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

## Author Info

• Yu-Min Yen
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## 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.

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File URL: http://arxiv.org/pdf/1005.5082

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1005.5082.

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Date of revision: Sep 2013
Handle: RePEc:arx:papers:1005.5082

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Web page: http://arxiv.org/

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## References

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1. Brodie, Joshua & Daubechies, Ingrid & De Mol, Christine & Giannone, Domenico & Loris, Ignace, 2008. "Sparse and stable Markowitz portfolios," Working Paper Series 0936, European Central Bank.
2. Victor DeMiguel & Lorenzo Garlappi & Francisco J. Nogales & Raman Uppal, 2009. "A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms," Management Science, INFORMS, vol. 55(5), pages 798-812, May.
3. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320.
4. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67.
5. Ravi Jagannathan & Tongshu Ma, 2002. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," NBER Working Papers 8922, National Bureau of Economic Research, Inc.
6. Jianqing Fan & Jingjin Zhang & Ke Yu, 2008. "Asset Allocation and Risk Assessment with Gross Exposure Constraints for Vast Portfolios," Papers 0812.2604, arXiv.org.
7. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
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