Study of optimization problems in the insurance industry

Abstract

In the insurance industry, optimization problems play a pivotal role in vari- ous aspects of business operations, such as risk management, pricing, and claim settlement. As competition in the market intensifies, insurance companies are increasingly turning to advanced analytical techniques and mathematical mod- eling approaches to optimize their strategies and maximize profitability. This thesis investigates the most significant and emerging optimization problems in the insurance industry, with a focus on cutting-edge techniques and method- ologies that enhance efficiency and effectiveness in these areas. An extensive review of the literature is presented, the latest trends and innovations in op- timization techniques are discussed, and novel solutions to some of the most pressing challenges faced by (re)insurance providers are proposed. Through case studies and empirical analyses, this research demonstrates the value of adopting advanced optimization methods and tools in the insurance industry, providing valuable insights for both academics and practitioners. The first part deals with the multi-constrained Pareto-optimal reinsurance problems based on general distortion risk measures, which become technically challenging and have only been solved usingad hoc methods for certain special cases. In this research, the method developed by Lo (2017) is extended by proposing a generalized Neyman-Pearson framework to identify the optimal forms of the solutions. Then a dual formulation is developed, which shows that the infinite-dimensional constrained optimization problems can be reduced to finite-dimensional unconstrained ones. With the support of the Nelder-Mead algorithm, the optimal solutions can be obtained efficiently. To illustrate the versatility of our approach, several detailed numerical examples are provided, many of which were only partially resolved in the literature. In the second part, an evolutionary game model is developed based on a cost-benefit analysis of the insurer, the managing general agency (MGA)/ broker, and the consumer in a digitalization setting. Our findings suggest that an MGA partnership could lead to higher social welfare, considering the underwriting cost and mismatching cost. The MGA partnership is particularly beneficial for small- and medium-sized insurers, and can also expand consumer demand. The evolutionary stable state of the inter-mediated insurance market is determined by the relationship between the consultation fee and its critical value. Finally, the empirical data and simulation results are consistent with the predictions of our model.