Almost nobody will deny that the stock market developed a bubble in the last years of the 20th century. Consider famous episodes such as the market's inability to figure out that 3Com was the majority owner of Palm, the maker of handheld computers, the dizzying heights reached by the Nasdaq Composite index as it exploded in late 1999, more than doubling in value in the year up to its late-winter 2000 peak, and the huge bath taken by investors in the Nasdaq from February 2000 to September 2002 as the index lost three-quarters of its value. All this happened, even though the long-awaited recession proved shallower than anyone had forecast and trend productivity growth proved faster than even the most zealous boosters of the "new economy" had dared project. It is next to impossible to interpret these events within the context of a rational-expectations model, in which stock prices provide the best possible forecasts of future values. Only those who want their colleagues to doubt their own rationality even try.

The interesting thing about the late-1990s stock market bubble, however, is how short it was. Conventional histories start with the excitement produced by the initial public offering of Netscape, the internet company, in 1995. But the stock market was not then in a bubble. Those who invested in the Nasdaq in March 1995 and then shadowed the index have earned real returns averaging 9.3 per cent per year; those who invested in the Nasdaq in September 1995 - the month after Netscape's IPO - have earned 7.3 per cent annually since. Compare this with the average 8.8 per cent real return on the Nasdaq since the start of the 1970s, and it is clear there is no evidence that the Netscape-mad market of 1995 was significantly overvalued.

By the end of 1996, Alan Greenspan, chairman of the US Federal Reserve, was worrying about the stock market. "How do we know," he asked an audience at the American Enterprise Institute, the Washington think-tank, "when irrational exuberance has unduly escalated asset values?" If you had invested in the Nasdaq while Mr Greenspan was writing his speech, you would have realised a real return from then until now of 8.1 per cent per year. The answer to the question Mr Greenspan implied in December 1996 - "Are asset prices unduly escalated by irrational exuberance?" - is no.

When, then, did the late-1990s bubble begin? That is a surprisingly hard question to answer. Since the start of 2000, the Nasdaq has been hit by two particular pieces of bad news: the September 11 2001 terror attack on the US, and the fact that it has turned out to be much harder than anyone expected to transform technical excellence in information and communications technology into durable profits. The beneficiaries of high-tech innovation since 2000 have not been the workers, entrepreneurs and financiers of Silicon Valley, but users of their products and ideas, such as Wal-Mart, the US retailer, its shareholders and its customers.

Given that these two pieces of fundamental bad news were unknown (and unknowable) in the late 1990s, it seems reasonable to set up three yardsticks to assess the beginning of the bubble. Because the high-tech stocks in the tech-heavy Nasdaq are particularly risky, a rational market would be expected to price the Nasdaq to produce returns higher than the stock market's long-run historic average of 6.5 per cent per year. But we also suspect that actual returns have fallen behind rationally expected returns. If these two factors cancel each other out, and the Nasdaq was in a bubble, then returns from the start of that bubble period until now will have been less than the 6.5 per cent per year we expect from stocks. As a second, more stringent, yardstick, a bubble can be defined as a period during which real returns do not match the 3 per cent per year expected from investment in bonds. The third and most stringent yardstick would define a bubble as a period in which returns are negative.

According to the second and third yardsticks the bubble was remarkably short. Measured from then until now, the realised Nasdaq real return drops below 3 per cent per year in October 1998, and in November 1998 it drops below zero. According to these measures, it was overvalued for less than a year and a half before its peak in March 2000. According to the first yardstick, the bubble was a year and a half longer. Cumulative real returns on the Nasdaq have lagged behind the 6.5 per cent per year we expect from a diversified portfolio of stocks since April 1997. Using this yardstick, the bubble lasted for less than three years before the Nasdaq peaked.

Fischer Black, the economist, offered another definition of a "large" bubble: a time when assets are twice their fundamental values or more. On this definition, the Nasdaq bubble became a "large" one only in September 1999, less than half a year before its peak.

You can use this analysis to give an optimistic reading of stock market performance: when the great and good - Mr Greenspan, for example - were puzzled and cautious, the market was still cranking out valuations that look, in retrospect, remarkably close to the fundamental values of the underlying assets. In spite of unprecedentedly rapid technological change with very uncertain long-run consequences and the fact that humans are prone to irrational exuberance, the stock market kept doing its job of feeding the real economy an appropriate shadow value of capital, and did not succumb to irrationality until 1997 or 1998.

Today many people - including us - are worried about high levels of bond prices and real estate in the US and elsewhere that seem incompatible with likely scenarios for the world economy. This look back at the bubble of the 1990s is somewhat reassuring: financial markets were not as dumb as we feared then, and so are probably not as dumb as we fear now. Like the late 1990s, any bond or real-estate bubble this decade will probably be of short duration.

However, short duration does not necessarily mean small magnitude. The size of the late 1990s stock market bubble at its peak still supports a very pessimistic reading of stock market rationality. Those who followed the advice of the "hyper-bulls" - the Kevin Hassetts, James Glassmans, George Gilders and their ilk - and bought the Nasdaq at the end of February 2000 expecting it to rise much further, have seen a realised real rate of return of minus 16 per cent per year. In spite of the recovery since late 2002, they are still down by more than half.

Brad DeLong is professor of economics at the University of California, Berkeley, and Konstanin Magin is a post-doctoral fellow at the Center of Integrated Nanomechanical Systems

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