BhlmannStraub model This model was formulated by Bhlmann and Straub; see Bhlmann and Straub (1970). The model we have discussed in 4.6 (EBCT Model 1) clearly shows similarities with a pure Bayesian approach and is a necessary and useful introduction to empirical credibility methods. However, it involves rather restrictive assumptions and is not very useful in practice. EBCT Model 2 the BhlmannStraub model encompasses a major generalisation of Model 1 by allowing for changing levels of business (changing risk volumes). It is easy to see why this is such an important and practical 185 186 Model based pricing setting premiums

The risk during one year may relate to cover for a small business with four shops and three delivery vans on the road the business may do well, expand, and next year have six shops and four vans on the road. The increased estate (buildings, vans, stock) and general activity is not taken account of by Model 1 but is taken account of by Model 2.

With this recognition of changing risk volumes, it is inappropriate now to assume that, given the risk parameter, the claims variables are identically distributed. The assumptions we do make for Model 2 are most conveniently expressed in a manner which makes them less restrictive than was the case for Model 1 and these assumptions are made not about the claims variables themselves, but about the variables representing claims per unit of risk volume. So, let Y1, Y2, . . . , Yn represent the aggregate claims in n successive years for a risk, and let P1, P2, . . . , Pn be corresponding risk volumes. These risk volumes are known numbers (not random variables) and can be quantied in various ways for example, numbers of policies in a changing portfolio, numbers of shops in a chain, numbers of vehicles in a eet, etc. A sensible general measure which can be used perhaps obvious once mentioned is the annual premium income the insurer has charged to cover the risk over recent years (provid

We introduce Xi to represent the aggregate claims in year i scaled to take account of the volume of business, that is Xi = Yi/Pi , i = 1, 2, . . . , n, so Xi is the aggregate claims per unit of risk volume in year i. The basic structure of this model is that the distribution of each variable Xi, i = 1, 2, . . . , n, depends on the value of a risk parameter , which is xed for that risk but unknown, and is regarded as a random variable with unknown distribution function. It is not appropriate to assume that the Xi are identically distributed, either conditionally (given ), or unconditionally. Assumptions (1) Given , the Xi, i = 1, 2, . . . , n, are independent. (2) E[Xi | ] does not depend on i. (3) PiVar[Xi | ] does not depend on i. Under these assumptions we dene m() = E[Xi|] and s2() = PiVar[Xi|].