Risk-return portfolio modelling

Master Thesis


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University of Cape Town

Markowitz introduced the concept of modelling the risk associated with a given security as the variance of the expected return and showed how under certain conditions an investors portfolio can be managed by balancing the expected return of the portfolio and its variance. Building on Markowitz original framework, William Sharpe, extended these ideas by connecting a portfolio to a risky asset. This extension became known as the Sharpe Index Model. There are number of assumptions governing the residuals of the Sharpe index model, one being that the error terms of the stocks are uncorrelated. The Troskie-Hossain innovation to the Sharpe Index model relaxes this assumption. We evaluate the Troskie-Hossain model relative to the Sharpe Index Model and Markowitz portfolio, and find that the Troskie-Hossain model approximates the Markowitz efficient frontier and optimal portfolio very closely. Further examining the residuals, we find evidence of autocorrelation and heteroskedasticity. Using ARMA to model the autocorrelation of the residuals has very little impact on the efficient frontier when working with log returns. However when working with simple returns the ARMA shifts the efficient frontier to the left. We find that GARCH(l , 1) models capture most of the autocorrelation in the squared residuals for both simple returns and log returns and shifts the efficient frontier to the left. Modelling a non-constant conditional mean and non-constant conditional variance (ARMA and GARCH) has proven difficult. The more complex a model becomes the more difficult the estimation. We investigate the effects of dividend yields on the efficient frontier, as well as using simple returns vs log returns in portfolio construction. Including dividend yields in our return data shifts the efficient frontier upwards. However only the a's are increased, and the f3's and f3 t-statistics of the shares remain the same. This shift effect of dividends has no impact on the time series or heteroskedastic models. The simple returns efficient frontier lies above that of the log returns efficient frontier. The a 's for simple returns are very different to those of log returns, however the f3's lie in a similar region to those of log returns.