Book Description

Businesses face various risks that may negatively influence their operations, therefore implementing strategies to deal with risks is important. In recent years, risk management has become an active area of research in finance and insurance. The primary goals of risk management include identifying, assessing and controlling risks to minimize their potential impact. For insurance companies, reinsurance is an effective risk management tool to control risks. As a natural measure of risk, we consider the ruin probability of an insurance business. Our ultimate objective is to evaluate the impact of reinsurance in risk management, particularly in minimizing the ruin probability, and to find the corresponding optimal reinsurance policies. We first study the problem of minimizing the ruin probability in a discrete-time risk model with unknown parameters. A proportional reinsurance is purchased to control the ruin probability. We formulate the problem as a Markov decision process and solve this problem by means of discrete-time dynamic programming. The Bayesian approach is applied to address the issue of parameter uncertainty. We obtain the explicit expressions of minimum ruin probabilities and the corresponding optimal reinsurance strategies. Some structural properties of ruin probabilities are investigated under certain conditions. We also consider an optimization problem by joint decisions of excess-of-loss reinsurance and investment in a continuous-time financial market. The reserve may be invested in a financial market consisting of a risk-free asset and a risky asset with the price process follows geometric Brownian motion. Borrowing is allowed, however, the interest rate of borrowing is higher than the return rate of risk-free. Meanwhile, an excess-of-loss reinsurance is purchased. We apply stochastic control theory and Hamilton-Jacobi-Bellman equation to find the optimal strategy of joint reinsurance and investment decisions, and derive the closed form expression of the minimum ruin probability function. Our results are illustrated numerically. Both theoretical and numerical results show that reinsurance has a significant effect in alleviating the risk of ruin.