Abstract:
Markowitz's (1952) seminal work on Modern Portfolio Theory (MPT) describes a methodology to construct an optimal portfolio of risky stocks. The constructed portfolio is based on a trade-off between risk and reward, and will depend on the risk- return preferences of the investor. Implementation of MPT requires estimation of the expected returns and variances of each of the stocks, and the associated covariances between them. Historically, the sample mean vector and variance-covariance matrix have been used for this purpose. However, estimation errors result in the optimised portfolios performing poorly out-of-sample. This dissertation considers two approaches to obtaining a more robust estimate of the variance-covariance matrix. The first is Random Matrix Theory (RMT), which compares the eigenvalues of an empirical correlation matrix to those generated from a correlation matrix of purely random returns. Eigenvalues of the random correlation matrix follow the Marcenko-Pastur density, and lie within an upper and lower bound. This range is referred to as the "noise band". Eigenvalues of the empirical correlation matrix falling within the "noise band" are considered to provide no useful information. Thus, RMT proposes that they be filtered out to obtain a cleaned, robust estimate of the correlation and covariance matrices. The second approach uses L-moments, rather than conventional sample moments, to estimate the covariance and correlation matrices. L-moment estimates are more robust to outliers than conventional sample moments, in particular, when sample sizes are small. We use L-moments in conjunction with Random Matrix Theory to construct the minimum variance portfolio. In particular, we consider four strategies corresponding to the four different estimates of the covariance matrix: the L-moments estimate and sample moments estimate, each with and without the incorporation of RMT. We then analyse the performance of each of these strategies in terms of their risk-return characteristics, their performance and their diversification.
Reference:
Ushan, W. 2015. Portfolio selection using Random Matrix theory and L-Moments. University of Cape Town.
Includes bibliographical references