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Recent trends and future directions.
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The concepts of portfolio optimization and diversification have been instrumental in the understanding of financial markets and the development of financial decision-making. The major breakthrough came in 1952 with the publication of Harry Markowitz's theory of portfolio selection. Markowitz suggested that sound financial decision-making was a quantitative trade-offbetween risk and return. His work spurred a vast amount of research on quantifying market behavior-one of the main practical consequences was the acceptance of the notion that diversification reduces portfolio risk.
More than 50 years after Markowitz's seminal work, substantial advances have been made in both the theory and the practice of portfolio management. Today, quantitative techniques for forecasting asset returns, and for portfolio allocation, risk measurement, trading, and rebalancing, to mention a few, have a major presence in the financial industry. Their proliferation has been facilitated by the reduced cost of computing power and the increased availability of sophisticated and specialized software, allowing investors to incorporate their forecasts about the future direction of markets into disciplined analytical frameworks.
As the use of quantitative techniques has become widespread in the investment industry, the consideration of estimation risk and model risk has grown in importance. Bayesian techniques and robust estimation of model parameters, for example, are now common in financial applications. Most recently, practitioners have begun incorporating the uncertainty introduced by estimation errors directly into the portfolio optimization process by mathematical techniques referred to as robust optimization.
Unlike the traditional approach, where inputs to the portfolio allocation framework are treated as deterministic, robust portfolio optimization incorporates the notion that inputs have been estimated with errors. In this case, the inputs are not the traditional forecasts, such as expected returns and asset covariances, but rather uncertainty sets including these point estimates (e.g., confidence intervals around the forecasts).
We review major trends in robust optimization and its applications in portfolio management. We begin by explaining the main ideas behind the robust optimization approach, and discuss the relation between robust optimization and other robust methods for portfolio management. We describe some of the latest developments in robust optimization applications, and discuss future directions in robust portfolio management.
THE ROBUST OPTIMIZATION APPROACH
Introduced in the operations research literature by Ben-Tal...