abstract

The spectral properties of financial correlation matrices can show features known from completely random matrices. A major reason is noise originating from the finite lengths of the financial time series used to compute the correlation matrix elements. In recent years, various methods have been proposed to reduce this noise, i.e.  to clean the correlation matrices. This is of direct practical relevance for risk management in portfolio optimization. In this contribution, we discuss in detail the power mapping, a new shrinkage method. We show that the relevant parameter is, to a certain extent, self-determined. Due to the “chirality” and the normalization of the correlation matrix, the optimal shrinkage parameter is fixed. We apply the power mapping and the well-known filtering method to market data and compare them by optimizing stock portfolios. We address the rôle of constraints by excluding short selling in the optimization.

abstract

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. In the case of the Generalised Random Energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies \(E_N\lt \beta _{\rm c} N\), where \(\beta _{\rm c}\) is the critical temperature. We also explain the more complex behaviour that sets in at higher energies.

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We discuss the spectral density for standard and free random Lévy matrices in the large \(N\) limit. The eigenvalue spectrum is unbounded with power law tails in both cases.

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We describe a method to determine the eigenvalue density of empirical covariance matrix in the presence of correlations between samples. This is a straightforward generalization of the method developed earlier by the authors for uncorrelated samples (Z. Burda, A. Görlich, J. Jurkiewicz, B. Waclaw, cond-mat/0508341). The method allows for exact determination of the experimental spectrum for a given covariance matrix and given correlations between samples in the limit \(N\rightarrow \, \infty \) and \(N\) / \(T=r=\textrm {const}\) with \(N\) being the number of degrees of freedom and \(T\) being the number of samples. We discuss the effect of correlations on several examples.

abstract

We compare some methods recently used in the literature to detect the existence of a certain degree of common behavior of stock returns belonging to the same economic sector. Specifically, we discuss methods based on random matrix theory and hierarchical clustering techniques. We apply these methods to a portfolio of stocks traded at the London Stock Exchange. The investigated time series are recorded both at a daily time horizon and at a 5-minute time horizon. The correlation coefficient matrix is very different at different time horizons confirming that more structured correlation coefficient matrices are observed for long time horizons. All the considered methods are able to detect economic information and the presence of clusters characterized by the economic sector of stocks. However, different methods present a different degree of sensitivity with respect to different sectors. Our comparative analysis suggests that the application of just a single method could not be able to extract all the economic information present in the correlation coefficient matrix of a stock portfolio.

abstract

This note is a short review of recent results concerning the fluctuation behavior of the largest eigenvalue of a class of random covariance matrices. We also present a concrete application of these results to a model checking problem in time series analysis to highlight their practical relevance.

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Finding the mean of the total number \(N_{\rm tot}\) of stationary points for \(N\)-dimensional random Gaussian landscapes can be reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. First such a reduction is illustrated for a class of models describing energy landscapes of elastic manifolds in random environment, and a general method of attacking the problem analytically is suggested. Then the exact solution to the problem [Y.V. Fyodorov, Phys. Rev. Lett. 92, 240601 (2004) and Phys. Rev. Lett. 93, 149901(E) ( 2004)] for a class of landscapes corresponding to the simplest, yet nontrivial “toy model” with \(N\) degrees of freedom is described. For \(N\gg 1\) our asymptotic analysis reveals a phase transition at some critical value \(\mu _{\rm c}\) of a control parameter \(\mu \) from a phase with finite landscape complexity: \(N_{\rm tot}\sim e^{N{\mit \Sigma }},\,\, {\mit \Sigma }(\mu \lt \mu _{\rm c})\gt 0\) to the phase with vanishing complexity: \({\mit \Sigma }(\mu \gt \mu _{\rm c})=0\). This is interpreted as a transition to a glass-like state of the matter.

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We discuss the wealth condensation mechanism in a simple toy economy in which individual agent’s wealths are distributed according to a Pareto power law and the overall wealth is fixed. The observed behaviour is the manifestation of a transition which occurs in Zero Range Processes (ZRPs) or “balls in boxes” models. An amusing feature of the transition in this context is that the condensation can be induced by increasing the exponent in the power law, which one might have naively assumed penalised greater wealths more.

abstract

The use of multi-antenna arrays has been predicted to provide substantial throughput gains for wireless communication systems. However, these predictions have to be assessed in realistic situations, such as correlated channels and in the presence of interference. In this review, we show results obtained using methods borrowed from statistical physics of random media for the average and the variance of the distribution of the mutual information of multi-antenna systems with arbitrary correlations and interferers. Even though the methods are asymptotic in the sense they are valid in the limit of large antenna numbers, the results are accurate even for small arrays. We also optimize over the input signal covariance with channel covariance feedback and calculate closed-loop capacities. This method provides a simple tool to analyze the statistics of throughput for arrays of any size.

