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The most common formula for the ex-ante tracking error is $\sqrt{w^{T}Cw}$, where $w$ is a vector of excess weights relative to the benchmark and $C$ a forecast of covariance matrix. The sums of both $w_p$ (the vector of portfolio weights) and $w_b$ (the vector of benchmark weights) are set to 1 and they share as many rows as $C$.

In the litterature, the size of $C$ seems limited to that of the benchmark.

What are the implications, in terms of bias of the ex-ante tracking error, for an active investor that buys stocks outside her benchmark? And what are the possible solutions to limit this?

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  • $\begingroup$ Interesting question. The "conventional answer" wd prob be that in this case the Benchmark is wrong and needs to be extended. But I am looking forward to reading some more creative answers. $\endgroup$ – noob2 Aug 21 '17 at 15:18
  • $\begingroup$ Maybe I'm missing something, but what do you think is the problem with the benchmark putting zero weight on some Apple while the portfolio puts positive weight on Apple? Your estimate of the tracking error would depend on how well you estimate the covariance between Apple and the benchmark, but I don't see any particularly special issues? $\endgroup$ – Matthew Gunn Aug 21 '17 at 20:30
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There's nothing in the math that says a portfolio can only put non-zero weights on securities where the benchmark puts positive weights. So I'm not sure I understand your problem?

Quick math review

Let $R$ be a $k \times 1$ random vector denoting next period returns.

Let $\mathbf{w}_b$ and $\mathbf{w}_b$ be $k \times 1$ vectors denoting weights of the portfolio.

The return of the portfolio and the benchmark are given by given by: $$ r_p = R \cdot \mathbf{w}_p \quad \quad r_b = R \cdot \mathbf{w}_b$$

If one defines tracking error as the standard deviation of the difference:

\begin{align*} \sqrt{ \operatorname{Var}\left( r_p - r_b \right) } &= \sqrt{ \operatorname{Var}\left( \left(\mathbf{w}_p - \mathbf{w}_b \right) \cdot R \right) } \\ &= \sqrt{ \left(\mathbf{w}_p - \mathbf{w}_b \right)' \operatorname{Var}(R) \left(\mathbf{w}_p - \mathbf{w}_b \right) } \end{align*}

Example:

Let's say $k=2$ and I have two stocks, Apple and Google.

$$R = \begin{bmatrix} R_{AAPL} \\ R_{GOOG} \end{bmatrix} $$.

Let's say my portfolio is 100% Apple and my benchmark is Google. Hence:

$$ \mathbf{w}_p = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \mathbf{w}_b = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \mathbf{w}_p - \mathbf{w}_b = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$$

And hence tracking error is given by: $$ \sqrt{ \operatorname{Var}(R_{AAPL}) - 2 \operatorname{Cov}( R_{AAPL}, R_{GOOG}) + \operatorname{Var}(R_{GOOG})}$$

How good my estimate of the tracking error is will depend on how good is my estimate of the variance of Apple and Google and how good is my estimate of the covariance between them.

What can go wrong?

A broad category of problems comes from using an estimate $\Sigma$ instead of the true covariance matrix $\operatorname{Var}(R)$.

Another perhaps more subtle source of problems come from generating weights $\mathbf{w}_p$ directly or indirectly based upon estimate $\Sigma$. If you choose weights to minimize tracking error based upon sample covariance matrix $\Sigma$, you're almost certainly going to get a downward biased estimate of your true tracking error. Sample covariance matrix $\Sigma$ may have eigenvalues near zero while the true covariance matrix doesn't (indeed if time number of time periods $T$ used to estimate covariance is less than the number of securities $N$, this mechanically must be true).

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  • $\begingroup$ I just wanted to make sure the covariance matrix could be of a different size than the benchmark, thank you for making that clear. $\endgroup$ – user1627466 Aug 22 '17 at 7:37
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As a simple answer, the covariance matrix should not represent only assets in the benchmark. It should include the universe of assets. As an example, a benchmark might be 60% US Large Stocks and 40% US Aggregate Bonds. A manager might also buy emerging market stocks. One can just use a larger covariance matrix that includes emerging market stocks. The benchmark will have zero exposure to this just as a manager may have zero exposure to assets in the benchmark. As you point out tracking error is a function of excess weights which is the difference.

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