Tracking error

Many portfolios are managed to a benchmark, such as an index (i.e. the S&P500 in the USA or the CAC40 in France). Some portfolios are expected to replicate the returns of an index exactly (an index fund), while others are expected to deviate slightly from the index in order to generate excess returns or to lower transaction costs. Tracking error (a.k.a. Active Risk) is a measure of how closely the portfolio follows the index, and is measured as the standard deviation of the difference between the portfolio and index returns.

There is two ways to express the objective of a fund manager: either minimizing the tracking error for a given expected return over a predefined benchmark or maximizing the expected return for a given tracking error.

Ex-ante vs ex-post tracking error

If tracking error is measured historically, it is called 'realised' or 'ex post' tracking error. A model is usually used to predict tracking error (a.k.a. ex-ante tracking error). The former is more useful for reporting or analysis purposes, whereas ex ante is generally used by portfolio managers to control risk to satisfy client guidelines.

Fabozzi et al. (2006) cite, and briefly describe, three different multifactor models used in equity portfolio management to predict tracking error:

• Statistical Factor Models
• Macroeconomic Factor Models
• Fundamental Factors Models

In the finance industry, some models are well-known and widely used by practitionners. Among them, we can notice the MSCI Barra Models (based on factors such as country, industry, style or currency) and the Northfield Fundamental Equity Model.

Hwang and Satchell (2001) have argued that ex-ante and ex-post tracking error must differ, as portfolio weights are ex-post stochatic in nature. Futhermore, they showed that ex-port tracking error must be higher than ex-ante tracking error.

Mathematical definition

As defined in Chincarini and Daehwan (2006), most portfolio managers when using tracking error define it as beeing the standard deviation of the returns of the portfolio minus the returns of the benchmark.

It can be expressed as:

$TE=\sigma (r_{P}-r_{B})={\sqrt {r_{P}-r_{B}}}$

where $r_{P}$ is the returns of the portfolio $r_{B}$ is the return of the benchmark

As the porfolio manager is working with a sample (and not the full history of datas), we have to adjust that formula for degrees of freedom (see Shein (2000)). In that case, the tracking error formula can be written as:

$TE={\sqrt {\frac {\sum _{p=1}^{N}(R_{P}-R_{B})^{2}}{N-1}}}$

where N is the number of return period.

References

Chincarini, L. and Daehwan K. (2006), Quantitative Equity Portfolio Management, Mc Graw Hill

Hwang, S, and Satchell, SE "Tracking error: Ex ante versus ex post measures", Journal of Asset Management, Volume 2, Number 3, 1 December 2001, pp. 241-246(6)

Fabozzi, Focardi and Kolm, Financial Modeling of the Equity Market, Wiley Finance, 2006