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A short review on multiuser communication systems is given. System design for iterative multiuser decoding is improved by means of large system results from statistical physics and random matrix theory. With the application of multiuser detection for wireless communications in mind, it is shown how a system of linear equations with random coefficients can be solved efficiently exploiting the asymptotic convergence of its eigenvalue spectrum. In addition, the conditional convergence of the diagonal elements of a power of a random matrix drawn from a Marchenko–Pastur ensemble is established.

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The degree of convergence of the business cycles of the economies of the European Union is a key policy issue. In particular, a substantial degree of convergence is needed if the European Central Bank is to be capable of setting a monetary policy which is appropriate to the stage of the cycle of the Euro zone economies. I consider the annual rates of real GDP growth on a quarterly basis in the main economies of the EU (France, Germany, Italy, UK, Spain, Belgium and the Netherlands) over the period 1980Q1–2004Q4. An important empirical question is the degree to which the correlations between these growth rates contain true information rather than noise. The technique of random matrix theory is able to answer this question, and has been applied successfully in the physics journals to financial markets data. I find that the correlations between the growth rates of most of the core EU economies contain substantial amounts of true information, and exhibit considerable stability over time. Even in the late 1970s and early 1980s, these economies moved together closely over the course of the business cycle. There was a slight loosening at the time of German re-unification, but the economies have moved back into close synchronisation. The same result holds when Spain is added to the group of core EU countries. However, the problems of the German economy which arose from the early 1990s onwards has led to Germany becoming increasingly less synchronised with the rest of the core EU. Further, the results obtained with a data set of the converged EU core plus the UK show no real convergence between the UK and this group of economies.

abstract

We study empirical covariance matrices in finance. Due to the limited amount of available input information, these objects incorporate a huge amount of noise, so their naive use in optimization procedures, such as portfolio selection, may be misleading. In this paper we investigate a recently introduced filtering procedure, and demonstrate the applicability of this method in a controlled, simulation environment.

abstract

This contribution to the proceedings of the Cracow meeting on ‘Applications of Random Matrix Theory’ summarises a series of studies, some old and others more recent on financial applications of Random Matrix Theory (RMT). We first review some early results in that field, with particular emphasis on the applications of correlation cleaning to portfolio optimisation, and discuss the extension of the Marčenko–Pastur (MP) distribution to a non trivial ‘true’ underlying correlation matrix. We then present new results concerning different problems that arise in a financial context: (a) the generalisation of the MP result to the case of an empirical correlation matrix (ECM) constructed using exponential moving averages, for which we give a new elegant derivation (b) the specific dynamics of the ‘market’ eigenvalue and its associated eigenvector, which defines an interesting Ornstein–Uhlenbeck process on the unit sphere and (c) the problem of the dependence of ECM’s on the observation frequency of the returns and its interpretation in terms of lagged cross-influences.

abstract

We consider a portfolio of stocks whose returns conform to a stationary, multivariate distribution whose all integer moments are finite. For this portfolio we derive the distribution of eigenvalues of various sample covariance matrices and the moments of the eigenvalue distribution, for a particular type of distribution, in terms of the parameters of the portfolio distribution.

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The bipartite graph connecting products and reviewers of that product is studied empirically in the case of amazon.com. We find that the network has power-law degree distribution on the side of reviewers, while on the side of products the distribution is better fitted by stretched exponential. The spectrum of normalised adjacency matrix shows power-law tail in the density of states. Establishing the community structures by finding localised eigenstates is not straightforward as the localised and delocalised states are mixed throughout the whole support of the spectrum.

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On the basis of a local-projective with nonlinear constraints (LPNC) approach (see K. Urbanowicz, J.A. Hołyst, T. Stemler and H. Benner, Acta Phys. Pol. B35, 2175 (2004)) we develop a method of noise reduction in time series that makes use of constraints appearing due to the continuous character of flows. As opposed to local-projective methods in our method we do not need to determine the Jacobi matrix. The approach has been successfully applied for separating a signal from noise in the Lorenz model and in noisy experimental data obtained from an electronic Chua circuit. The method was then applied for filtering noise in human voice.

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We review a recent formulation of the RNA folding problem as an \(N\times N\) matrix field theory. It is based on a systematic classification of the terms in the partition function according to their topological character. In particular large-\(N\) terms yield the secondary structures, whereas pseudo-knots are obtained by calculating the 1 / \(N^{\,2}\) corrections. We also describe a Monte Carlo algorithm for the prediction of RNA secondary structures with pseudoknots, based on this topological approach.

abstract

I review the approach of using large \(N\) matrix field theory to fold RNA and then discuss a recent simplified model that could be solved analytically. I then outline how entropic contributions could be included starting from first principles.