Estimating Portfolio Volatility with Exponentially Weighted Covariance

Summary

The document asks how the sample length used to estimate portfolio volatility affects the result, and whether recent returns can receive greater weight. Portfolio volatility is computed from asset weights and a covariance matrix, so changing the return sample changes the covariance estimate and therefore the volatility estimate. The answer describes calculating covariance with weighted observations, centering returns on their weighted means, and extending the calculation to a matrix form using a diagonal matrix of observation weights.

This provides a general method for emphasizing recent observations, but it does not specify a decay schedule, explain how to choose the weighting parameter, or compare the approach with an unweighted estimate. The cited response also cautions against confusing observation weights in the covariance calculation with portfolio asset weights. No empirical results or guidance on sample length are provided, so practical performance and sensitivity to the weighting scheme remain open questions.

Key ideas

• Portfolio volatility depends on the covariance matrix estimated from the chosen return sample.
• Weighted covariance can give more influence to selected observations, including more recent returns.
• The weighted calculation centers observations on weighted means and can be written in matrix form.
• Observation weights in covariance estimation are distinct from portfolio asset weights.

Tags

# portfolio volatility over time


# portfolio volatility over time












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

## Answer by Attack68 (score 1, accepted)

https://quant.stackexchange.com/a/51141

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