Expected portfolio return maximization under the mean-variance framework or other risk measure minimization are examples of active investment strategies, that is, strategies that identify a universe of attractive investments, and ignore inferior investments opportunities. A different approach, referred to as a passive investment strategy, argues that in the absence of any superior forecasting ability, investors might as well resign themselves to the fact that they cannot beat the market. From a theoretical perspective, the analytics of portfolio theory tell them to hold a broadly diversified portfolio anyway. Many mutual funds are managed relative to a particular benchmark or stock universe, such as the S&P 500 or the Russell 1000. The portfolio allocation models are then formulated in such a way that the tracking error relative to the benchmark is kept small.

Standard Definition of Tracking Error

To incorporate a passive investment strategy, we can change the objective function of the portfolio allocation problem so that instead of minimizing a portfolio risk measure, we minimize the tracking error with respect to a benchmark that represents the market, such as the Russell 3000, or the S&P 500. Such strategies are often referred to as indexing. The tracking error can be defined in different ways. However, practitioners typically mean a specific definition: the variance (or standard deviation) of the difference between the portfolio ...

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