Abstract
The thesis consists of two independent topics. Chapter 1 gives the overall introduction to lead the reader from a general subject area to two particular topics. Chapter 2 addresses financial portfolio management, while Chapter 3 studies asymptotic behavior of sparse random networks. Overall conclusion is made in Chapter 4. In Chapter 2, we look at three types of regularizations for the Markowitz mean variance portfolio selection model: single-norm regularizations on individual stocks, mixed-norm regularizations on stock groups, and composite regularizations that combine single-norm and mixed-norm regularizations. With mixed-norm regularizations, our model can do both group and stock choices at the same time. Using both US and global equities market data, we find that practically all regularized portfolios outperform the traditional mean-variance portfolio in terms of out-of-sample risk-adjusted performance as assessed by the Sharpe ratio. Furthermore, stock selection and group screening using l1 and l2, 1 regularizations can reduce volatility, turnover rate, and leverage ratio effectively. However, due to various means of grouping methods, there are times when diversification among multiple groups is preferable. Further to that, we discover that portfolio turnover and leverage are positively correlated. Higher transaction costs are incurred when portfolios are largely leveraged with consequently high turnover rates. In Chapter 3, we use sparse exchangeable graphs to construct random network models that could be used to analyze a wide range of complicated real-world networks. These models are particularly useful for exploring power-law features of degree distributions, number of edges, and other key network metrics that support networks’ scalefree structure. Earlier researches on such graphs usually impose a marginal condition of univariate regular variation (e.g., power-law tail) on the graphex function. While in this chapter, we apply multivariate regularly varying graphex functions that yield sparse exchangeable graphs. In the study, we focus on a rarely investigated network metric: the distribution of the number of common vertices (connections) shared by a pair of vertices. We discover that the distribution of the number of common connections is also regularly varying, with the tail indices of regular variation being determined by the type of graphex function utilized. Our findings are validated on simulated graphs generated from regularly varying graphex functions with estimated tail index parameters. The regularized portfolio selection model has provided various improvements to the classical mean-variance model, and the study in sparse random networks adopts a different methodology in analyzing asymptotic behaviors in large-scale complex networks