Model Uncertainty with Applications in Finance and Insurance

Abstract

This thesis aims to develop rigorous and practically relevant optimization models for risk management in finance and insurance, with particular attention to robust portfolio selection under distributional model uncertainty and optimal reinsurance design.

First, in Chapter 1 we introduce the background and motivations for the research questions and models studied in this thesis, provide the preliminaries necessary for understanding and analyzing them, and describe the overall structure of the thesis.

In Chapter 2, we propose a reward-penalty mechanism and incorporate it into portfolio management. We consider a robust portfolio selection problem, where the joint distribution of the underlying asset losses is assumed to be uncertain and belongs to a prescribed multivariate distribution set. The objective is to determine optimal portfolios by minimizing the worst-case conditional value-at-risk (CVaR) of the portfolio loss under both distributional uncertainty and the reward-penalty mechanism. We first derive a closed-form expression for the worst-case CVaR, which generalizes several existing results, including those of Jagannathan(1977), Chen et. al (2011), and Cai et. al (2024). Then apply this expression to obtain optimal portfolios under a classical mean-covariance-based distribution set and a generalized mean-covariance-based distribution set. Empirical studies based on real market data show that the proposed models can outperform several related portfolio strategies. The results also demonstrate that incorporating downside risk into portfolio loss can improve risk management and investment performance, while revealing the trade-off between enhancing expected portfolio returns and controlling worst-case portfolio CVaR.

In Chapter 3, we study robust enhanced index tracking portfolio selection models under distributional uncertainty. The objective is to construct portfolios that can outperform a benchmark index while controlling the risk of underperformance. Since the joint distribution of asset and index losses is typically unknown in practice, we consider robust models based on partial distributional information. Guided by Pareto optimality, the proposed models balance worst-case mean loss and downside risk, subject to a constraint on worst-case mean return. Two types of uncertainty sets are considered. For the uncertainty set with a known mean vector and covariance matrix, closed-form optimal solutions are derived. For a more general mean-covariance-based uncertainty set, the problem is reformulated as a tractable convex optimization problem. Empirical results based on real market data show that the proposed models outperform the tracked index, the equally weighted portfolio, and several robust benchmark models in terms of cumulative wealth and risk-adjusted performance. The results also show that balancing mean loss and downside risk can lead to better portfolio performance than minimizing downside risk alone.

In Chapter 4, we introduce a performance-based premium principle for optimal reinsurance design. Under this premium principle, the reinsurance premium is adjusted according to the realized ceded loss relative to a baseline premium: the insurer receives a reward when the realized ceded loss is below the baseline level, and pays an additional premium when it exceeds the baseline level. Based on this pricing mechanism, we first study optimal reinsurance problems from the insurer’s perspective and derive the optimal retention levels for quota-share and stop-loss contracts under VaR and TVaR minimization. We then consider the joint perspective of the insurer and the reinsurer by maximizing their joint survival probability and obtain the corresponding optimal quota-share and stop-loss retentions. More generally, we investigate optimal reinsurance design under a broad class of reinsurance contracts and a general premium principle for determining the baseline premium. Numerical examples are provided to illustrate how the performance-based premium principle affects the optimal limited quota-share and limited stop-loss retentions.

Finally, in Chapter 5, we conclude the thesis by summarizing the key findings and discussing potential directions for future research and development that build upon the models and results presented in this thesis.