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A Krylov subspace approach to large portfolio optimization

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  • Bajeux-Besnainou, Isabelle
  • Bandara, Wachindra
  • Bura, Efstathia

Abstract

With a large number of securities (N) and fewer observations (T), deriving the global minimum variance portfolio requires the inversion of the singular sample covariance matrix of security returns. We introduce the Break-Down Free Generalized Minimum RESidual (BFGMRES), a Krylov subspaces method, as a fully automated approach for deriving the minimum variance portfolio. BFGMRES is a numerical algorithm that provides solutions to singular linear systems without requiring ex-ante assumptions on the covariance structure. Moreover, it is robust to illiquidity and potentially faulty data. US and international stock data are used to demonstrate the relative robustness of BFGMRES to illiquidity when compared to the “shrinkage to market” methodology developed by Ledoit and Wolf (2003). The two methods have similar performance as assessed by the Sharpe ratios and standard deviations for filtered data. In a simulation study, we show that BFGMRES is more robust than shrinkage to market in the presence of data irregularities. Indeed, when there is an illiquid stock shrinkage to market allocates almost 100% of the portfolio weights to this stock, whereas BFGMRES does not. In further simulations, we also show that when there is no illiquidity, BFGMRES exhibits superior performance than shrinkage to market when the number of stocks is high and the sample covariance matrix is highly singular.

Suggested Citation

  • Bajeux-Besnainou, Isabelle & Bandara, Wachindra & Bura, Efstathia, 2012. "A Krylov subspace approach to large portfolio optimization," Journal of Economic Dynamics and Control, Elsevier, vol. 36(11), pages 1688-1699.
  • Handle: RePEc:eee:dyncon:v:36:y:2012:i:11:p:1688-1699
    DOI: 10.1016/j.jedc.2012.04.009
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    References listed on IDEAS

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    1. Tola, Vincenzo & Lillo, Fabrizio & Gallegati, Mauro & Mantegna, Rosario N., 2008. "Cluster analysis for portfolio optimization," Journal of Economic Dynamics and Control, Elsevier, vol. 32(1), pages 235-258, January.
    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. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    4. William F. Sharpe, 1963. "A Simplified Model for Portfolio Analysis," Management Science, INFORMS, vol. 9(2), pages 277-293, January.
    5. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    6. Chan, Louis K C & Karceski, Jason & Lakonishok, Josef, 1999. "On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model," Review of Financial Studies, Society for Financial Studies, vol. 12(5), pages 937-974.
    7. Ledoit, Oliver & Wolf, Michael, 2008. "Robust performance hypothesis testing with the Sharpe ratio," Journal of Empirical Finance, Elsevier, vol. 15(5), pages 850-859, December.
    8. Louis K.C. Chan & Jason Karceski & Josef Lakonishok, 1999. "On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model," NBER Working Papers 7039, National Bureau of Economic Research, Inc.
    9. Golosnoy, Vasyl & Okhrin, Yarema, 2009. "Flexible shrinkage in portfolio selection," Journal of Economic Dynamics and Control, Elsevier, vol. 33(2), pages 317-328, February.
    10. Gilli, Manfred & Pauletto, Giorgio, 1998. "Krylov methods for solving models with forward-looking variables," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1275-1289, August.
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    Cited by:

    1. Fotis Papailias & Dimitrios Thomakos, 2015. "Covariance averaging for improved estimation and portfolio allocation," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 29(1), pages 31-59, February.

    More about this item

    Keywords

    Krylov subspaces; Singular systems; Algorithm; Sample covariance matrix; Global minimum portfolio;

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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