Random Matrix Filtering in Portfolio Optimization

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.Comment: 9 pages with 3 EPS figure

Portfolio optimization under expected shortfall: contour maps of estimation error

The contour maps of the error of historical resp. parametric estimates for large random portfolios optimized under the risk measure Expected Shortfall (ES) are constructed. Similar maps for the sensitivity of the portfolio weights to small changes in the returns as well as the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-the-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the optimization for a given number of different assets in the portfolio, at a given confidence level and a given level of relative estimation error. The necessary sample sizes invariably turn out to be unrealistically large for any reasonable choice of the number of assets and the confi dence level. These results are obtained via analytical calculations based on methods borrowed from the statistical physics of random systems, supported by numerical simulations

Analytic approach to variance optimization under an (1) constraint

The optimization of the variance of a portfolio of N independent but not identically distributed assets, supplemented by a budget constraint and an asymmetric (1) regularizer, is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions differently, so the present treatment includes the no-short-constrained portfolio optimization problem as a special case. Results are presented for the out-of-sample and the in-sample estimator of the regularized variance, the relative estimation error, the density of the assets eliminated from the portfolio by the regularizer, and the distribution of the optimal portfolio weights. We have studied the dependence of these quantities on the ratio r of the portfolio's dimension N to the sample size T, and on the strength of the regularizer. We have checked the analytic results by numerical simulations, and found general agreement. Regularization extends the interval where the optimization can be carried out, and suppresses the large sample fluctuations, but the performance of (1) regularization is rather disappointing: if the sample size is large relative to the dimension, i.e. r is small, the regularizer does not play any role, while for r's where the regularizer starts to be felt the estimation error is already so large as to make the whole optimization exercise pointless. We find that the (1) regularization can eliminate at most half the assets from the portfolio (by setting their weights to exactly zero), corresponding to this there is a critical ratio r = 2 beyond which the (1) regularized variance cannot be optimized: the regularized variance becomes constant over the simplex. These facts do not seem to have been noticed in the literature

Formation and Stabilization of Noble Metal Nanoparticles

The kinetics of homogeneous and heterogeneous nucleation processes of metal (Ag, Pd) nanoparticles was studied by UV-VIS spectrometry. Silver nanoparticles were prepared in aqueous solution by homogeneous nucleation using poly(vinylpyrrolidone) (PVP) and sodium citrate as sterical stabilizers. Reduction was ensured by adding hydroquinone. According to kinetic functions, reduction is an autocatalytic process: a slow, continuous nucleation is followed by a fast, autocatalytic growth. The presence of polymer inhibits nucleation and retards the rate of particle growth. Formation of palladium nanoparticles was investigated in aqueous medium via reduction by hydrazine, using PVP and the clay mineral hectorite as stabilizers. Effects of the polymer and concentration of silicate and palladium ions on the particle formation rate were analyzed. The rate of reduction is decreased by increasing amounts of stabilizing agents and increased by increasing concentrations of precursor ions. The kinetics of heterogeneous nucleation was determined based on the adsorption of the palladium species at the clay mineral particles and the viscosity of the hectorite dispersion