Dynamic and robust estimation of risk and return in modern portfolio theory

The portfolio selection method developed by Markowitz gives a rational investor a way of evaluating different investment options in a portfolio using the expected return and variance of the returns. Sharpe uses the same optimization approach but estimates the mean and covariance in a regression framework using the index models. Sharpe makes a crucial assumption that the residuals from different assets are uncorrelated and that the beta estimates are constant. When the Sharpe model parameters are estimated using ordinary least squares, the regression assumptions are violated when there is significant autocorrelation and heteroskedasticity in the residuals. Furthermore, the presence of outlying observations in the data leads to unreliable estimates when the ordinary least squares method is used. We find significant correlation in the residuals from different shares and thus we use the Troskie-Hossain model which relaxes this assumption and ultimately produces an efficient frontier that is almost identical to the Markowitz model. The combination of the GARCH and AR models to remove both autocorrelation and heteroskedasticity is used on the single index model and it causes the efficient frontier to shift significantly to the left. Using dynamic estimation through the Kalman filter, it is noticed that the beta coefficients are not constant and that the resulting efficient frontiers significantly outperform the Sharpe model. In order to deal with the problem of outlying observations in the data, we propose using the Minimum Covariance Determinant, (MCD) estimator as a robust version of the Markowitz formulation. Robust alternatives to the ordinary lea.st squares estimator are also investigated and they all cause the efficient frontier to shift to the left. Finally, to solve the problem of collinearity in the multiple index framework, we construct orthogonal indices using principal components regression to estimate the efficient frontier.

Grantor dc:publisher.institution

Department of Mathematics and Applied Mathematics

Year dc:date.issued

2008

Author dc:creator

• Mupambirei, Rodwel

Advisor dc:contributor.advisor

• Troskie, Casper G

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/11427/4913

OAI identifier oai:identifier

oai:open.uct.ac.za:11427/4913