In this post, we introduce a new model for a company in the energy S&P 500 subcategory – Pepco Holdings (POM). The importance of any energy related company is dictated by the influence of growing energy price on the overall economic performance. For many companies presented in this blog during the past week the forecast for the second quarter of 2011 is not promising. But before presenting the model we would like to refresh the overall approach.
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.
For POM, the defining indices are as follows: the index of food away from home (SEFV) and the index of owners' equivalent rent of residence (ORPR). The CPI components are leading by 4 and 5 months, respectively. Figure 1 depicts the evolution of both indices which provide 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
where POM(t) is the share price in US dollars, t is calendar time.
The predicted curve in Figure 2 leads the observed one by 4 months. The residual error is of $0.95 for the period between July 2003 and March 2011. In the second quarter of 2011, the model foresees a rise by $1.5.
Figure 1. Evolution of the price indices ORPR and SEVF.
Figure 2. Observed and predicted POM share prices.