Are we Thinking about Correlation Wrong?

One of the fascinating things about writing in the financial space is that new ideas and theories emerge each and every day. Write well enough, with enough memes and GIFs, and you become a thought leader in the space. We write about alternative investments (and particularly Managed Futures) alongside some other great publications: CTA Intelligence, Futures Magazine, and ValueWalk to name a few. But very rarely do we see an actual CTA (Commodity Trading Advisor) step up to the financial blogosphere plate and share their research and strategy. Here’s calling all CTAs to put out more of your thought leadership… trust us, there’s an appetite for it.

Along those lines – we were excited to come across ReSolve Asset Management’s blog with a double feature on trends in finance research! These guys are smart, and there’s a lot to unpack in both of these articles (article 1 & article 2) but one thing we found interesting was their challenge to reconsider what it means for investments to be positively and negatively correlated. Take a look:

Consider two strategies A and B, and their returns over a 10-year period. Their return series is depicted in the table below.

Both strategies have an annual average return of 10. Whenever A’s return is above its average of 10, B’s return is below its average of 10. And whenever A’s return is below its average of 10, B’s return is above its average of 10. Thus, regressing strategy A’s returns on strategy B’s returns over this period will conclude they are negatively correlated. Note that they are negatively correlated even though they both always produced positive returns.

Now imagine that the same strategies produced the following returns in a different 10-year period.

Over this period, the same strategies have an average annual return of 0 percent. Perhaps the styles went out of favor. However, whenever A’s return is above its average of 0, B’s return is below its average of zero. And whenever A’s return is below its average of 0, B’s return is above its average of zero. Thus, regressing A on B will render the conclusion that they are negatively correlated.

Now let’s string together the two 10-year periods so that we have a 20-year period. Thus, the return series looks like this:

Recall that both A and B had average returns in the first 10 years of 10 percent, and average returns of 0 percent in the second 10 years. Thus, their average return for the full 20 years in both cases is 5 percent. Now: Are A and B positively or negatively correlated?

A closer inspection reveals that, over the full 20-year period whenever A’s return was above its average of 5, B’s return was also above its average of 5. And whenever A’s return was below its average of 5, B’s return was also below its average of 5. Thus, we see that despite the fact that A and B were negatively correlated over each of the two 10-year periods independently, over the full 20-year period they were positively correlated.

This example highlights an omnipresent by rarely discussed challenge with financial time-series. Specifically, that the measured relationship between variables will almost always change dramatically across time. This effect is not isolated to observations over two distinct periods of time; rather, we observe similar dynamics at play when time series are observed at different frequencies. In fact, variables can appear to be negatively correlated at one frequency – say daily – and yet be positively correlated at another frequency – say monthly!

So, correlations matter, but like everything in life, context should come to play. It’s mathematically possible that things not correlated over long periods are highly correlated over shorter periods (see our recent post), and likewise possible for negatively correlated time series over short time periods to be highly correlated over longer ones. This is why it’s always wise to check out daily returns in addition to monthlies to ferret out any of these seeming anomalies.  To learn more about the paradox, check out more of Resolve’s thoughts and research on their blog here.

And to the rest of you CTAs out there – start blogging!

Disclaimer
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