Rational decisions, random matrices and spin glasses

Galluccio, Stefano; Bouchaud, Jean‐Philippe; Potters, Marc

doi:10.1016/s0378-4371(98)00332-x

Cited by 107 publications

(95 citation statements)

References 10 publications

(16 reference statements)

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“…In this way we obtain "return series" y it that have a distribution characterized by the "true" covariance matrix σ (0) ij , while σ (1) ij will correspond to the "empirical" covariance matrix. Of course, in the limit T → ∞ the noise disappears and σ (1) ij → σ (0) ij . The main advantage of this simulation approach over empirical studies is that the "true" covariance matrix is exactly known.…”

Section: Resultsmentioning

confidence: 99%

“…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 [1,2,3] and illustrated later by [4,5] 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.…”

mentioning

confidence: 76%

“…This long known difficulty has been put into a new light by [1,2,3] 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.…”

Section: Introductionmentioning

confidence: 99%

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Noisy covariance matrices and portfolio optimization II

Pafka

Kondor

2003

Physica A: Statistical Mechanics and its Applications

110

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Recent studies inspired by results from random matrix theory [1,2,3] 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 [1,2,3] and illustrated later by [4,5] 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.

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Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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Noisy covariance matrices and portfolio optimization II

Pafka

Kondor

2003

Physica A: Statistical Mechanics and its Applications

110

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“…Recently, the problem of understanding the correlations among the returns of different stocks has been addressed by applying methods of random matrix theory to the cross-correlation matrix [47][48][49]. Aside from scientific interest, the study of correlations between the returns of different stocks is also of practical relevance in quantifying the risk of a given portfolio [29].…”

Section: Correlations Among Different Unitsmentioning

confidence: 99%

Financial time series: A physics perspective

Gopikrishnan¹,

Plerou²,

Amaral³

et al. 2000

AIP Conference Proceedings

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Abstract. Physicists in the last few years have started applying concepts and methods of statistical physics to understand economic phenomena. The word "econophysics" is sometimes used to refer to this work. One reason for this interest is the fact that Economic systems such as financial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that economic problems viewed from a different perspective might yield new results. This article reviews the results of a few recent phenomenological studies focussed on understanding the distinctive statistical properties of financial time series. We discuss three recent results -(i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes -from tiny fluctuations to very drastic events, such as market crashes, eg., the crash of October 19th 1987, sometimes referred to as " Black Monday". The distribution of price fluctuations decays with a power-law tail well outside the Levy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days, (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months, (iii) Correlations among different companies: The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. Prom a study of the eigenvalue statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks, we find that the largest 5-10% of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix, whereas the rest of the eigenvalues conform to random matrix behavior -suggesting that these 5-10% of the eigenvalues contain system-specific information about correlated behavior.

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“…With the nonlinear constraint, the portfolio problem is equivalent to ÿnding the ground state of a spin system instead of the mean ÿeld solution. If one uses the empirical cross-correlation matrix C to calculate the interaction matrix of the portfolio free energy, the optimization problem has a spin glass type complexity [20]. When keeping only the ferromagnetic couplings describing the market and cluster correlations, the complexity of the optimization problem is reduced that of a random ÿeld ferromagnet [19].…”

mentioning

confidence: 99%

Random magnets and correlations of stock price fluctuations

Rosenow

Gopikrishnan

Plerou

et al. 2002

Physica A: Statistical Mechanics and its Applications

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Random magnets provide a paradigm for the study of competing interactions and frustration in physics. Here, we suggest that this paradigm is also useful for the study and explanation of correlations between stock price changes of di erent companies: it (i) provides for a mechanism to explain the origin of correlations, (ii) allows to understand the occurrence of power-law correlations in the time series of highly correlated eigenmodes, and (iii) is a useful framework for the analysis of optimal investment strategies where the knowledge of (anti-)correlations is an important prerequisite for the reduction of risk. The theoretical analysis of a physical system often starts by setting up a Hamiltonian describing both the dynamics of the basic constituents and the interactions between them. Then, one proceeds to calculate correlation functions relating physical observables to the microscopic Hamiltonian and compares them to experimental data. This procedure is possible because one has an a priori idea about the description of the system. The situation is di erent when describing economical systems, as there exists no microscopic theory which is applicable to all types of systems. For this reason, an inductive procedure starting with the analysis of empirical correlation function is more adequate. After getting an intuition for the system under consideration, one can proceed

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Rational decisions, random matrices and spin glasses

Cited by 107 publications

References 10 publications

Noisy covariance matrices and portfolio optimization II

Noisy covariance matrices and portfolio optimization II

Financial time series: A physics perspective

Random magnets and correlations of stock price fluctuations

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