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Noisy Covariance Matrices and Portfolio Optimization II

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  • Szilard Pafka
  • Imre Kondor
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    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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    File URL: http://arxiv.org/pdf/cond-mat/0205119
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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. 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 wpe_270, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
    2. 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, vol. 389(16), pages 3250-3253.
    3. 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.
    4. 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.
    5. 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.
    6. 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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