Essays on Portfolio Selection, Continuous-time Analysis, and Market Incompleteness

Abstract

This thesis consists of three self-contained essays evaluating topics in portfolio selection, continuous-time analysis, and market incompleteness.

The two opposing investment strategies, diversification and concentration, have often been directly compared. Despite the less debate regarding Markowitz's approach as the benchmark for diversification, the precise meaning of concentration in portfolio selection remains unclear. Chapter 1, coauthored with Jiawen Xu, Kai Liu, and Tao Chen, offers a novel definition of concentration, along with an extreme value theory-based estimator for its implementation. When overlaying the performances derived from diversification (in Markowitz's sense) and concentration (in our definition), we find an implied risk threshold, at which the two polar investment strategies reconcile -- diversification has a higher expected return in situations where risk is below the threshold, while concentration becomes the preferred strategy when the risk exceeds the threshold. Different from the conventional concave shape, the estimated frontier resembles the shape of a seagull, which is piecewise concave. Further, taking the equity premium puzzle as an example, we demonstrate how the family of frontiers nested inbetween the estimated curves can provide new perspectives for research involving market portfolios.

Parametric continuous-time analysis for stochastic processes often entails the generalization of a predefined discrete formulation to a continuous-time limit. However, unknown convergence rates of the frequency-dependent parameters can destabilize the continuous-time generalization and cause modelling discrepancy, which in turn leads to unreliable estimation and forecast. To circumvent this discrepancy, Chapter 2, coauthored with Tao Chen and Renfang Tian, proposes a simple solution based on functional data analysis and truncated Taylor series expansions. It is demonstrated through a simulation that our proposed method is superior in both fitting and forecasting continuous-time stochastic processes compared with parametric methods that encounter troubles uncovering the true underlying processes.

When the markets are incomplete, perfect risk sharing is impossible and the law of one price no longer guarantees the uniqueness of the stochastic discount factor (SDF), resulting in a set of admissible SDFs, which complicates the study of financial market equilibrium, portfolio optimization, and derivative securities. Chapter 3, coauthored with Tao Chen, proposes a discrete-time econometric framework for estimating this set of SDFs, where the market is incomplete in that there are extra states relative to the existing assets. We show that the estimated incomplete market SDF set has a unique boundary point, and shrinks to this point only when the market completes. This property allows us to develop a novel measure for market incompleteness based upon the Wasserstein metric, which estimates the least distance between the probability distributions of the complete and incomplete market SDFs. To facilitate the parametrization of market incompleteness for implementation, we then consider in detail a continuous-time framework, in which the incompleteness specifically arises from stochastic jumps in asset prices, and we demonstrate that the theoretical results developed under the discrete-time setting still hold true. Furthermore, we study the evolution of market incompleteness in four of the world's major stock markets, namely those in China, Japan, the United Kingdom, and the United States. Our findings indicate that an increase in market incompleteness is usually caused by financial crises or policy changes that raise the likelihood of unanticipated risks.