Category Archives: stock prices

The Logistic Map and the Emergence of Complexity

What's the connection between a dripping faucet, the Mandelbrot Set, a population of rabbits, thermal convection in a fluid, and the firing of neurons in your brain?

That's the lead question asked in the following under-19 minute video by Veritasium exploring the logistic map, in which very complex and chaotic outcomes follow from very seemingly simple math relationships.

If you'd like to play around with the logistic map for modeling the population of rabbits over time, using the equation:

Xn+1 = rXn(1 - Xn)

We've built a simple tool to make it easier to cycle through the outputs for whatever parameter values you choose to enter. If you're reading this article on a site that republishes our RSS news feed, please click here to access a working version of this tool our site.

Logistic Map Parameter Values
Input DataValues
Percentage of Maximum Population (Xn)
Growth Rate (r)

For added fun, we'll note that bifurcated behavior has also been observed in stock prices. A 2014 paper by David Nawrocki and Tonis Vaga describes that scenario:

We propose a bifurcation model of market returns to describe transitions between an 'over-reaction' mean regressive state and 'under-reaction' trend persistent states. Since July 1929, the Dow Jones Industrial Average has exhibited non-stationary state transition behavior, including: (1) mean regressive behavior during crisis situations during the Great Depression of the 1930s and again in the crisis of 2008 when the availability of credit was interrupted; (2) strongly bifurcated, or trend persistent behavior from the 1940s through 1975; and (3) more efficient behavior since 1975. The bifurcation dynamic evident in the pre-1975 era is somewhat enhanced by conditional volume and moderate volatility. The bifurcation model is used to develop a quantitative measure of the degree of market efficiency, which indicates that the market has become more efficient, i.e. less trend persistent, since 1975 with the advent of negotiated commissions and computerized trading techniques. Similar findings are presented for the S&P 500 index and the CRSP Value Weighted Index, which represent large capitalization markets.

There's also a 2016 paper by Marzena Kozlowska et al. that points to the "flickering" behavior in stock prices as an early warning signal when the market nears a "bifurcated catastrophic transition", or "tipping point".

We came across both these papers while investigating potential explanations for why long winning streaks and especially why long losing streaks have become less frequent over time. One thing led to another and suddenly we were immersed in the math of rabbit population growth and Mandelbrot sets, which is pretty interesting in and of itself.

Welcome to the world of complexity!...

Iron Mountain as a Successful Bet on the COVID-19 Economic Recovery

Back on 18 November 2020, we described the stock of document storage giant Iron Mountain (NYSE: IRM) as a "COVID-19 play, or rather, a bet on the future for the coronavirus pandemic's real world impact on business activities".

By that, we recognized that the future for IRM's stock price would very much depend on the extent to which the global economy recovered from the coronavirus pandemic and Europe's economy in particular, since Iron Mountain was actively expanding its business in that region. We're following up that observation today, comparing how the stock price of IRM compares with the benchmark of the S&P 500 (Index: SPX). The following chart shows that comparison over the seven months from 18 November 2020 to 18 June 2021:

IRM vs S&P 500, 18 November 2020 - 18 June 2021

Nice to see the bet played out well. As for our coverage, unless we find Iron Mountain back in the kind of circumstances that originally drew our attention to it, we're closing out our short series on future prospects of the company's stock price today.

Previously on Political Calculations

Here are the two previous posts where we've discussed the prospects for Iron Mountain's stock price, presented below in chronological order:

What Is m?

What is the value of m?

Before we answer that question, we should probably take a moment to explain what m is:

The basic relationship we've observed that exists between changes in the rate of growth of stock prices and changes in the rate of growth of their dividends per share, or in our terminology, their accelerations, is given by the equation:


Where Ap is the change in the rate of growth of stock prices and Ad is the change in the rate of growth of dividends per share. m is an amplification factor that varies over long periods of time but can be nearly constant for short-to-intermediate periods of time, which we'll focus more upon in future posts.

That explanation of what m is has been there from the very beginning of when we first formulated the dividend futures-based model we use to forecast stock prices from the observations that preceded it just over 11 years ago, during the stock market crash of 2008-2009.

Finding the value of m is something that can only be done by empirical observation, where we can use historic stock price and dividend futures data to solve that simple relationship for m. Doing that was especially challenging 11 years ago, because quarterly dividend futures as we know them today didn't exist at the time. It wasn't until the Chicago Board of Exchange first introduced them in March 2010, which were subsequently discontinued and replaced by the CME Group's quarterly dividend futures in 2015 that we could fully flesh out the potential of the dividend futures-based model we had defined.

What we found when the quarterly dividend futures data came into existence is that the value of m was 5. And that value held constant through Friday, 20 March 2020.

That's when things really started to go haywire for the dividend futures-based model's forecasts of the alternative trajectories the S&P 500 would be most likely to take based on how far forward in time investors were looking as they made their current day investment decisions, with the actual value of the S&P 500 moving outside the levels the model indicated it would most likely go with the amplification factor set to 5, the first time that has happened in the model's history.

Going back to our week-by-week news archive of market-moving events, we think the trigger that effectively reset the value of the amplification factor m was the Fed's over-the-weekend firing of its 'bazooka' to backstop the commercial paper market, which companies typically use to borrow money, but which had nearly completely broken down as part of the economic impact of the coronavirus pandemic.

After trading resumed on Monday, 23 March 2020, investors appear to be using a different amplification factor. The question is what is the new value for m?

As best as we can tell, the new value of m since 20 March 2020 is somewhere between 1 and 2. In the following animated chart, we're showing what the dividend futures-based model would forecast for the period from 23 March 2020 onward looks like when m = 5, when m = 2, and when m = 1, with all other values based on data available through the close of trading on 8 April 2020.

