Optimal reinsurance minimizing the ruin probability in a diffusion setting

Reinsurance is one of the key risk management tools used by insurance companies to spread risk and receive financial protection against large losses. This comes at the price of the reinsurance premium which reduces the insurer’s profits in exchange for safety. This thesis focuses on analytically finding the optimal retention levels under three different excess of loss contracts, with the purpose of minimizing the ruin probability in infinite time and from the point of view of the insurance company. The expected value premium principle is used for the calculation of both the insurer's and reinsurer's premiums. The same analysis is developed when considering two dependent classes of risk. A diffusion approximation of the classical Crámer-Lundberg risk process with reinsurance is considered. After building the model, the ruin probability function is characterized and conclusions regarding the optimal strategies are drawn. For the dependent case, the optimal strategy depends not only on the marginal distributions of the underlying risk, but also on the distribution of the sum of the claim severities. To better contextualize the analytical results, a numerical analysis is developed in each case, using the R software, considering different distributions and several values for its parameters. The analytical results show that, for some particular cases of the excess of loss treaty, it is always optimum to transfer part of the risk to the reinsurer; for other cases, the optimal strategy is to retain all the risk and, for the remaining cases, it depends on the distribution of the underlying risk. The numerical results corroborate the analytical ones. In particular, the optimal reinsurance strategy under dependences is different if the two classes of risk are considered independently.