Correlation: A Misunderstood Concept in Finance

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3 mins read

Let's start out with a simple test. In the image below, are returns of the Stock 1 (cyan) correlated with the returns of Stock 2 (green) positively or negatively?

If you answered postively, you are hardly alone. In fact, the returns of Stock 1 and Stock 2 have a perfect correlation of negative one in their returns. Even though they are both increasing, and start and end at the same value.

A common misconception is that if two series are moving the same direction that means they are positively correlated. There are plenty of articles in financial literature that claim exactly this. See Investopedia, and Robinhood here for instance:

A positive correlation indicates that two variables have a relationship with each other and move in the same direction — when one rises or falls, so does the other.

That is not true, and unfortunately an all-too-common belief that misses a crucial point about correlation. You cannot generally tell the correlation of two stocks, or any time-series for that matter, by looking at a chart.

In the chart above, Stock 1 and Stock 2 are both increasing every day together in the same direction, yet they are perfectly negatively correlated. You can see the values here in this Google Sheets. How can this be then?

It is because correlations are calculated around the mean, and do not indicate direction whatsoever. If you look at the correlation formula below, you can see that the mean of each series in deducted in the numerator. That means that if two series are positively correlated, they are moving together with respect to their own individual means.

ρX,Y=cov(X,Y)σXσY=E[(XμX)(YμY)σXσY\rho_{X,Y} = \frac{cov(X, Y)}{\sigma_X \sigma_Y}=\frac{E[(X-\mu_X)(Y-\mu_Y)}{\sigma_X \sigma_Y}

Correlation on its own has no information about the direction of time-series. The only metric indicating long-term direction is the average.

In the chart above, Stock 1 and Stock 2 are both increasing at the average rate of 2.5 units every step, however they are increasing in alternating [1, 4] and [4, 1] increments. This means that when Stock 1 is increasing by 1 unit (below its average of 2.5), Stock 2 is increasing by 4 units (above its average of 2.5), thus the perfect negative correlation between the two.

Now can you guess the correlation between Stock 1 and Stock 3?

Note that throughout this article we discussed the most common measure of correlation, which is the Pearson Correlation Coefficient.