A Note On The Multiplicative Ratemaking Model

Abstract

The multiplicative ratemaking, model we have in mind is the following one. Within a certain branch of insurance we have, say for simplicity, two tarif arguments U and V. For example, in motor insurance we could think of U and V as being make of car and geographical district respectively. In fire insurance U could be class of construction for buildings and V could relate to fire defense capacities.The arguments are of a qualitative nature and argument U has r levels, while argument V has k levels. To our disposal we have statistical experience of the business for a certain period of time, consisting of—risk exposures nij (i = 1 … r, j = 1 … k).Risk exposure nij thus corresponds to the ith U-level and the jth V-level. It could be e.g. number of policy years or sum insured during the period of observation for objects belonging simultaneously to U-level i and V-level j.The nijS are known non-random quantities.—(relative) risk measures pij(i = 1 … r, j = 1 …k).Risk measure pij could be e.g. claims frequency, i.e. number of Claims divided by number of policy years, or claims cost per policy year or claims cost as a percentage of sum insured. In general pij is thus the observed number or the observed amount of claims belonging simultaneously to U-level i and V-level j, divided by the corresponding risk exposure nij.

Estimation and Testing for Functional Form in Pure Premium Regression Models

“…An estimation method to minimize weighted squared prediction errors subject to balancing constraints later was suggested by BAILEY (1963) and analyzed by JUNG (1968) and AJNE (1974). The motivation for the balance criterion evidently is that it is reasonable to assume full credibility for certain partitions of the data.…”

A Comparison of Risk Classification Methods for Claim Severity Data

“…The author also thanks the General Insurance Association of Malaysia (PIAM), in particular Mr. Carl Rajendram and Mrs. Addiwiyah. (1960) suggested the minimum chi-squares, Bailey (1963) proposed the zero bias, Jung (1968) produced a heuristic method for minimum modified chi-squares, Ajne (1975) applied the method of moments also for minimum modified chi-squares, Chamberlain (1980) used the weighted least squares, Coutts (1984) produced the method of orthogonal weighted least squares with logit transformation, Harrington (1986) suggested the maximum likelihood procedure for models with functional form, and Brown (1988) proposed the bias and likelihood functions.…”

“…Bailey and Simon (1960) compared systematic bias and found that the multiplicative model overestimates the high risk classes, Jung (1968) and Ajne (1975) also found that the estimates for multiplicative model are positively biased, Bailey (1963) compared the models by producing two statistical criteria, i.e., minimum chi-squares and average absolute difference, Freifelder (1986) predicted the pattern of over and under estimation of the models if they were misspecified, Brown (1988) discussed the additive and multiplicative models which were derived from the maximum likelihood and minimum bias approaches, Jee (1989) compared the predictive accuracy of the models, Holler et al (1999) compared their initial values sensitivity, andMildenhall (1999) identified the GLMs with the additive and multiplicative models.…”

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

“…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:' Pij = Poaibj + eij (1) where Pij is the claim proportion for class ij; Po is overall mean; ai and bj are the effects of the levels i and j, for rating factors a and b respectively; and eij 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:…”

“…where cij is normally distributed with zero mean rather than a fixed parameter. The model (5) implies that without data analysis using observed values for Pij, it is impossible to get information regarding departure of the model (5) from its reduced form, that is, the loglinear or multiplicative model without interaction terms expressed as (1).…”