École Nationale Supérieure de
Techniques Avancées

Résumé

Within the framework of Markowitz mean/variance optimization, the estimation of the covariance matrix of the returns of the assets is of paramount importance for the investor who wants to select a portfolio or for the asset manager who wishes to control the risks of his portfolio. Investors who rely on diversification tend to hold a large number of different assets, typically hundreds of assets. For example to estimate a covariance matrix of size 500, one needs around four or five years of daily financial data. Also to have a more realistic view of the market behavior, investors tend to consider recent data, therefore the amount of available data to estimate the covariance matrix is limited: although bigger than the size of the object to be estimated it is not much bigger. In this case it is well known that the empirical estimator has limited efficiency especially when combined with Markowitz theory. The goal of this work is to study different estimators and to compare their efficiency. We will focus in particular on two estimators coming from random matrix theory. We will study Ledoit-Péché’s estimator for covariance matrices which are "rotationally invariant" and a recently developed estimator for crosscovariance matrices which enables to study the correlations between assets of different nature (eg: stocks, bonds, change rates, commodities...). Estimation by crossvalidation, a common technique of machine learning will also be considered. We will try to show the interest and efficiency of these estimators for risk management of large financial portfolios.