Noisy Covariance Matrices and Portfolio Optimization

According to recent findings [1,2], empirical covariance 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 [1], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [1,2] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions.Comment: 7 pages, 3 figure

Normal Helium 3: a Mott-Stoner liquid

A physical picture of normal liquid $^3$He, which accounts for both ``almost localized'' and ``almost ferromagnetic'' aspects, is proposed and confronted to experiments.Comment: 4 pages, RevTeX3.0, 1 EPS figur

Note on log-periodic description of 2008 financial crash

We analyze the financial crash in 2008 for different financial markets from the point of view of log-periodic function model. In particular, we consider Dow Jones index, DAX index and Hang Seng index. We shortly discuss the possible relation of the theory of critical phenomena in physics to financial markets.Comment: 13 pages, 7 figures; references and few comments added

Random Matrix Theory and Fund of Funds Portfolio Optimisation

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.Comment: 17 Page

Estimated Correlation Matrices and Portfolio Optimization

Financial correlations play a central role in financial theory and also in many practical applications. From theoretical point of view, the key interest is in a proper description of the structure and dynamics of correlations. From practical point of view, the emphasis is on the ability of the developed models to provide the adequate input for the numerous portfolio and risk management procedures used in the financial industry. This is crucial, since it has been long argued that correlation matrices determined from financial series contain a relatively large amount of noise and, in addition, most of the portfolio and risk management techniques used in practice can be quite sensitive to the inputs. In this paper we introduce a model (simulation)-based approach which can be used for a systematic investigation of the effect of the different sources of noise in financial correlations in the portfolio and risk management context. To illustrate the usefulness of this framework, we develop several toy models for the structure of correlations and, by considering the finiteness of the time series as the only source of noise, we compare the performance of several correlation matrix estimators introduced in the academic literature and which have since gained also a wide practical use. Based on this experience, we believe that our simulation-based approach can also be useful for the systematic investigation of several other problems of much interest in finance

Random Matrix Theory and the Failure of Macroeconomic Forecasts

By scientific standards, the accuracy of short-term economic forecasts has been poor, and shows no sign of improving over time. We form a delay matrix of time-series data on the overall rate of growth of the economy, with lags spanning the period over which any regularity of behaviour is postulated by economists to exist. We use methods of random matrix theory to analyse the correlation matrix of the delay matrix. This is done for annual data from 1871 to 1994 for 17 economies, and for post-war quarterly data for the US and the UK. The properties of the eigenvalues and eigenvectors of these correlation matrices are similar, though not identical, to those implied by random matrix theory. This suggests that the genuine information content in economic growth data is low, and so forecasting failure arises from inherent properties of the data.Comment: 15 Pages, 2 Figure