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.

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.

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.

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:

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?

References

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.