The Importance of Correlation

Earlier this week The Wall Street Journal ran a piece on, of all things, the importance of the correlation coefficients between the returns of investments. I have mixed feelings about it.

On the one hand, correlation between asset returns is a neglected subject of great importance. The mid-Twentieth Century realization of its central role was the start of modern financial theory as we now know it. A professional level understanding of risk begins and ends with correlations, so it would make some sense for amateur investors to know at least the basics.

On the other hand, the article serves as a good reminder of why they know so little. Despite being called Why the Math of Correlation Matters, it contains no math. This might be because the author worried that her readers would find the math scary and hard, but I fear it is because the author herself finds it scary and hard.

The correlation coefficient (known simply as correlation to his friends) is a basic statistical measure that gauges the relatedness of two sets of numbers. It ranges from –1 to 1. As the WSJ explains well, a correlation of 0 means that the two sets of numbers are unrelated, that knowing the values of set X gives you no information at all about set Y.

Positive values tell you that the numbers run together to some degree, with a perfect score of 1 meaning that they are in perfect lock step. Negative numbers denote an inverse relationship, with –1 meaning that they are perfectly counterbalanced.

However, perfect lockstep does not mean identical, only that knowing the value for one set of numbers is enough to work out the other. In algebraic terms, a correlation of 1 means that there exists an equation to calculate Y from X of the form:

Y = a + bX (or Y = a – bX when correlation is –1)

Where a and b (a.k.a the intercept and slope) are constants. So, for example, Fahrenheit and Celsius temperature readings are correlated at 1.0, with a = 32 and b = 1.8. The WSJ explains b, but forgets about a, which is a larger oversight than you might think.

The importance of correlation in the investing world comes from the simple (and Nobel Prize winning) insight that since investors naturally seek to minimize risk, what they should do is construct portfolios with assets that have as low a correlation with each other as possible. Buying two assets with a correlation of 1.0 is pointless. They will both do well or poorly at exactly the same time.

But buying assets with low correlations to each other makes you better off, since their returns will tend to balance each other out. That means that the portfolio of assets will have a lower volatility and/or a higher return than any individual asset within it. Thus the power of diversification, sometimes referred to as the only free lunch in investing.

The WSJ article provides an enlightening table of asset class correlations with the S&P 500 over the past ten years. The top three assets are non-US developed market stocks, US small cap stocks, and emerging market stocks, which come in at correlations to the S&P 500 of 0.89, 0.88, and 0.82 respectively.

Those numbers are likely a surprise to the many sophisticated investors who cleverly diversified themselves away from US large caps by shifting a little into small caps and non-US stocks. There is some diversification value to be had there, but not as much as was available in such things as real estate, commodities, and bonds. Stocks are stocks the world over, and a bad month on Wall Street is likely to be a bad month in Tokyo and Rio de Janeiro too.

Which is not to say that putting a chunk of your money in emerging markets ten years ago would not have been very clever. The MSCI EM index was up an average of 11% a year over that period while the S&P 500 was close to flat. Being highly (or even perfectly) correlated does not mean that one investment is not habitually better, just as Fahrenheit temperature readings are systematically higher than Celsius ones. This is the importance of the a (or alpha) part that the WSJ skipped over.

Overall, the article is an unsatisfying tease. It tells us that correlation is important, and hints at why, but never quite gets down to brass tacks. The last section, subtitled “Where can I find information on correlation?” is particularly disappointing.

Most online correlation calculators are available only for financial advisers. So, one option is to ask your financial planner.

Tools for individuals include assetcorrelation.com, which finds correlations between assets and between asset classes.

R-squared, a measure found on Morningstar.com, shows strength of correlations between funds and benchmark indexes, but not directions of movement. The scale ranges from 0 to 100.

If you get yourself lists of the returns from two assets, Excel will calculate the correlation. (=CORREL() ) It’s help function even has a decent explanation of how the calculation is performed. Assetcorrelation.com turns out to be a useful, if not very flexible, site.

R-squared is another statistical measure commonly used in finance. In the context of measuring the risk of a mutual fund, it is the square of the correlation between the fund and the benchmark. R is stats notation for an estimated correlation. R-squared runs from 0 to 1, not 0 to 100.

