The essence of any quantitative model consists in the accuracy of prediction or predictive power. The higher is signal/noise ratio in a given data set the better can be the estimate of model parameters or the uncertainty of corresponding prediction. For the S&P 500 stock market index, the most prominent signals were measured during the periods of the fastest change: 1987, 2001 and 2008.

We have developed a model [1] linking the S&P 500 and its returns to the population of some characteristic age. The original model links the S&P 500 annual returns, Rp(t), to the number of nine-year-olds, N9:

Rp(t) = AdlnN9(t) + B (1)

where Rp(t) is the S&P 500 yearly return, A and B are empirical coefficients to be determined by some fitting procedure. They may change depending on the approximation used to represent N9. In the previous post on the S&P 500 returns we have approximated N9 by the number of three-year-olds, N3, six years before. Accordingly, we have obtained a prediction of the S&P return at a six year horizon, i.e. in 2010 one can foresee the returns in 2016. Relevant empirical relationship is as follows”

Rp(t+6) = 100dlnN3(t) - 0.23 (2)

Figure 1 depicts the S&P 500 returns, both actual one and that predicted by relationship (2). Both curves are coinciding in practical terms between 2008 and the middle of 2010.

In 1987 and 2001 abrupt falls in the returns were also observed. In this respect, the model also demonstrates an excellent predictive power, as Figures 2 and 3 depict. There are obvious differences between the measured S&P 500 returns. In 1987, the fall was very fast but not deep, from +0.3 to -0.1. In 2001, the returns declined gradually from +0.3 to -0.3 in 2002. In 2008, the observed curve fell from 0 to -0.5, i.e. approximately same as in 1987.

In all cases the model gives a good prediction of the timing and amplitude of the observed returns. So, the model has a good predictive power, considering that the prediction can be obtained at a nine year horizon with the birth rate used as a proxy to N9.

Therefore, one might treat our prediction of the 2011 fall as a reliable one.

During the last two weeks, the S&P 500 has been growing at a healthy pace. Currently, it exceeds the predicted level by approximately 50 to 100 points. This is a good reason to suggest that a significant force, which must eventually return the index to the trend line, has been developing in September 2010. If the growth continues into the second half of September one might expect a dramatic drop in October 2010. However, this will be just a part of the overall decrease to the level of -0.5 expected in July-August 2011.

Figure 1. The prediction of the S&P 500 annual return for the period between 2008 and 2012. We tentatively put the September's closing level at 1030.

Figure 2. The prediction of the S&P 500 annual return for the period between 1985 and 1989.

Figure 3. The prediction of the S&P 500 annual return for the period between 1998 and 2003.

**References**

1. Kitov, I., Kitov, O. (2010). S&P 500 returns revisited, http://ideas.repec.org/p/pra/mprapa/21733.html.

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