A Unified Micro-Model for Loss Reserves, IBNR and Unearned Premium Risk with Dependence, Inflation, and Discounting

This paper introduces a unified micro-level stochastic framework for the joint modeling of loss reserves (RBNS), incurred but not reported (IBNR) reserves, and unearned premium risk under dependence, inflation, and discounting. The proposed framework accommodates interactions between indemnities, expenses, reporting delays, and settlement delays, while allowing for flexible parametric dependence structures and dynamic financial adjustments. An Aggregate Trend Renewal Process (ATRP) is used as one possible implementation of the joint model for payments, expenses, and delays; however, the methodological contribution of the paper lies in the unified micro-level reserving architecture rather than in the ATRP itself. The framework produces forward-looking reserve and premium risk measures with direct applications to pricing, reserving, and capital management. We implement the framework using an aggregate trend renewal process at the individual claim level, which can be applied to the usual run-off triangle to obtain predictions for each accident-development year. Closed-form expressions for the first two raw and joint conditional moments of predicted payments are derived, together with approximations of their distribution functions. A detailed case study on medical malpractice insurance illustrates the practical relevance of the approach and its calibration on real-world data. We also investigate data heterogeneity, parameter uncertainty, distributional approximations, premium risk, UPR sensitivity to operational delays and inflation, and risk capital implications under alternative assumptions. The results highlight the advantages of unified micro-level modeling for dynamic liability and premium risk assessment in long-tailed lines of business.

💡 Research Summary

The paper proposes a unified micro‑level stochastic framework that simultaneously models three core insurance liabilities: reported‑but‑not‑settled (RBNS) loss reserves, incurred‑but‑not‑reported (IBNR) reserves, and unearned premium reserves (UPR). Unlike traditional macro‑level reserving that works on aggregated run‑off triangles, the authors work at the individual claim level, capturing the full life‑cycle of each claim—occurrence, reporting delay, and settlement delay—through a flexible counting‑process structure.

The implementation vehicle is the Aggregate Trend Renewal Process (ATRP), an extension of the Trend Renewal Process (TRP) originally developed for reliability engineering. In the ATRP, three independent TRPs generate (i) the occurrence times (T_k) of claims, (ii) the reporting delays (\xi_k) from occurrence to insurer notification, and (iii) the settlement delays (\zeta_k) from notification to payment of indemnity and expenses. Claim‑specific loss amounts (X_k) and expense amounts (Y_k) are modeled as non‑negative i.i.d. variables, while allowing for dependence between (X_k) and (Y_k) (and optionally across claims) via copulas or multivariate distributions. Inflation and discounting are incorporated by attaching to each claim a constant inflation force (\alpha_k) and a constant interest force (\beta_k); the resulting discount factor (\exp