برآورد ماتریس همبستگی و بهینه سازی سبد سرمایه گذاری

Correlations of returns on various assets play a central role in financial theory and also in many practical applications. From a theoretical point of view, the main interest lies in the proper description of the structure and dynamics of correlations, whereas for the practitioner the emphasis is on the ability of the models to provide adequate inputs for the numerous portfolio and risk management procedures used in the financial industry. The theory of portfolios, initiated by Markowitz, has suffered from the “curse of dimensions” from the very outset. Over the past decades a large number of different techniques have been developed to tackle this problem and reduce the effective dimension of large bank portfolios, but the efficiency and reliability of these procedures are extremely hard to assess or compare. In this paper, we propose a model (simulation)-based approach which can be used for the systematical testing of all these dimensional reduction techniques. To illustrate the usefulness of our framework, we develop several toy models that display some of the main characteristic features of empirical correlations and generate artificial time series from them. Then, we regard these time series as empirical data and reconstruct the corresponding correlation matrices which will inevitably contain a certain amount of noise, due to the finiteness of the time series. Next, we apply several correlation matrix estimators and dimension reduction techniques introduced in the literature and/or applied in practice. As in our artificial world the only source of error is the finite length of the time series and, in addition, the “true” model, hence also the “true” correlation matrix, are precisely known, therefore in sharp contrast with empirical studies, we can precisely compare the performance of the various noise reduction techniques. One of our recurrent observations is that the recently introduced filtering technique based on random matrix theory performs consistently well in all the investigated cases. Based on this experience, we believe that our simulation-based approach can also be useful for the systematic investigation of several related problems of current interest in finance.

Correlation matrices of financial returns play a crucial role in several branches of modern finance such as investment theory, capital allocation and risk management. For example, financial correlation matrices are the key input parameters to Markowitz's classical portfolio optimization problem [1], which aims at providing a recipe for the selection of a portfolio of assets so 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 correlations of returns (of the assets making up the portfolio), which are usually obtained from historical return series data. However, if one estimates a n×n correlation matrix from n time series of length T each, with T bounded for evident practical reasons, one inevitably introduces estimation error, which for large n can become so overwhelming that the whole applicability of the theory becomes questionable. This difficulty has been well known by economists for a long time (see e.g. [2] and the numerous references therein). Several aspects of the effect of noise (in the correlation matrices determined from empirical data) on the classical portfolio selection problem have been investigated e.g. in Ref. [3]. One way to cope with the problem of noise is to impose some structure on the correlation matrix, which may certainly introduce some bias in the estimation, but by effectively reducing the dimensionality of the problem, could, in fact, be expected to improve the overall performance. Such a best-known structure is that imposed by the single-index (or market) model, which has stimulated strong interest in the academic literature (see e.g. Ref. [2] for an overview and references) and has also become widely used in the financial industry (the coefficient “beta”, relating the returns of an asset to the returns of the corresponding wide market index, has long been a widespread tool in the financial community). On economic or statistical grounds, several other correlation structures have been experimented with in the academic literature and financial industry, for example multi-index models, grouping by industry sectors, macroeconomic factor models, models based on principal component analysis, etc. Several studies (see e.g. Ref. [4]) attempt to compare the performance of these correlation estimation procedures as input providers for the portfolio selection problem, although all these studies have been restricted to the use of given specific empirical samples. More recently, additional procedures to impose some structure on correlations (e.g. Bayesian shrinkage estimators) or bounds directly on the portfolio weights (e.g. no short selling) have been explored, see e.g. Ref. [5]. The general conclusion of all these studies is that reducing the dimensionality of the problem by imposing some structure on the correlation matrix may be of great help for the selection of portfolios with better risk-return characteristics. The problem of estimation noise in financial correlation matrices has been put into a new light by the application of results from random matrix theory [6], [7] and [8]. 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. [7], e.g., it is reported that about 94% of the spectrum of correlation matrices determined from return series of the S&P 500 stocks can be fitted by that of a random matrix. Furthermore, two subsequent studies [9] and [10] have shown that the risk-return characteristics of optimized portfolios could be improved, if prior to optimization one filtered out the lower part of the eigenvalue spectrum of the correlation matrix in an attempt to remove (at least partially) the noise, a procedure similar to principal component analysis. Other approaches inspired by physics and aimed at extracting information from noisy correlation data have been introduced in [11] and [12]. It is important to note that all the above studies have used (given) empirical datasets, which in addition to the noise due to the finite length of the time series, also contain several other sources of error (caused by non-stationarity, market microstructure etc.). The motivation of our previous study [13] came from this context. In order to get rid of these additional sources of error, 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 correlation matrix is exactly known. The key observation of Ref. [13] is that the effect of noise, e.g. in the context of a simple portfolio optimization framework, strongly depends on the ratio T/n, where n is the size of the portfolio while T is the length of the available time series. Moreover, in the limit n→∞, T→∞ but T/n=const. the suboptimality of the portfolio optimized using the “noisy” correlation matrix (with respect to the portfolio obtained using the “true” matrix) is (1−n/T)−1/2 exactly. Therefore, since the length of the time series T is limited in any practical application, any bound one would like to impose on the effect of noise translates, in fact, into a constraint on the portfolio size n. The aim of this paper (besides extending the analysis of the previous study) is to introduce a model (simulation)-based approach that can be generally used for systematically testing and comparing the various noise (or dimension) reduction techniques that have been introduced in the literature and applied in practice. As an illustration of the usefulness of this approach, we introduce several toy models with the goal to progressively incorporate the relevant features of real-life financial correlations and, in the world of these models, we study the effect of noise (here solely due to the estimation error caused by the finiteness of the surrogate time series generated by the models) on a very simple form of the classical portfolio optimization problem. More precisely, we compare the performance of different correlation matrix “estimation” methods (the filtering procedure introduced in [9] and [10] among them) in providing inputs for the selection of portfolios with optimal risk characteristics. Our findings re-confirm the usefulness of techniques that effectively reduce the dimensionality of the correlation matrix for portfolio optimization. The approach we adopt here is, in fact, very common in physics, where one starts with some bare model and progressively adds finer and finer details in order to study the behavior of the “world” embodied by the model by comparing it to real-life (experimental) results. We believe that our model-based approach can be useful for the systematic study of several other problems in which financial correlation matrices play a crucial role.

In this paper, we described a model (simulation)-based approach which can be used for a systematic investigation of the performance of various noise reduction procedures applied in portfolio selection and risk management. To demonstrate the usefulness of this approach we developed several toy models for the structure of financial correlations and, by considering only the noise arising from the finite length of the model-generated time series, we analyzed the performance of several correlation matrix estimation procedures in a simple portfolio optimization context. Our results agree well with the findings of previous empirical studies. The effect of noise in correlation matrices determined from financial series can indeed be large. However, most practitioners use techniques that, by generally reducing the effective dimensionality of the problem, can very efficiently suppress the effect of noise. We found that the filtering based on random matrix theory is particularly powerful in this respect. The success of dimensional reduction procedures goes a long way to explain how correlation matrices that contain a huge amount of noise can nevertheless remain useful in practice.