Thoughts on Correlation

Mean reversion is aided by volatility, and momentum trading is aided by relative strength.  I have been exploring the distribution of relative strength.  Market strength and breadth are handily characterized by the mean and standard deviation of this distribution.  It is almost a measure of market correlation, but not quite.  I am tempted to call it “collimation,” because it measures how tightly the body of stocks hew to their long term trends.

I can find only one published measurement of market correlation.  The CBOE computes implied correlation, based on option prices for fifty stocks.  See the CBOE website.  Below is a chart showing correlation, along with VIX and the SPX.

CBOE CorrelationThe CBOE data is hard to work with, because each of the series is based on options expiring in January of a different year.  For example, the nearest dated series just became KCJ, which is based on options expiring January 2015.  I produced a long series for this chart by splitting each year at July 1.  So, instead of 2-14 months, the concatenated series uses options expiring in 6-18 months.  This avoids the jolt of switching series at yearend.

There are times when index option implied volatility moves and there is no corresponding shift in implied volatilities of options on those components. This outcome is due to the market’s changing views on correlation.

It would be nice to measure historical correlation among stocks, directly, so I charted the average 20 day pairwise correlation among the nine sector ETFs.  I think this is a pretty good indicator, but it doesn’t predict JCJ.  Instead of the 36 pairs, you can simply compare each ETF to SPY.  The results are about the same.

CorrelETFSome people like to combine correlation and volatility into a single ratio, JCJ/VIX.  What does this number mean?  I’ll come back to that, but first let’s consider how we might use correlation and volatility to characterize market conditions:

Low correlation and volatility – This is an emerging bull market, which rewards good stock selection.  If it persists, though, the herd will follow and correlation will increase.

Low volatility, and correlated – This is a mature bull market, in which all stocks are rising together.  You might as well just buy the index.

Uncorrelated, and volatile – Simmering volatility with low correlation means that diversification is protecting the VIX but weak stocks are suffering.

High correlation and volatility – Elevated VIX means the market is in trouble, and all stocks are going down.  In a crisis, as Dr. Bandy says, all correlations go to 100% (or at least 80%)

I find it best to look at JCJ (or KCJ) and VIX independently.  So, why do people use the ratio?  Here’s the best explanation I can think of.  Start with the formula for index volatility:

Fig1Recognizing that ρ = 1 where i = j, we can write a more compact formula:

Fig2You can see how VIX is affected by the pairwise correlation term. The formula for average correlation is:

Fig3Since:

Fig4Then:

Fig5This makes intuitive sense.  It’s the sum of the weighted variance terms with pairwise correlations, divided by the same sum without them.  In fact, it’s a lot like:

Fig6This is different from JCJ because it includes the same-stock (i = j) cases.  Still, since:

Fig7You could make a case for VIX2/JCJ as a gauge of total volatility “under the surface.”  I am not a fan of this, although it would be interesting if the CBOE were to publish an absolute volatility index without the diversification effect.

I can think of one other reason to look at JCJ/VIX.  Since SPX is negatively correlated with VIX (over short timeframes) you might want to chart SPX versus 1/VIX to show a divergence.  I do this on a daily chart with one minute bars.

VIXIn this case, using JCJ/VIX might be useful, but you still have to run a second chart to see which one is causing the divergence.  The short answer is, look at volatility and look at correlation, but don’t combine them.

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