Category Archives: stock prices

Another View of the Bottom Dropping Out of the S&P 500

When we say the bottom dropped out of the S&P 500 (Index: SPX) last week, what does that really mean?

It seems obvious that we're referring to the decline in stock prices, but there's much more to it than that. After all, the stock market has weeks when its value rises and has weeks when its value falls from the previous week, but we don't claim the bottom has dropped out of the index every time the latter situation happens. What makes the trading week ending 21 January 2022 different?

It wasn't the rate at which stock prices declined. On no single day of the holiday-shortened trading week did the S&P 500 drop by more than 2% of their previous day's closing value, the volatility threshold we use for any given day's trading to qualify as being interesting.

Nor did the overall 8.3% decline of the index since its 3 January 2022 record high qualify as interesting, falling less than the 10% threshold that it would take to qualify as a correction in stock prices according to professional traders. By both these definitions, what happened in the Nasdaq composite index (Index: COMP) during the past three weeks qualifies as interesting, but not so much for the more diverse S&P 500.

Instead, our observation has a lot to do with how stock prices were behaving in recent months. Here, let's start with the latest update to our chart showing the S&P 500's periods of relative order and chaos, which now covers the full 30-year period from December 1991 through December 2021.

S&P 500 Average Monthly Index Value vs Trailing Year Dividends per Share, December 1991 to December 2021

According to a long-standing set of technical definitions we developed, here's how we define when a period of order exists in the market:

Order exists in a market whenever the change in the price of assets in the market are closely coupled with the change in the income that might be realized from owning or holding the assets, within a band of approximately normal variation about a central tendency.

In the case of the stock market, the assets are stocks, the price of which is given by the value of the S&P 500 index. The income that might be realized from owning or holding the assets are dividends, represented in the chart by the index' trailing year dividends per share.

When we refer to these two things being closely coupled, we mean when both the value of the S&P 500 and the index' trailing year dividends per share are either rising or falling together in a general trend. When that coupling exists, the variation of stock prices about a mean trend line can be approximately described by a normal distribution.

Looking at the most recent period of our 30-year chart, we see the period from March 2021 through December 2021 could reasonably qualify as a period of order according to our definition. Both stock prices and dividends per share have both been rising during these months.

Our next chart zeroes in on this period with daily data, where we confirm a close coupling exists in the period from 30 June 2021 through the end of 2021 and into January 2022.

S&P 500 Index Value vs Trailing Year Dividends per Share, 31 March 2021 to 21 January 2022

We've added statistical control chart-style thresholds to the chart to visualize a statistical hypothesis test. Here, when stock prices fall within three standard deviations of the mean trend, order can be said to exist for the S&P 500. Once stock prices fall outside that range indicated by the red dashed lines, we reject the hypothesis that order exists in the market.

On 21 January 2022, we see the level of stock prices drop below that key statistical threshold. We confirm order has broken down for the S&P 500 in this analytical approach and has done so by breaking through the lower limit of the range we would expect to find stock prices had the short established period of relative order in the U.S. stock market not broken down. This backward-looking approach confirms the similar assessment we arrived at using a forward-looking model.

To put it more colorfully, the bottom has dropped out of the S&P 500.

And now you know why we can say that! Now the question has become "is what happened on 21 January 2022 an outlier event and the trend still holds, or has it truly broken down?" We'll learn the answer to that question as early as this week.

Previously on Political Calculations

The Definition of a Bubble

Some months ago, Joakim Book put his finger on a problem that has bedeviled financial professionals and Nobel prize-winning economists for a much longer period of time.

A seemingly simple question has bothered the discipline of finance for decades: what is a bubble? In a theoretical sense it's a banal question: if the price of an asset is trading much higher than what it's actually worth, it's a bubble; if not, then it isn't.

That begs the question of how the great professors of finance and economics have attempted to define what a bubble is for their discipline. Here's what Book found when he surveyed the field:

The financial historians William Quinn and John Turner published a book on bubbles last year – Boom and Bust: A Global History of Financial Bubbles – that I reviewed for CapX. They acknowledge this definitional problem of finance's vast bubble literature – and sidestep the issue by offering a practical definition of their own: an asset's price has to increase by 100% and then fall by at least 50%. While completely arbitrary numbers (why, a 99% increase or a 49% collapse is not a bubble...?), it at least allows them to investigate the rich history of our financial past. And it's a lot more useful than entirely empty ones we may get from Nobel Laureates Robert Shiller or Joseph Stiglitz (where "'fundamental' factors do not seem to justify such a price").

It seems far from right for the distinguished professors from Yale University and from Columbia University to give their opinions on bubbles in the stock and housing markets without having ever established how they ought to be defined, especially as they possessed the building blocks for a proper and useful definition. To their credit, the financial historians from Queen's University, Belfast have a practical rule of thumb, if not a definition, though a limited one that can only be seen in a rear view mirror.

