Determining the insurance premium for the generalized exponential utility function in the collective risk model

Abstract

In this article, we investigate the determination of insurance premium in the collective risk model taking into account economic aspects. To conduct such research, we borrow well-known tools from utility theory and probability theory. The initial analyses in the literature (Rothschild and Stiglitz, 1976; Stiglitz, 1977) focused mainly on examining different effects (information asymmetry, market behavior). Instead, we focus on the calculation of insurance premiums, for which we can also find many examples in the literature. We wish to emphasize that our one-period model has many simplifications in terms of opportunities for expansion, which we will give some ideas to solve. The first option involves choosing between the individual risk model and the collective risk model in which the nature of the insurance product helps. In the case of the individual risk model, the existence of a claim to the policy is of interest to the insurer (e.g., due to the fixed amount insured). On the contrary, when it comes to the collective risk model, the number of claims is also interesting. In Verrall (1989) article we shall find a possible mathematical treatment of the individual risk model, moreover, it also supplements both approaches to the different risk models by describing similarities and pathways. Because of the greater possibility of application, we use the collective risk model, but we assume that the various claims events are independent of each other. The second expansion option is also to allow insurance companies to carry out investment activities, that is, insurance companies should not necessarily transfer all their assets to insurance businesses. Of course, compared to daily practice, we simplify taking into account investment activity in our analysis. Numerical analyzes in this article also reflect this fact. We want to draw the attention to the fact that insurance companies need to apply different regulations in their investment policy in order to be able to impose this simplification. Additionally, our primary goal remains to determine insurance premiums in a broadly acceptable setting. Our third extension in this article is cost introduction. One of the simplest choices could be the direct cost variable associated with the insurance contract we are working on. We do not use fixed costs as we primarily focus on calculating premiums for a single period.

Finally, contrary to the possibilities of expansion so far, it is not the insurance company's financial trajectory that we want to further refine its characterization but rather its valuation. Throughout the article, we assume that the insurance company's only decision variable is the insurance premium. Therefore, our job is to find out the insurance premium level at which an insurer enters the insurance market. When solving a problem, we focus on the expected utility level rather than the expected value level. Much more important is the introduction of the risk attitude and measure by Pratt (1964), where we also find some calculations for specific utility functions. The generalized exponential utility function is presented as a combination of two possible evaluation methods. Therefore, it can manage both standard approaches based on expected value and risk aversion nature.

The model described in this article is currently more theoretical than applied, but we believe that it can be developed relatively easily by a particular insurer. We introduce a new metric to add to standard measures (such as the ratio of return) that may be a bit more complex, but in addition to standard measures (such as rate of return), insurance premiums can be determined. The determination of optimal premiums differs from other economic methods because of the use of collective risk models and generalized exponential utility functions. We managed to write a premium formula using the Lambert {\small $W$} function whose main field of application is not exactly the solution of microeconomic problems.