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Abstract In this paper we show that ex ante and ex post tracking errors must necessarily differ, since portfolio weights are ex post stochastic in nature. In particular, ex post tracking error is always larger than ex ante tracking error. Our results imply that fund managers always have a higher ex post tracking error than their planned tracking error, and thus unless our results are considered, any performance fee based on ex post tracking error is unfavourable to fund managers.
Keywords: tracking error, risk and performance measurement, active management
Introduction
Portfolio performance is usually evaluated against a prespecified benchmark portfolio. One most frequently used measure is tracking error (TE), sometimes defined as differences between portfolio returns and the benchmark portfolio returns. TE is simple and easy to calculate as well as a powerful tool in structuring and managing index funds. Two common sources of tracking errors come from the attempts to outperform the benchmark and the passive portfolio replication of the benchmark by a sampled portfolio.
In the analysis of TE, outperforming the benchmark is equivalent to having a positive expected TE; we call the mean TE 'expected relative return' in this study. The risk related to TE is measured by the volatility of the difference between managed portfolio returns and benchmark returns. The volatility is called TE throughout our study.1 Thus, minimising TE as well as maximising expected relative return is a sensible goal for investors.
Most studies on TE have concentrated on how to minimise TE, or how to maximise expected relative return for a given TE; see Larsen and Resnick (1998) and Baierl and Chen (2000). Roll (1992) derived an efficient portfolio in `TE -- expected relative return' space and showed that a Markowitz efficient frontier dominates the efficient frontier derived with TE.
Pope and Yadav (1994), on the other hand, showed that serial correlation of the returns differences between an index fund portfolio and the underlying benchmark portfolio results in a biased estimate of TE. For example, the annual TE calculated with the daily TE will not be a good estimate of the true annual TE in the presence of serial correlation.
In this paper, we suggest a different source of bias in the TE, which arises from the...