Insurance Pricing, and Optimal Strategy Design under Distributional Model Uncertainty

Abstract

The objective of this thesis is to develop theoretically sound and practically applicable solutions to two optimal reinsurance models, and to investigate the properties of the premium with background risk. Specifically, Chapter 1 introduces model uncertainty into the optimal reinsurance problem, while Chapter 2 focuses on solving the optimal reinsurance problem within the framework of a performance-based variable premium. Chapter 3 examines the properties of the indifference premium for an insurable risk in the presence of a background risk, considering both scenarios with and without model uncertainty.

In Chapter 2, we explore a non-cooperative optimal reinsurance problem incorporating likelihood ratio uncertainty, aiming to minimize the worst-case risk of the total retained loss for the insurer. We establish a general relation between the optimal reinsurance strategy under the reference probability measure and the solution in the worst-case scenario. This relation can be generalized to insurance design problems quantified by tail risk measures. We provide a sufficient and necessary condition for when the problem using reference measure has at least one common optimal solution with its worst-case counterpart. As an application of this relation, optimal policies for the worst-case scenario quantified by the expectile risk measure are determined. Additionally, we explore the corresponding cooperative problem and compare its value function with that of the non-cooperative model.

In the literature, insurance and reinsurance pricing is typically determined by a premium principle, characterized by a risk measure that reflects the policy seller's risk attitude. Building on the work of Meyers (1980) and Chen et al. (2016), Chapter 3 propose a new performance-based variable premium scheme for reinsurance policies, where the premium depends on both the distribution of the ceded loss and the actual realized loss. Under this scheme, the insurer and the reinsurer face a random premium at the beginning of the policy period. Based on the realized loss, the premium is adjusted into either a "reward" or "penalty" scenario, resulting in a discount or surcharge at the end of the policy period. We characterize the optimal reinsurance policy from the insurer's perspective under this new variable premium scheme. In addition, we formulate a Bowley optimization problem between the insurer and the monopoly reinsurer. Numerical examples demonstrate that, compared to the expected-value premium principle, the reinsurer prefers the variable premium scheme as it reduces the reinsurer's total risk exposure.

Chapter 4 investigates the properties of the indifference premium for an insurable risk in the present of a background risk. We establish conditions that guarantee the uniqueness of the premium and provide general properties of the indifference premium without imposing specific assumptions on the dependence structure between the insurable risk and the background risk. Furthermore, we demonstrate that the dependence structure between these risks significantly affects the behavior of the indifference premium. In particular, we analyze the indifference premium under various dependence structures, including independence, stochastic monotonicity, and mutual exclusivity. We further explore the impact of model uncertainty on the indifference premium by deriving equivalent conditions for the dependence structures between the insurable and the background risks and their corresponding uniform random variables. Finally, we illustrate our findings using expected utility preference operators as representative examples.