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.

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 Market Crash?.

NEW: (Evidence of Fueling of the 2000 New Economy Bubble by Foreign Capital Inflow: Implications for the Future of the US Economy and its Stock Market ), this new paper attempts to construct a coherent analysis of the US stock market linking technical analysis of the type presented below to macroeconomic thinking. We combine a macroeconomic analysis of feedback processes occurring between the economy and the stock market with a technical analysis of more than two hundred years of the DJIA to investigate possible scenarios for the future, three years after the end of the bubble and deep into a bearish regime. We also detect a log-periodic power law(LPPL) accelerating bubble on the EURO against the US dollar and the Japanese Yen. In sum, our analyses is in line with our previous work on the LPPL "anti-bubble" representing the bearish market that started in 2000.

This figure shows 9 years of the evolution of the Japanese Nikkei index and more than 7 years of the USA S&P500 index, compared to each other. In previous updates, we applied a simple translation of 11 years between the two indices, paralleling many analysts procedures. To compare with our new procedure discussed below, this translation can be written mathematically as

(1) time(S&P500) = 1*time(Nikkei) + 11

where the 1 in front of time(Nikkei) means that there is neither a contraction nor a dilation of one time with respect to the other. It is however frustrating to perform such a match by eye-balling and a more rigorous approach is called for, called a cross-correlation described in the next figure caption. Our cross-correlation analysis shows that the best match between the Nikkei and S&P500 indices is obtained by the following mapping between the two times:

(2) time(S&P500) = 0.9375*time(Nikkei) + 16.25.

Notice that the `1' has been replaced by the factor 0.9375, which means that the intrinsic time scale of evolution of the S&P500 is flowing faster than its Nikkei counterpart. As references, this expression matches the time pairs (1984 for Nikkei and 1995 for S&P500) corresponding to a local translation of 11 years and (1988 for Nikkei and 1998.75 for S&P500) corresponding to a local translation of 10.75 years. The shrinking value of the shift expresses the contraction of the US time versus the Japanese time.

When time(Nikkei) = 1990 (that is, Jan-1-1990) which is the very top of the Nikkei bubble, we have time(S&P500) = 2000.6250 (that is, Aug-15-2000) which is the exact onset of the US antibubble found in our papers.

Thus, in this figure , the times of the S&P500 and of the Nikkei here are no more mapped to each other through the 11-year translation as done in previous updates but have in addition a contraction, given by the factor 0.9375 in equation (2). 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 termed log-periodic "anti-bubbles". (The term antibubble was inspired by the concept of "antiparticle" in physics. Just as an antiparticle is identical to its sister particle except that it carries exactly opposite charges and destroys its sister particle upon encounters, an antibubble is both the same and the opposite of a bubble; it's the same because similar herding patterns occur, but with a bearish vs. bullish slant). 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.

This color plot shows the value of the cross-correlation C(t1, t2) between the S&P500 in the time interval [t1, t1+5 years] and the Nikkei in the time interval [t2, t2+5 years], where t1 and t2 are varied over large time intervals shown in the figure. t1 is the abscissa and t2 is the ordinate. The colored contours plot the value of the cross-correlation coefficient C(t1, t2) as a function of t1 and t2. Regions in red mean large cross-correlations and thus good matching between the two indices in their respective intervals. We can observe a line of crests outlined by the violet straight line which expresses statistically through C(t1, t2) the remarkable matching between the two indices. The violet line is the best linear fit to this line of crests and corresponds mathematically to equation (2). This figure thus confirms the visual matching by a simple and robust statistical analysis and uncovers the novel feature of a significant time contraction of the patterns of the S&P500 compared with those of the Nikkei, as explained above. The implication is clear: it is naive to expect a perfect superposition described by a simple translation.

Discussion of the Nikkei-S&P500 matching patterns: The matching between the Nikkei and the S&P500 time series obtained here by a rigorous quantitative analysis of the cross-correlation of the two shifted time series should not lead to the belief that the S&P500 index is bound to follow blindly this correlation in the future. In contrast with chartism or technical analysis, we try to develop a scientific understanding of these bubble-antibubble phases. The similitude between the Nikkei and US markets are part of the search for "universal" properties, that allow us to establish a theory (in short, a theory is a story of repeatable/reproducible occurrences). Using this theory then allows us to describe idiosyncratic behaviors, that is, deviations from one case to another, or in other words, the parts of the evolutions that are not universal. This is what should give us an hedge for predictions.

