Statistical and Implied Correlation: Sector ETFs

by: Don Fishback

Statistical Correlation

Correlation: In statistics, it’s the degree to which two or more variables show a tendency to move relative to each other. In finance, it’s the tendency of two or more securities, sectors or asset classes to move together.

Correlation values range from -1 to +1. If one stock moves in perfect lockstep with another, the correlation is going to be +1. If they move exactly opposite one another, the correlation is going to be -1.

Needless to say, a perfect correlation of + or -1 is nearly impossible. But recently, we’ve seen a few examples where correlations have climbed to unusually high levels. The correlations I’m speaking of are the various Sector SPDRs and the S&P 500 SPDR.

Sector SPDRs are nine Exchange Traded Funds (ETFs) based on the different industry groups represented in the S&P 500. If the different industry groups are moving together, then correlation should be high. If the industry groups are moving in opposite directions, then correlation should be low.

For instance, this time last year, the market was actually recovering from a drop in early July. But there was extreme weakness in two industry groups: energy and materials. Meanwhile, consumer staples and health care names were moving higher. Correlation was relatively low.

Move forward to October 2008, when everything was plunging together. Correlation was extremely high. The same thing happened in late February, early March.

What’s interesting is that, in the latest rally, correlation eclipsed the level it was at when everything was collapsing, hitting an all-time high since the industry group data became available in 1997. In other words, nearly everything has been going up in unison.

IMAGE Correl2 Small In Depth: Correlation

Click image to enlarge

Take a look at the chart above and you’ll see the clear graphical representation. The chart measures the average of the one-month correlation of each of the nine Sector SPDRs compared to the S&P 500 SPDR. Because it’s measuring price action over the past month, it’s telling us what has happened.

Implied Correlation

There is, however, another type of correlation. It’s called implied correlation. The CBOE recently launched a new Implied Correlation Index, and it measures something completely different than historical connectedness.

Implied correlation measures the implied volatility of the at-the-money options of the constituent securities in the S&P 500 and compares those many implied volatility readings to the implied volatility of the S&P 500 index options. In layman’s terms, it tells us whether the individual stock options are cheap or expensive relative to the index options. The higher the implied correlation index, the more expensive index options are relative to individual stock options. If the index ever did climb above 100 (which it did in November 2008), it means that, generally speaking, the S&P 500 index options have a higher implied volatility than the index’s constituent stock options.

The CBOE has put together a “white paper” describing the index in great detail, including how to calculate it, along with sample data from May 29, 2009. I couldn’t resist. I built a spreadsheet using the data so that anyone can see how the calculation process works. The VIX that day was 28.17; the Implied Correlation Index was 59.46. In the spreadsheet I created, you can change the implied volatility values to see how changes in the index or stock volatility impact the implied correlation index. The spreadsheet is available here.

The CBOE also put together a spreadsheet. Theirs has historical data on the Implied Correlation Index. The data only goes back to 2007 though. Here’s a chart of the various implied correlation indexes. They’re distinguished by the expiration month of the S&P 500 options each index is tracking. For instance, the 2010 version of the index is tracking S&P 500 index options that expire in January 2010. That particular index has a ticker symbol of ICJ.

IMAGE ICI1 Small In Depth: Correlation

Click image to enlarge

I’d like the index history to go back further. But that’s alright. Because I’ve figured out a way to get a great estimate of the Implied Correlation Index using data from our software ODDS Online.

You may recall that we gather the average at-the-money implied volatility of the front-month and next-month options, weighted so that the average implied volatility measures a consistent 30-calendar-day duration. We do this every single night for every individual stock and index. We then take these individual implied volatility readings for the 500 stocks in the S&P 500, add them up, and divide by 500 to get the average implied volatility of the options on the stocks that are in the S&P 500. We designate this as SPXIV, as shown in the chart below.

