Bubbles and Scale Invariance

In yesterday’s post I mentioned that bubbles were exponential, scale-invariant and self-similar, making it virtually impossible to time their collapse.

Let’s flesh out this assertion by looking at a particular market index.

For the first 17 years of its existence, this index had a mean of 100 and a standard deviation of 56. (Prices have been scaled to avoid easy recognition). That’s a pretty stable time series.

Then something happened. Over the next 8.5 years, the index went from a starting value of 200 (already near the upper end of its previous range) to a value of 700. What’s more, this rise took on exponential, maybe even super-exponential characteristics, as the graph below makes clear.


Would you sell? If you did, you’d be out of luck. Because over the next 25 months, the index went from 700 to 1100. Once again the rise looked exponential or better:


(Note that this graph has the same start date as the previous one, but different scales on each axis).

Would you sell? If you did, you’d be out of luck again. Because over the next 15 months the index went from 1100 to 1500, with the pace of expansion growing ever higher


(Once again, this graph has the same start date as the previous two, but different scales on each axis).

Now would you sell? How much further and faster can the market rise? The answer is, quite a bit. Over the next 5 months the index rocketed from 1500 to 2800. If you had sold the index at any of the previous junctures – and note that at each of those points, the graph looked convincingly bubbly – you would almost certainly have been carried out at a loss.

2800 was, in fact, the high; over the next 31 months the index dropped all the way back to 600. Here’s the full graph, with dates and true (unscaled) values.


I’ve marked the extrema of each of the previous graphs onto the composite graph, to demonstrate how scale-invariance works. Although zooming in on any sub-graph gives the impression that it’s an exponential curve about to pop, these curves just get lost in the main graph. It’s not easy to time bubbles.

Postscript: The index in question is of course the Nasdaq composite in the days of the dot-com expansion. Interestingly, Alan Greenspan warned about ‘irrational exuberance’ in December 2006, shortly after the first of the graphs above. Three years later he had changed his tune (‘capitulated’?) quite substantially.