Marginal Tracking Error Contribution from Portfolio Covariance

Summary

The document considers how to calculate an asset’s marginal contribution to tracking error relative to a benchmark. Its answer reframes active risk as the volatility of a combined position consisting of the portfolio and a short position in the benchmark. In covariance terms, benchmark variance and portfolio-benchmark covariance are included alongside the portfolio’s own variance.

Differentiating tracking variance with respect to an asset weight gives its marginal effect on active variance, accounting for its covariance with other holdings and with the benchmark. This provides a route to marginal tracking error contribution through a covariance matrix augmented with the benchmark. The response gives a variance derivative rather than fully specifying conversion to marginal tracking error or contribution in tracking-error units, so the precise definition and scaling should be checked against the portfolio-risk convention in use.

Key ideas

• Tracking error can be viewed as the volatility of a portfolio combined with a short benchmark position.
• The active-risk calculation includes portfolio variance, benchmark variance, and portfolio-benchmark covariance.
• An asset’s marginal effect on tracking variance depends on its weight and covariances with holdings and the benchmark.
• The response derives a marginal tracking variance expression but does not fully specify the scaling for tracking error contribution.

Tags

# Marginal contribution to Tracking error


# Marginal contribution to Tracking error












I'm trying to calculate Marginal contribution to Tracking error. I would use the following formula:

MCTE(asset i)=TE(excess return asset i vs.benchmark)* Beta(excess return asset i vs. benchmark AND excess return Portfolio vs. benchmark).

Am I right?

## Answer by Charles Fox (score 1)

https://quant.stackexchange.com/a/48870

Conceptually, I think of this as the volatility of a portfolio with a -100% position in the benchmark. Then you can just add a row and column to the portfolio's co-variance matrix.

$Tracking Variance = \sum \sum w_i w_j \sigma_i \sigma_j \rho_{i,j} + \sigma^2_{bench} + 2\sum w_i (w_{bench} = -1)\sigma_i \sigma_{bench}\rho_{i,bench}$

Taking the first derivative with respect to some $w_*$:

$\Delta Tracking Variance = 2(w_*\sigma^2_*+ \sum_{i\not= *} w_i \sigma_i \sigma_* \rho_{i,*} - \sigma_* \sigma_{bench}\rho_{*,bench})$

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