# A Linear Programming Model for Selecting Sparse High-Dimensional Multi-period Portfolios

European Journal of Operational Research, Volume 273, Issue 2, 1 March 2019, Pages 754-771

45 Pages Posted: 18 May 2015 Last revised: 16 Dec 2018

See all articles by Chi Seng Pun

## Chi Seng Pun

Nanyang Technological University (NTU) - School of Physical and Mathematical Sciences

## Hoi Ying Wong

The Chinese University of Hong Kong (CUHK) - Department of Statistics

Date Written: May 17, 2015

### Abstract

This paper studies the mean-variance (MV) portfolio problems under static and dynamic settings, particularly for the case that the number of assets ($p$) is larger than the number of observations ($n$). We prove that the classical plug-in estimation seriously distorts the optimal MV portfolio in the sense that the probability, that the plug-in portfolio will outperform the bank deposit, tends to 50\% for $p\gg n$ and a large $n$. We investigate a constrained $\ell_1$ minimization approach for directly estimating effective parameters appearing in the optimal portfolio solution. The proposed estimator is efficiently implemented with linear programming and the resulting portfolio is called the linear programming optimal (LPO) portfolio. We derive the consistency and the rate of convergence for the LPO portfolios. The LPO procedure essentially filters out unfavorable assets based on the MV criterion, resulting in a sparse portfolio. The advantages of the LPO portfolio include its computational superiority and its applicability for dynamic settings and non-Gaussian distributions of asset returns. Simulation studies validate the theory and illustrate its finite-sample properties. Empirical studies show that the LPO-based portfolios outperform the equally weighted portfolio, and the estimated optimal portfolios using shrinkage and other competitive estimators.

Keywords: Investment analysis; High-dimensional portfolio selection; Dynamic mean-variance portfolio; $\ell_1$ minimization; Sparse portfolio

JEL Classification: G11; C13; C16

Suggested Citation

Pun, Chi Seng and Wong, Hoi Ying, A Linear Programming Model for Selecting Sparse High-Dimensional Multi-period Portfolios (May 17, 2015). European Journal of Operational Research, Volume 273, Issue 2, 1 March 2019, Pages 754-771, Available at SSRN: https://ssrn.com/abstract=2607324 or http://dx.doi.org/10.2139/ssrn.2607324