estimated correlation matrices and portfolio optimization abstract 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

dynamics of competition between collectivity and noise in the stock market abstract detailed study of the financial empirical correlation matrix of the 30 companies which deutsche aktienindex dax comprised during the period of the last 11 years using the time window of 30 trading days is presented this allows clear identification of a nontrivial time dependence of the resulting correlations in addition as a rule the drawdowns are always accompanied by a sizable separation of one strong collective eigenstate of the correlation matrix which at the same time reduces the variance of the noise states the opposite applies to drawups in this case the dynamics spreads more uniformly over the eigenstates which results in an increase of the total information entropy analogous study of the market corresponding to daw jones industrial average djia leads to similar conclusions in the latter case however the correlations are weaker on average one possible reason for this effect is that the market represented by djia is less susceptible to various external factors than the one represented by dax

random matrix theory and fund of funds portfolio optimisation abstract the proprietary nature of hedge fund investing means that it is common practise for managers to release minimal information about their returns the construction of a fund of hedge funds portfolio requires a correlation matrix which often has to be estimated using a relatively small sample of monthly returns data which induces noise in this paper random matrix theory rmt is applied to a cross correlation matrix c constructed using hedge fund returns data the analysis reveals a number of eigenvalues that deviate from the spectrum suggested by rmt the components of the deviating eigenvectors are found to correspond to distinct groups of strategies that are applied by hedge fund managers the inverse participation ratio is used to quantify the number of components that participate in each eigenvector finally the correlation matrix is cleaned by separating the noisy part from the non noisy part of c this technique is found to greatly reduce the difference between the predicted and realised risk of a portfolio leading to an improved risk profile for a fund of hedge funds

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

how are rescaled range analyses affected by different memory and distributional properties a monte carlo study abstract financial data has been extensively studied for correlations using pearson s cross correlation coefficient as the point of departure we employ an estimator based on recurrence plots the correlation of probability of recurrence c p r to analyze connections between nine stock indices spread worldwide we suggest a slight modification of the c p r approach in order to get more robust results we examine trends in c p r for an approximately 19 month window moved along the time series and compare them to trends in binning c p r into three levels of connectedness strong moderate and weak we extract the trends in number of connections in each bin over time we also look at the behavior of c p r during the dot com bubble by shifting the time series to align their peaks c p r mainly uncovers that the markets move in and out of periods of strong connectivity erratically instead of moving monotonically towards increasing global connectivity this is in contrast to which gives a picture of ever increasing correlation c p r also exhibits that time shifted markets have high connectivity around the dot com bubble of 2000 we use significance tests using twin surrogates to interpret all the measures estimated in the study

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

cross correlation dynamics in financial time series abstract the dynamics of the equal time cross correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows empirical results for the s p 500 and the dow jones euro stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross correlation matrix over these time windows oppose those of the largest eigenvalue this behaviour is shown to be independent of the size of the time window and the number of stocks examined a basic one factor model is then proposed which captures the main dynamical features of the eigenvalue spectrum of the empirical data through the addition of perturbations to the one factor model leading to a market plus sectors model additional sectoral features are added resulting in an inverse participation ratio comparable to that found for empirical data by partitioning the eigenvalue time series we then show that negative index returns drawdowns are associated with periods where the largest eigenvalue is greatest while positive index returns drawups are associated with periods where the largest eigenvalue is smallest the study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

phase synchronization based minimum spanning trees for analysis of financial time series with nonlinear correlations abstract the cross correlation coefficient has been widely applied in financial time series analysis in specific for understanding chaotic behaviour in terms of stock price and index movements during crisis periods to better understand time series correlation dynamics the cross correlation matrices are represented as networks in which a node stands for an individual time series and a link indicates cross correlation between a pair of nodes these networks are converted into simpler trees using different schemes in this context minimum spanning trees mst are the most favoured tree structures because of their ability to preserve all the nodes and thereby retain essential information imbued in the network although cross correlations underlying msts capture essential information they do not faithfully capture dynamic behaviour embedded in the time series data of financial systems because cross correlation is a reliable measure only if the relationship between the time series is linear to address the issue this work investigates a new measure called phase synchronization ps for establishing correlations among different time series which relate to one another linearly or nonlinearly in this approach the strength of a link between a pair of time series nodes is determined by the level of phase synchronization between them we compare the performance of phase synchronization based mst with cross correlation based mst along selected network measures across temporal frame that includes economically good and crisis periods we observe agreement in the directionality of the results across these two methods they show similar trends upward or downward when comparing selected network measures though both the methods give similar trends the phase synchronization based mst is a more reliable representation of the dynamic behaviour of financial systems than the cross correlation based mst because of the former s ability to quantify nonlinear relationships among time series or relations among phase shifted time series

