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Noisy Covariance Matrices and Portfolio Optimization II

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  • Szilard Pafka
  • Imre Kondor

Abstract

Recent studies inspired by results from random matrix theory [1,2,3] 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 [1,2,3] and illustrated later by [4,5] 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

Paper provided by arXiv.org in its series Papers with number cond-mat/0205119.

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Date of creation: May 2002
Date of revision: May 2002
Handle: RePEc:arx:papers:cond-mat/0205119

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Web page: http://arxiv.org/

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Cited by:
  1. Schäfer, Rudi & Guhr, Thomas, 2010. "Local normalization: Uncovering correlations in non-stationary financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 389(18), pages 3856-3865.
  2. Lan Liu & Hao Lin, 2010. "Covariance estimation: do new methods outperform old ones?," Journal of Economics and Finance, Springer, Springer, vol. 34(2), pages 187-195, April.
  3. 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.
  4. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 391(1), pages 187-208.
  5. Takashi Shinzato, 2014. "Self-Averaging Property of Minimal Investment Risk of Mean-Variance Model," Papers 1404.5222, arXiv.org, revised Apr 2014.
  6. Imre Kondor & Szilard Pafka & Gabor Nagy, 2006. "Noise sensitivity of portfolio selection under various risk measures," Papers physics/0611027, arXiv.org.
  7. 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.
  8. Hirschberger, Markus & Qi, Yue & Steuer, Ralph E., 2007. "Randomly generating portfolio-selection covariance matrices with specified distributional characteristics," European Journal of Operational Research, Elsevier, Elsevier, vol. 177(3), pages 1610-1625, March.
  9. Wilcox, Diane & Gebbie, Tim, 2007. "An analysis of cross-correlations in an emerging market," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 375(2), pages 584-598.
  10. Lisewski, Andreas Martin & Lichtarge, Olivier, 2010. "Untangling complex networks: Risk minimization in financial markets through accessible spin glass ground states," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 389(16), pages 3250-3253.
  11. Beno\^it Collins & David McDonald & Nadia Saad, 2013. "Compound Wishart Matrices and Noisy Covariance Matrices: Risk Underestimation," Papers 1306.5510, arXiv.org.
  12. Diane Wilcox & Tim Gebbie, 2004. "An analysis of Cross-correlations in South African Market data," Papers cond-mat/0402389, arXiv.org, revised Sep 2006.
  13. Rosenow, Bernd, 2008. "Determining the optimal dimensionality of multivariate volatility models with tools from random matrix theory," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 32(1), pages 279-302, January.
  14. Sandoval, Leonidas Junior & Bruscato, Adriana & Venezuela, Maria Kelly, 2012. "Building portfolios of stocks in the São Paulo Stock Exchange using Random Matrix Theory," Insper Working Papers, Insper Working Paper, Insper Instituto de Ensino e Pesquisa wpe_270, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
  15. Leonidas Sandoval Junior & Adriana Bruscato & Maria Kelly Venezuela, 2012. "Building portfolios of stocks in the S\~ao Paulo Stock Exchange using Random Matrix Theory," Papers 1201.0625, arXiv.org, revised Mar 2013.

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