Multivariate Credibility for Aggregate Loss Models

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. It is a special type of experience rating that employs a weighted average of claims experience and a previously established price to determine a new price for each risk class under consideration. This article extends traditional credibility formulas in two aspects. The new procedures are called “multivariate credibility” because both aspects make use of additional sources of data when compared to traditional formulas. Specifically, the first portion of the paper considers data from both the claims number and claims amount processes. Assuming an aggregate loss model for total claims, optimal insurance pricing formulas are derived. The insurance prices turn out to be an intuitively appealing weighted average of the overall mean claim, the claims number experience, and the claims amount experience. The second portion of the paper considers data from claims number and amount processes from multiple lines of business. By using covariances among lines of business (that are conditional on the unobserved heterogeneity), this article shows how to derive more efficient insurance prices. Accounting for covariance among different random quantities (securities) is standard practice in the investment industry. It is more difficult in an insurance context because of the heterogeneity associated with different risk classes. Nonetheless, ignoring this covariance has important ramifications, both theoretically and practically. For an illustrative sample of Massachusetts automobile claims, we show that the relative differences in accounting for and ignoring the covariance range from −3.9% to 14.5% for a selected bundle of insurance coverages.