https://doi.org/10.1140/epjb/e20020153
Noisy covariance matrices and portfolio optimization
1
Department of Physics of Complex Systems, Eötvös University,
Pázmány P. sétány 1/a, 1117 Budapest, Hungary
2
Market Risk Research Department, Raiffeisen Bank,
Akadémia u. 6, 1054 Budapest, Hungary
Corresponding author: a syl@complex.elte.hu
Received:
31
December
2001
Published online: 15 May 2002
According to recent findings [1,2], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [1], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [1,2] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions.
PACS: 87.23.Ge – Dynamics of social systems / 05.45.Tp – Time series analysis / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2002