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Heterogeneity and the need for capital in the individual model

Loosely speaking, actuaries believe that the heterogeneity of the risks comprised in a given insurance portfolio tends to increase its dangerousness. This is turn leads to requiring more capital. This paper aims to formalize this intuitive idea in the individual model of risk theory. The impact of the heterogeneity (for claim occurrences and/or claim sizes) will be studied with the help of various stochastic orders. The concept of majorization, allowing for comparing the dispersion of the components of two vectors of real numbers, and the closely related Schur-increasingness, turn out to be the appropriate tools to deal with this problem. The method proposed in this paper distinguishes between diversification and dispersion: spreading maturities by dispersing claim occurrences lessens the need for capital, while increasing uncertainty in claim amounts increases the need for capital.