It’s time to revisit the price model for Allergan (AGN) first estimated in January 2011. As before, we model the monthly (adjusted for dividends and splits) closing prices between July 2003 and March 2011. It is found that the best fit model obtained in January is still valid in April 2011 with almost the same coefficients and time lags (see Appendix for details of our deterministic share pricing concept).
Briefly, we decompose a share price into a weighed sum of two individual CPI components, linear time trend component and constant free term. We allow positive and negative time lags between variables and seek to minimize the RMS model error by varying all involved coefficients. The set of CPI components consists of 92 independent price indices of different level: from major (overall and core CPI) to very small (photo and related materials). When both defining components lead the modeled price, one can predict future evolution of the stock; at least in the near future. The bets-fit two-component (2-C) model for AGN is as follows:
AGN(t)= -1.90FH(t-3) – 1.63THI(t-0) +16.56(t-1990) + 337.58
where AGN(t) is the price of a share in US dolars, FH in the index of food at home leading the stock price by 3 months, THI is the index of tenants’ and household insurance, (t-1990) is the elapsed time. Quantitatively, the best fit model provides RMSE=$3.37 for the period between July 2003 and March 2011. Also, it has been valid during the past seventeen months and we expect it to be valid in the first half of 2011, at least. The defining CPI indices are displayed in Figure 1. The FH index has been growing at a high rate during since December 2011. This effect did not overcome the positive time trend of $16.6 per year and the fall in the THI during the same time. The share may grow in the second quarter if all the observed trends hold.
Figure 1. The price indices THI and FH between 2002 and 2011
Figure 2. Observed and predicted share prices AGN.
In its general form, our pricing model is as follows:
sp(tj) = Σbi∙CPIi(tj-ti) + c∙(tj-2000 ) + d + ej (1)
where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-ti) is the i-th component of the CPI with the time lag ti, 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 eleven months. Apparently, this is an artificial limitation and might be changed in a more elaborated model.
System (1) contains J equations for I+2 coefficients. For POM we use a time series from July 2003 to March 2011, i.e. 94 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 POM model, we use 92 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics.
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 and its components. 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.