model based on hidden Markov models (HMMs). Initially formulated as a continuous-time model, it addresses the complexity of incorporating temporal dependencies and policyholder risk attributes. However, the continuous-time model faces significant challenges in maximizing the likelihood and fitting right-truncated reporting delays. To overcome these issues, we introduce two discrete-time versions: one incorporating unsystematic randomness in reporting delays through a Dirichlet distribution and one without. We provide the EM algorithm for parameter estimation for all three models and apply them to an auto-insurance dataset to estimate IBNR claim counts. Our results show that while all models perform well, the discrete-time versions demonstrate superior performance by jointly modeling delay and frequency, with the Dirichlet-based model capturing additional variability in reporting delays. This approach enhances the accuracy and reliability of IBNR reserving, offering a flexible framework adaptable to different levels of granularity within an insurance portfolio.