In this study, we investigate the competition among insurers under the mean–variance criterion. The optimization problems are formulated for finite and infinite insurers. The surplus processes of the insurers are characterized by jump-diffusion processes with common and idiosyncratic insurance risks. The insurers can purchase a reinsurance business to divide the insurance risk. In the financial market, the insurers decide the proportion of their fund to be retained as cash and to be invested in a stock characterized by a jump-diffusion process with common and idiosyncratic financial risks. The insurers compete with each other and are concerned with the relative performance. By the extended Hamilton-Jacobi-Bellman equation, the explicit forms of the n-agent equilibrium and mean-field equilibrium (MFE) are obtained in the n-agent case and mean-field case, respectively. Our results show that the MFE of the reinsurance strategy is composed of two parts, one part associated with the individual preference and the other associated with the common insurance shock. Meanwhile, the MFE of the investment strategy is composed of three parts: the individual preference, common market risks, and common shocks. Numerical examples are presented at the end of this article to demonstrate the effects of different parameters on the MFE. The results reveal that the insurers become convergent in a competitive environment.

This paper considers a discrete-time risk model by introducing a temporal dependence structure between the number of claims for each period. The risk model is based on the first-order integer-valued moving average (INMA(1)) process with compound Poisson distributed innovations. We derive the explicit expression for the moment generating function of the aggregate claim amount, which can be used for the calculation of some related quantities. We examine the properties of the adjustment coefficient for measuring the dangerousness of an insurance portfolio. Some special cases are included and numerical examples are provided to illustrate the results obtained in the paper.

This paper considers an extension of the classical discrete-time risk model for which an INAR(1) process is utilized to model a temporal dependence between the number of claims. We apply a recursive method for deriving the Laplace transform of the aggregate claims with or without discounting in this framework. This methodology is implemented for the class of INAR(1) processes with an arbitrary innovations' distribution. Three risk models via specific INAR(1) processes are studied when the distribution of the individual claim sizes belongs to the class of mixed Erlang distributions. These different models allow us to discuss the frequent manifestations of equidispersion, overdispersion and zero inflation, and to evaluate the distribution of the (discounted) aggregate claims. Numerical examples are performed in order to illustrate the results obtained in this paper.

This paper investigates risk aggregation and capital allocation problems for an insurance portfolio consisting of several lines of business. The class of multivariate INAR(1) processes is proposed to model different sources of dependence between the number of claims of the portfolio. The total capital required for the whole portfolio is evaluated under the TVaR risk measure, and the contribution of each line of business is derived under the TVaR-based allocation rule. We provide the risk aggregation and capital allocation formulas in the general case of continuous and strictly positive claim sizes and then in the case of mixed Erlang claim sizes. The impact of both time dependence and cross-dependence on the behavior of risk aggregation and capital allocation is numerically illustrated.