In April 2011, we introduced a new model for Pepco Holdings (POM). The defining CPI indices were as follows: the index of food away from home (SEFV) and the index of owners' equivalent rent of residence (ORPR). (See model details in Appendix). Figure 1 depicts the evolution of these CPIs which lead the POM share price by 4 and 5 months, respectively. The best fit model, i.e. the lowermost RMS residual error, between July 2010 and March 2011:
POM(t) = -2.66SEVF(t-4) +1.06ORPR(t-5) +11.83(t-1990) + 101.35, March 2011
where POM(t) is the share price in U.S. dollars, t is calendar time. The upper panel in Figure 2 displays the observed monthly closing price and that predicted by the above relationship.
In April, we predicted that “In the second quarter of 2011, the model foresees a rise by $1.5.” Actual monthly closing price has increased from $18.55 in March to $19.63 in June 2011. The predicted price is well within the high/low monthly bounds, i.e. practically within the uncertainty bounds of the POM price.
Here we revisit the initial model with new data available through January 2012. The model is stable and is defined by the same CPIs with similar coefficients. Time delays are also similar but the SEVF leads the share price by 5 months:
POM(t) = -2.19SEVF(t-5) +0.76ORPR(t-5) +10.57(t-1990) + 95.97, January 2012
In the lower panel of Figure 2, we show the current model and the uncertainty bounds as presented high and low monthly prices. The overall fit is good and we expect the current price to fall in the 2012Q1. Figure 3 demonstrates that the model error in January 2012 is positive and it must fall back to 0 in the near future.
Figure 1. The evolution of defining CPI.
Figure 2. Observed and predicted POM share prices.
Figure 3. The model error, stdev= $0.89
In its general form, our pricing model is as follows:
sp(tj) = Σbi∙CPIi(tj-Di) + c∙(tj-2000 ) + d + ej (1)
where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-Di) is the i-th component of the CPI with the time lag Di, 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 January 2012, i.e. 105 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. Usually, 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.