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Heterogeneity and the need for capital in the individual model. (English) Zbl 1142.91039

The authors consider the individual model of risk theory, where the claim amount for policy \(i, i=1,\dots,n,\) has the form \(X_{i}=J_{q_{i}}C_{i}\). The random variable \(J_{q_{i}}\) is Bernoulli distributed with the mean \(q_{i}\); it indicates whether at least one claim occurred for policy \(i\). The random variable \(C_{i}\) is then the total cost of all these claims. The impact of the heterogeneity is studied with the help of various stochastic orders. The concept of majorization arising as a measure of diversity of the components of a vector is described. The authors discuss various stochastic orders and related risk measures. The main results of this paper are devoted to the study of the impact of an increase in the heterogeneity of the portfolio either at the occurrence level \(q_{i}\) or at the cost level \(C_{i}\), or both.

MSC:

91B30 Risk theory, insurance (MSC2010)
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