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(ProQuest: ... denotes formulae omitted.)
The performance of a fund relative to a chosen benchmark has traditionally been a key determinant of the fund's attractiveness as an investment. Hence, a commonly implemented strategy throughout the investment community is to minimize the tracking error relative to a given benchmark, subject to a given expected level of positive excess return.
In spite of their wide use, expected excess returns and tracking errors have been difficult to estimate, especially in mean-variance optimized portfolios, in part due to the statistical noise contained in the means and variances of asset returns estimated from historical data (see Scherer and Martin [2007] for a review). Chan et al. [1999] look at forecasting the second moments of the returns in order to reduce the tracking error, while Pachamanova [2006] suggests using robust optimization.
Our research looks at the effect of estimation error on the expected excess return and tracking error for portfolios optimized using mean-variance optimization. We derive asymptotic formulae that eliminate dominant terms of the bias arising from noisy estimates, hence improving the implementation power of the tracking measures. We extend to the case of tracking-erroroptimized portfolios the methods developed by Siegel and Woodgate [2007] for meanvariance optimized portfolios. We use the method of statistical differentials to find Taylor-series approximations to expectations of random variables, obtaining theoretical results that are asymptotically correct when the number of time periods is large and that remain statistically consistent when estimated values are substituted for unknown parameters. Our approach is non-Bayesian, not relying upon prior assumptions on the unknown parameters.
We perform the following thought experiment. Suppose an investor forms classical sample estimates of asset means, variances, and covariances and uses them as if they were the true assets' distribution parameters to form a tracking-error-optimized portfolio that achieves a specified expected excess return with respect to a given benchmark. We refer to these classical sample performance estimates as "naïve" estimates, since they do not take into account the estimation error stemming from using a sample rather than the population in determining these parameters. If the estimation error wrongly suggests that an asset will have a high expected return, then an optimized portfolio heavily invested in this asset will be disappointing. In particular, estimates of the naïve...