IMAGE SPXIV5 Small In Depth: Correlation

Click image to enlarge

I then take VIX (28.17 on May 29) and divide it by SPXIV (48.79 on May 29), and then multiply the result by 100. I did that, and the answer is 57.74. Why did I choose May 29? Because, as noted earlier, that’s the data that the CBOE uses in their white paper, which means I can examine how my index approximates or deviates from their index. What we find is that the 57.74 of my index comes extremely close to the CBOE’s Implied Correlation Index value of 59.46 for that day.

One key difference between my way of calculating the index and the CBOE’s is that they use a constant expiration date. Right now, the nearest expiration option series they’re tracking is the January 2010 (the ticker symbol for the Implied Correlation of this series is ICJ). We’re tracking the August and September 2009 expirations, and rolling forward.

That said, when you plot my VIX/SPXIV compared to the shortest duration Implied Correlation Index, you’ll find that these two factors have a very high correlation coefficient of nearly 77%. That is, they tend to move together a lot, as shown below!

IMAGE Correl3 Small In Depth: Correlation

Realizing that we can create our own index that closely mimics the behavior of the CBOE Implied Correlation Index over the 2-1/2-year time-frame for which we have index price history, we can use that knowledge and our proprietary 9-1/2-year SPXIV data and VIX to peer back in history much further than you could if all you had access to was the free CBOE data. Below is a 9-1/2-year chart showing VIX/SPXIV that we were able to create, plotted with the S&P 500 and the VIX. At the top is the S&P 500, in the middle is the VIX (log scale so that the spike in 2008 doesn’t distort things), and in the middle is my version of the Implied Correlation Index.

IMAGE Correl4 Small In Depth: Correlation

Click image to enlarge

There’s more — an added kicker: What these relationships mean is that we can reverse the process. Instead of having to go about and make all those complicated calculations to determine SPXIV like me, you at home can get a very good estimate of what that number is simply by dividing VIX by the Implied Correlation Index (realizing that the index is a percent). For instance, take 28.17 and divide it by 0.5946 and you get 47.38, which is strikingly close to our SPXIV value of 48.79!

Okay, so we’ve got a ton of data, and it’s obviously providing unique information. So now what? What does the data tell us? Those are good and valid questions, and they’re the type of questions you should ask of every indicator you contemplate using. Generally speaking, the indicator is providing a measure of expected movement, only the movement being measured is the relative size of movement coming from a wide variety of groups: index options traders, and options traders in 50 of the largest cap stocks. From these many participants, the Implied Correlation Index measures expectations for the magnitude of the movements of the 50 largest cap stocks in the S&P 500 compared to the expectations for the magnitude of the move of the S&P 500. When the Implied Correlation Index is high, options market participants think that the index will have large moves relative to the size of the moves of individual stocks. When the index is low, options market participants think that individual stocks will have bigger moves than index traders are anticipating.

At first, the latter scenario seems illogical. How could you see individual stocks go crazy while the S&P 500 remains subdued? Not only can it happen, it has happened! Here’s how. Let’s say you have an index based on two stocks: Microsoft (NASDAQ:MSFT) and Johnson & Johnson (NYSE:JNJ). They both have about the same market cap. MSFT goes up 20% while JNJ goes down 20%. Both stocks are wild. But what did your two-stock index do? It stayed flat. The statistical correlation between the two was well below 1.00. In fact, the statistical correlation was a perfect -1.00. This is similar to what happened in July 2008 when the oil and material sectors crumbled while consumer staples and health care stocks rallied. Statistical correlation dropped because some stocks plunged while others rallied. Meanwhile, stock indexes stayed relatively steady.

Confused? I wouldn’t be surprised, because it’s not like statistical volatility versus implied volatility. In that case, both versions of volatility are measuring magnitude: actual vs. expected.

Correlation, however, now has two very different definitions. The first definition is the one most people think of: correlation in the statistical sense, which is connectedness in terms of direction. Implied correlation, on the other hand, measures something completely different: expected magnitude of stock movement versus expected magnitude of index movement.

Look for much more on this topic, including how to use the indicator for trading purposes, in the very near future.

P.S. – If after reading this, your head didn’t explode, then you might want to get one of these!