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Norm constrained minimum variance portfolios with short selling

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Abstract

Short selling is a wealth-building trading procedure which, when included in the portfolio construction, not only helps increase the return on investment but also reduces the investor’s overall exposure to the market risk. In this study, we incorporate it in the minimum variance model by analyzing several constraints that aptly consider the different practical settings of a short sale transaction. We propose to utilize the short-rebate gain by maximizing this additional interest due to short sales in the objective function. In constraints, we impose the bounds on 1- and 2-norm that respectively generate sparse and diversified portfolios. Along with the norm constraints, we also bound the budget constraint to homogenize the allocations in long and short and avoid the dominance of one strategy over the other. We present empirical results highlighting the effect of the specific choice of constraints and, thereafter, conduct a comparative analysis of our proposed models vis-a-vis several related models from literature across eight global data sets using the rolling window scheme. We observe that our proposed models outperform the others in terms of several performance measures. In particular, the 1-norm constrained model generates statistically significant portfolios as compared to other related models in terms of variance and Sharpe ratio. Additionally, the threshold parameter of the 1-norm constraint provides the flexibility to tune the short sale budget, proving more favorable for a short sale scenario.

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Data availability

The data sets for the current study have been fetched by the authors from the Bloomberg Database Management software, accessed through the Department of Management Sciences, IIT Roorkee, India. The data is privately fetched and hence not publicly available; however, can be made available from the corresponding author on a reasonable request.

Notes

  1. For evidence of the poor performance of mean-variance model based on sample estimates of the mean vector and covariance matrix, see Merton (1980), Best and Grauer (1991), Chopra and Ziemba (1993), Jagannathan and Ma (2003) and Kondor et al. (2007).

  2. The authors in DeMiguel et al. (2009) determine the parameter value \(\delta\) using two elaborate techniques, namely, cross-validation over portfolio variances and maximization of returns in the last period.

  3. Here \(\lfloor \cdot \rfloor\) denotes the greatest integer function or the floor function.

  4. We use the R codes of Ledoit and Wolf (2008, 2011) available at https://www.econ.uzh.ch/en/people/faculty/wolf/publications.html to perform the statistical tests.

  5. Note that the value of threshold parameters \(\zeta _1\) and \(\zeta _2\) for the models RF-NC1W and RF-NC2W are the same as that of RF-NC1 and RF-NC2.

  6. A portfolio with no allocation in any asset.

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Acknowledgements

The authors sincerely thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions, which have considerably improved the presentation and quality of the paper. The authors acknowledge the support of the Department of Management Studies, IIT Roorkee, India, for providing access to the Bloomberg Database to collect data. The first author would also like to thank the Ministry of Human Resource and Development (MHRD), New Delhi, India, for financial support.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, 247667, India

    Vrinda Dhingra & Shiv Kumar Gupta

  2. Department of Mathematics, Netaji Subhas University of Technology, New Delhi, 110078, India

    Amita Sharma

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  1. Vrinda Dhingra
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  2. Shiv Kumar Gupta
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Correspondence to Shiv Kumar Gupta.

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Dhingra, V., Gupta, S.K. & Sharma, A. Norm constrained minimum variance portfolios with short selling. Comput Manag Sci 20, 6 (2023). https://doi.org/10.1007/s10287-023-00438-2

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