Predicting Multivariate Insurance Loss Payments Under the Bayesian Copula Framework

ABSTRACT The literature of predicting the outstanding liability for insurance companies has undergone rapid and profound changes in the past three decades, most recently focusing on Bayesian stochastic modeling and multivariate insurance loss payments. In this article, we introduce a novel Bayesian multivariate model based on the use of parametric copula to account for dependencies between various lines of insurance claims. We derive a full Bayesian stochastic simulation algorithm that can estimate parameters in this class of models. We provide an extensive discussion of this modeling framework and give examples that deal with a wide range of topics encountered in the multivariate loss prediction settings. INTRODUCTION In his 1987 article, Buhlmann made the comment that the casualty actuaries have to master the skills of probabilistic thinking in order to deal with a variety of risky situations. Because of the complexity and diversity of risks arising in the property-casualty insurance, probability distributions of unknown quantities are often required for the formal solutions of decision problems. One such important example is the practice of insurance loss reserving. Due to reasons such as late reported claims, judicial proceedings, or schedules of benefits for employer's liability claims, many types of property-casualty insurance claims often have lengthy settlement periods, with liability claims often taking years or even decades to complete. In order to be able to respond to outstanding claims, every insurance company must set aside a provision, known as a loss reserve. The loss reserve is typically a property-casualty insurance company's largest balance sheet liability. Its proper prediction is therefore a matter of vital importance to the company and the research around loss reserve prediction has become a central subject in modern actuarial science. The loss reserving literature has undergone rapid and profound changes in the past three decades, where a large number of innovative applications of statistical methods have led to the proliferation of a great variety of stochastic loss reserving models. As the body of the literature continues to grow, some new trends have been observed in recent years. One such important trend is the increasing reliance on Bayesian methods (e.g., de Alba, 2002; Ntzoufras and Dellaportas, 2002; Antonio and Beirlant, 2008; Zhang, Dukic, and Guszcza, 2012). Bayesian analysis has gained some of its popularity due to its inherent advantages in the applied loss forecasting context. For example, the posterior predictive distribution conveys a wealth of information well beyond the uncertainty estimators commonly used in the traditional stochastic loss reserving analysis. This full measure of liability risk can be further incorporated into the determination of economic capital for the purpose of capital allocation and making strategic and tactical decisions. The second trend is developments in the realm of multivariate methods, where the major focus lies in the creation of models that enable integration of correlated multidimensional claim information for simultaneous statistical processing in order to improve, with respect to inference efficiency and prediction and risk assessment accuracy, upon univariate stochastic methods that could potentially fail due to the negligence of the inter-relationship within the multidimensional data. For example, substantial effort (e.g., Braun, 2004; Merz and Wuthrich, 2008; de Jong, 2012; Shi and Frees, 2011) has been devoted to making the extension of standard stochastic reserving models to account for correlations among various lines of insurance, with the aim to depict a more accurate picture of the liability risks faced by the insurance company. Although considerable advances have been achieved in both of the above fields, Bayesian analysis in the multivariate setting has rarely been found in the loss reserving literature (see Shi and Frees, 2011). …