نظریه ماتریس تصادفی برای بهینه سازی سبد سرمایه گذاری: یک رویکرد پایدار

We apply random matrix theory (RMT) to an empirically measured financial correlation matrix, C, and show that this matrix contains a large amount of noise. In order to determine the sensitivity of the spectral properties of a random matrix to noise, we simulate a set of data and add different volumes of random noise. Having ascertained that the eigenspectrum is independent of the standard deviation of added noise, we use RMT to determine the noise percentage in a correlation matrix based on real data from S&P500. Eigenvalue and eigenvector analyses are applied and the experimental results for each of them are presented to identify qualitatively and quantitatively different spectral properties of the empirical correlation matrix to a random counterpart. Finally, we attempt to separate the noisy part from the non-noisy part of C. We apply an existing technique to cleaning C and then discuss its associated problems. We propose a technique of filtering C that has many advantages, from the stability point of view, over the existing method of cleaning.

Random matrix theory (RMT), originally developed for use in nuclear physics, has been described by, among others, Dyson in a series of papers beginning with Dyson [1] and subsequently in collaboration with Mehta (Mehta and Dyson [2], Dyson and Mehta [3], Mehta [4]) as the matrix representation of the average of all possible interactions in a nucleus. It can be used to identify non-random properties which are deviations from the universal predictions of RMT; properties that are specific to the considered system. Close agreement between the distribution of the eigenvalues of a matrix M, with those from a matrix made up of random entries implies that M has entries that contain a considerable degree of randomness as has been shown in the literature [5], [6], [7] and [8]. This matrix consisting of random elements with unit variance and zero mean is called a random matrix [4]. In the case of a correlation matrix C, the level of agreement between its eigenvalue distribution and those of a random matrix, represents the amount of randomness (or noise) in C and thus, deviations from RMT represent genuine correlation (cf. [5], [6] and [7]). This is precisely the problem that we wish to address, i.e., the identification of the true information (correlated assets) in a noisy financial correlation matrix. The method tests the null hypothesis that the distribution of eigenvalues of the correlation matrix is random. Since the correlation matrix is symmetric, the random matrix, with which it is compared, should also be symmetric [8]. Before applying the cleaning method to real correlation matrices, we need to determine the role of the amount of random noise on the spectral properties of a random matrix. This is done by examining the difference between eigenspectrum of a correlation matrix made up of simulated data with different amounts of random noise and that of a random matrix. We can then proceed with confidence to examine the stability of real correlation matrices using real data. The empirical data set we use consists of 30-min intraday prices from the S&P500 Index from the the beginning of April 1997 to the beginning of April 1999. This provides about 1500 data points for about 450 companies. In this paper our initial objective, therefore, is to separate the noisy part from non-noisy part in C. Removal of the noise makes the optimization process more reliable, leaving the analyst in a better position to estimate the risk associated with the constructed portfolio. However, the techniques for removing noise from C should be considered carefully. A standard technique is initially applied to clean C but assessment of the results achieved reveal that it is not particularly satisfactory on the grounds of stability. We therefore, go on to discuss a filtering technique that takes account of the stability in a more precise way. Advantages of the new approach are validated by application to a financial data set from the S&P500.

We have applied RMT to determine the noise in an empirically measured correlation matrix, C. As a preliminary, we examined RMT results on simulated data with varying volumes of noise. The independence of the number of deviated eigenvalues from the volume of added noise implies that results of RMT are also independent of the amount of noise in the data. For a set of actual data from S&P500 we deduced that less than 5% of the eigenvalues carry useful information with the rest reflecting noise. This is in agreement with previous work of Bouchaud and Potters [14] which indicates that at most 6% of C is information. These results were obtained principally by eigenvalue analysis and confirmed in outline by complementary eigenvector analysis which indicated that the market eigenvector (the eigenvector corresponding to the largest eigenvalue) has a different construction to the other eigenvectors, implying that most information in C is measured by this quantity. Finally we have examined the well-known and commonly used technique for noise removal from a correlation matrix [14]. But, however, find that it decreases the level of stability of C. We have alternatively applied the Krzanowski [25] model to study the stability of the financial correlation matrix after removal of noise and conclude that this offers real improvement on the usual method. The improvement was tested by comparing the realised and the predicted optimal portfolios [14], with expectation of a shorter distance between the realised and the predicted risk for Cclean than that of the original C (attributed by the authors to the higher stability of Cclean). We show that this is not the case and in fact there is a negative relationship between the stability of C and the closeness of the predicted and realised risks. This assertion is also demonstrated by the experiments of filtering C, based on the Krzanowski [25] model. The commonly used technique of noise removal not only fails to assure stability, but can actually lead to a considerable deterioration. This finding offers valuable insight for portfolio optimisation.