ماتریس کوواریانس پر سر و صدا و بهینه سازی مجموعه دوم

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.

Covariance matrices of financial returns play a crucial role in several branches of finance such as investment theory, capital allocation or risk management. For example, these matrices are the key input parameters to Markowitz's classical portfolio optimization problem [6], which aims at providing a recipe for the composition of a portfolio of assets such that risk (quantified by the standard deviation of the portfolio's return) is minimized for a given level of expected return. For any practical use of the theory it would therefore be necessary to have reliable estimates for the volatilities and correlations of the returns on the assets making up the portfolio (i.e., for the elements of the covariance matrix), which are usually obtained from historical return series. However, the finite length T of the empirical time series inevitably leads to the appearance of noise (measurement error) in the covariance matrix estimates. It is clear that this noise becomes stronger and stronger with increasing portfolio size n, until at a certain n one overexploits the available information to such a degree that the positive definiteness of the covariance matrix (and with that the meaning of the whole exercise) is lost. This long known difficulty has been put into a new light by Galluccio et al. [1], Laloux et al. [2] and Plerou et al. [3] where the problem has been approached from the point of view of random matrix theory. These studies have shown that empirical correlation 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 Ref. [2], e.g., it is reported that about 94% of the spectrum of correlation matrices determined from return series on the S&P 500 stocks can be fitted by that of a random matrix. One wonders how, under such circumstances, covariance matrices can be of any use in finance. Indeed, in Ref. [2] the authors conclude that “Markowitz's portfolio optimization scheme based on a purely historical determination of the correlation matrix is inadequate”. Two subsequent studies [4] and [5] found that the risk level of optimized portfolios could be improved if prior to optimization one filtered out the lower part of the eigenvalue spectrum of the covariance matrix, thereby removing the noise (at least partially). In both of these studies, portfolios have been optimized by using the covariance matrix extracted from the first half of the available empirical sample, while risk was measured as the standard deviation of the return on these portfolios in the second half of the sample. Laloux et al. [4] and Plerou et al. and Rosenow et al. [5] found a significant discrepancy between “predicted“ risk (as given by the standard deviation of the optimal portfolio in the first half of the sample) and “realized” risk (given by its actual realization in the second half), although this discrepancy could be diminished by the use of the filtering technique. While these results suggest potential applications of random matrix theory, they also reinforce the doubts about the usefulness of empirical covariance matrices. On the other hand, Markowitz's theory is one of the pillars of present day finance. For example, the Capital Asset Pricing Model (CAPM), which plays a kind of benchmark role in portfolio management, was inspired by Markowitz's approach; various techniques of capital allocation are based on similar ideas. Furthermore, over the years, covariance matrices have found industry-wide applications also in risk management. For example, RiskMetrics [7], which is perhaps the most widely accepted methodology for measuring market risk, uses covariance matrices as its fundamental inputs. The presence of such a high degree of noise in empirical covariance matrices as suggested by Gallucio et al. [1], Laloux et al. [2], Plerou et al. [3], Laloux et al. [4], and Plerou et al. and Rosenow et al. [5] and the fact that these matrices are so widely utilized in the financial industry constitute an intriguing paradox. The motivation for our previous study [8] stemmed from this context. In addition to the noise due to the finite length of time series, real data always contain additional sources of error (non-stationarity, changes in the composition of the portfolio, in regulation, in fundamental market conditions, etc.). In order to get rid of these parasitic effects, we based our analysis on data artificially generated from some toy models. This procedure offers a major advantage in that the “true” parameters of the underlying stochastic process, hence also the statistics of the covariance matrix are exactly known. Furthermore, with a comparison to empirical data in mind, where the determination of expected returns becomes an additional source of uncertainty, we confined ourselves to the study of the minimal risk portfolio. Our main finding was that for parameter values typically encountered in practice the “true” risk of the minimum-risk portfolio determined in the presence of noise (i.e., based on the covariance matrix deduced from finite time series) is usually no more that 10–15% higher than that of the portfolio determined from the “true” covariance matrix. In the present work we continue and extend our previous analysis, but keep to the same toy-model-based approach as before. These models can be treated both numerically and, in the limit when n and T go to infinity with r=n/T=fixed, analytically. Varying the ratio r=n/T we show that the difference between “predicted” and “realized” risk can, indeed, reach the high values found in [4] and [5] when r is chosen as large as in those papers, but decreases significantly for smaller values of this ratio. This observation eliminates the apparent contradiction between Laloux et al. [4] and Plerou et al. and Rosenow et al. [5] and our earlier results [8]. Since in the simulation framework we know the exact process, not only its finite realizations, we can compare the “predicted” and “realized” risk to the “true” risk of the portfolio. We find that “realized” risk is a good proxy for “true” risk in all cases of practical importance and that “predicted” risk is always below, whereas “realized” risk is above the “true” risk. For asymptotically small values of n/T all the noise vanishes, but the value of T is, for evident reasons, limited in any practical application, therefore any bound one would like to impose on the effect of noise translates, in fact, into a constraint on the portfolio size n. Regarding one other aspect of the problem, we find that the effect of noise is very different depending on whether we wish to optimize the portfolio, or merely want to measure the risk of a given, fixed portfolio. While in the former case the effect of noise remains important up to relatively small values of n/T, in the latter case it becomes insignificant much sooner. This explains why covariance matrices could have remained a fundamental risk management tool even to date.

In this paper, we have studied the implications of noisy covariance matrices on portfolio optimization and risk management. The main motivation for this analysis was the apparent contradiction between results obtained on the basis of random matrix theory and the fact that covariance matrices are so widely utilized for investment or risk management purposes. Using a simulation-based approach we have shown that for parameter values typically encountered in practice the effect of noise on the risk of the optimal portfolio may not necessarily be as large as one might expect on the basis of the results of [1], [2] and [3]. The large discrepancy between “predicted” and “realized” risk obtained in [4] and [5] can be explained by the low values of T/n used in these studies; for larger T/n the effect becomes much smaller. The analytic formulae derived in this paper provide a lower bound on the effect of noise, or, conversely, an upper bound on the size of the portfolios whose risk can be estimated with a prescribed error. Finally, we have shown that for portfolios with weights determined independently from the covariance matrix data, the effect of noise on the risk measurement process is quite small. A very interesting topic for further research would be to analyze the magnitude of these effects for portfolios constructed by techniques really used in practice, for example by an asset manager or hedge fund, since these methods often combine sophisticated optimization schemes with more subjective expert assessments.