By by Bjørn Sundt.

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Let ~1 1 -62- be _the aggregate claim amount of year j. We assume that N ,N ,... are conditional1 2 ly mdependent and identically distributed given a random variable e, and that the -63- This exercise is based on Sundt (1991a). 9 and mdependent of e and the claim numbers. a) Discuss the model assumptions. 2C we concluded somewhat informally that because of symmetry, the credibility estimator depended on the observations only through their average. In the present exercise we shall look more closely at this sort of symmetry.

Let P .. be the new pure pre'J mium in cell ( i,J), that is, level i for Criterion 1 and level j for Criterion 2. To determine the premiums we apply data from the insurance portfolio the previous year. In cell (i,J) we observed n .. lJ introduce the average claim amount per policy Y. =X ~J ' policies are independent. 'J It is assumed that the We consider first multiplicative rating models, that is, (i=l, ... ,I; j=l, ... ,J) We shall look at some different methods for determination of the factors a.

Tn. 1 d) How would you estimate the value of e? 3 Let N be the number of claims incurred in an insurance portfolio during a given year, and let Yi be the amount of the ith of these claims. We assume that the Y/s are mutually independent and identically distributed and independent of N. We introduce the accumulated claim amount Var X= ENVar Y + E2 YVar N. b) Find expressions for EX and Var X when N is Poisson distributed with parameter 0, that is, on-{} Pn = Pr(N=n) = 1iT e . 1B. s. for all n, then the structure distribution U has to be either continuous with density of the form r( a+f1 oa--1(1-0)~1 u( ())-r(a)l' ) (0<0<1; a,{J>O) or concentrated at one point.

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