The issue of estimation risk is of particular interest to the decision-making processes of portfolio managers who use long-short investment strategies. Accordingly, our paper explores the question of whether a VaR constraint reduces estimation risk when short sales are allowed. We find that such a constraint notably decreases errors in estimates of the expected return, standard deviation, and VaR of optimal portfolios. Furthermore, optimal portfolios in the presence of the constraint are substantially closer to the 'true' efficient frontier than those in its absence. Finally, we provide VaR bounds and confidence levels for the constraint that lead to the best out-of-sample performance.

In Das, Markowitz, Scheid, and Statman (2010), an investor divides his or her wealth among mental accounts with short selling being allowed. For each account, there is a unique goal and optimal portfolio. Our paper complements theirs by considering estimation risk. We theoretically characterize the existence and composition of optimal portfolios within accounts. Based on simulated and empirical data, there is a wide range of account goals for which such portfolios notably outperform those selected with the mean-variance model for plausible risk aversion coefficients. When short selling is disallowed, the out performance still typically holds but to a considerably lesser extent.

Portfolio selection models generally assume that the investor knows the parameters of the probability distribution of security returns. In practise the investor must, however, employ estimates of the necessary parameters. In this paper we investigate the effect of estimation risk on the efficient frontier in the lower partial moment framework. The results of the average difference between the actual and estimated portfolios show that the estimated portfolios are biased predictors of the actual portfolios. However, the estimates of the optimal portfolios can be improved. If our concern is the uncertainty in the optimal portfolio weights, then a bootstrap approach should be used to improve the optimizations. On the other hand, if our concern is related to the risk and portfolio mean returns of the optimized portfolios, then a James-Stein approach should be used.

We propose a new adjustment in mean-variance portfolio weights to incorporate uncertainty caused by the fact that, in general, we have to use estimated expected returns when determining optimal portfolios. The adjustment amounts to using a higher pseudo risk-aversion rather than the actual risk-aversion and has a straightforward interpretation. The difference between the actual and the pseudo risk-aversion depends on the sample size, the number of assets in the portfolio, and the curvature of the mean-variance frontier. We show how short sales constraints and time-varying expected returns are incorporated in our framework. Applying the adjustment to international portfolios, we show that the adjustments are nontrivial for G5 country portfolios and that they are even more important when emerging markets are included. The exclusion of short sales is found to have a further important impact on the adjusted portfolio weights. In case expected country returns are time- varying, our adjustment induces a significantly smaller variability in portfolio weights that is commonly found.

The estimation of the inverse covariance matrix plays a crucial role in optimal portfolio choice. We propose a new estimation framework that focuses on enhancing portfolio performance. The framework applies the statistical methodology of shrinkage directly to the inverse covariance matrix using two non-parametric methods. The first minimises the out-of-sample portfolio variance while the second aims to increase out-of-sample risk-adjusted returns. We apply the resulting estimators to compute the minimum variance portfolio weights and obtain a set of new portfolio strategies. These strategies have an intuitive form which allows us to extend our framework to account for short-sale constraints, high transaction costs and singular covariance matrices. A comparative empirical analysis against several strategies from the literature shows that the new strategies generally offer higher risk-adjusted returns and lower levels of risk.

Tail Mean-Variance (TMV) has emerged from the actuarial community as a criterion for risk management and portfolio selection, with a focus on extreme losses. The existing literature on portfolio optimization under the TMV criterion relies on the plug-in approach that substitutes the unknown mean and covariance of asset returns in the optimal portfolio weight with their sample counterparts. The plug-in method inevitably introduces estimation risk and usually has poor out-of-sample performance. We propose an optimal combination of the plug-in and 1/N rules to improve out-of-sample performance. Our proposed combined portfolio consistently outperforms both the plug-in and 1/N portfolios on both simulated and real-world datasets.