Scientific literature definitely has a positive bias with successful examples more often published than failures. We also add to this bias selecting only successful models. But it is always a pleasure to describe a very stable and accurate model. Harley-Davidson (HOG) is one of the best illustrations of our concept linking stock prices to CPI components. Here we present two models for HOG (see a brief description of the concept in Appendix). One was obtained in September 2009 and covered the period from October 2008. The most recent HOG model uses the monthly closing price for March 2001 and the CPI estimates published on April 14, 2011. (Through 2010, the model was the same as in 2011.)
The importance of the HOG model for our concept is obvious – it validates deterministic character of stock pricing. For investors, it is also important to have a long-term reliable prediction of stock prices.
For HOG, the defining indices are as follows: the index of rent of primary residence (RPR) and the index of owners' equivalent rent of residence (ORPR). Both CPI components are leading the share price. Figure 1 depicts the evolution of the indices which provide the best fit model, i.e. the lowermost RMS residual error, between July 2008 and March 2011. The models are as follows:
HOG(t) = -13.82RPR(t-3) +12.77ORPR(t-4) +17.82(t-1990) – 163.94, before September 2009
HOG(t) = -11.30RPR(t-3) + 9.83ORPR(t-3) +17.53(t-1990) – 36.34, after September 2009
where HOG(t) is the share price in US dollars, t is calendar time.
Both models are depicted in Figure 2. The predicted curves lead the observed ones by 3 months. The residual error is of $4.10 for the period between July 2003 and March 2011. In the second quarter of 2011, the model foresees a fall to the level of $30 per share.
Figure 1. Evolution of the price indices ORPR and RPR.
Figure 2. Observed and predicted POM share prices. Upper panel – the model for September 2009. Lower panel – the model for march 2011.
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.