Based on a theory of cooperative herding and imitation working both in bullish as well as in bearish regimes, 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).

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

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This figure shows 8 years of the evolution of the Japanese Nikkei index and 7 years of the USA S&P500 index, compared to each other after a translation of 11 years 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 two following figures.

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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 May.16,2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter of each year.

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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 May.16,2003, 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" discussed in the figure caption of figure 1, 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 May.16,2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

In mid-January 2003, we proposed an extension of our previous log-periodic power law model of the "anti-bubble" regime of the USA market since the summer of 2000, in terms of the renormalization group framework to model critical points. We are thus able to accurately model the five "crashes" that punctuated the overall market descent since August 2000 in a fully consistent way with no additional parameters (actually with one parameter less than the most parsimonious formula used previously).

Please refer to the following paper for a detailed technical description and for more detailed results: W.-X. Zhou and D. Sornette, Renormalization group analysis of the 2000-2002 anti-bubble in the US S&P 500 index: Explanation of the hierarchy of 5 crashes and Prediction (eprint at http://arXiv.org/abs/physics/0301023).

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Fig. 3 shows the predictions of the future of the US S&P 500 index applying the so-called 'zero-phase' Weierstrass-type function, which is another child for 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 May.16,2003, 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 May.16,2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

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Fig. 4 extends figures 1 and 2 by performing a sensitivity analysis on the simple log-periodic formula (continuous lines in figures 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 figure 2, we have generated 10 realizations of an artificial S&P500 by adding white noise to the black solid line. The white noise (shown as the blue dots) is drawn from a student distribution with 3 degrees of freedom and 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 by our log-periodic formula. This yields the bundle of 10 curves shown here 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.

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Fig. 5 extends figures 1 and 2 by performing a sensitivity analysis on the log-periodic formula with a second log-periodic harmonic (dashed lines in figures 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 figure 2, we have generated 6 realizations of an artificial S&P500 by adding white noise to the dashed solid line. The white noise (shown as the blue dots) is drawn from a student distribution with 3 degrees of freedom and with a variance equal to that of the residuals of the fit of the real data by the dashed continuous curve. We have then fitted each of these 6 synthetic noisy clones of the S&P500 by our log-periodic formula. This yields the 6 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 providing quite unstable scenarios: the precise timing of the highs and lows seem to be quite sensitive to the realization of the noise. This suggests that the predictions based on the dashed lines shown in figures 1 and 2 are not reliable. The real S&P500 price trajectory is shown as the red wiggly line.

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Fig. 6 extends figure 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 figure 3, we have generated 10 realizations of an artificial S&P500 by adding white noise to the black solid line. The white noise (shown as the blue dots) is drawn from a student distribution with 3 degrees of freedom and with a variance equal to that of the residuals of the fit of the real data by the black continuous curve of figure 3. We have then fitted each of these 10 synthetic noisy clones of the S&P500 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.

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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 figures 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 white 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. Last, note that the VIX is predicted to continue to decrease (smaller expected risks) well in the regime when the price have started their descent, with a bottom in the last quarter of 2003. The VIX is expected to turn up and increase significanly only later in 2004. This dephasing of the log-periodic fits of the S&P500 and of the VIX is reminiscent of the well-known stock market proverb: "Bear markets decline on a slope of hope".