60 Years of portfolio optimization: Practical challenges and current trends

•We review approaches for implementing Markowitz mean–variance analysis in practice.•Review covers inclusion of transaction costs, constraints, sensitivity to inputs.•We selectively highlight new trends and developments. The concepts of portfolio optimization and diversification have been instrumental in the development and understanding of financial markets and financial decision making. In light of the 60year anniversary of Harry Markowitz’s paper “Portfolio Selection,” we review some of the approaches developed to address the challenges encountered when using portfolio optimization in practice, including the inclusion of transaction costs, portfolio management constraints, and the sensitivity to the estimates of expected returns and covariances. In addition, we selectively highlight some of the new trends and developments in the area such as diversification methods, risk-parity portfolios, the mixing of different sources of alpha, and practical multi-period portfolio optimization.