On automobile insurance ratemaking

Suppose that an automobile insurance plan is characterized by a double classification. The risks are thus divided into classes, i = 1, 2, …, p (e.g. defined by use of car and age of operator), and groups j = 1, 2, …, q (e.g. defined by licence and by accidents during the last three years). The experience of the company is described by the observed “relative loss ratios” r ij and some measure of exposure n ij . A general model, often used, is that the r ij : s are observations of random variables with the expected values g ij = g (α i , β j ), where the relativities α i are parameters representing the classes i and the relativities β j represent the influence of the groups j . One of the ratemaker's problems is to find a realistic function g (α, β) and to obtain estimates a i of α i and b j of β j . In their paper “Two Studies in Automobile Insurance Rate-making” (ASTIN Bulletin Vol. I, Part IV, page 192-217) Robert Bailey and LeRoy Simon have thoroughly analyzed this problem for private passenger automobiles in Canada. They have principally studied three different types of the function g (α β), namely g (α β) = αβ (Method 2), g (α β) = α + β (Method 3) and g (α β) = 3αβ— 2 (Method 4). The authors show in an appendix, that the variance of r ij is approximately g (α i β j )/ Kn ij where K ≅ 0.005 for the Canadian data. They estimate the relativities α i and β j ; by making χ 2 = K. Σ n ij ( r ij — g ij ) 2 / g ij a minimum. For the Canadian material, the “method 4” agrees best with the observations. This method gives an observed χ 2 value of about 8 for 11 degrees of freedom.

[1]  R. Bailey,et al.  Two Studies in Automobile Insurance Ratemaking , 1960, ASTIN Bulletin.