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When estimating portfolio vol. with:

$\sigma = \sqrt{w^T \cdot cov \cdot w}$

How does the sample length of returns affect $\sigma$?

Is it possible to exponentially weight something to give more weight to recent vol?

Any references greatly appreciated.

Thanks

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The empirical covariance matrix is $cov = \frac{1}{N-1}(X-\bar{X})^T(X-\bar{X})$ where $X$ is your array of sample returns.

You can estimate an empirical covariance matrix with weighted observations e.g. with:

$$ \frac{\sum_i w_i (x_i-\mu_x(x;w))(y_i-u_y(y;w))}{\sum_i w_i} $$

reference is top of google: https://doc-archives.microstrategy.com/producthelp/10.10/FunctionsRef/Content/FuncRef/WeightedCov__weighted_covariance_.htm

I believe the vector notation for the above if you want to implement it with linear algebra is:

$$ weighted cov = \frac{1}{\delta^Tw} (X - \bar{X}_w)^TW(X-\bar{X}_w) $$ where $W$ is a diagonal matrix of the weights

edit: dont confuse the weights $w$ here for your notation where the weights are those of your portfolio assets.

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  • $\begingroup$ Thanks is this a standard technique in portfolio volatility? I had tried to google it, but didn't manage to find any references. $\endgroup$ – chris Feb 13 at 16:59

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