10 June 2020 – The title of this essay sounds like a new Argentinian dance craze, but it’s not. It’s a pattern of stock-price fluctuations that has been repeated over, and over, since folks have been tracking stock prices. It doesn’t get the attention it deserves because people who pretend that they have power (i.e., the People In Charge – PIC), and can wisely dispense it, don’t like things that show how little power they actually have. So, they ignore the heck out of them, thereby proving themselves dumb, as well as powerless.

There’s been a lot of blather in the news media recently about some hypothetical “V-shaped recovery,” which a lot of pundits, especially those of the Republican-Party persuasion (notably led by that master of misinformation, Donald Trump), want you to believe the U.S. economy is experiencing. In an attempt to prove their case, they point to the performance over approximately the past three months of all three major equity-market indices, those being the Dow-Jones Industrial Average (DJIA), the Standard and Poor’s 500-Stock Composite Index (S&P), and the National Association of Securities Dealers Automated Quotations index (NASDAQ),. Those three indices do tell a consistent story, but it’s not the one the V-shaped-recovery fans want you to believe. The story is actually much more complicated. It’s what’s called the dead-cat bounce.

To understand the dead-cat bounce that has been going on since the U.S. equities market crashed in March, you have to understand what I was driving at in this space on 18 March 2020. That was about the time the market bloodbath hit bottom. By the way, I’d been mostly out of the market, and into cash, for several months at that point. I could see that something evil was bound to happen in the near future. I just didn’t know what it would be. It turned out to be a pandemic coming out of the blue.

In that 18 March essay, I spent a whole lot of space developing the chaotic-market theory, which visualizes markets as having an equilibrium value based on classical efficient-market theory, with a free-roaming chaotic component riding on it. The chaotic component arises as millions of investors jostle to control prices of thousands of equity instruments (stocks). One of the first things those of us who have been responsible for designing and building feedback control systems run into is a little phenomenon called pilot-involved oscillation (PIO), named after an instability all pilots have to deal with when learning to land an airplane. PIO arises from the inescapable fact that feedback response comes some time after the system moves off equilibrium. Obviously, the response can’t come before the movement, it has to come after. That’s why they call it a “response!” That time lag is what causes the PIO.

A feedback-controlled system’s behavior follows what’s called a inhomogeneous time-dependent linear differential equation. Let me break that name down a bit. The “inhomogeneous” part just means there is something driving the system. In the case of equities markets, that’s the underlying economy setting the equilibrium in accordance with Adam Smith’s supply and demand. The “time-dependent” part just means that things change over time. As Jim Morrison said: “The future’s uncertain and the end is always near.” A “linear differential equation” means that what happens next depends on what happened before, and the rate at which things are changing, now. Without going into the applied mathematics of finding a solution, I’ll just skip to the end, and tell you that there’s only one solution: the dratted things oscillate. That is, they go up and down, always overshooting and undershooting the equilibrium point.

Do you see the connection, now?

That solution is called a damped harmonic oscillator, which simply means that the thing’s overshooting and undershooting follows a regular sinusoidal (you’ll have to look that one up, yourself) pattern, but it dies out over time. The rate at which the oscillation dies out is controlled by something called the damping ratio, which can take on any value from zero to infinity. Zero damping means the oscillation doesn’t die out. A damping ratio exactly equal to one means the system over- or undershoots once, then comes back to its equilibrium value. A damping ratio much over one makes the system respond sluggishly, and not oscillate at all.

Now, with that explanation in mind, look at the market behavior depicted in the graph above. The graph starts at the beginning of March 2020. Investors started to realize that the pandemic was going to trash the U.S. economy around mid-February, so you see that I’ve cut off some of the start of the crash that happened before 1 March. By 1 March, stock prices were falling like a stone until 23 March. That’s when the dead cat hit the pavement, and bounced. It bounced too high and, around 27 March, it started falling back down, only to undershoot again. Around 2 April, it bottomed and started back up, again. Looking at these movements quantitatively, we can see the clear pattern of a damped oscillation with a period of about 12 days, and a damping ratio of between 0.2 and 0.4.

To bring out the underlying pattern, I’ve filtered the data by averaging over three days for each point in the data set to get the smoother red line. The three-day filter (called a Butterworth filter, by the way) does little to suppress the slower 12-day oscillations, or the even slower smack from the pandemic’s economic hit. I does, however, pretty well filter out the daily noise from the fast-moving day-trading fluctuations.

Clearly, we are in a recovery. There’s no doubt about that! The economy is coming back to life after being practically shut down for a short period of time. The initial shock from the pandemic is largely over. Look for a gradual return to the three-to-five-percent-per-year long-term growth rate we’ve seen over the century-and-a-quarter history of the DJIA.