60 Years Of Portfolio Optimization: Practical Challenges And Current Trends

European Journal of Operational Research 234 (2014) 356–371 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor 60 Years of portfolio optimization: Practical challenges and current trends Petter N. Kolm a,⇑, Reha Tütüncü b, Frank J. Fabozzi c a Courant Institute, New York University, 251 Mercer St., New York, NY 10012, United States Goldman Sachs Asset Management, 200 West St., New York, NY 10282, United States c EDHEC Business School, 393, Promenade des Anglais BP3116, 06202 Nice Cedex 3, France b a r t i c l e i n f o Article history: Available online 1 November 2013 Keywords: Black–Litterman Estimation errors Mean–variance optimization Multi-period optimization Portfolio constraints Portfolio optimization a b s t r a c t 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 60 year 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. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction The concepts of portfolio optimization and diversification have been instrumental in the development and understanding of financial markets and financial decision making. The major breakthrough came in 1952 with the publication of Harry Markowitz’s theory of portfolio selection (Markowitz, 1952). The theory, popularly referred to as modern portfolio theory, provided an answer to the fundamental question: How should an investor allocate funds among the possible investment choices? First, Markowitz quantified return and risk of a security, using the statistical measures of its expected return and standard deviation. Second, Markowitz suggested that investors should consider return and risk together, and determine the allocation of funds among investment alternatives on the basis of their return-risk trade-off. Before Markowitz’s seminal article, the finance literature had treated the interplay between return and risk in an ad hoc fashion. The idea that sound financial decision-making is a quantitative trade-off between return and risk was revolutionary for two reasons. First, it