Rational Bubbles, Part 2: Why Investors Should Understand What's Known in Statistics as the "Exploding" Process

## Originally studied to model physical events like the collapse of structures, the "exploding" process can be used to explain price trajectories for certain securities.

MINYANVILLE ORIGINAL Last week I described a simple price process that could underlie a rational bubble: That is, a price process such that each investor individually had a positive expected return, but that all investors collectively knew had to crash.

In statistics this is known as an “exploding” process. They were originally studied to model physical events like the collapse of structures. They share the characteristic that the variance of the process increases faster than its magnitude. This makes the process unstable at any finite value, so it has some probability of going to infinity in finite time (this is for continuous exploding processes, the example I gave was a discrete exploding process which cannot go to infinity but is otherwise similar). But because the process has a finite expected value, the probability of going to infinity has to be infinitesimal, and the process is nearly certain to get closer to zero.

Consider, for example, a microscopic crack in a bridge support. The larger the crack is, the faster it grows, and as it grows its potential for further growth grows faster than its size. This means a support can look fine, with no observable cracks, and an instant later collapse.

Imagine a security whose price was equal to the length of a specific crack. Its price would have bubble-like characteristics; the higher the price went (that is the longer the crack), the more the potential for further increases, so the more demand for the security which forces the price even higher. However, this would be an entirely rational bubble, a microscopic crack really could grow large enough to bring down the entire structure. In this case, the process is limited by the size of the structure, the crack cannot literally grow to infinity. But when it is microscopic, the growth potential is so huge, it would behave almost as if it had infinite growth potential.

This could explain exploding price processes for securities that bet on bridge collapses, but why would Internet stocks or housing prices explode? You might see an analogy with Internet stocks at least, the more the Internet grew, the more potential for growth. But we're talking real economics, not physical propagation of cracks. If people get excited about the Internet, why bid up the price of existing Internet stocks, why not use the money to make real investments in new Internet companies? Since there's no barrier to entry to creating Internet businesses, you would think the price would be limited by the replacement cost.

Of course, there was quite a bit of investment in real Internet resources like fiber optic cables and networking equipment, and feverish creation and IPO of new Internet businesses. But prices of existing stocks soared nonetheless. Moreover, the increases seemed unrelated to any rational estimate of future cash flows. The stocks that soared the most were not the ones best-positioned to profit from explosive Internet growth and while some individual valuations could be defended by partisans, no one could defend collective valuations of the sector.

For other bubbles, the economic case is even harder to see. You can come up with plausible reasons that an Internet company might grow in a few years to 100 or 1,000 times its current value, but what about a house or a bar of gold? When Internet stocks double on scant news, you can argue that it resulted from a slightly increased probability of a hundred-fold price appreciation—it's hard to find statistical evidence for or against that claim. But in 2006, US residential housing was valued at over \$25 trillion, or about 175% of US GDP. An optimist might think it could grow to 200% or 250% of GDP, but not 2,000% or 10,000% . Without even a tiny probability of massive appreciation, it's hard to see justification for 10% or 20% housing price increases without obvious changes in fundamentals.

Next week we'll take a look at the economic factors that can support a bubble-like price process, without explosions.

(Also see: Yes, There Is Such a Thing as a Rational Bubble -- We're In One Now)
No positions in stocks mentioned.

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