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Asset Allocation and Risk Assessment with Gross Exposure Constraints for Vast Portfolios


  • Jianqing Fan
  • Jingjin Zhang
  • Ke Yu


Markowitz (1952, 1959) laid down the ground-breaking work on the mean-variance analysis. Under his framework, the theoretical optimal allocation vector can be very different from the estimated one for large portfolios due to the intrinsic difficulty of estimating a vast covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to the accumulation of estimation errors. We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the theoretical optimal portfolios have similar performance to the empirically selected ones based on estimated covariance matrices and there is no error accumulation effect from estimation of vast covariance matrices. This gives theoretical justification to the empirical results in Jagannathan and Ma (2003). We also show that the no-short-sale portfolio is not diversified enough and can be improved by allowing some short positions. As the constraint on short sales relaxes, the number of selected assets varies from a small number to the total number of stocks, when tracking portfolios or selecting assets. This achieves the optimal sparse portfolio selection, which has close performance to the theoretical optimal one. Among 1000 stocks, for example, we are able to identify all optimal subsets of portfolios of different sizes, their associated allocation vectors, and their estimated risks. The utility of our new approach is illustrated by simulation and empirical studies on the 100 Fama-French industrial portfolios and the 400 stocks randomly selected from Russell 3000.

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  • Jianqing Fan & Jingjin Zhang & Ke Yu, 2008. "Asset Allocation and Risk Assessment with Gross Exposure Constraints for Vast Portfolios," Papers 0812.2604,
  • Handle: RePEc:arx:papers:0812.2604

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    References listed on IDEAS

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    Cited by:

    1. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    2. Jianqing Fan & Yingying Li & Ke Yu, 2012. "Vast Volatility Matrix Estimation Using High-Frequency Data for Portfolio Selection," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 412-428, March.
    3. Xing, Xin & Hu, Jinjin & Yang, Yaning, 2014. "Robust minimum variance portfolio with L-infinity constraints," Journal of Banking & Finance, Elsevier, vol. 46(C), pages 107-117.
    4. Yu-Min Yen, 2010. "A Note on Sparse Minimum Variance Portfolios and Coordinate-Wise Descent Algorithms," Papers 1005.5082,, revised Sep 2013.

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