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.

Blackboard Lecturing Crop 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, which finds correlations between assets and between asset classes.

R-squared, a measure found on, 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. 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.

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