Quantitative management on heterogeneous insurance portfolios

Abstract

In the context of insurance, the annual premium is the amount paid by the policyholder on an annual basis to cover the cost of the insurance policy being purchased. It is the primary cost to the policyholder of transferring the risk to the insurer which depends on the type of insurance. To this regard, the extreme claim amounts, claim ranges as well as the aggregate claim amounts become important for insurance analysis since they provide useful information for insurers to determine the annual premium.

The first part focuses on stochastic comparisons on the largest claim amounts and claim ranges arising from two sets of heterogeneous insurance portfolios in the sense of the usual stochastic order, the hazard rate order, the reversed hazard rate order and the likelihood ratio order. First, the usual stochastic ordering is established between the largest claim amounts from two sets of independent and heterogeneous claims when the matrix of interested parameters changes in the sense of the row weak majorization order. Second, the ordering properties of the largest claim amounts are discussed when the occurrence levels are equal and have left tail weakly stochastic arrangement increasing (LWSAI) dependence structure. Third, both the hazard rate ordering and the reversed hazard rate ordering are established for the largest claim amounts from two sets of heterogeneous independent claims under appropriate conditions. Fourth, in the setting of independent multiple-outlier claim models, the effects of heterogeneity among sample sizes are studied on the stochastic properties of the largest claim amounts in the sense of the likelihood ratio order. Finally, the usual stochastic ordering is employed to compare the claim ranges from two sets of heterogeneous portfolios.

The second part deals with stochastic properties of aggregate claim amounts for two sets of heterogeneous insurance portfolios in the sense of the usual stochastic order and the stop-loss order. First, the usual stochastic order is established when the matrix of interested parameters changes according to the row weak majorization order. Second, the effects of heterogeneity among occurrence probabilities on aggregate claim amounts are investigated by means of the usual stochastic order when the claim severities are dependent but the dependence structure is unknown or characterized by an arrangement increasing joint density function. Third, the usual stochastic order is established between aggregate claim amounts when the claim occurrence levels are LWSAI. Fourth, stochastic properties of aggregate claim amounts are investigated in terms of the stop-loss order when the claim severities are comonotonic or right tail weakly stochastic arrangement increasing (RWSAI). One practical application in assets allocation for risk-seeking investors is also presented to illustrate the established theoretical results.