not all that glitters is rmt in the forecasting of risk of portfolios in the brazilian stock market abstract using stocks of the brazilian stock exchange bm f bovespa we build portfolios of stocks based on markowitz s theory and test the predicted and realized risks this is done using the correlation matrices between stocks and also using random matrix theory in order to clean such correlation matrices from noise we also calculate correlation matrices using a regression model in order to remove the effect of common market movements and their cleaned versions using random matrix theory this is done for years of both low and high volatility of the brazilian stock market from 2004 to 2012 the results show that the use of regression to subtract the market effect on returns greatly increases the accuracy of the prediction of risk and that although the cleaning of the correlation matrix often leads to portfolios that better predict risks in periods of high volatility of the market this procedure may fail to do so the results may be used in the assessment of the true risks when one builds a portfolio of stocks during periods of crisis

the reaction of stock markets to crashes and events a comparison study between emerging and mature markets using wavelet transforms abstract we study here the behaviour of the first three eigenvalues 1 2 3 and their ratios 1 2 1 3 2 3 of the covariance matrices of the original return series and of those rebuilt from wavelet components for emerging and mature markets it has been known for some time that the largest eigenvalue 1 contains information on the risk associated with the particular assets of which the covariance matrix is comprised here we wish to ascertain whether the subdominant eigenvalues 2 3 hold information on the risk of the stock market and also to measure the recovery time for emerging and mature markets to do this we use the discrete wavelet transform which gives a clear picture of the movements in the return series by reconstructing them using each wavelet component our results appear to indicate that mature markets respond to crashes differently to emerging ones in that emerging markets may take up to two months to recover while major markets take less than a month to do so in addition the results appears to show that the subdominant eigenvalues 2 3 give additional information on market movement especially for emerging markets and that a study of the behaviour of the other eigenvalues may provide insight on crash dynamics

one factor model for the cross correlation matrix in the vietnamese stock market abstract random matrix theory rmt has been applied to the analysis of the cross correlation matrix of a financial time series the most important findings of previous studies using this method are that the eigenvalue spectrum largely follows that of random matrices but the largest eigenvalue is at least one order of magnitude higher than the maximum eigenvalue predicted by rmt in this work we investigate the cross correlation matrix in the vietnamese stock market using rmt and find similar results to those of studies realized in developed markets us europe japan 9 18 as well as in other emerging markets 20 21 19 22 importantly we found that the largest eigenvalue could be approximated by the product of the average cross correlation coefficient and the number of stocks studied we demonstrate this dependence using a simple one factor model the model could be extended to describe other characteristics of the realistic data

signal and noise in correlation matrix using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator this relation can be used in practice to compute eigenvalue invariants of the covariance correlation matrix results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom like for instance those in statistical physics lattice measurements of field theory genetics quantitative finance and other applications of multivariate statistics

model for correlations in stock markets we propose a group model for correlations in stock markets in the group model the markets are composed of several groups within which the stock price fluctuations are correlated the spectral properties of empirical correlation matrices reported recently are well understood from the model it provides the connection between the spectral properties of the empirical correlation matrix and the structure of correlations in stock markets

topological properties of stock networks based on minimal spanning tree and random matrix theory in financial time series abstract we investigated the topological properties of stock networks constructed by a minimal spanning tree we compared the original stock network with the estimated network the original network is obtained by the actual stock returns while the estimated network is the correlation matrix created by random matrix theory we found that the consistency between the two networks increases as more eigenvalues are considered in addition we suggested that the largest eigenvalue has a significant influence on the formation of stock networks

