A basket half full: sparse portfolios

Abstract

The existing approaches to sparse wealth allocations (1) are limited to low-dimensional setup when the number of assets is less than the sample size; (2) lack theoretical analysis of sparse wealth allocations and their impact on portfolio exposure; (3) are suboptimal due to the bias induced by an ${\ell }_1$-penalty. We address these shortcomings and develop an approach to construct sparse portfolios in high dimensions. Our contribution is twofold: from the theoretical perspective, we establish the oracle bounds of sparse weight estimators and provide guidance regarding their distribution. From the empirical perspective, we examine the merit of sparse portfolios during different market scenarios. We find that in contrast to non-sparse counterparts, our strategy is robust to recessions and can be used as a hedging vehicle during such times.

Keywords:

• High dimensionality
• Portfolio optimization
• Factor investing
• De-biasing
• Post-Lasso
• Approximate factor model

JEL Classifications:

• C13
• C55
• C58
• G11
• G17

Acknowledgments

I greatly appreciate thoughtful comments and immense support from Tae-Hwy Lee, Jean Helwege, Jang-Ting Guo, Aman Ullah, Matthew Lyle, Varlam Kutateladze and UC Riverside Finance faculty. I also thank seminar participants at the 14th International CFE Conference (virtual), 2021 Southwestern Finance Association Annual Meeting, and Vilnius University.

The author would like to thank the editor and two anonymous referees for their helpful and constructive comments on the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 MSCI ITALY, MSCI SPAIN, MSCI PORTUGAL, MSCI FRANCE, MSCI GERMANY, MSCI AUSTRIA, MSCI DENMARK, MSCI FINLAND, MSCI NETHERLANDS, MSCI SWEDEN, MSCI SWITZERLAND, MSCI TURKEY, MSCI CANADA, MSCI BRAZIL, MSCI MEXICO, MSCI COLOMBIA, MSCI ARGENTINA, MSCI PERU, MSCI CHILE, MSCI CHINA, MSCI INDIA, MSCI INDONESIA, MSCI RUSSIA, MSCI JAPAN, MSCI MALAYSIA, MSCI SINGAPORE, MSCI TAIWAN, MSCI SOUTH AFRICA, MSCI AUSTRALIA, MSCI KOREA, MSCI US.

2 Since the optimization problem with a cardinality constraint is not convex, we find a solution using the Lagrangian relaxation procedure of Shaw et al. (2008)

3 Our empirical results suggest that the unbiased estimator $\hat{\theta }=\left(\right(T-p-2){\widehat{\mathbf{\text{m}}}}^{\prime }{\hat{\mathbf{\Sigma }}}^{-1}\widehat{\mathbf{\text{m}}}-p)/T$ is oftentimes negative even after using the adjusted estimator defined in Kan and Zhou (2007, p. 2906).

4 Note that we cannot directly apply Theorem 2.2 of van de Geer et al. (2014) since $\mathbf{\text{y}}$ needs to be estimated and we first need to show consistency of the respective estimator.

5 The results for larger degrees of freedom do not provide any additional insight, hence we do not report them here. However, they are available upon request.

6 The conclusions from using daily data are the same as those for monthly returns, hence we do not report them in the main manuscript text. However, they are available upon request.