We use the same pricing model as previously. In its general form, this pricing model is as follows:
sp(tj) = Σbi∙CPIi(tj-i) + c∙(tj-2000 ) + d + ej (1)
where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-i) is the i-th component of the CPI with the time lag i, i=1,..,I; bi, c and d are empirical coefficients of the linear and constant term; ej is the residual error, which statistical properties have to be scrutinized. By definition, the bets-fit model minimizes the RMS residual error. The time lags are expected because of the delay between the change in one price (stock or goods and services) and the reaction of related prices. It is a fundamental feature of the model that the lags in (1) may be both negative and positive. In this study, we limit the largest lag to fourteen months. Apparently, this is an artificial limitation and might be changed in a more elaborated model. In any case, a fourteen-month lag seems to be long enough for a price signal to pass through.
System (1) contains J equations for I+2 coefficients. For General Electric (GE) we use a longer time series from January 1995, i.e. 192 monthly readings. Due to the negative effects of a larger set of defining CPI components their number for all models is (I=) 2. To resolve the system, we use standard methods of matrix inversion. As a rule, solutions of (1) are stable with all coefficients far from zero. In the GE model, we use 73 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics (2011). All involved indices must start before 1993. That’s why we have excluded 19 indices from the previously used set, including such major ones as communication, education, and recreation. They started after 1994.
Due to obvious reasons, longer time series guarantee a better resolution between defining CPIS. In general, there are two sources of uncertainty associated with the difference between observed and predicted prices. First, we have taken the monthly close prices (adjusted for splits and dividends) from a large number of recorded prices: monthly and daily open, close, high, and low prices, their combinations as well as averaged prices. Second source of uncertainty is related to all kinds of measurement errors and intrinsic stochastic properties of the CPI. One should also bear in mind all uncertainties associated with the CPI definition based on a fixed basket of goods and services, which prices are tracked in few selected places. Such measurement errors are directly mapped into the model residual errors. Both uncertainties, as related to stocks and CPI, also fluctuate from month to month.
Currently, General Electric (GE) is the second biggest company in the S&P 500 list just shy from Exxon Mobil. The defining indices are as follows: the index of motor vehicle maintenance and repair (MVR) and the index of motor vehicle insurance (MVI), both are subcategories of the transportation index. Both CPI components are contemporary with the share price. Figure 1 depicts the evolution of both indices which provide the best fit model, i.e. the lowermost RMS residual error, between January and December 2010:
GE(t) = -1.72*MVR(t) – 0.46*MVI(t) +15.67(t-2000) + 292.6
The predicted curve in Figure 2 is in sync with the observed one. The residual error is of $2.84 for the period between January 1995 and December 2010. Both defining components, especially MVI, grew at a slightly higher than usual rate between 2007 and 2009. This effect has pushed down the share price in 2008 and in the beginning of 2009. Since 2009, both indices evolve at a lower rate and the price has been showing a weak increase. The GE price will hardly be growing at a healthy rate in 2011 and looks slightly overestimated, as Figure 3 demonstrates.
Figure 1. Evolution of the price indices MVR and MVI.
Figure 2. Observed and predicted GE share prices.
Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $2.84.