Academic journal article Journal of Risk and Insurance

A Comparative Analysis of Alternative Pure Premium Models in the Automobile Risk Classification System

Academic journal article Journal of Risk and Insurance

A Comparative Analysis of Alternative Pure Premium Models in the Automobile Risk Classification System

Article excerpt

A Comparative Analysis of Alternative Pure Premium Models in the Automobile Risk Classification System

In the insurance market, risks are traditionally divided into various classes whose prices differ according to various characteristics of those risks. The goal of a classification system is to group homogenous risks and charge each group a premium commensurate with the average expected loss of its members. Failure to achieve this goal may lead to adverse selection and, perhaps, moral hazard with welfare losses to consumers and economic losses to insurers. Though it is important to be able to estimate the pure premiums for a class using data for only that class because the experience for any given class may not be sufficiently credible. As a result, a large number of studies have developed and estimated models of pure premiums that utilize data for multiple classes.

The class rating process may be divided into two broad phases: (a) the optimal design of the system, which involves the selection of risk classification factors and the definition of associated risk classes; and (b) the estimation of pure premiums for the risk classes, which involves the choice of an appropriate functional form and estimation method. This article primarily attempts to answer a number of unanswered questions concerning the second phase: establishing a modeling approach to estimate pure premiums for risk classes structured into the cross-classified rating system, focusing on some recently developed estimation models. In addition, the article reviews the previous modeling work, identifies important questions that remain unanswered, and looks for empirical evidence concerning these unanswered questions using various methods.


An early study on modeling of pure premiums within the risk classification system was conducted by Almer [4], who suggested a multiplicative model for use with cross-classified data with the following general form: (1)

[P.sub.ij] = [P.sub.0.a.sub.i.b.sub.j] + [e.sub.ij]

where [P.sub.ij] is the claim proportion for class ij: [P.sub.0] is overall mean; [a.sub.i] and [b.sub.j] are the effects of the levels i and j, for rating factors a and b respectively; and [e.sub.ij] is the error term. Bailey and Simon [6] analyzed loss ratios by comparing the multiplicative model and the additive model which may be expressed as follows:

[P.sub.ij] = [P.sub.0] + [a.sub.i] + [b.sub.j] + [e.sub.ij].

Since these two works were published, a considerable number of studies have focused on comparing additive and multiplicative models. In addition, many researchers have suggested various statistical procedures to estimate model parameters. A set of parameter values may be estimated so that it gives a best fit of the observed data by minimizing some function of deviations of actual from expected results. As Bennett [8] suggested, a weighted sum of squares of deviations can be considered an appropriate function to minimize under various assumptions about the value of the weight. Bennett also discussed some alternative sets of weights.

While the iteration procedure was used to derive the parameters for the multiplicative model in previous studies, a loglinear form of function has been employed to estimate the multiplicative model in the studies by Chang and Fairley [11], Sant [41], Fairley, Tomberlin, and Weisberg [19], Weisberg and Tomberlin [49], and Samson and Thomas [39]. Seal [42] constructed an additive model using the logit transformation, and Coutts [12] adopted Seal's approach in a study of automobile premium rating in the United Kingdom.

Chang and Fairley [11] and Fairley, Tomberlin, and Weisberg [19] noted that a loglinear model tends to overestimate high risks by double-counting classification factors. The dispute about whether risk classes for high risks are charged excessive premiums goes back to the findings of Bailey and Simon [61], where the results of analyzing Canadian automobile insurance data indicated that the multiplicative models produced systematic overestimates for the highest risk merit rating and driver classes. …

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