This is a quarterly report on the performance of our pricing model. Harley-Davidson (HOG) is one of the best illustrations of our concept (see a brief description of the concept in Appendix) linking stock prices to CPI components. For HOG, the model is stable for many years. The first model was obtained in September 2009 and covered the period from October 2008. Here we revisit the HOG model using the monthly closing price for December 2001 and the CPI estimates published in January 2012.
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 December 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
HOG(t) = -11.27RPR(t-3) + 9.55ORPR(t-3) +19.35(t-1990) – 8.57, September 2011
HOG(t) = -11.09RPR(t-4) + 9.40ORPR(t-4) +18.93(t-1990) – 7.53, December 2011
where HOG(t) is the share price in US dollars, t is calendar time. The model is characterised by standard deviation of $4.28 for the period between July 2003 and December 2011.
Two recent models are depicted in Figure 2. The predicted curve in December 2011 leads the observed ones by 4 months. We do foresee a further fall in the stock price to $26 per share in the first quarter of 2012. Figure 3 displays the residual error.
Figure 1. Evolution of the price indices ORPR and RPR.
Figure 2. Observed and predicted HOG share prices. Model for March, September, and December 2011.
Figure 3. The model residual error.
In its general form, our pricing model is as follows:
sp(tj) = Σbi∙CPIi(tj-ti) + c∙(tj-1990 ) + 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.