The Existence Of A Bubble vs. The Timing Of Its Crash

Journalists love to talk about bubbles. The Wall Street Journal has hinted at bubbles in both the Chinese stock market and the market for Bitcoin during the past month alone. But, financial economists are much more reluctant to call something a bubble. There’s debate about whether bubbles even exist. And, much of this debate revolves around whether it’s possible to predict the timing of the resulting crash. If bubbles really do exist, then it seems like there should be some theory of when they’re going to pop (Fama, 2016).

There’s a good reason why financial economists care so much about the timing of the crash: this is what matters most to traders. Put yourself in the shoes of a trader in mid August 1987. The DJIA has risen roughly 12% since the beginning of July. What you want to know is, “When’s this party going to end?” Should you get out now? Or, should you keep dancing for another few months? It would be really useful to have a theory that answered these questions. And, it would be super useful to know which variables help predict the timing of the crash.

But, here’s the thing: traders aren’t the only people who care about bubbles. And, the crash ain’t the only thing worth modeling. To illustrate, put yourself in the shoes of a market regulator in August 1987. You’re staring at the same data as before. But now, what you really want to know is, “Is this party going to end? Does it represent a bubble?” If it does, then it doesn’t matter to you when the crash happens. Black Monday or Black Thursday; October 1987 or May 1988; it’s all the same to you. The same number of people will be harmed in all cases.

There are good reasons to worry about the existence of a bubble even if you can’t predict the timing of the crash. What are these reasons? That’s the topic of today’s post.

A Simple Model

One way to think about a speculative bubble is as a kind of Ponzi scheme (e.g., see here). Here’s a simple model. Imagine a group of traders that tend to build enormous long positions whenever they enter the market—i.e., these traders each have excess demand. And, whenever they build a position, suppose that they hold this position for a limited period of time–e.g., six months or a year. Once this time is up, they cash out. A bubble starts when one of these traders enters the market and drives up the price a little bit with his excess demand. If the market doesn’t crash before this first trader’s time is up, then the resulting price increase attracts additional traders. So, when the first trader finally cashes out, two more traders take his place. If the market doesn’t crash before these two traders exit the market, then the further price increase caused by these two traders’ excess demand attracts four additional traders, and so on… Thus, the $N$ th round of a bubble contains $2^{N-1}$ traders.

Let $\theta \in (0, \, 1]$ denote the probability that the market crashes during the $N$ th round. The probability that a bubble lasts exactly one round—i.e., that the bubble ends immediately—is $\theta$ . The probability that a bubble lasts exactly two rounds is $(1-\theta) \times \theta$ . The probability that a bubble lasts exactly three rounds is $(1-\theta)^2 \times \theta$ . In general, we have that $\mathrm{Pr}[N=n|\theta] = (1 - \theta)^{n-1} \times \theta$ . Thus, larger values of $\theta$ imply that a bubble will crash sooner:

$\begin{equation*} \mathrm{E}[N|\theta] = {\textstyle \sum_{n=1}^\infty} \, (1-\theta)^{n-1} \times \theta = (1-\theta)/\theta \end{equation*}$

The number of rounds in any given bubble episode represents a draw from a negative binomial distribution.

Regulator’s Viewpoint

Here’s the important thing from a regulator’s point of view: once it crashes, most traders involved in a bubble will have lost money. If the bubble immediately crashes, then the first trader will be its last. $1/1=100\%$ of all traders involved will have lost money. If the bubble collapses during the second wave of traders, then there will be three traders in total. And, after the crash, two of them will have lost money, which corresponds to $2/(1+2) \approx 67\%$ of all traders involved. If the bubble collapses during the third round, then there will be seven traders in total. Four of them will have lost money after all is said and done. So, casualty rate will be $4/(1+2+4) \approx 57\%$ . In general, if the bubble lasts $N$ rounds, then after the bubble bursts a fraction

$\begin{equation*} \mathrm{F}[N] = 2^{N-1} \big/ \, \big({\textstyle \sum_{n=1}^N} \, 2^{n-1} \big) \end{equation*}$

of all traders will have lost money.

It’s easy to see that $\mathrm{F}[N] > 50\%$ for all $N \in \{1, \, 2,\, 3, \ldots\}$ . The figure to the right presents one way of looking at it. It shows the fraction of traders that will have lost money after a bubble ends, $\mathrm{F}[N]$ (y-axis), as a function of how long the bubble lasts, $N$ (x-axis). The fraction of traders who wind up suffering losses asymptotes towards $50\%$ from above as $N$ gets larger and larger, but it never quite gets there. Here’s another way to look at it. Notice that $\sum_{n=1}^N 2^{n-1} = 2^N-1$ . In other words, we have that $1 + 2 = 4-1$ and $1 + 2 + 4 = 8-1$ and so on… Thus, the number of traders who will have lost money after a bubble pops, $2^{N-1}$ , is always a little more than half of all traders involved in the bubble, $2^N -1$ , regardless of how long the bubble episode lasts.

What I love about this example is that $\mathrm{F}[N] > 50\%$ is true for all choices of $N \geq 1$ . It’s a statement that holds pointwise. More than half of all traders will have been harmed by a bubble no matter how likely it is that the bubble ends next period. Changing $\theta$ doesn’t matter. This is the sense in which a regulator cares about the likelihood of a bubble taking place but not the timing of the crash. If the regulator is trying to maximize overall well-being, then he wants to make sure this doubling process doesn’t get started in the first place. Why does he care when it ends? It’s going to be socially harmful no matter the timing.

Trader’s Perspective

In addition, this regulator-indifference result is perfectly consistent with the idea that traders care a lot about the timing of the crash. To make things simple, suppose that a trader will lose $\mathdollar 1m$ if he belongs to the $N$ th and final round that experiences the crash. But, he will profit by $\mathdollar 1m$ if he’s one of the $(N-1)$ rounds who cash out before then. In this setting, each trader has expected profit given by:

$\begin{equation*} (1 - 2 \cdot \theta) \times \mathdollar 1m = (1-\theta) \times \mathdollar 1m + \theta \times (-\mathdollar 1m) \end{equation*}$

Thus, entering the market and trying to ride the bubble only makes sense for a trader if $\theta \in (0, \, 0.5)$ —i.e., if the probability of a market crash in the next round is less than $50\%$ . In order to profit from a bubble, you have to get out before the crash. So, traders clearly care about the timing of the crash. Whether $\theta = 0.49$ or $\theta = 0.51$ makes a big difference to them. It’s just that this difference doesn’t matter to a regulator.

A Simple Model

Regulator’s Viewpoint

Trader’s Perspective

Trackbacks