Already, as early as September 2002, in our paper [*] based on an analysis carried on the stock market time series available up to Aug. 25, 2002, we wrote that we could see a clear difference between the Nikkei and the SP500. Thus the qualitative analogy is there but, quantitatively, there are serious differences. Technically, after two years and a half after the top in Dec. 31, 1989, we find that the Nikkei has started to shift to another antibubble regime while no such shift is yet detectable after more than three years since the start of the antibubble in the US. In addition, the US markets have been characterized by much stronger crashes and rallies, modelled below by our "zero-phase-Weierstrass" functions. These two facts suggest to us that the herding forces are even stronger in the US and that investors react even more on hair-trigger to any "news".

To sum up, the similarities between the shifted Nikkei and the S&P500 are qualitative: bubble preceding antibubble, strong speculation and herding, similar fear and herding in the anti-bubble regime, some problems with bad loans or bad accounting, strong commitment from the central banks and governments to provide liquidity and cash... But there are differences and these differences can be detected. Thus, we are not proponents of a superposition of the two time series to predict the future evolution of the US stock markets. It is clear to us that their future will be different, according to the forecasts proposed below.

[*] D. Sornette and W.-X. Zhou, The US 2000-2002 Market Descent: How Much Longer and Deeper? Quantitative Finance 2 (6), 468-481 (2002) http://www.iop.org/EJ/S/1/NCA203394/RCM0rqd2bn5eBW0XZGGwvA/toc/1469-7688/2/6
(http://arXiv.org/abs/cond-mat/0209065)

Fig. 1 shows the predictions of the future of the US S&P 500 index performed on Aug. 24, 2002. The continuous line is the fit and its extrapolation, using our theory capturing investor herding and crowd behavior. The theory takes into account the competition between positive feedback (self-fulfilling sentiment), negative feedbacks (contrariant behavior and fundamental/value analysis) and inertia (everything takes time to adjust). Technically, we use what we call a "super-exponential power-law log-periodic function" derived from a first order Landau expansion of the logarithm of the price. The dashed line is the fit and its extrapolation by including in the function a second log-periodic harmonic. The two fits are performed using the index data from Aug. 9, 2000 to Aug. 24 2002 that are marked as black dots. The blue dots show the daily price evolution from Aug. 25, 2002 to Aug. 15, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

Fig. 2 shows the new predictions of the future of the US S&P 500 index using all the data from Aug. 9, 2000 to Aug. 15, 2003, illustrated by (continuous and dashed) black lines. Again, the continuous line is the fit and its extrapolation using the super-exponential power-law log-periodic function derived from the first order Landau expansion of the logarithm of the price, while the dashed line is the fit and its extrapolation by including in the function a second log-periodic harmonic. We also present the two previous fits (red lines) performed on Aug. 24, 2002 (shown in Fig. 1) for comparison, so as to provide an estimation of the sensitivity of the prediction and of its robustness as the price evolves. The blue dots show the daily price evolution from Aug. 9, 2000 to Aug. 15, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

Fig. 3 shows the predictions of the future of the US S&P 500 index obtained by applying the so-called 'zero-phase' Weierstrass-type function, which is another child of our general theory of imitation and herding between investors. As for the previous figures, our theory takes into account the competition between positive feedback (self-fulfilling sentiment), negative feedbacks (contrariant behavior and fundamental/value analysis) and inertia (everything takes time to adjust). This 'zero-phase' Weierstrass-type function adds one additional ingredient: it attempts to capture the existence of 'critical' points within the anti-bubble, corresponding to accelerating waves of imitation within the large scale unraveling of the herding anti-bubble. The continuous black line is the forward prediction using all the data from Aug. 9, 2000 to Aug. 15, 2003, while the dashed black line is the retroactive prediction using the data from Aug. 9, 2000 to Aug. 24, 2002. Both lines are reconstructed and extrapolated from the fits to a six-term zero-phase Weierstrass-type function. We also present the two previous fits (red lines) performed on Aug. 24, 2002 (shown in Fig. 1) for comparison. The blue dots show the daily price evolution from Aug. 9, 2000 to Aug. 15, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

The striking development observed in the update on June, 19, 2003 is again confirmed. The 'zero-phase' Weierstrass-type function, which up to May 18, 2003 selected a series of downward critical crashes, is now selecting as the dominant critical points the bullish accelerations. The formula is thus deciphering the coexistence of two sets of critical points: (i) the crashes previously recognized which have punctuated the descent in the last three years and (ii) the bursts of upward accelerating rallies. This formula is however not rich enough in its present version to capture these two sets simultaneously and has to choose between the two, as a result of their relative strengths. This new twist does not change fundamentally our prediction of a drastic turn towards a systematic downward trajectory till the summer of 2004.

