The Nobel laureate Harry Markowitz showed that an investor who cares only about the mean and variance of portfolio returns should hold a portfolio on the efficient frontier. To compute these portfolios, one needs to solve a quadratic program whose coefficients depend on the mean and the covariance matrix of asset returns. In practice, one needs to replace these quantities by their sample estimates, but due to estimation error the mean-variance portfolios that rely on sample estimates typically perform poorly out of sample; a difficulty that has been referred to in the literature as the ``error-maximization'' property of the portfolio optimization problem.
In this talk, we first illustrate the difficulties inherent to estimating optimal portfolios by comparing the out-of-sample performance of the mean-variance portfolio and its various extensions relative to the naive benchmark strategy of investing a fraction 1/N of wealth in each of the N assets available. The results show that none of the optimal portfolios is consistently better than the 1/N rule in terms of Sharpe ratio across seven empirical datasets. We then discuss several approaches proposed in the recent literature to overcome these difficulties in practice, including robust optimization and estimation, shrinkage estimation, Bayesian estimation, and norm constraints. Finally, we highlight open issues that offer opportunities for researchers in optimization to contribute to this area.