# PhD Project: Pricing in a competitive stochastic insurance market

There is a classic actuarial approach towards the pricing of insurance risk. In particular, the expected value premium principle, or the risk-based principle gained particular academic and practitioner’s interest (see for instance, Kaas et al. 2005). These principles need an approximation of the underlying distribution of insurance risk (claim sizes and timings), but they do not take into account the other insurers in the market. In economic theory on industrial organizations, competition, between alternative insurers in our case, may drive prices down, as observed in the well-known Bertrand competition. In this project, we aim to model the competition of multiple insurers in the market, where the insurance risk is observed and stochastic and all insurers aim to optimize a mean-variance objective function. In particular, we determine the prices as the Nash equilibria in the market. Focusing on a two-period model initially, we wish to see if competition, indeed, leads to lower prices and more insured risk for the insurer. This will allow us to simulate the distribution of the net asset value (also called basic own funds in Solvency II regulation) of the insurance companies. Also, we hypothesize that diversification of insurance policies yield an equilibrium where there is one insurer that attracts almost all policyholders. If the insured risk is too high, there is an interest for the policyholders (and thus the regulator) to introduce competition constraints (such as one preventing a monopole). This offset between regulation and diversification is non-trivial. Our approach extends the deterministic approaches of Taylor (1986) and Wu and Pantelous (2017), where there is no uncertainty in the pay-out of insurance policies. In a deterministic setting, the (risk-based) regulation is essentially irrelevant. Next, we aim to extend our results to a dynamic continuous-time setting. In this setting, open-loop Nash equilibria are studied to determine the premium profile, extending the approach of Boonen et al. (2018) to the case of stochastic insurance risk. This project is computationally more advanced than the first one, but it may allow us to understand what pricing profiles can be expected in stochastic insurance markets. For instance, Boonen et al. (2018) provide a deterministic example with premium cycles. Premium cycles are well-studied empirically, and exist in some insurance markets.