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

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