Mixing Collective Risk Models

The form of the collective risk model is S = X₁ + … + X_M, where X represents a common severity distribution and M is a claim-count random variable. The model for a second loss is T = Y₁ + … + Y_N. Let Z represent the mixed severity, i.e., the distributions of X and Y weighted according to their expected claim counts. How does the mixed model U = Z₁ + … + Z_M+N compare with S + T? Although the mean is unaffected, we will show that if the claim-count distributions are negative binomial with a common contagion, Var[U]? Var[S + T]. In other words, attendant to the reduction of homogeneity (upon mixing the severities) is a reduction of variance. An appendix reveals the conditions under which one may fully reduce a set of collective risk models to one mixed model. This should be of value to the task of modeling correlated exposures.

Keywords: collective risk model, mixed distribution, negative binomial, contagion, homogenous, moment generating function, multinomial.