Optimality, sparsity and regularization parameter analysis for a risk diversification portfolio selection model

Qian Li, Aining Kou and Hao Lin

Abstract

Global “black swan” events, such as the 2007–9 global financial crisis and the 2020–23 Covid-19 pandemic, caused significant disruption to financial markets, in turn leading to increased interest in risk diversification. Meanwhile, sparse portfolio selection is critical for reducing transaction costs and improving managerial efficiency. This paper introduces the risk diversification mean–variance (RDMV) model. By analyzing the structural properties of this model, we prove that under mild conditions on the regularization parameter, all Karush–Kuhn–Tucker (KKT) points of the model globally minimize its objective function. We validate this theoretical characterization by comparing our solutions with a global optimum obtained using Gurobi, the global optimization solver. Further, we provide theoretical results on the relationship between the regularization parameter and sparsity, indicating that the RDMV model exhibits a certain degree of sparsity in portfolio weights without explicit sparse regularization terms. To solve the RDMV model efficiently, we introduce a proximal gradient algorithm with guaranteed convergence. Moreover, an adaptive regularization-parameter-setting strategy is designed to balance risk diversification and sparsity in the RDMV model. Finally, empirical analysis demonstrates that the RDMV model outperforms classical mean–variance, sparse portfolio and risk-diversified portfolio models in terms of sparsity, stability and net profit during the global financial crisis and the Covid-19 pandemic.