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.

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.

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.