Cluster based portfolio optimization under uncertainty: Statistical and Robust approaches

Portfolio optimization refers to the process of determining the ideal fraction of an investment capital across a set of assets so that it meets specific requirements or constraints, while aiming for an objective such as minimizing risk or maximizing returns. Often in academic research, this problem is treated as a standalone task, with the portfolio of available securities assumed to be predefined and given as input. In a realistic scenario, the first step towards an optimal allocation is the choice of that portfolio of securities from a set of possible investment opportunities. This thesis aims at studying in detail the two fundamental steps of portfolio management, namely portfolio selection and asset allocation. The first step is addressed through the application of several clustering algorithms in order to extract useful knowledge from a vast set of investment assets. Methods like K-means and hierarchical clustering have proven useful for many different tasks including data summarization, enhanced decision-making and ultimately, efficient portfolio selection. The resulting clusters have been analyzed in detail and a well-diversified portfolio has been built. Later, it served as a dataset for studying and comparing different portfolio optimization strategies. Optimal asset allocation has been historically studied by Harry Markowitz when he first introduced the Mean Variance Portfolio: an optimization problem that aims at maximizing the expected return of the financial position while maintaining a desired level of risk, measured in volatility or variance of the resulting portfolio. However, this method suffers from several drawbacks especially during uncertain market conditions, and for this reason advanced mathematical methods have recently been developed to overcome the criticalities of Markowitz asset allocation. This project is focused on two main approaches: hierarchical clustering-based allocation and robust optimization allocation. For what concerns the first category, the main contributions in the literature have been implemented and a different way of estimating the similarity matrix has been tested: this new similarity matrix is based on the concept of lower tail dependence coefficient estimated with two non-parametric, copula-based estimators. The second approach, with the use of a quadratic-uncertainty set, consists of a two-step optimization problem that can be efficiently solved when reformulated as second-order cone programming. Since both approaches are relatively new, the impact of the optimization parameters involved has been studied in detail with a particular emphasis on their performance during periods of great recession, subsequently different combinations of parameters have been selected, representing different risk-aversion levels of an ideal investor. Finally, the chosen configurations have been tested in a simulation with real stock data and compared against several classical approaches and a real benchmark in order to assess the goodness of the developed methods.