Sparse and robust portfolio selection via semi-definite relaxation

Abstract

In investment management, especially for automated investment services, it is critical for portfolios to have a manageable number of assets and robust performance. First, portfolios should not contain too many assets in order to reduce the management fees, transaction costs, and taxes. Second, portfolios should be robust as investment environments change rapidly. In this study, therefore, we propose two convex portfolio selection models that provide portfolios that are sparse and robust. We first perform semi-definite relaxation to develop a sparse mean-variance portfolio selection model, and further extend the model by using $L_2$-norm regularization and worst-case optimization to formulate two sparse and robust portfolio selection models. Empirical analyses with historical stock returns demonstrate the effectiveness of the proposed models in forming sparse and robust portfolios.

Keywords:

• Portfolio selection
• sparse portfolio
• $L_2$-norm regularization
• robust optimization
• semi-definite relaxation
• robo-advisor

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 See Kim et al. (2019) for more detailed descriptions on automated portfolio managements.

2 Controlling portfolio cardinality can also be obtained by other approaches on measuring portfolio risk (e.g., Bruni, Cesarone, Scozzari, & Tardella, 2015).

3 Data available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

4 The selection of the risk preference parameter λ is critical to the investment, and Fabozzi, Kolm, Pachamanova, and Focardi (2007) discuss in detail about the risk-aversion formulation of mean-variance model in Chapter 2. Note that the meaning of λ is exactly the opposite in Fabozzi et al. (2007), since λ is attached to the variance term in Fabozzi et al. (2007) whereas it is attached to the mean term throughout this study.

Additional information

Funding

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2018R1C1B6004271) and the New Faculty Fund (1.180089.01) of UNIST (Ulsan National Institute of Science and Technology).