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