Understanding Bear Markets Mathematically

Buckle up, this is a long post that lays out exactly what is happening in the current bear market. If you are interested in understanding bear markets from a mathematical standpoint, then continue reading!

I see a lot of confusion from traders and even finance professionals about bear markets and the current market selling we are experiencing. Many traders and investors are attempting to determine where the bottom is, when we will get there and what exactly is happening.
Through the aggressive selling we have seen, many people attempt to rationalize and make sense of it via linking the market as reactive to current events (i.e. the atrocities happening in Ukraine, inflation, COVID, etc. etc.). This leads to some people anticipating a pessimistic outlook on the market and others having an optimistic outlook.
But is the selling the result of current events? Is it the result of Elliot Wave theory or Fibonacci retracement tests?

I will present my theory as a retail quantitative trader by job and as a statistician by education.
In order to understand why the market is selling off, or why the market behaves the way it does, you need to subscribe to a theory of the market. Many traders have their own theory of the market and may not even realize. Among quantitative based traders like myself, there are two generally opposing views held. That is, the market is simply random and illogical, operating on randomness alone. The alternative is the market is somewhere between random and organized in that it is actually a chaotic system that works under the laws of chaos theory. As a statistician, this is the theory I personally subscribe to. There are two major reasons for this. Firstly, true randomness is extremely rare. Naturally occurring phenomena tend to operate in systems that have a degree of randomness but still retain predictability (i.e. chaos theory). The second and most prominent reason that I subscribe to this theory is that when we look at the elements of a chaotic system, the market meets all characteristics. There are said to be six general elements of a chaotic system, these are a system that is:
1. Dynamic;
2. Deterministic
3. Intrinsically coherent,
4. Reactive,
5. Structured; and
6. Recursive.

So, why does this matter you may ask? Well, if you subscribe to the chaotic theory of the market, then you can quantify the market and market behaviour mathematically. The characteristics of a chaotic system are organized in such a way that it is possible to make inferences and conclusions via statistical and mathematical approaches because at the core of chaos theory, the laws of nature, probability and mathematics still apply.
Because the laws of probability and mathematics apply to chaotic systems, we can anticipate it to follow rules and phenomena within statistical theories (the recursive element of chaotic systems).
And alas, that brings us to our current bear market. So what is up with this current bear market? Well, let me tell you. And let me use SPY as the example; however, this applies to all stocks.
Any system operating on natural laws of mathematics and probabilities has an inclination to operate within a mean. And while the ideal mathematical world assumes such things as normal distribution and linear relationships, that doesn’t generally happen in the real world. In the real world, we see drifts away from means and linear relationships; however, overall, a linear or quadratic relationship will exist at its core. But the presumption of natural law, and as well, chaotic systems, is things will naturally tend to migrate back towards their mean the further away from their mean they deviate (intrinsically coherent).

If you follow my posts, you will have heard me speak of such things as regression towards the mean. This is precisely what I am referring to here. The further a stock, or any natural phenomena, gets from their mean, the more likely it is for you to observe a regression back towards their mean in repeating data. This phenomena is so omnipresent, that it is covered in many disciplines, like medicine, psychology, data management, etc.
Now taking SPY as an example, if you look at the image below, you will see SPY over its full lifetime of trading. The solid line represents a linear relationship and the curved lined represents a quadratic relationship. Again, things in nature tend to operate in a linear or quadratic function. And you can see this clearly in this image, with SPY staying along those lines. It sometimes drifts above or below the line, only to be pulled back towards that line (i.e. regress towards the mean).

We can actually express this relationship SPY is displaying in a linear equation. The equation is y=5.084 x 10^-6 + 63.124, where X is the squared value of SPY’s trading days. Using this formula, we can actually assess where SPY’s values should fall at any given time. For example, November 20, 2017 (or SPY’s 6256th day of trading), SPY traded around $262. If we apply this formula, here is the result:

Y = 5.084E-6(6256 x 6256) + 63.124
= 263.42
So SPY was operating well within its natural limits. However, around January of 2018, SPY drifted too far away from its mean. As a result, we saw a snap back that brought SPY back towards its range (regression towards the mean). However, in SPY’s regression towards the mean, it overshot it a little bit, and around December of 2018, SPY was trading too low.
For example, December 28th of 2018, SPY was trading around 251, when the normal value should have been around 280. We quickly saw SPY get back on track within the month, and progress normally for the remainder of the year up until COVID.
During the COVID crash, SPY dropped to 229 on March 23, 2020, when SPY’s price should have been around 310. Now, if you remember our 6 principles of a chaotic system is its reactive. That means, SPY will react to external stimuli. The market is not immune to news. It is reactive and we saw this. However, despite it being reactive, it still must operate within the natural order of things and so we saw the market quickly "correct" itself after this aggressive selling off.

It wasn’t until September 10th of 2020 that we started seeing problems. This exact date was the turning point for SPY, where SPY began growing faster than its natural growth. It kept growing to the point of an unnatural deviation from its mean. So what do you think needed to happen?
That’s right, a regression towards its mean.
So that begs the question, where should SPY be trading at currently? Well, I gave you the tools to figure it out, but I will do it for you ;).
SPY is currently on its 7331th day of trading. This brings SPY’s natural price to 355.47. We are still not there yet. And rest assured, in all of history during these bear markets, SPY has corrected itself back to its natural state. As you have seen, I have done the math.
So what can we expect? Well, as you can see, this doesn’t just happen over night. Because SPY has deviated away from its mean at record levels, it will take some time. But if we do some prospective calculations, SPY’s natural trading range for day 7400 or 69 trading days away, is 362.01.

So what does this all mean? Well, I mean, I hope you took away what is happening here. But if you want the cliff notes, here it is:
SPY will drop below 400.
SPY, and the market in general, is simply obeying mathematical rules and principles and regressing back towards its healthy mean and expected growth level.
Bear markets, corrections, sell-offs, these are all terms of Finance people and don’t actually describe what is happening here. The phenomenon we are witnessing is a regression towards natural levels. You can call it whatever you like, bear market or dump or sell-off, but what it is, is a natural and expected outcome and a mathematical certainty.
Hope you found this helpful!

As always, not financial advice, trade safe and take care.