Animation: Alternative Futures - SP 500 - 2020Q1 and 2020Q2 - Standard Model with m = 5, m = 2, and m = 1 - Snapshot on 8 April 2020

Given the ongoing elevated level of volatility in stock prices, it may be a while before we can get to a relatively quiet point where we can properly calibrate the model and determine what the value of m has become. We can however answer a question we've had since 23 April 2009: A "short-to-intermediate period of time" in the U.S. stock market may last for a decade!

Whether that's "up to a decade", "at least a decade", or "a decade on average", will take a lot longer to discover.

25/2/2020: No, 2019-nCov did not push forward PE ratios to 2002 levels

Markets are having a conniption these days and coronavirus is all the rage in the news flow.  Here is the 5 days chart for the major indices:

And it sure does look like a massive selloff.

Still, hysteria aside, no one is considering the simple fact: the markets have been so irrationally priced for months now, that even with the earnings being superficially inflated on per share basis by the years of rampant buybacks and non-GAAP artistry, the PE ratios are screaming 'bubble' from any angle you look at them.

Here is the Factset latest 20 years comparative chart for forward PEs:

You really don't need a PhD in Balck Swannery Studies to get the idea: we are trending at the levels last seen in 1H 2002. Every sector, save for energy and healthcare, is now in above 20 year average territory.  Factset folks say it as it is: "One year prior (February 20, 2019), the forward 12-month P/E ratio was 16.2. Over the following 12 months (February 20, 2019 to February 19, 2020), the price of the S&P 500 increased by 21.6%, while the forward 12-month EPS estimate increased by 4.1%. Thus, the increase in the “P” has been the main driver of the increase in the P/E ratio over the past 12 months."

So, about that 'Dow is 5.8% down in just five days' panic: the real Black Swan is that it takes a coronavirus to point to the absurdity of our markets expectations.

The Lévy Stable Distribution of Stock Prices

We often use the phrase "Lévy flight events" to describe the outsized movements of stock prices, but we've never really addressed the reason why we use that terminology!

Let's start by looking at the day-to-day volatility of stock prices for the S&P 500 (Index: SPX) since 3 January 1950. In the following chart, we've shown that volatility as the percentage change from the previous trading day's closing value for the index, where we've also presented the mean and standard deviation of that variation all the way through the close of trading on Friday, 20 September 2019.

S&P 500 Daily Volatility, 3 January 1950 - 20 September 2019

If the volatility of stock prices followed a normal Gaussian distribution, we would expect that:

  • 68.3% of all observations would fall within one standard deviation of the mean trend line
  • 95.5% of all observations would fall within two standard deviations of the mean trend line
  • 99.7% of all observations would fall within three standard deviations of the mean trend line

But that's not what we see with the S&P 500's data, is it? For our 17,543 daily observations, we instead find:

  • 78.7% of all observations fall within one standard deviation of the mean trend line
  • 95.3% of all observations fall within two standard deviations of the mean trend line
  • 98.6% of all observations fall within three standard deviations of the mean trend line

Already, you can see that the day-to-day variation in stock prices isn't normal, or rather, is not well described by normal Gaussian distribution. While there are about as many observations between two and three standard deviations of the mean as we would expect in that scenario, there are way more observations within just one standard deviation of the mean than we would ever expect if stock prices were really normally distributed.

Additional discrepancies also show up the farther away from the mean you get. There are more outsized changes than if a normal distribution applied, which is to say that the real world distribution of stock price volatility has fatter tails than would be expected in such a Gaussian distribution.

There are other kinds of stable distributions that also have a central tendency of for variation in data to appear near the mean that do a better job of describing the variation in stock prices. One of these was developed by French mathematician Paul Lévy and is now known as the Lévy distribution.

The Lévy distribution's applicability for describing the variation of stock prices was recently validated in a paper posted at arXiv by Takumi Fukunaga and Ken Umeno, who found that the Lévy distribution does a better job than the normal Gaussian distribution for the S&P 500, the Nikkei 225, the Dow 30, and the Shanghai Stock Exchange (SSE) indices. Figure 1 from the paper compares the probability density functions of the standardized raw data, the Lévy’s stable distribution with estimated parameters (α, β), and the Gaussian distribution, while Table 1 gives the Lévy’s stable distribution's parameters for each stock market index:

Fukunaga and Umeno, Universal Lévy’s stable law of stock market and its characterization, Table 1 and Figure 1

For all four stock market indices, the Lévy distribution outperforms the normal Gaussian distribution in describing the variation of stock prices. What's more, Fukunaga and Umeno find all four indices share very similar parameter values for their Lévy distributions in their period of interest from 2 January 1975 to their arbitrary cutoff date of 31 May 2017.

In terms of the generalized central limit theorem, Lévy’s stable distribution is theoretically more suitable than the Gaussian distribution for fitting the log-returns of the stock markets. The stock prices with power-law tails would not converge to the Gaussian distribution, since the classical central limit theorem cannot be applied in this case.

The parameters (α, β) show a similar value regardless of the stock index. The stability parameter α of all the stock indices were around α = 1.6 which seems to be universal, and lower than the Gaussian distribution corresponding to α = 2. As β has a negative value, it is shown that the stock market has a skewness. Then, the parameters fluctuate by dividing the analyzing time-windows. There is a correlation between the price and (α, β), especially when the financial crisis occurred.

Overall, we see that while there still more observations concentrated around the means for each index than would be expected by either the normal Gaussian distribution or the Lévy’s stable distribution, the Lévy distribution comes much closer to accounting for that greater concentration than the Gaussian distribution does, while more accurately reflecting the frequency of large, outsized changes in stock prices.

And that's why we use the phrase "Lévy flight events" whenever we discuss days where stock prices changed by very large percentages from the previous day's closing value.