Title: Bertrand oligopoly in insurance markets with Value at Risk Constraints

Abstract: Since 2016 the operation of insurance companies in the European Union is regulated by the Solvency II directive. According to the EU directive the capital requirement should be calculated as a 99.5\% of Value at Risk. In this study, we examine the impact of this capital requirement constraint on equilibrium premiums and capitals. We discuss the case of the oligopoly insurance market using Bertrand's model, assuming profit maximizing insurance companies facing Value at Risk constraints. First we analyze companies' decision on premium level. The companies strategic behavior can result positive as well as negative expected profit for companies. The desired situation where competition eliminate positive profit and lead the market to zero-profit state is rare. Later we examine ex post and ax ante capital adjustments. Capital adjustment does not rule out market anomalies, although somehow changes them. Possibility of capital adjustment can lead the market to a situation where all of the companies suffer loss. Allowing capital adjustment results monopolistic premium level or market failure with positive probabilities.