Mean-variance portfolio optimization is more popular than optimization procedures that employ downside risk measures such as the semivariance, despite the latter being more in line with the preferences of a rational investor. We describe strengths and weaknesses of semivariance and how to minimize it for asset allocation decisions. We then apply this approach to a variety of simulated and real data and show that the traditional approach based on the variance generally outperforms it. The results hold even if the CVaR is used, because all downside risk measures are difficult to estimate. The popularity of variance as a measure of risk appears therefore to be rationally justified.

Mean-semivariance and minimum semivariance portfolios are a preferable alternative to mean-variance and minimum variance portfolios whenever the asset returns are not symmetrically distributed. However, similarly to other portfolios based on downside risk measures, they are particularly affected by parameter uncertainty because the estimates of the necessary inputs are less reliable than the estimates of the full covariance matrix. We address this problem by performing PCA using Minimum Average Partial on the downside correlation matrix in order to reduce the dimension of the problem and, with it, the estimation errors. We apply our strategy to various datasets and show that it greatly improves the performance of mean-semivariance optimization, largely closing the gap in out-of-sample performance with the strategies based on the covariance matrix.

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