No One Actually Understands Correlation, But (Hopefully) Now You Will

It is a common refrain these days among investment managers that “fundamentals don’t matter.” The market does what it does because of ________. (possible answers include: easy monetary policy; tech bubble v2.0; passive investing bubble) This makes all stocks go up together. With stocks so highly correlated it is impossible for a stock picker to succeed because “bad stocks” get rewarded just as much as “good stocks.” That is why passive investing is so popular. It is all one big, self-reinforcing bubble. When the pointy reckoning finally arrives all of us fancy active manager types will laugh our way to the bank.

I admit I am guilty of saying some of this stuff myself. Which I suppose makes me extra guilty because I understand correlation and how to interpret the statistic and am still using the term imprecisely.

Here is a good analysis by Aaron Brask on the Alpha Architect blog refuting this argument. He argues from first principles and even conducts a simulation to show that mathematically, correlation does not impact the expected returns of individual stocks. I will steal his chart because it is a convenient summary:


Mathematically, Brask’s argument is irrefutable. That’s the nice thing about mathematics. When you’re right, you’re right.

I will go further and argue that all this confusion about correlation stems from the fact that many finance professionals don’t actually understand it. They use a heuristic: “correlation = perform the same.” That is how they are used to explaining correlation to clients (and each other). The heuristic is fine for generic spiels about portfolio diversification but it can be dangerous when applied to actual portfolio management decisions.

I have encountered this on several occasions. For example, a colleague once asked if there was a mistake in a chart that showed the S&P 500 correlated nearly perfectly with a 50/50 blend of T-Bills and the S&P 500. My colleague was using that heuristic of “correlation = perform the same.” The two portfolios are indeed perfectly correlated. However, the historical return of the blended portfolio is much lower because T-Bills tend to return much less than stocks over time. This is exactly what Brask illustrates in his simulation.

Look again at his chart. Instead of focusing your attention on the return, compare the shapes of the line graphs. Pretty close, right? That visual similarity is indicative of high correlation. That’s because correlation measures similarity in the variation of returns, not similarity in returns themselves.

To illustrate further, here are three more visuals, graphing relative outperformance/underperformance of different portfolios over time.

The first is the S&P 500 versus itself (perfect correlation = 1):


The next is the S&P 500 versus the Bloomberg Barclays Aggregate Bond Index (correlation = basically 0 but in fact is slightly negative):


The final comparison is S&P 500 versus a 50/50 blend of itself and T-Bills (correlation = very near 1) :


Notice how in the first and third charts, the points plot in a straight line, while in the middle chart they are an uncoordinated blob. In this visualization, the more the plotted points resemble a straight line, the higher the correlation. The third chart shows a strong linear relationship but with much higher returns for the S&P 500 over the blended portfolio.

An improved heuristic for correlation is to think about the extent to which two assets share common risk factors. An investment grade bond index is mostly exposed to reinvestment and interest rate risk (seasoned with a pinch of credit risk). A stock index has little direct interest rate risk and almost no reinvestment risk. It is more exposed to the business cycle and economic variables such as real wage growth. Intuitively, you would expect very little correlation between stock index returns and bond index returns.

T-Bills have very little risk of any kind. Some people think of them as risk free. That is not entirely accurate but for this exercise it is a safe assumption. When you combine T-Bills with the S&P 500 the only relevant risk exposures are those of the S&P 500. They will drive 100% of the variation in portfolio returns over time, despite the fact that 50% of the portfolio is risk free. You can therefore expect high correlation with the S&P 500.

So when someone in the investment business says, “high correlations are bad for stock pickers,” she isn’t actually talking about correlation. What she actually means is, “market environments where investors don’t care whether they own good businesses or bad businesses make it difficult for active managers to outperform their benchmarks on a relative basis.” Hopefully she understands the difference intellectually and is just speaking imprecisely, using the “correlation = perform the same” heuristic for convenience. But you never know. People will trot out some pretty silly stuff when billions of dollars of fee revenue are on the line.

(Incidentally, if you are the type of person who likes to give prospective financial advisors quizzes before hiring them, this is a good topic to add to your list)

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