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Forecast Covariance and Portfolio Ex-Ante Tracking Error

Article Quant Q&A · Author: GT213
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Summary

The document explains two ways to estimate tracking error for a portfolio of funds relative to a benchmark. For an ex-ante estimate, form a forecast covariance matrix and apply it to the portfolio’s active weights, meaning portfolio weights minus benchmark weights. The square root of the resulting quadratic form is the forecast tracking error. Historical covariances can serve as forecasts if past relationships are expected to persist; factor models are offered as another approach.

A second response describes a historical, or realized, calculation: construct portfolio and benchmark return series, take their period-by-period differences, and calculate the standard deviation of those active returns. The distinction depends on timing: an ex-ante forecast uses information available before the measurement period, while an ex-post estimate uses returns observed during it. The document does not prescribe a covariance forecasting method, estimation window, return frequency, or adjustments for changing weights, so results depend on those choices.

Key ideas

  • Ex-ante tracking error is the square root of active weights multiplied through a forecast covariance matrix.
  • Active weights are portfolio weights measured relative to the benchmark.
  • Historical covariance estimates can be used as forecasts if past dependence is expected to continue, while factor models are another option.
  • Realized tracking error can be estimated as the standard deviation of portfolio returns minus benchmark returns over the period.
  • An ex-ante estimate uses information available before the period, whereas an ex-post estimate uses returns observed within it.

Tags

Portfolio constructionRisk managementStatistics
Full text
# How to calculate ex Ante Tracking Error


# How to calculate ex Ante Tracking Error












I'm looking to find the correct way to calculate the ex ante tracking error of a portfolio.

If say I have 10 funds, and their historical returns series (used to calculate mean return, standard deviation and correlations/ covariances or anything else needed) how would I calculate the ex ante tracking error for a portfolio made up of the 10 funds, with weights fixed?

Thank you

## Answer by Comp_Warrior (score 5)

https://quant.stackexchange.com/a/31496

For ex-ante tracking error, you need a forecast covariance matrix $C$. Then the quantity you require is $\sqrt{w^{T}Cw}$, where $w$ is a vector of excess weights relative to the benchmark. You can construct a forecast covariance matrix from realized covariances if you think historical relationships will persist, or you use other methods, for example factor models.

## Answer by user18663 (score 1)

https://quant.stackexchange.com/a/31471

Here are the steps:- 1.First you need to calculate the historical return series for your portfolio from the historical return series for each fund by adding the returns for each fund on a daily basis taking weights into consideration. 2. Calculate the historical return series for your benchmark. 3. Calculate the difference between the returns of your portfolio and the benchmark for each day. 4. Once you get the return series for the difference between the two, calculate the standard deviation of this series.

I hope this helps.

## Answer by APOSTATE LOYAL (score -3)

https://quant.stackexchange.com/a/58964

both of them are right，and Danish gives one way to compute ex ante tracking error。the importance is，at a given period （t1，t2），ex ante use information before t1（such as return series to compute cov matrix），however，ex post uses information between this period，and this is what “realized” means。

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.