Hierarchical Clustering as a Dimension Reduction Technique for Markowitz Portfolio Optimization

Abstract

Optimal portfolio selection is a common and important application of an optimization problem. Practical applications of an existing optimal portfolio selection methods is often difficult due to high data dimensionality (as a consequence of the large number of securities available for investment). In this paper, a method of dimension reduction based on hierarchical clustering is proposed. Clustering is widely used in computer science, a lot of algorithms and computational methods have been developed for it. As a measure of securities proximity for hierarchical clustering Pearson pair correlation coefficient is used. Further, the proposed method’s influence on the quality of the optimal solution is investigated on several examples of optimal portfolio selection according to the Markowitz Model. The influence of hierarchical clustering parameters (intercluster distance metrics and clustering threshold) on the quality of the obtained optimal solution is also investigated. The dependence between the target return of the portfolio and the possibility of reducing the dimension using the proposed method is investigated too. For each considered example in the paper graphs and tables with the main results of the proposed method - application which are the decrease of the dimension and the drop of the yield (the decrease of the quality of the optimal solution) - for a portfolio constructed using the proposed method compared to a portfolio constructed without the proposed method are given. For the experiments the Python programming language and its libraries: scipy for clustering and cvxpy for solving the optimization problem (building an optimal portfolio) are used.