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

## Yu-Min Yen

May 10, 2010

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., 2009), 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.

Number of Pages in PDF File: 27

Keywords: Sparse Portfolio, Coordinate-wise Descent Algorithm, Regularisation, Group Selection

JEL Classification: G11, C61, C40

working papers series

#### Suggested Citation

Yen, Yu-Min, A Note on Sparse Minimum Variance Portfolios and Coordinate-Wise Descent Algorithms (May 10, 2010). Available at SSRN: http://ssrn.com/abstract=1604093 or http://dx.doi.org/10.2139/ssrn.1604093

#### Contact Information

##### Yu-Min Yen (Contact Author)
Institute of Economics, Academia Sinica ( email )
Nankang
Taipei, 115
Taiwan
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