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Dr. Alex Castaldo

10/10/04
The Theoretical Link Between Volatility and Correlation

In the international economics literature much used to be made of the fact that correlations between country stock markets increase during so called "crisis periods," this was thought to show that investors behave differently at crisis times than at normal times.

More recently it has been realized by Rigobon and others that correlations increase during times of high volatility for mechanical reasons having to do with the definition of correlation, and that this does not necessarily mean a shift in behavior or in the underlying stochastic process.

(Why do I bring this up? Big Al reports that the correlation between CSCO and MSFT increased from 0.313 to 0.415 between two periods. He also shows that the VIX went from 18.53 to 23.22, in other words the second period was more volatile than the first. The latter result may help explain the former).

A simple simulation experiment illustrates the theoretical link:

"To see the link between volatility and correlation, consider the unrealistically simple case of two random variables, x and y, that are independently and identically distributed bivariate normal, with means equal to zero, variances equal to unity and a correlation of 0.5. A large sample of draws of such (x,y) pairs is shown in Figure 1. Now consider splitting the full sample into two subsamples based on the value of x: a low volatility subsample, including all (x,y) pairs with an absolute value of x less than 1.96; and a high volatility subsample, including all (x,y) pairs with an absolute value of x greater than or equal to 1.96. Intuitively, the effect of trimming the ends off the joint distribution in the low volatility subsample [will be] to reduce the sample correlation between x and y. By contrast, the correlation for the high volatility subsample should be enhanced because the support of its distribution is disjointed, with one portion picking up the large positive values of both variables while the other portion of the distribution picks up the large negative values of the variables. Indeed, as noted in the figure [not shown], the correlation for the high volatility subsample is 0.81, while that for the low volatility sample is 0.45. Note that the correlation in the latter subsample is close to the population value of 0.5; this latter result may not be surprising since the low volatility subsample includes 95% of the data."

Source: http://www.bis.org/publ/confer08k.pdf

In other words, when you focus on a high volatility subperiod, you EXPECT that the correlations will increase. After reading such things, I have become very cautious about what changing correlations really mean in a world where we know that the volatility is constantly changing.

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