Multivariate longitudinal modeling of insurance company expenses

Abstract

Insurers, investors and regulators are interested in understanding the behavior of insurance company expenses, due to the high operating cost of the industry. Expense models can be used for prediction, to identify unusual behavior, and to measure firm efficiency. Current literature focuses on the study of total expenses that consist of three components: underwriting, investment and loss adjustment. A joint study of expenses by type is to deliver more information and is critical in understanding their relationship.This paper introduces a copula regression model to examine the three types of expenses in a longitudinal context. In our method, elliptical copulas are employed to accommodate the between-subject contemporaneous and lag dependencies, as well as the within-subject serial correlations of the three types. Flexible distributions are allowed for the marginals of each type with covariates incorporated in distribution parameters. A model validation procedure based on a t-plot method is proposed for in-sample and out-of-sample validation purposes. The multivariate longitudinal model effectively addresses the typical features of expenses data: the heavy tails, the strong individual effects and the lack of balance.The analysis is performed using property–casualty insurance company expenses data from the National Association of Insurance Commissioners of years 2001–2006. A unique set of covariates is determined for each type of expenses. We found that underwriting expenses and loss adjustment expenses are complements rather than substitutes. The model is shown to be successful in efficiency classification. Also, a multivariate predictive density is derived to quantify the future values of an insurer’s expenses.