3/3/11

Modeling S&P 500 returns. March 2011

We restart (or continue) reporting on the evolution of the S&P 500 and our prediction made in the beginning of 2009. Between March 2009 and September 2010, the prediction based on the number of nine-year-olds, N9, fitted the observed S&P 500 with minor deviations. All in all, sixteen months in a raw we were right and did not see any source which might disturb our prediction. However, there is one source of problem for many economic and econometric models we have built – population distributions provided by the US Census Bureau.

Here, we reintroduce the original model which links the S&P 500 annual returns, Rp(t), to the number of nine-year-olds, N9. To obtain a prediction we use the number of three-year-olds, N3, as a proxy to N9 at a six-year horizon:

Rp(t+6) = 100dlnN3(t) - 0.23 (1)

where Rp(t+6)is the S&P 500 return at a six-year horizon. Figure 1 depicts relevant S&P 500 returns, both actual one and that predicted by relationship (1). The latter curve has been deviating the latter one since October 2010. Currently, this deviation is very big and put our model under strong doubt.

Figure 1. Observed and predicted S&P 500 returns.

This is not the end of the model, however. We continue using the link between real GDP and N9, as described in this paper. It was shown that one can exchange them when one of these two is not well estimated (usually N9). In that sense, one can use real GDP instead on N9.

As discussed in our working paper on S&P 500, there exists a trade-off between the growth rate of real GDP, G(t), and the S&P 500 returns, R(t). The predicted returns, Rp(t), can be obtained from the following relationship:

Rp(t) = 0.0062dlnG(t) - 0.01 (2)

where G(t) is represented by the Q/Q (annualized) growth rate, because only quarterly readings of real GDP are published by the BEA.

With a small correction of the coefficients in (2), Figure 2 displays the observed S&P 500 returns and those obtained using real GDP, as presented by the US Bureau of Economic Analysis. As before, the observed returns are MA(12) of the monthly returns. The period after 2003 is relatively well predicted, including that not predicted by (1). Therefore, it is reasonable to assume that G(t) can be used for modeling of the S&P 500 index and returns. Reciprocally, current S&P 500 may be used for the estimation of real GDP.

Figure 2. Observed S&P 500 return and that predicted from real GDP. For a given quarter, all monthly values of the growth rate relative to the previous quarter are equal.

To understand the deviation associated with N9 we are waiting for the final results of the 2010 census. This is also crucial for many economic models we have developed for the U.S. Other developed countries do not demonstrate such big deviations.