Index-tracking optimal portfolio selection

Abstract

This paper considers a portfolio selection problem with multiple risky assets where the portfolio is managed to track a benchmark market barometer, such as the S&P 500 index. A tracking optimization model is formulated and the tracking-efficient (TE) portfolios are shown to inherit interesting properties compared with Markowitz mean-variance (MV) optimal portfolios. In comparison to an MV-portfolio, both the beta and the variance of a TE-portfolio are higher by fixed amounts that are independent of the expected portfolio return. These differences increase with index variance, are convex quadratic in the asset betas, and depend on the asset means and covariance matrix. Furthermore, a TE portfolio is obtained by simply extending an MV portfolio by constant adjustments to portfolio weights, independent of the specified mean return of the portfolio, but dependent on the index variance and asset return parameters. Consequently, at lower thresholds of risk, TE portfolios are better-diversified than MV portfolios.