Optimization, model uncertainty, and testing in risk and insurance

Abstract

This thesis focuses on three important topics in quantitative risk management and actuarial science: risk optimization, risk sharing, and statistical hypothesis testing in risk.

For the risk optimization, we concentrate on risk optimization under model uncertainty where only partial information about the underlying distribution is available. One key highlight, detailed in Chapter 2, is the development of a novel formula named the reverse Expected Shortfall (ES) optimization formula. This formula is derived to better facilitate the calculation of the worst-case mean excess loss under two commonly used model uncertainty sets – moment-based and distance-based (Wasserstein) uncertainty sets. Further exploration reveals that the reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. Chapter 3 considers a different approach to derive the closed-form worst-case target semi-variance by including distributional shape information, crucial for finance (symmetry) and insurance (non-negativity) applications. We demonstrate that all results are applicable to robust portfolio selection, where the closed-form formulas greatly simplify the calculations for optimal robust portfolio selections, either through explicit forms or via easily solvable optimization problems.

Risk sharing focuses on the redistribution of total risk among agents in a specific way. In contrast to the traditional risk sharing rules, Chapter 4 introduces a new risk sharing framework - anonymized risk sharing, which requires no information on preferences, identities, private operations, and realized losses from the individual agents. We establish an axiomatic theory based on four axioms of fairness and anonymity within the context of anonymized risk sharing. The development of this theory provides a solid foundation for further explorations on decentralized and digital economy including peer-to-peer (P2P) insurance, revenue sharing of digital contents and blockchain mining pools.

Hypothesis testing plays a vital role not only in statistical inference but also in risk management, particularly in the backtesting of risk measures. In Chapter 5, we address the problem of testing conditional mean and conditional variance for non-stationary data using the recent emerging concept of e-statistics. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. To demonstrate the practical application of these methodologies, empirical studies using financial data are conducted under several settings.