Equity portfolio optimization under parameter uncertainty

Abstract

For investors seeking to use equity markets to make money, allocating funds to assets in a given portfolio requires knowledge of the asset return means and covariance. Neither one of these parameters are known when such allocation is made. Traditionally, historical returns are used to estimate these parameters. Literature reveals that such estimates are not without errors that impact on the performance of the portfolio. The errors are attributable to the fact that the true values are not known and are difficult to predict. A number of models have been constructed to attempt to address such estimation errors. From the models that exist in theory, three groups are observed according to similarity in attributes. The first group seeks to address parameter uncertainty by only using covariance, instead of both return means and covariance to compute optimal portfolio weights. The second group uses Bayesian statistics to address parameter uncertainty. The third group uses diversification by constructing an additional fund to address parameter uncertainty. Simple two-asset portfolios are constructed to simplify the comparison between the models. Assets are arbitrarily selected from the Johannesburg Securities Exchange when demonstrating how the models can be implemented. Hypothetical portfolios are used to demonstrate the differences between the models when applied on assets with a variety of attributes such as whether or not they are positively correlated, perfectly or partially correlated. The research confirms that performance of any two rules against real data sets does not provide proof that either rule is better than the other. This observation is made when comparing the results against what other researchers have found. There is however, merit in choosing one model over another, depending on the context that an investor is investing in. Each of the groups that seek to address estimation errors demonstrates ability to outperform the Classical rule (where estimates are treated as though they are actuals)