Investing, Media | Frank Curmudgeon | October 6, 2010 11:59 am

No Comments

By Steve, October 6, 2010 @ 12:38 pm

Speaking of financial planners, I caught one just yesterday saying that a correlation of -1 means unrelated.
By jim, October 6, 2010 @ 1:45 pm

Correlation has been topic I don’t know much about myself. I don’t have a finance background and correlation is not something I’ve seen discussed in personal finance stuff at all. So its just something I haven’t learned yet.

I think the WSJ article was a pretty decent introduction to the topic. Its a pretty complex topic to cover in any detail and a newspaper article is limited in length and has to keep things appropriate for a broad audience.
By Stagflationary Mark, October 6, 2010 @ 2:15 pm

Steve,

That’s just sad.

I’d love to find a financial planner who offered financial advice with a perfect -1 correlation. He or she would be SO bad that you could actually make money off of the advice by *always* doing the exact opposite!

Perhaps you have stumbled upon such a person.
By Stagflationary Mark, October 6, 2010 @ 2:22 pm

One more thought on what a chart showing the lack of correlation looks like. I created the chart to make a point.

http://illusionofprosperity.blogspot.com/2010/09/failed-keynesian-phillips-curve.html

It shows the average unemployment over the previous 12 months vs. the year over year inflation rate (from 1948 to 2010). We’re told that inflation helps us all. I’m not seeing the correlation there. It looks more like an ink blot test to me.
By Stagflationary Mark, October 6, 2010 @ 2:29 pm

Out of curiosity, I just checked the r-squared value in the chart I just offered.

It was 0.02. On a scale of -1 to 1, that’s about as close to zero correlation as you can get.
By David Harper, October 6, 2010 @ 8:16 pm

The actual math is strangely always avoided, but any CFA/MBA knows: correlation = covariance(X,Y/[volatility X]*[volatility Y]; this allows such that correlation is a function of volatility and teases out the weakness of covariance/variance.

No two assets exhibit a “single correlation” as such; each sample produces a different historical correlation. Like volatility, current correlation is only an estimate of an unobserved relationship – going forward will time-vary.

(Also, correlation is a linear dependence; non-linear dependence may be more realistic, handled with copulas)
By bex, October 6, 2010 @ 11:08 pm

Another thing I wish the article talked about was the problem of herd mentality… Once too many people think two assets are not co-related, they sometimes BECOME correlated.

This was one of the downfalls of the “Long Term Capital Management” folks. Once some hedge fund starts making a lot of money, people (sometimes) try to deduce the pattern… the end result being that they all start to buy the same things, because they think they are hedged.

Now… assume asset A goes up, and asset B goes down… also assume that there’s suddenly a recession, or some other problem where these guys don’t have enough liquidity. What will happen?

Well, a lot of folks will start selling off asset A while it’s up to get some cash… which will probably make the price fall… just at the time that asset B is falling too.

Suddenly, all that fancy hedging ain’t worth squat if too many people think it’s a good idea
By Rob Bennett, October 7, 2010 @ 11:19 am

The importance of correlation in the investing world comes from the simple (and Nobel Prize winning) insight that since investors naturally seek to minimize risk

This is why things are such a mess today.

Investors seek to minimize risk? You could have fooled me.

A regression analysis of the historical data showed that the most likely 10-year return for stocks in 2000 was a negative 1 percent real. TIPS were paying a certain 4 percent real. Most investors (and most “experts”!) went with high stock allocations.

This is minimizing risk?

What most investors (and most “experts”!) do is construct convoluted theories for rationalizing their strong emotional preference for Get Rich Quick investing strategies.

I mean no personal insult with these words. My intent is to get some people to take a look at core flaws in the Modern Financial Theory mumbo jumbo that is in the process of eating us all alive (in my view!).

Rob
By Mt, October 7, 2010 @ 11:31 am

There is a person who made all these spreadsheets:

http://www.financialwebring.org/gummy-stuff/gummy_stuff.htm

In the list, there is one that I like called “Correlation” which pulls the information for 30 stocks from the internet and charts the correlations. There are others, too.

If you sift through the site, it is a treasure trove. I haven’t seen software available for purchase which can do everything he has done for free with spreadsheets.

The guy is like the Canadian Da Vinci of Excel investment data.
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