By intuition, the subdivision of an insurance portfolio into a number of classes is said to be good if it reflects the heterogeneity of the portfolio in an efficient way. To illustrate this rather vague statement we take the following very simple example:

The portfolio consists of 20 independent risks, 10 of them producing an expected loss ratio of say 30% each (type A risks) and 80% each (type B risks) respectively.

This “natural” subdivision is certainly better than, for instance, no subdivision at all;

or, the finest possible subdivision with 20 classes consisting of only 1 risk each (because there is no point in differentiating between risks of the same type);

or, 5 classes each containing two A- and two B-risks (here the number of classes is unnecessarily high and the heterogeneity has been completely wiped out; statistics based on this subdivision would even make us believe that the portfolio is totally homogeneous).

As a matter of fact, the above “natural” subdivision is—of course!—the best of all subdivisions, it is the optimal subdivision in this case.

In practice, however, as we all know, it is not easy to find the optimal subdivision. For one thing, the inherent structure or “natural” subdivision is not known a priori and secondly, for many different reasons, we can only choose from a limited number of subdivisions and not from all theoretically possible solutions. Note that even with only 20 risks there are 58.1012 possibilities of subdividing the portfolio. Thus, in practice, there is only a relatively small number of admissible subdivisions and the optimal one may not be among them, but we still need some sort of statistical criterion to choose the best one from these admissible subdivisions.