Tracking error: Difference between revisions

Revision as of 13:17, 5 May 2007

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. If a model is used to predict tracking error, it is called '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.

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

References

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

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-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 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.
+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.  If a model is used to predict tracking error, it is called '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.
-'''Tracking error''' is mathematically the same as [[Active Risk]], and has historically been used in the context of index portfolio or fund management, but, especially in Europe, is now typically used to describe the [[standard deviation]] of returns, either active or passive. The active return is the difference in the return of a portfolio and its benchmark.
+==Mathematical definition==
-An index manager aiming to match the return of a benchmark index seeks to minimize realised tracking error, i.e. the standard deviation of returns about the benchmark. Using the tracking error will allow him to slightly decrease the replication (because of beeing allowed to deviate from the benchmark) with an important decrease in transaction costs. Optimizers, based on 'ex-ante' (see below) tracking will help the manager in the construction of his portfolio.
+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.
-An active portfolio manager, on the other hand, aims to achieve a positive active return with a low active risk.
+It can be expressed as:
+<math>TE=\sigma(r_P-r_B)=\sqrt{ r_P - r_B } </math>
+where <math>r_P</math> is the returns of the portfolio
+<math>r_B</math> is the return of the benchmark
-It is important to distinguish between observed, or realised, tracking error and predicted tracking error.  For active portfolios 'ex ante' tracking error measures are necessarily lower than 'ex post'.  This is because future portfolio weights are randomly, but normally distributed around a mean (‘ex post [[stochastic]]’).   Some studies have suggested a factor of as high as two (see ''Tracking error : ex ante versus ex post measures'', Satchell and Hwang, Journal of Asset Management, Vol 2. No 3, December 2001).
+==References==
+Chincarini, L. and Daehwan K. (2006), ''Quantitative Equity Portfolio Management'', Mc Graw Hill
-'''Tracking Error = stdev(RETURN(portfolio) - beta * RETURN(index))'''
 [[Category:Economics Workgroup]]

Tracking error: Difference between revisions

Revision as of 13:17, 5 May 2007

Ex-ante vs ex-post tracking error

Mathematical definition

References

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