7Hierarchical Credibility

# 7Hierarchical Credibility - 6 Hierarchical Credibility 6.1...

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6 Hierarchical Credibility 6.1 Motivation In practice, we often encounter hierarchical structures, both in statistical data analysis and in the calculation of premiums. Insurance data frequently have a hierarchical structure. The individual risks are classi f ed according to their “tari > positions”, tari > positions are grouped together into “subgroups”, subgroups into “groups”, groups into “main groups”, which together make the total of a line of business. In the statistical interpretation, this structure will be constructed in reverse order. One is f rst interested in the development of a line of business as a whole, then in the development of the main groups, and so on. Not infrequently, a hierarchical procedure is also used in the calculation of premiums, whereby the reasoning follows a hierarchical tree. By this is meant a “top down” procedure, in that f rst the expected aggregate claim amount for the whole line of business is ascertained and then this amount is successively “distributed” over the lower levels. Consider the following two examples: Example 1: Group accident insurance in Switzerland, premium for acci- dents at work (Figure 6.1). In Switzerland, each f rm has to buy a workers’ compensation insurance. The premium calculation scheme used by the private insurers is basically as follows: the individual f rms are grouped together according to their type of business (e.g. banks, hospitals), and these types of business are grouped into danger classes (groups of types of business). The premium calculation is carried out by f rst f xing the “tari > level” for the various danger classes, then within the danger classes for the types of business.

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144 6 Hierarchical Credibility group accident insurance danger classes data types of business Fig. 6.1. Example 2: Industrial f re insurance in Germany (Figure 6.2). For a long time, the premium calculation of industrial f re insurance in Germany followed a hierarchical procedure: f rst the expected aggregate claim amount for the whole line of business is calculated and this is then successively broken down into “books”, within the “books” into “groups of statistical accounts”, and then within the groups into the individual accounts. industrial fire insurance books groups of statistical accounts individual accounts data Fig. 6.2. Such a hierarchical system has the advantage of leading to a well-founded, properly balanced distribution of the burden of claims, in particular of large claims, within the collective. In the following we will incorporate the idea of a hierarchical structure into the credibility framework along the lines presented in the paper by Bühlmann and Jewell [BJ87].
6.2 The Hierarchical Credibility Model 145 6.2 The Hierarchical Credibility Model For didactic reasons, we will consider in the following a model with f ve levels.

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## This note was uploaded on 12/02/2011 for the course ACTSC 432 taught by Professor Davidlandriault during the Spring '09 term at Waterloo.

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7Hierarchical Credibility - 6 Hierarchical Credibility 6.1...

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