The Basics of Pricing with GLMs

Abstract

As described in the previous section, the goal of a tariff analysis is to determine how one or more key ratios Y vary with a number of rating factors. This is reminiscent of analyzing how the dependent variable Y varies with the covariates (explanatory variables) x in a multiple linear regression. Linear regression, or the slightly larger general linear model, is not fully suitable for non-life insurance pricing, though, since: (i) it assumes normally distributed random errors, while the number of insur-ance claims follows a discrete probability distribution on the non-negative integers, and claim costs are non-negative and often skewed to the right; (ii) in linear mod-els, the mean is a linear function of the covariates, while multiplicative models are usually more reasonable for pricing, cf. Sect. 1.3. Generalized linear models (GLMs) is a rich class of statistical methods, which generalizes the ordinary linear models in two directions, each of which takes care of one of the above mentioned problems: • Probability distribution. Instead of assuming the normal distribution, GLMs work with a general class of distributions, which contains a number of discrete and continuous distributions as special cases, in particular the normal, Poisson and gamma distributions. • Model for the mean. In linear models the mean is a linear function of the covari-ates x. In GLMs some monotone transformation of the mean is a linear function of the x's, with the linear and multiplicative models as special cases. These two generalization steps are discussed in Sects. 2.1 and 2.2, respectively. GLM theory is quite recent—the basic ideas were introduced by Nelder and Wed-derburn [NW72]. Already in the first 1983 edition of the standard reference by Mc-Cullagh and Nelder there is an example using motor insurance data; in the second edition [MN89] this example can be found in Sects. 8.4.1 and 12.8.3. But it was not until the second half of the 90's that the use of GLMs really started spreading, partly in response to the extended needs for tariff analysis due to the deregulation of the insurance markets in many countries. This process was facilitated by the publication of some influential papers by British actuaries, such as [BW92, Re94, HR96]; see also [MBL00], written for the US Casualty Actuarial Society a few years later.