Robust Decision-Making in Finance and Insurance

Abstract

Traditional finance models assume a decision maker (DM) has a single view on the stochastic dynamics governing the price process but, in practice, the decision maker (DM) may be uncertain about the true probabilistic model that governs the occurrence of different states. If only risk is present, that is, the DM fully relies on a single probabilistic model P. When the DM is ambiguous, he holds different views on the precise distributions of the price dynamics. This type of model uncertainty due to the multiple probabilistic views is called ambiguity.

In the presence of model ambiguity, Maccheroni et al. (2013) propose a novel robust mean-variance model which is referred to as the mean-variance-variance (M-V-V) criterion in my thesis. The M-V-V model is an analogue of the Arrow-Pratt approximation to the well-known smooth ambiguity model, but it offers a more tractable structure and meanwhile separates the modeling of ambiguity, ambiguity aversion, and risk aversion.

In Chapters 3 and 4, we study the dynamic portfolio optimization and the dynamic reinsurance problem under the M-V-V criterion and derive the equilibrium strategies in light of the issue of time inconsistency. We find the equilibrium strategies share many properties with the ones from smooth ambiguity, but the time horizon appears inconsistently in the objective function of the M-V-V criterion, in turn causing the equilibrium strategies to be non-monotonic with respect to the risk aversion. To resolve this issue, we further propose a mean-variance-standard deviation (M-V-SD) criterion. The corresponding equilibrium investment strategy exhibits the appealing feature of limited stock market participation, a well-documented stylized fact in empirical studies. The corresponding equilibrium reinsurance strategy also displays the property of restricted insurance retention.

Chapter 5 analyzes optimal longevity risk transfers, focusing on differing buyer and seller risk aversions using a Stackelberg game framework. We compare static contracts, which offer long-term protection with fixed terms, to dynamic contracts, which provide short-term coverage with variable terms. Our numerical analysis with real-life mortality data shows that risk-averse buyers prefer static contracts, leading to higher welfare gains and flexible market conditions, while less risk-averse buyers favor dynamic contracts. Ambiguity, modeled as information asymmetry, reduces welfare gains and market flexibility but does not change contract preferences. These findings explain key empirical facts and offer insights into the longevity-linked capital market.

In the rest of the chapters, Chapter 1 introduces the background literature and main motivations of this thesis. Chapter 2 covers the mathematical preliminaries for the sub-sequent chapters. The core analysis and findings are presented in the following chapters. Finally, Chapter 6 concludes the thesis and suggests potential directions for future research.