Variety of Covariance Ellipses

I found it helpful to look at covariance ellipses for a variety of tame distributions so that I would have a better feel for those I come across in the wild.

Following is a quick tour that I hope will show the effect of changing the parameters of the distributions of the series in question on the covariance ellipses:

1. For uncorrelated series, what is the effect of changing the standard deviations of the series?
2. For series with identical standard deviations, what is the effect of changing the correlation coefficients?
3. For series with different standard deviations, what is the effect of changing the correlation coefficients?

Varying Standard Deviation of Uncorrelated Series

The ellipses are stretched in the direction of the distribution with the largest standard deviation. The axes of the ellipses are theoretically parallel to the axes of the plots: The maximum variance portfolio of two uncorrelated series is made up only of the series with the largest variance. The minimum variance portfolio contains only the series with the smaller variance.

I say "theoretically" because, in the case of distributions with very similar variances and very low correlations (which is what we have in the first chart: since the series are randomly generated they DO NOT have exactly the same variances nor are their correlation coefficients exactly zero) the axes will take on a direction randomly dictated by the particular sample.

Similarly, in the case where the series have different variances, but zero correlation, the direction of the ellipse axes (the eigen vectors of the covariance matrix) will be flipped one way or the other depending on the slight deviation from zero in the measured correlation. Notice how, in the second chart the measured correlation is just slightly negative and the semi-axes are -ve and + ve respectively. In the third chart where the correlation comes out just +ve, both the semi-axes are negative.

A final point: if I had plotted correlation ellipse rather than covariance ellipses, all the charts would be the same aside from random variations.

Varying Correlation

This is another of those "aha" situations. Intuitively, I always expect to see varying correlation cause the ellipse to rotate. Of course, it doesn't rotate at all it causes the ellipse to change shape. Notice that the axes of all the three ellipses in the following chart have the same orientation of 45 degrees. The eigen vectors have coefficients of \inline \pm \frac {1} {\sqrt{2}} = \pm sin(45) = \pm cos(45) in all positions.

As the magnitude of the correlation coefficient increases the ellipse changes from a perfect circle (\inline \rho = 0) elongating in the direction dictated by the sign of \inline \rho and getting narrower perpendicular to this direction until it ultimately ends up as a straight line.

Different Variance, Different Correlation

Finally we see some rotation in the covariance ellipses. IF the distributions have different variances (which, in the wild, they mostly will) then when the magnitude of the correlation coefficient increases, the ellipse BOTH stretches AND rotates away from the axis of the series with the larger variance. The final resting angle of the ellipse (compressed to a line) that represents the fully correlated series is dictated by the relationship of the variances of the two series.

This brief synopsis gives me enough understanding to be able to look at changes in covariance ellipses over time, or as a result of partitioning data either by time frame or some other criteria, and make a quick qualitative interpretation of what has happened to the correlation relationships.

Edit: LaTeX issues fixed