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# Yes, There Is Such a Thing as a Rational Bubble -- We're in One Now

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## Double, double, toil and trouble, fire burn and caldron bubble. Why financial bubbles are not as crazy as they seem.

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Therefore, rational bubble models from the 1980s are coming back into fashion, including one of mine. I like it because it's mine, of course, but also because I think it's the simplest one. In my opinion, the basic logical argument is the same for all rational bubble models, and more elaborate mathematical structures don't add insight. The key to applying these models is the empirical methods used to investigate them, and the quality of data, not the sophistication of the underlying mathematics.

To understand the sense in which a bubble can be rational, consider an asset whose price each quarter either doubles or goes to \$2 with equal probability. I'm not telling you why the price does this for now; just assume that it does. If the price of the asset today is P, its expected price in one quarter is (2P + 2)/2 = P + 1. So buying the asset at any price has a positive expected value, and therefore is arguably rational.

On the other hand, the price of this asset must eventually go to \$2, so buying it at any price above \$2 will lead to certain loss.

If you simulated this price path and graphed it on a computer, you would see bubble-like behavior. The price would go up steadily for a while, sometimes to relatively high values like \$64 or more, but always fall back to \$2 at some point.

Think about buying this asset at \$16 and holding it for 25 years. Since its expected value increases \$1 per quarter, the expected value at the end of the period is \$116, or an expected compounded annual return of 8.25%. That is all true, and all misleading. For 0.01% of the time, you end up with over \$10,000; in those cases, you have an average of \$843,776. The other 99.99% of the time when you end up with less than \$10,000, your average terminal value is \$13. \$16 of the \$116 expected value comes from the one in nonillion (one followed by 30 zeros) chance of getting over \$16 nonillion by having all 100 coin flips come up in your favor. This kind of distribution, with a high expected value driven by microscopic chances of astronomical payouts, even meaningless payouts, was discussed in Whodunit?

This feels like a bubble. Every investor rationally expects a return greater than his or her investment, yet also knows that the price must eventually crash below his or her purchase price. Before resolving this dilemma, we have to discuss why a price might follow this kind of process. I'll tackle that next week.
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No positions in stocks mentioned.
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