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

Aging and diffusion in low dimensional environments

We study out of equilibrium dynamics and aging for a particle diffusing in one dimensional environments, such as the random force Sinai model, as a toy model for low dimensional systems. We study fluctuations of two times $(t_w, t)$ quantities from the probability distribution $Q(z,t,t_w)$ of the relative displacement $z = x(t) - x(t_w)$ in the limit of large waiting time $t_w \to \infty$ using numerical and analytical techniques. We find three generic large time regimes: (i) a quasi-equilibrium regime (finite $\tau=t-t_w$) where $Q(z,\tau)$ satisfies a general FDT equation (ii) an asymptotic diffusion regime for large time separation where $Q(z) dz \sim \bar{Q}[L(t)/L(t_w)] dz/L(t)$ (iii) an intermediate ``aging'' regime for intermediate time separation ($h(t)/h(t_w)$ finite), with $Q(z,t,t') = f(z,h(t)/h(t'))$. In the unbiased Sinai model we find numerical evidence for regime (i) and (ii), and for (iii) with $\bar{Q(z,t,t')} = Q_0(z) f(h(t)/h(t'))$ and $h(t) \sim \ln t$. Since $h(t) \sim L(t)$ in Sinai's model there is a singularity in the diffusion regime to allow for regime (iii). A directed model, related to the biased Sinai model is solved and shows (ii) and (iii) with strong non self-averaging properties. Similarities and differences with mean field results are discussed. A general approach using scaling of next highest encountered barriers is proposed to predict aging properties, $h(t)$ and $f(x)$ in landscapes with fast growing barriers. We introduce a new exactly solvable model, with barriers and wells, which shows clearly diffusion and aging regimes with a rich variety of functions $h(t)$.Comment: 43 pages, 19 .eps figures, RevTe

Signal and Noise in Financial Correlation Matrices

Using Random Matrix Theory one can derive exact relations between the eigenvalue spectrum of the covariance matrix and the eigenvalue spectrum of its estimator (experimentally measured correlation matrix). These relations will be used to analyze a particular case of the correlations in financial series and to show that contrary to earlier claims, correlations can be measured also in the ``random'' part of the spectrum. Implications for the portfolio optimization are briefly discussed.Comment: 6 pages + 2 figures, corrected references, Talk at Conference: Applications of Physics in Financial Analysis 4, Warsaw, 13-15 November 200

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

Risk Minimization through Portfolio Replication

We use a replica approach to deal with portfolio optimization problems. A given risk measure is minimized using empirical estimates of asset values correlations. We study the phase transition which happens when the time series is too short with respect to the size of the portfolio. We also study the noise sensitivity of portfolio allocation when this transition is approached. We consider explicitely the cases where the absolute deviation and the conditional value-at-risk are chosen as a risk measure. We show how the replica method can study a wide range of risk measures, and deal with various types of time series correlations, including realistic ones with volatility clustering.Comment: 12 pages, APFA5 conferenc