This state of affairs is all the more aggravating because there is a suitably workable definition that's been around now for more than a decade. Here it is:

An economic bubble exists whenever the price of an asset that may be freely exchanged in a well-established market first soars then plummets over a sustained period of time at rates that are decoupled from the rate of growth of the income that might be realized from owning or holding the asset.

Let's put this definition to practical use. Let's go back to turn of the millennium to consider the Dot Com Bubble. The assets involved here are shares of stocks so in addition to working with stock prices, we'll be considering dividends as well, which represents the income that might be realized from owning or holding shares of stock.

The following chart shows the trajectory of the S&P 500 index with respect to its underlying trailing year dividends per share. It shows the periods of relative order preceding and following the Dot Com Bubble, during which stock prices and dividends were generally coupled with both following rising trends while they lasted. In between is the chaotic event of the Dot Com Bubble, when they became decoupled, which is a very visibly different period.

S&P 500 Index Value vs Trailing Year Dividends per Share, 17 December 1991 through 31 December 2007

During relative periods of order in the stock market, we can borrow some basic techniques from statistical analysis to help determine when these periods begin and end. We've done that in the following two charts, one for the period of relative order that ran from 17 December 1991 until the seminal event of 7 May 1997, the other for the period from 30 June 2003 through 31 December 2007, which ended with the onset of the "Great Recession" in January 2008.

S&P 500 Index Value vs Trailing Year Dividends per Share, 17 December 1991 through 30 June 1997
S&P 500 Index Value vs Trailing Year Dividends per Share, 31 March 2003 through 31 March 2008

In these charts, we've mapped the main trend curve using a power law relationship between stock prices and their trailing year dividends per share. The standard deviation for each is based upon the residual variation of the data points with respect to the main trend curve. We've overlaid a statistical control chart-style thresholds to visualize where we expect to find the data assuming a normal distribution and to set up a statistical hypothesis test.

In that test, we can say that stock prices and dividends are coupled in a relatively stable period of order while their trajectory stays within three significant deviations of the mean trend curve. If it moves outside that range and stays outside of it, the odds are that relatively stable relationship no longer applies. In these two charts, you can easily see how quickly those relative periods of order broke down after the dates marking the end of each.

Bubble Photo by Raspopova Marina on Unsplash -

What this means is that we have effective tools for determining when the inflation phase of a bubble in stock prices has begun. It can only occur when stock prices become decoupled from their underlying trailing year dividends per share and begin to soar. We still have the problem of knowing whether a true bubble has formed until it might enter into its deflation phase, but we're on fairly safe ground in assuming a bubble is inflating until a new relative period of order develops to confirm otherwise.

The power law relationship between stock prices and trailing year dividends per share for an index like the S&P 500 during periods of order gives us some insight into how bubbles form. The exponent is a ratio, with the exponential growth rate of stock prices in the numerator and the exponential growth rate of dividends per share in the denominator. That means when the growth rate for dividends becomes small, the potential for decoupled growth in stock prices becomes large.

Structurally, the power law relationship exists because the index is composed of two different kinds of companies: those that pay dividends to their shareholding owners and those that do not. If the index were only made up of dividend paying firms, the relationship between stock prices and dividends per share would be linear. The power law math most often kicks in when companies that do not pay dividends either see rapid growth or become heavily weighted within the index, which contributes stock price movements that are not coupled with changes in dividends.

While non-dividend paying firms are always present in the index, it is only when market conditions develop that favor share price gains in these firms without proportionate gains in dividend paying firms that bubbles develop by the definition. For example, the inflation phase of the Dot Com Bubble took hold in the S&P 500 index shortly after the tax rate for capital gains was set lower than the tax rate for dividends on 7 May 1997, giving investors a very strong incentive to start weighting these firms much more heavily in their investing portfolios and causing them to be relatively bid up in value as a result. Order did not return to the U.S. stock market until after the end of the quarter in which the tax rates for dividends and capital gains were reunified on 21 May 2003.

We've done the most work in developing or applying these definitions and tools for stock prices, but the logic holds in the prices of other assets where investors can earn dividend-like income from simply owning or holding the asset. That includes assets like housing, where we can assess the value of housing prices with respect to the income that can be earned from owning a house: rent.

The following chart tracks the median asking sale prices of vacant homes against the median asking annualized rent for vacant homes. In it, we find our definition of a bubble works once again, even though we didn't have sufficient data to confirm the deflation portion of the U.S. housing bubble at the time we drafted it:

U.S. Median Asking Sales Price vs Annualized Asking Rent for Vacant For-Sale or For-Rent Units, 1988-Q1 through 2021-Q3

Unlike stock prices and dividends, we see the relationship between home sale prices and rents is linear. For this basic example, we treated the pre- (1988-Q1 through 2005-Q1) and post-bubble (2009-Q4 through 2019-Q4) periods as if they share the same general trajectory, which appears to be an okay initial assumption. We've also omitted data since 2019 from the analysis because of the impact of the coronavirus pandemic on data collection during 2020 and early 2021 and also because of what initially appears to be the formation of a new bubble with respect to the main trendline in 2021. The latter is a topic for a different day.

