These analyses are researched by D.Sornette and W.-X. Zhou.

Based on a theory of cooperative herding and imitation working both in bullish as well as in bearish regimes that we have developed in a series of papers, we have detected the existence of a clear signature of herding in the decay of the US S&P500 index since August 2000 with high statistical significance, in the form of strong log-periodic components. Since August 2000, the USA as well as most other western markets have depreciated almost in synchrony according to complex patterns of drops and local rebounds. We have proposed to describe this phenomenon using the concept of a log-periodic power law (LPPL) antibubble, characterizing behavioural herding between investors leading to a competition between positive and negative feedbacks in the pricing process. This work was motivated by the similitude between the evolutions of Nikkei 225 and S&P 500 Index.

Fig. 1 shows 9 years of the evolution of the Japanese Nikkei index and almost 8 years of the USA S&P500 index, compared to each other after a translation described in the update of September 17, 2003 has been performed. The years are written on the horizontal axis (and marked by a tick on the axis where January 1 of that year occurs). This figure illustrates an analogy noted by several observers that our work has made quantitative. The oscillations with decreasing frequency which decorate an overall decrease of the stock markets are observed only in very special stock markets regimes, that we have terms log-periodic "anti-bubbles". By analyzing the mathematical structure of these oscillations, we quantify them into one (or several) mathematical formula(s) that can then be extrapolated to provide the prediction shown in the following figures. Note that extrapolating is often a risky endeavour and needs to be justified. In our case, the extrapolations, which give the forecasts, are based on the belief that these equations offered below embody the major forces in the market at the macroscopic scale. This leads to the possibility of describing several probable scenarios. We do not believe in the existence of deterministic trajectories but we aim at targeting the most probable future paths.

Please refer to the following paper for a detailed description: D. Sornette and W.-X. Zhou, The US 2000-2002 Market Descent: How Much Longer and Deeper? Quantitative Finance 2 (6), 468-481 (2002) (e-print at http://arXiv.org/abs/cond-mat/0209065).

Why Stock Markets Crash: For a general presentation of the underlying concepts, theory, empirical tests and concrete applications, with a discussion of previous predictions, see the recent book, Why Stock Markets Crash.

Several of our readers have suggested to us that we should analyze the data from the perspective of foreigners, by converting the market price in euro, British pound or Yen, for instance. This makes sense if one takes into account (1) the artificial and distorting input of the liquidity input by the Fed (which amounts to an effective inflation in dollar terms, hence its depreciation, to be naively simple) and (2) the importance of foreign investors on the US stock markets recycling their surplus dollars. As the figures below show, the picture from the vantage of a foreigner is very different than for an US investor. The tentative conclusion of this new study is that the strong impact of the Fed intervention has perturbed the fingerprints of the antibubble, so that we conclude that it has ended in the US, while maybe in reality the herding bearish-bullish oscillations are still present but are hidden by the distorting feedback actions of the Fed. Then, the antibubble signature could be better observed from the different reference frame of foreigners.

Figure 2 shows the S&P 500 index denominated in EUR from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas. Recall that we started to show this time series and its fits with the LPPL formulas in our update on 2003/10/20 (last figure).

Figure 3 shows the S&P 500 index denominated in GBP from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Figure 4 shows the S&P 500 index denominated in Gold Fixes FM from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas. Note that, in our previous predictions, we have used XAU.

London Gold Fixing: We use the price of gold fixed twice a day in London by the five members of the London gold pool, all members of the London Bullion Market Association. The fixes start at 10.30 a.m., and 3.00 p.m. London time. There are many websites that provide historical Fix data, for instance,

http://www.lbma.org.uk/statistics_historic.htm,

http://www.kitco.com/gold.londonfix.html, and

http://www.amark.com/archives/fixes.asp.

XAU: The Philadelphia Stock Exchange (PHLX) Gold&Silver Sector (XAU) is a capitalization-weighted index composed of the common stocks of 12 companies involved in the gold and silver mining industry. XAU was set to an initial value of 100 in January 1979; options commenced trading on December 19, 1983. We obtained the data from http://www.oanda.com/convert/fxhistory.

Remark: the XAU and London Gold Fixing are very close to each other and the analyses using one or the other do not exhibit noticeable differences.

Figure 5 shows the r.m.s. (root-mean-square, an inverse measure of the quality of the fits) of the residuals of the respective fits, such as those shown in Figs. 2-4. This figure shows very clearly the change of regime around February 2003, materialized by the jump in r.m.s. in ALL fits. Note that the same occurs for the S&P 500 in US dollar. It is also very interesting to see that the first-order fit and the second-order fit separate around Feb 2003, that is, there is a bifurcation in the data (the bifurcation is slightly earlier for AU). It is also clear from the figure that there is another change of regime around the beginning of 2004.

Fig. 5 shows that the r.m.s. of the fit residuals of the second-order Landau formula keep decreasing as a function of time (the quality of the fits improves), in contrast with those of the first-order formula. This confirms the visual impression that the second-order Landau fits capture very well the LPPL oscillations when compared with the first-order fits, as shown in Figs. 2-4. Beyond the quality and predictive power of the proposed fits, we would like to stress the importance of identifying "regime switches". The present values confirm that we are living in a well-established LPPL anti-bubble regime.

Several of our readers suggested that we should also investigate the stock indexes in Europe since those are traded directly in EUR and in GBP. We thus show below the counterparts of the S&P 500 in EUR and in GBP with DAX of Germany, CAC 40 of France, and FTSE 100 of the United Kingdom. We see that the antibubble is right on track in these stock markets.

Figure 6 shows the DAX index of Germany from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Figure 7 shows the CAC 40 index of France from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Figure 8 shows the FTSE 100 of the United Kingdom from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Some of our readers also suggested that we should look at the Russell 3000 index which represents approximately 98% of the U.S. market. We follow this suggestion and provide the following three figures. The results for the SP500 and the Russell 3000 are quite similar. These figures confirm what we have found in Figs. 2-4. Again, we see that the second-order Landau formula outperforms the first-order and provides quite convincing fits.

Figure 9 shows the Russell 3000 index denominated in EUR from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Figure 10 shows the Russell 3000 index denominated in GBP from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Figure 11 shows the Russell 3000 index denominated in Gold Fixes FM from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

Figure 12 shows the Russell 3000 index denominated in USD from 2000/08/09 to 2004/08/16 and its fits using the first-order and second-order Landau formulas.

**THIS IS AN EXPERIMENT PERFORMED IN REAL TIME AND WE WILL CONTINUE UPDATING EVERY MONTH.**

**REMEMBER THAT** this analysis is for academic purposes only and must not be construed as investment or trading advice.