Dear all,

The first RSS Lancashire and East Cumbria local group seminar of the year will take place at 4pm on Thu 27 Oct in Lab 2 of the Postgraduate Statistics Centre at Lancaster University.  It will be followed by refreshments.

Title: High dimensional portfolio optimization by wavelet thresholding

Speaker: Sébastien Van Bellegem


Abstract:

A financial portfolio is a selection of securities. The aim of optimal allocation is to choose among all possible portfolios the one that minimizes some risk subject to a given expected return. The statistical problem in portfolio allocation is to find the optimal linear combination of random variables (e.g. assets). The risk often takes the form of the variance of this linear combination and we recall in this talk that portfolio optimization is a quadratic programming problem under a linear constraint.

The inputs of this optimization problem are the mean return vector of securities, and their cross-covariance matrix. Classical theory of portfolio allocation assumes these moments to be known. However in practice they need to be estimated and we show that this estimation step has a significant impact on the out-of-sample performance of the selected portfolio. In this talk we are particularly interested by the situation where the number of securities available is large relative to the sample size, a situation we call ``high dimensional''. However, the empirical covariance matrix of high dimensional data is known to be badly conditioned, meaning that its inverse is unstable. We show in this talk that portfolio optimization behaves poorly in this case, since the inverse of the (empirical) covariance matrix is an important tool to select the optimal portfolio.

In this talk we argue that, under realistic assumptions, wavelet bases are well suited to concentrate the information of the covariance matrix on a small number of coefficients. In other words, wavelets achieve some decorrelation among the securities. This phenomenon is extremely useful since it implies that inversion of high dimensional covariance matrices is simpler in the wavelet domain. We exploit this property and introduce a new thresholding rule of the empirical covariance matrix in the wavelet domain, based on a generalization of the so-called "Tree Structured Wavelet (TSW) denoising". In contrast to standard wavelet thresholding approaches, this denoising do not operate on each wavelet coefficient at a time but on groups of coefficients. We show that this method ensures the denoised empirical matrix to be a valid covariance matrix and also demonstrated the good performance of these decision rules on simulations when compared to benchmarks methods.

This is a joint work with Daniel Koch (UCLouvain).


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