As for the U.S. housing bubble of the early 2000s, though we're omitting the statistical control-chart lines to provide a statistical hypothesis test, we find its inflation phase clearly took hold after 2005-Q1 and peaked in 2007-Q2. Its deflation phase then endured through 2009-Q4, after which asking sale prices and asking rents for vacant units in the U.S. recoupled in a new period of relative order.

That assessment generally agrees with the findings of an August 2021 NBER working paper by Gabriel Chodorow-Reich, Adam M. Guren, and Timothy J. McQuade, which Tyler Cowen commented upon shortly after its publication.

We reevaluate the 2000s housing cycle from the perspective of 2020. National real house prices grew steadily between 2012 and 2019, with the largest price growth in the same areas that had the largest booms between 1997 and 2006 and busts between 2006 and 2012. As a result, the areas that had the largest booms also had higher long-run price growth over the entire 1997-2019 period. With “2020 hindsight,” the 2000s housing cycle is not a boom-bust but rather a boom-bust-rebound.

We argue that this pattern reflects a larger role for fundamentals than previously thought.

As I see it, there was a “negative bubble” circa 2008-2009, based on panic about the shadow banking system that was at the time reasonable but also turned out to be wrong. You can argue however that there was a small bubble at the time (see Figure 1 in the paper, and compare that say to the Japanese stock market), or a bubble in a few particular regions. And do you know who got this right at the time? Our own Alex T., perhaps he will tell you the story in more detail.

The authors continue:

A few papers ascribe a role to fundamental factors in the 2000s cycle as we do. Writing near the peak of the boom, Himmelberg et al. (2005) found “little evidence of a housing bubble” because of fundamental growth, undervaluation in the 1990s, and low interest rates. Ferreira and Gyourko (2018) estimate the timing of the boom across cities and show that the beginning of the boom was “fundamentally based to a significant extent” but that fundamentals revert in roughly three years. We similarly conclude that fundamentals played a significant role in the boom, but based on different methods that focus on long-term fundamentals rather than short-term income growth. More recently, Howard and Liebersohn (2021) propose an explanation for housing cycles based on divergence in regional income growth, in which fluctuations in fundamentals fully explain the cycle, and Schubert (2021) identifies spillovers of fundamentals across cities via migration networks.

We excerpted all this text because it illustrates how economists have been stumbling without an effective definition of what a bubble is. Even though we've had at least two very noticeable bubble events in the 21st century, analysts continue to struggle to determine when they began to inflate, how big they became and even when they ended. All of which stop being problems after an appropriate framework is established for evaluating whether a bubble exists.

That's true even of Chodorow-Reich, Guren, and McQuade, who though more than a decade removed from the housing bubble, haven't fully accounted for its dynamics in their excellent work updating the building general consensus for the event.

We've gone on in this discussion long enough, but before we conclude, let's talk about where our own definition is incomplete. We still don't have good proxies to use as the equivalent of stock share dividends or shelter rents in dealing with the prices of commodities like oil. Or copper. Or lumber. Or Bitcoin. All of which have been proposed to be in bubbles at one time or another, including the present. What do you suppose those equivalents might be for each of these things?


Joakim Book. The Bubble That Never Was: Finance’s Definition Problem. American Institute of Economic Research. [Online Article]. 22 June 2021.

Gabriel Chodorow-Reich, Adam M. Guren, and Timothy J. McQuade. The 2000s Housing Cycle with 2020 Hindsight: A Neo-Kindlebergerian View. National Bureau of Economic Research. NBER Working Paper 29140. [PDF Document]. August 2021.

Jack Ewing. Shiller's List: How to Diagnose the Next Bubble. New York Times (DealBook). [Online Article]. 27 January 2010.

Howard Silverblatt. Standard and Poor S&P 500 Earnings and Estimates [Excel Spreadsheet]. Accessed 29 November 2021.

Joseph Stiglitz. Symposium on Bubbles. Journal of Economic Perspectives, Vol. 4, No. 2. Spring 1990. pp. 13-18. DOI: 10.1257.jep.4.2.13. [PDF Document].

Mark Twain. Fenimore Cooper's Literary Offenses. Project Gutenberg. [EBook version of original 1895 publication]. Release Date: 20 August 2006. Last Updated 24 February 2018.

U.S. Census Bureau. Housing Vacancies and Homeownership (CPS/HVS). Table 11A/B. Quarterly Median Asking Rent and Sales Price of the U.S. and Regions: 1988 to Present. [Excel Spreadsheet]. Accessed 29 November 2021.

Yahoo! Finance. S&P 500 Historical Prices. [Online Database]. Accessed 29 November 2021.

Image credit: Photo by Raspopova Marina on Unsplash.

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 Data Values
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.