A Note on Sparse Minimum Variance Portfolios and Coordinate-Wise Descent Algorithms

TL;DR

This paper explores how coordinate-wise descent algorithms can efficiently solve sparse minimum variance portfolio problems with $l_q$ norm constraints, demonstrating improved out-of-sample performance on benchmark datasets.

Contribution

It introduces an efficient algorithm for sparse MVPs with $l_q$ norm constraints and extends to group asset selection, enhancing portfolio optimization methods.

Findings

Sparse portfolios have lower variance and turnover with higher Sharpe ratios.

Coordinate-wise descent algorithms effectively solve $l_q$ regularized MVP problems.

Extensions include group asset selection for improved portfolio diversification.

Abstract

In this short report, we discuss how coordinate-wise descent algorithms can be used to solve minimum variance portfolio (MVP) problems in which the portfolio weights are constrained by $l_{q}$ norms, where $1\leq q \leq 2$. A portfolio which weights are regularised by such norms is called a sparse portfolio (Brodie et al.), since these constraints facilitate sparsity (zero components) of the weight vector. We first consider a case when the portfolio weights are regularised by a weighted $l_{1}$ and squared $l_{2}$ norm. Then two benchmark data sets (Fama and French 48 industries and 100 size and BM ratio portfolios) are used to examine performances of the sparse portfolios. When the sample size is not relatively large to the number of assets, sparse portfolios tend to have lower out-of-sample portfolio variances, turnover rates, active assets, short-sale positions, but higher Sharpe ratios than the unregularised MVP. We then show some possible extensions; particularly we derive an efficient algorithm for solving an MVP problem in which assets are allowed to be chosen grouply.