Calculating Tracking Error from Active Weights and Return Covariances
Summary
The note addresses how to attribute portfolio tracking error when holdings differ from benchmark weights. It frames the portfolio’s return relative to the benchmark as the sum of each asset’s active weight multiplied by that asset’s return. For a two-asset example, the variance of this active return is calculated from the squared active weights times each asset’s variance, plus a covariance term that captures how the two returns move together.
This provides a direct calculation for total tracking error by taking the square root of the active-return variance. The answer recommends estimating variances and covariance over the period relevant to the analysis when a formal risk model is unavailable. It does not give a general marginal or standalone contribution formula for each position, and it cautions that suitable covariance estimates are not straightforward. The example is limited to two assets; larger portfolios require the corresponding full covariance calculation.
Key ideas
- Active weights are the differences between portfolio and benchmark weights.
- Portfolio active return is the sum of active weights multiplied by asset returns.
- Tracking error variance includes both individual asset variances and cross-asset covariances.
- Historical estimates can be used when no risk model is available, but the chosen period matters.
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# How to calculate a position's contribution to its portfolio's tracking error?
# How to calculate a position's contribution to its portfolio's tracking error?
Say we have assets X (with weight $w_a$) and Y (with weight $w_y$) in a portfolio. X and B returns are correlated: $Cov(R_x, R_y)\neq 0$.
The portfolio's tracking error is: $std(R_p - R_b) = std((w_x*(R_x-R_b)+w_y*(R_y -R_b))$.
How can I calculate, based on the asset's tracking error ($std(R_i-R_p)$) and its normalised weight $w_i$, this asset's contribution to the portfolio's tracking error?
Remarks:
- I saw this Quant question but I don't think it answers my question.
- Bloomberg has something called "tracking error contribution", but I don't know which formula they are using.
## Answer by mark leeds (score 1)
https://quant.stackexchange.com/a/57156
Hi: You can calculate the weights in the index of the two stocks. $w_{A}$ and $w_{B}$ and the weights of the stocks in the portfolio, $w^{\prime}_A$ and $w^{\prime}_B$. Then, the return contribution due to the mis-weighting, is $(w_{A} - w^{\prime}_{A}) R_{A} + (w_{B} - w^{\prime}_{B}) R_{B}$.
Then, assuming you don't have a risk model such as Barra, you can use brute force in order to obtain the variance of the return contribution above. You get
$(w_{A} - w^{\prime}_{A})^2 \times Var(R_{A}) $ +
$(w_{B} - w^{\prime}_{B})^2 \times Var(R_{B}) $ +
$ 2 \times (w_{A} - w^{\prime}_{A})( w_{B} - w^{\prime}_{B}) \times Cov(R_{A}, R_{B})$
Getting estimates of Var and Cov without a risk model is not straightforward. One way is to just use estimates from the time period that you are concerned with.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.