۱٫۱٫ Motivation

This paper constructs two models for determining the premium of general policies in competitive, non-cooperative, insurance markets. In the corresponding literature, there is still little research done on how the insurance premium follows from competition, and responds to changes initiated by competitors (Taylor, 1986; Daykin and Hey, 1990; Emms, 2012; Pantelous and Passalidou, 2015; Wu and Pantelous, 2017). Moreover, despite the fact that in many lines of insurance the presence of underlying cycles has been observed empirically, there is a constant endeavour to understand the dynamics of insurance premiums (Cummins and Outreville, 1987; Rantala, 1988; Doherty and Kang, 1988; Daykin et al., 1994; Winter, 1994; Cummins and Danzon, 1997; Lamm-Tennant and Weiss, 1997; Taylor, 2008; Malinovskii, 2010). Furthermore, as it is observed in practice, the competition among insurance companies is getting stronger. Moreover, the domination of oligopoly markets is reflected often in the determination of insurance premiums (Friedman, 1982). Thus, a fair, but also commercially attractive premium is not any more a product of a simple risk assessment exercise, but a highly challenging decision-making process. Consequently, the demand of mathematical models is more essential than ever to investigate the interconnectivity among the competitors in the corresponding markets1 and to understand the formulation of premium cycles.

۱٫۲٫ Developments in competitive insurance markets

Over the last three decades, academics have been interested in investigating on how competition might affect insurance premiums and how insurers respond to changes in the premium levels that being offered by competitors. In actuarial science, Taylor (1986) pointed out that competition is a key component in insurance premium pricing. He embedded the law of demand to analyse the change of exposure volume through a comparison between the insurer’s and market average premiums. Later, Taylor (1987) noted that the optimum underwriting strategies might be substantially affected by the expense rates.2 Emms and his co-authors were able to extend significantly Taylor (1986, 1987)’s ideas developing a series of models in continuous time by implementing a variety of optimal control theory techniques (Emms and Haberman, 2005; Emms, 2007a, b; Emms et al., 2007; Emms and Haberman, 2009). In this paper, we defer from these approaches by modelling the competition via the open-loop Nash equilibrium.

۱٫۳٫ Game-theoretic approaches for insurance markets

In a control theory framework as presented in Section 1.2, a single insurer’s objective function is optimized considering the market information as inputs. However, as a key assumption, a premium strategy does not cause any reaction to the rest of the market’s competitors.