Sparse and risk diversification portfolio selection

Abstract

Portfolio risk management has become more important since some unpredictable factors, such as the 2008 financial crisis and the recent COVID-19 crisis. Although the risk can be actively managed by risk diversification, the high transaction cost and managerial concerns ensue by over diversifying portfolio risk. In this paper, we jointly integrate risk diversification and sparse asset selection into mean-variance portfolio framework, and propose an optimal portfolio selection model labeled as JMV. The weighted piecewise quadratic approximation is considered as a penalty promoting sparsity for the asset selection. The variance associated with the marginal risk regard as another penalty term to diversify the risk. By exposing the feature of JMV, we prove that the KKT point of JMV is the local minimizer if the regularization parameter satisfies a mild condition. To solve this model, we introduce the accelerated proximal gradient (APG) algorithm [Wen in SIAM J. Optim 27:124-145, 2017], which is one of the most efficient first-order large-scale algorithm. Meanwhile, the APG algorithm is linearly convergent to a local minimizer of the JMV model. Furthermore, empirical analysis consistently demonstrate the theoretical results and the superiority of the JMV model.

Keywords: Accelerated proximal algorithm; Linear convergence; Non-convex regularization; Sparse portfolio selection.

Fig. 1

No, Sh, $s^2$ and MMR for Optimal Portfolios Generated by JMV

Fig. 2

The smallest eigenvalue of Hessian matrix for different ${\lambda }_1$ and ${\lambda }_2$

Fig. 3

Evolution of portfolio values for S &P 500

Fig. 4

Evolution of portfolio values for FF 48

Fig. 5

Evolution of portfolio values for FF 100

Fig. 6

Comparisons of iterations of three first-order algorithms