THESIS
2010
xii, 54 p. : ill. ; 30 cm
Abstract
In modern portfolio theory, the covariance matrices of portfolio asset returns are always needed for different reasons in the design process. Indeed, the problem of estimating the covariance matrix has been studied since very early. But most of the classical estimators only work well when there are enough observations. When the sample size is comparable to the observation dimensions, however, these estimators will performs poorly or breakdown....[
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In modern portfolio theory, the covariance matrices of portfolio asset returns are always needed for different reasons in the design process. Indeed, the problem of estimating the covariance matrix has been studied since very early. But most of the classical estimators only work well when there are enough observations. When the sample size is comparable to the observation dimensions, however, these estimators will performs poorly or breakdown.
In recent years, some new estimators based on random matrix theory have been developed and used in wireless communication systems. They are proved to be consistent, not only when the sample size increases without bound for a fixed observation dimension, but also when the observation dimension increases to infinity at the same rate as the sample size. This property guarantees the good performance of the estimators even with limited sample size.
We first study the global minimum variance portfolio model. After reviewing different estimators, we focus on the shrinkage estimators, in which the shrinkage intensity is the difficult part to choose. A new RMT based estimator is introduced into this problem, to estimate the shrinkage intensity. Simulations base on both synthetic data and real market data have been done to show that the new estimator performs better than all other known estimators.
Secondly, we turn our interest to the portfolio hedging problem. An estimator developed by Mestre, which is used to estimate the eigenvalues and eigenvectors of the covariance matrix, is found to fit perfectly with our single factor model in hedging problem. Thus we compare the new estimator and the classical PCA method, and show that the new estimator has an excellent performance in small sample size scenarios, where the observation dimension and the sample size are comparable in magnitude, and can be used to improve the hedging strategy.
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