# 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

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:

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

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.

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