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

This article introduces a unified microlevel stochastic framework for the joint modeling of reported but not settled (RBNS) reserves, 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 article’s methodological contribution lies in the unified microlevel reserving architecture rather than in the ATRP itself. The framework produces forward-looking reserve and premium risk measures that support reserving analysis, uncertainty assessment, and scenario testing. 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 approach’s calibration and practical relevance. We investigate data heterogeneity, distributional approximations, model validation against aggregate and microlevel benchmarks, and calibrated IBNR and unearned premium reserves (UPR) projections. Additional sensitivity analyses, including parameter uncertainty, unequal financial assumptions, and alternative trend specifications, are reported in Supplementary Appendices B, F, and G. The results highlight the value of unified microlevel modeling for dynamic liability and premium risk assessment in long-tailed lines of business.