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Noisy covariance matrices and portfolio optimization II

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  • Pafka, Szilárd
  • Kondor, Imre
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    Abstract

    Recent studies inspired by results from random matrix theory (Galluccio et al.: Physica A 259 (1998) 449; Laloux et al.: Phys. Rev. Lett. 83 (1999) 1467; Risk 12 (3) (1999) 69; Plerou et al.: Phys. Rev. Lett. 83 (1999) 1471) found that covariance matrices determined from empirical financial time series appear to contain such a high amount of noise that their structure can essentially be regarded as random. This seems, however, to be in contradiction with the fundamental role played by covariance matrices in finance, which constitute the pillars of modern investment theory and have also gained industry-wide applications in risk management. Our paper is an attempt to resolve this embarrassing paradox. The key observation is that the effect of noise strongly depends on the ratio r=n/T, where n is the size of the portfolio and T the length of the available time series. On the basis of numerical experiments and analytic results for some toy portfolio models we show that for relatively large values of r (e.g. 0.6) noise does, indeed, have the pronounced effect suggested by Galluccio et al. (1998), Laloux et al. (1999) and Plerou et al. (1999) and illustrated later by Laloux et al. (Int. J. Theor. Appl. Finance 3 (2000) 391), Plerou et al. (Phys. Rev. E, e-print cond-mat/0108023) and Rosenow et al. (Europhys. Lett., e-print cond-mat/0111537) in a portfolio optimization context, while for smaller r (around 0.2 or below), the error due to noise drops to acceptable levels. Since the length of available time series is for obvious reasons limited in any practical application, any bound imposed on the noise-induced error translates into a bound on the size of the portfolio. In a related set of experiments we find that the effect of noise depends also on whether the problem arises in asset allocation or in a risk measurement context: if covariance matrices are used simply for measuring the risk of portfolios with a fixed composition rather than as inputs to optimization, the effect of noise on the measured risk may become very small.

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    Bibliographic Info

    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 319 (2003)
    Issue (Month): C ()
    Pages: 487-494

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    Handle: RePEc:eee:phsmap:v:319:y:2003:i:c:p:487-494

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    Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

    Related research

    Keywords: Noisy covariance matrices; Random matrix theory; Portfolio optimization; Risk management;

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    Cited by:
    1. Beno\^it Collins & David McDonald & Nadia Saad, 2013. "Compound Wishart Matrices and Noisy Covariance Matrices: Risk Underestimation," Papers 1306.5510, arXiv.org.
    2. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 187-208.
    3. Giacomo Livan & Jun-ichi Inoue & Enrico Scalas, 2012. "On the non-stationarity of financial time series: impact on optimal portfolio selection," Papers 1205.0877, arXiv.org, revised Jul 2012.
    4. Kondor, Imre & Pafka, Szilard & Nagy, Gabor, 2007. "Noise sensitivity of portfolio selection under various risk measures," Journal of Banking & Finance, Elsevier, vol. 31(5), pages 1545-1573, May.
    5. Diane Wilcox & Tim Gebbie, 2004. "An analysis of Cross-correlations in South African Market data," Papers cond-mat/0402389, arXiv.org, revised Sep 2006.
    6. Hirschberger, Markus & Qi, Yue & Steuer, Ralph E., 2007. "Randomly generating portfolio-selection covariance matrices with specified distributional characteristics," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1610-1625, March.
    7. Rosenow, Bernd, 2008. "Determining the optimal dimensionality of multivariate volatility models with tools from random matrix theory," Journal of Economic Dynamics and Control, Elsevier, vol. 32(1), pages 279-302, January.
    8. Imre Kondor & Szilard Pafka & Gabor Nagy, 2006. "Noise sensitivity of portfolio selection under various risk measures," Papers physics/0611027, arXiv.org.
    9. Wilcox, Diane & Gebbie, Tim, 2007. "An analysis of cross-correlations in an emerging market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 375(2), pages 584-598.
    10. Takashi Shinzato, 2014. "Self-Averaging Property of Minimal Investment Risk of Mean-Variance Model," Papers 1404.5222, arXiv.org, revised Apr 2014.
    11. Jushan Bai & Shuzhong Shi, 2011. "Estimating High Dimensional Covariance Matrices and its Applications," Annals of Economics and Finance, Society for AEF, vol. 12(2), pages 199-215, November.
    12. Schäfer, Rudi & Guhr, Thomas, 2010. "Local normalization: Uncovering correlations in non-stationary financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3856-3865.
    13. Lan Liu & Hao Lin, 2010. "Covariance estimation: do new methods outperform old ones?," Journal of Economics and Finance, Springer, vol. 34(2), pages 187-195, April.

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