 2011 Royal Statistical Society 0964–1998/12/175637 J. R. Statist. Soc. A (2012) 175, Part 2, pp. 637–656 A Bayesian non-linear model for forecasting insurance loss payments Yanwei Zhang, CNA Insurance Company, Chicago, USA Vanja Dukic University of Colorado—Boulder, USA and James Guszcza University of Wisconsin—Madison, USA [Received August 2010. Revised July 2011] Summary. We propose a Bayesian non-linear hierarchical model that addresses some of the major challenges that non-life insurance companies face when forecasting the outstanding claim amounts for which they will ultimately be liable.This approach is distinctive in several ways. First, data from individual companies are treated as repeated measurements of various cohorts of claims, thus respecting the correlation between successive observations. Second, non-linear growth curves are used to model the loss development process in a way that is intuitively appeal- ing and facilitates prediction and extrapolation beyond the range of the available data.Third, a hierarchical structure is employed to reflect the natural variation of major parameters between the claim cohorts, accounting for their heterogeneity. This approach enables us to carry out inference at the level of industry, company and/or accident year, based on the full posterior distribution of all quantities of interest. In addition, prior experience and expert opinion can be incorporated in the analyses through judgementally selected prior probability distributions. The ability of the Bayesian framework to carry out simultaneous inference based on the joint posterior is of great importance for insurance solvency monitoring and industry decision making. Keywords: Bayesian estimation; Generalized linear model; Hierarchical model; Insurance loss reserving; Longitudinal data; Non-linear growth curve 1. Background A distinctive feature of insurance is that it is a product whose cost to the supplier is unknown at the time of sale. Indeed, the loss payments for many types of liability insurance claims can take many months or even years to complete. Late reported claims, judicial pro- ceedings and schedules of benefits for serious employer’s liability claims are among the many reasons for lengthy claim settlement periods. This fundamental fact presents insurers with some unique analytical challenges relating to the valuation of outstanding liabilities and is one of the primary reasons for the existence of the actuarial profession. The need for statistical methods in insurance was first exemplified by Thomas Bayes’s intellectual executor Richard Price, who worked as a consultant for London’s Equitable life insurance company (Hacking, 1990). Every insurance company must set aside a provision, which is known as a loss reserve, to Address for correspondence: Yanwei Zhang, Statistical Research, CNA Insurance Company, 30th Floor, 333 South Wabash Avenue, Chicago, IL 60604, USA. E-mail: actuary - zhang@hotmail.com