Fig. 3bis is a modification of the 'zero-phase' Weierstrass-type function shown in Fig. 3, which contains only odd-terms in the expansion (this will be elaborated upon in a future technical communication). By this trick, the odd-zero-phase Weierstrass-type function is able to describe simultaneously the two sets of critical points mentioned in the caption of Fig. 3. The continuous black line is the forward prediction using all the data from Aug. 9, 2000 to Aug. 15, 2003, while the dashed black line is the retroactive prediction using the data from Aug. 9, 2000 to Aug. 24, 2002. Both lines are reconstructed and extrapolated from the fits to a six-term odd-zero-phase Weierstrass-type function. We also present the two previous fits (red lines) performed on Aug. 24, 2002 (shown in Fig. 1) for comparison. The blue dots show the daily price evolution from Aug. 9, 2000 to Aug. 15, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

In conclusion, the coexistence of the strong downward crashes and upward rallies in the overall anti-bubble regime suggests to us that the market is completely dominated by sentiment, confidence and lack thereof and by herding. These mechanisms are amplifying any news, perturbation or rumor spreading in the network of investors.

Fig. 4 extends Figs. 1 and 2 by performing a sensitivity analysis on the simple log-periodic formula (continuous lines in Figs. 1 and 2), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in black solid lines in Fig. 2, we have generated 10 realizations of an artificial S&P500 by adding GARCH noise to the black solid line. GARCH means "generalized auto-regressive conditional heteroskedasticity". It is a process often taken as a benchmark in the financial industry and describes the fact that volatility is persistent. The innovations of the used GARCH noise have been drawn from a Student distribution with 3 degrees of freedom with a variance equal to that of the residuals of the fit of the real data to ensure the agreement between the statistical properties of these synthetic time series and the known properties of the empirical distribution of returns. The fits are shown as the bundle of 10 curves in magenta. This bundle of predictions is coherent and suggests a good robustness of the prediction. The typical width of the blue dots give a sense of the variability that can be expected around this most probable scenario. The real S&P500 price trajectory is shown as the red wiggly line.

Fig. 5 extends Figs. 1 and 2 by performing a sensitivity analysis on the log-periodic formula with a second log-periodic harmonic (dashed lines in Figs. 1 and 2), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in dashed solid lines in Fig. 2, we have generated 10 realizations of an artificial S&P500 by adding the GARCH noise (described in the previous caption of Fig. 4) to the dashed solid line. We have then fitted each of these 10 synthetic noisy clones of the S&P500 by our log-periodic formula. This yields the 10 curves shown here in magenta. This test shows that the log-periodic formula with a second log-periodic harmonic (dashed lines in figures 1 and 2) is also providing stable scenarios: the precise timing of the highs and lows remain robust with respect to the realization of the noise. The real S&P500 price trajectory is shown as the red wiggly line.

Fig. 6 extends Fig. 3 by performing a sensitivity analysis on the 'zero-phase' Weierstrass-type function, in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in black solid lines in Fig. 3, we have generated 10 realizations of an artificial S&P500 by adding A GARCH noise to the black solid line. The innovations of the used GARCH noise have been drawn from a Student distribution with 3 degrees of freedom with a variance equal to that of the residuals of the fit of the real data by the black continuous curve. We have then fitted each of these 10 synthetic noisy clones of the S&P500 (shown as the blue dots) by our 'zero-phase' Weierstrass-type function. This yields the narrow bundle of 10 curves shown here in magenta. This bundle of predictions is very coherent and suggests a good robustness of the prediction. The typical width of the blue dots give a sense of the variability that can be expected around this most probable scenario. The real S&P500 price trajectory is shown as the red wiggly line.

Fig. 6bis is the same as Fig. 6 but for the odd-zero-phase Weierstrass function shown in Fig. 3bis.

Fig. 7 analyses the VIX index by fitting it with our simple log-periodic formula. The VIX index is one of the world's most popular measures of investors' expectations about future stock market volatility (that is, risk). See http://www.cboe.com/micro/vixvxn/introduction.asp. For historical data, see http://www.cboe.com/micro/vixvxn/specifications.asp. The VIX time series is shown as the red wiggly curve. We have followed the same procedure as for Figs. 4-6: (i) we fit the real VIX data with our simple log-periodic formula; (ii) we then generate 10 synthetic time series by adding GARCH noise to the fit; (iii) we redo a fit of each of the 10 synthetic time series by the simple log-periodic formula and thus obtain the bundle of 10 predictions shown as the magenta lines. Strikingly, we first observe that our log-periodic formula is able to account quite well for the behavior of the VIX index, strengthening the evidence that the market is presently in a strong herding (anti-bubble) phase. Note also the rather good stability of the predictions, suggesting a reasonable reliability.

Announcement: As this is obviously of strong interest from a theoretical and practical point of view, we have developed a technique to forecast the end of the anti-bubble whose results and conclusions will be presented in the next update.

Cautionary note: Note that extrapolating is often a risky endeavor and needs to be justified. In our case, the extrapolations, which give the forecasts, are based on the belief that the theory and equations used above 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.