on the analysis of cross correlations in south african market data we report briefly on an application of random matrix theory to the analysis of sa financial market data an analysis of cross correlations in south african financial market data e print cond mat 0402389 correlation matrices c are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange from january 1993 to december 2002 spectral properties of c are tested against random matrix predictions we highlight some quantitative differences which arise when treating prices as existing only when measured as opposed to interpolating missing or illiquid trading days with a zero order hold

dynamics of cross correlations in the stock market co movements of stock price uctuations are described by the cross correlation matrix c the application of random matrix theory rmt allows to distinguish between spurious correlations in c due to measurement noise and true correlations containing economically meaningful information by calculating cross correlations for di3erent time windows we study the time dependence of eigenvectors of c which are related to economic sectors and the time evolution of the largest eigenvalue which describes the average correlation strength we use these results to forecast cross correlations and test the quality of our forecast by constructing investments in the stock market which expose the invested capital to a minimum level of risk only c

reply to comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable in a previous paper sk la and c zek 1996 a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable was suggested this method has been analysed in a comment by guardiola and ros 1996 the present comment is the reply to this previous comment the aim is to give a simple and straightforward proof of the method and clarify some other points important for its application

identifying complexity by means of matrices abstract complexity is an interdisciplinary concept which first of all addresses the question of how order emerges out of randomness for many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related characteristics based on several natural complex dynamical systems like the strongly interacting quantum many body systems the human brain and the financial markets by relating empirical observations to the random matrix theory and quantifying deviations in terms of a reduced dimensionality we present arguments in favour of the statement that complexity is a phenomenon at the edge between collectivity and chaos

cross correlations in an emerging market financial data abstract we report the results of a study of the spectral properties of the correlation matrix of price fluctuations in the stock prices in an emerging market istanbul stock exchange market by using random matrix theory although properties such as the distributions of correlation matrix elements eigenvalue spacings and components of the eigenvectors show similar trends to those observed for the other markets the eigenvalue distribution is found to be markedly different compared to similar studies

fractional matching preclusion of product networks abstract the matching preclusion number of a graph introduced in 2 as a fault analysis is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings as a generalization liu and liu 14 recently introduced the concept of fractional matching preclusion number the fractional matching preclusion number of graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matching if the sets of edges of the graph attaining the minimum are precisely those incident to a single vertex of minimum degree we say such graph is fractional super matched in this paper the upper and lower bounds for the fractional matching preclusion number for cartesian product direct product strong product and lexicographic product are obtained and we give sufficient conditions for such graphs to be fractional super matched

rational decisions random matrices and spin glasses we consider the problem of rational decision making in the presence of nonlinear constraints by using tools borrowed from spin glass and random matrix theory we focus on the portfolio optimisation problem we show that the number of optimal solutions is generally exponentially large and each of them is fragile rationality is in this case of limited use in addition this problem is related to spin glasses with levy like long ranged couplings for which we show that the ground state is not exponentially degenerate

comparing the structure of an emerging market with a mature one under global perturbation a b s t r a c t in this paper we investigate the tehran stock exchange tse and dow jones industrial average djia in terms of perturbed correlation matrices to perturb a stock market there are two methods namely local and global perturbation in the local method we replace a correlation coefficient of the cross correlation matrix with one calculated from two gaussian distributed time series whereas in the global method we reconstruct the correlation matrix after replacing the original return series with gaussian distributed time series the local perturbation is just a technical study we analyze these markets through two statistical approaches random matrix theory rmt and the correlation coefficient distribution by using rmt we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks we find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with gaussian distributed time series the mean values of correlation coefficients the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the rmt prediction fall sharply in both markets by comparing these two markets we can see that the djia is more sensitive to global perturbations these findings are crucial for risk management and portfolio selection

data clustering and noise undressing of correlation matrices we discuss an approach to data clustering we find that maximum likelihood leads naturally to an hamiltonian of potts variables that depends on the correlation matrix and whose low temperature behavior describes the correlation structure of the data for random uncorrelated data sets no correlation structure emerges on the other hand for data sets with a built in cluster structure the method is able to detect and recover efficiently that structure finally we apply the method to financial time series where the low temperature behavior reveals a nontrivial clustering