A Division of Windham Capital Management, LLC

Efficient Portfolios for Multiple Objectives

The theoretical foundation of our analysis is based on portfolio theory, which was introduced in 1952 by Harry Markowitz. His innovation, which is sometimes called mean-variance optimization, requires estimates of expected returns, standard deviations, and correlations. With this information, we combine assets efficiently so that for a particular level of expected return the efficiently combined assets offer the lowest level of expected risk, usually measured as standard deviation or its squared value, variance. A continuum of these portfolios plotted in dimensions of expected return and standard deviation is called the efficient frontier. We identify portfolios along the efficient frontier by maximizing a measure of investor satisfaction defined by the following quantity:

Expected Return - Risk Aversion x Standard Deviation²

Some investors also care about relative risk; that is, performance relative to a benchmark. Relative risk is measured as tracking error. Just as standard deviation measures dispersion around an average value, tracking error also measures dispersion, but instead around a benchmark’s returns. It is the standard deviation of relative returns. In this case we identify efficient portfolios by substituting tracking error aversion for risk aversion and tracking error for standard deviation, as shown:

Expected Return - Tracking Error Aversion x Tracking Error²

In many situations, investors care about both absolute and relative performance. They typically deal with concern about relative performance by employing ad hoc constraints to mean-variance optimization in order to prevent the solutions from deviating too far from the benchmark. We address this dual focus more rigorously by augmenting the definition of investor satisfaction to include both measures of risk explicitly:

Expected Return - Risk Aversion x Standard Deviation²
- Tracking Aversion x Tracking Error²

This approach produces an efficient surface in three dimensions - expected return, standard deviation, and tracking error. The efficient surface is bounded on the upper left by the traditional mean-variance efficient frontier. The right boundary of the efficient surface is the mean-tracking error efficient frontier. It comprises portfolios that offer the highest expected return for varying levels of tracking error. The lower boundary of the efficient surface represents combinations of the minimum risk portfolio and the benchmark portfolio.


This approach typically yields an expected result that is superior to constrained mean-variance optimization.

  • For a given expected return, it typically produces a portfolio with a lower standard deviation and less tracking error
  • For a given standard deviation, it typically produces a portfolio with a higher expected return and less tracking error
  • For a given tracking error, it typically produces a portfolio with a higher expected return and a lower standard deviation

Category:Understanding the Software -> Optimization