Chaos Theory, Market Cycles and Macro Regimes

The market environment is a critical element to making investment decisions.  But knowing that environment is confusing.  Why?  The markets don’t work in a vacuum.  They are not stand alone entities, and are influenced by many things.  In fact there are so many elements to think about that it overwhelms the typical investor so that they largely ignore them, or go with what they read in the media. We know there is a market cycle, and an economic cycle. But how can we measure them? How can we know where we are so we can plan for the future?

Chaos Theory

The market system is actually simpler than it looks, but also being quite complex, and it relates to chaos theory. It sounds like a joke to say the market is chaotic (duh!) but in chaos theory a system can look random but be completely deterministic. And a chaotic system can have multiple dimensions which also look unrelated to one another though they are directly tied. Trying to understand one dimension without taking into account the others is a losing game.  Look at a well known chaotic attractor, the Lorenz attractor. It’s one of the poster children of chaos theory and usually shown in two dimensions so it looks like owl eyes.

This system is three interrelated sub-systems (in the x,y, and z dimensions) but the output of one becomes the input to another. Because they are interconnected, all three evolve together through time.  If we were to look at the three equations separately as time series it would look like this:

While two of the series, x and y, look very similar to one another, the third one, z, looks completely different.  In fact, x and y have a 60% correlation with one another but have correlations of 1% and -2% respectively with z.  If you look at x and y they spend a fair amount of time either above or below the zero line.  In the “owl eyes” graph this is when the system shifts from one eye to the other.  That shift is largely caused by z which looks statistically unrelated to x and y.   So it would be easy to overlook z if we didn’t know there was a direct link.

We can think of x, y, and z as following their own cycles. Cycles are when x, y and z go up and down. But x and y also have regimes, periods of time where they are above or below zero. 

x and y also show you the nature of “non-periodic cycles.” A periodic function like a sine wave would go above and below the zero line with great regularity. If we were to measure a cycle length from peak to peak they would always be equal which defines a “periodic” cycle. But x and y spend different times above and below zero and also from peak to trough. The regimes are of different lengths as are the cycles, but they do have an average length. This is similar to the investment horizon length I use in the Fractal Market Hypothesis.

While the Lorenz attractor does not model markets, it is a useful model for understand the mathematical concepts between cycles and regimes.  As you can see in the time series chart, cycles are shorter than regimes.  But both shift because of what might be considered internal and external influences. El Nino gave us a metaphor for long term regimes. The Lorenz attractor gives a mathematical example.

The real lesson is that an economic equivalent of the z dimension of the Lorenz attractor can cause a shift in the market regime though it’s impact is difficult to confirm with standard analysis. In the market climatology format I’ve set up here, the market and business cycles are kind of like x and y in that they are closely related most of the time. The regime changers are the financial instability cycle which is caused by excessive leverage, and/or inflation levels. These regimes are much longer than the market and business cycles. But when two or three of them coincide then the results are very dramatic. So a bear market coinciding with a financial crisis is quite different than a bear market without financial instability. Likewise high inflation changes the character of the crisis yet again. A common theme of these very long cycles is that there is such a long interval between occurrences that investors become complacent and think that those events will no longer happen. As Hyman Minsky says, that usually makes the crisis worse when it happens.