## Tuesday, April 13, 2010

### Correlation

Sometimes the seemingly simplest concepts give me the greatest grief. Correlation is a prime example. We know that in the world of finance correlation measures how two or more securities move in relation to one another. Web sites give us chart overlays and numbers, positive or negative (between +1 and -1), representing the correlation between two assets over a given period of time. It all seems so straightforward and yet I’ve always had the nagging suspicion that my understanding was superficial. Perhaps as a result I’ve always distrusted the concept.

I turned to Carol Alexander’s Market Models: A Guide to Financial Data Analysis (Wiley, 2001) to try to sort things out. Alexander gives a more elegant definition than I provided: “Correlation is a measure of co-movements between two return series.” (p. 5) It is a standardized form of covariance; that is, “for two random variables X and Y the correlation is just the covariance divided by the product of the standard deviations.” (p. 7) Thus far we’re in the realm of elementary statistics.

But now things get dicey, and I’m starting to understand why I was always uncomfortable with the notion of correlation. Let me quote from Alexander rather than sound pretentious summarizing something I just learned. “The greater the absolute value of correlation, the greater the association or ‘co-dependency’ between the series. If two random variables are statistically independent then a good estimate of their correlation should be insignificantly different from zero. We use the term orthogonal to describe such variables. However, the converse is not true. That is, orthogonality (zero correlation) does not imply independence, because two variables could have zero covariance and still be related (the higher moments of their joint density could be different from zero). In financial markets, where there is often a non-linear dependence between returns, correlation may not be an appropriate measure of co-dependency.” (p. 7)

Since I am not a statistician, I’m not going to try to follow the various threads of Alexander’s claims here. For those who are interested, there’s a good analysis of linear correlation and the problems of measuring nonlinear relations at
statsoft.com.

But, leaving the world of statistics briefly, it makes intuitive sense that the returns of two assets can be uncorrelated but not independent of one another, especially if we view markets as adaptive systems. For instance, I pulled up a spread sheet from February of commodity inter-market correlations (compliments of MRCI) that showed a zero correlation over a 180 trading day period between orange juice and soybean meal returns. But in the world of managed futures, where portfolio managers work hard to get just the right mix, who would say that these two return series were truly independent of one another?

Returning to the world of statistics, we are hit with the distinction between constant and time-varying correlation models. The constant model assumes that the relationship between the two assets is stable over time; correlation is independent of the time at which it is measured. But aren’t financial markets quintessentially temporal? Alexander writes that “in currency markets, commodity markets and equity markets it is not uncommon for time-varying correlation estimates to jump around considerably from day to day. Commonly, cross-market correlation estimates are even more unstable.” (pp. 15-16)

So how in the world does one hedge correlation risk? The answer, in a nutshell, is that one can’t. Adjusting “mark-to-model values for uncertainty in correlation estimates” hasn’t worked out all that well. We know that in a crisis everything becomes highly correlated.

Correlation, Alexander concludes, is unobservable and can only be estimated within the context of a model. Therefore, she says, the analysis of correlation is a very complex subject. (p. 19) At least I now feel justified in my sense of unease with the concept.

Nonetheless, in a future post I’m going to overcome my theoretical squeamishness to look at how a trader can use correlation analysis to dissect his trading performance.

1. In basic terms -- we learned on the CFA exam that correlations rise in times of crisis.

In mathematical terms, the correlation between 2 assets is a function of their Betas and the Variance of the market.

If we assume both assets have a beta of 1.0 and identical residual risk, then correlations rise as market volatility rises.

Thus, a crisis causes overall variance to rise -- and rising variance mathematically causes correlations to rise.

By the way, more 'ETF eye candy' :)

http://www.etfreplay.com/backtest_ma.aspx

chris
www.ETFreplay.com

2. Brenda,

I analyzed the relationship between Roubini's popularity in Google search results and S&P 500 index. The results are pretty interesting. Roubini's popularity is significantly negatively correlated with S&P 500, and a 1 point increase in popularity results in a 114 point decline in S&P 500 index. You can find the article at the following address: