We would like to introduce another deterministic model for a share price from the S&P 500 index. This time we present a company from consumer goods category - Pitney Bowes (NYSE: PBI) which “provides software, hardware, and services to enable physical and digital communications”. The model is deterministic since it has been obtained by decomposition of a share price into a weighted sum of two consumer price indices. One may follow up our simple assumption that the growth in a CPI related to PBI (e.g. information technology) relative to some independent but dynamic reference (e.g. food away from home) should be seen in a higher pricing power for the studied company. Hence, our stock price model tries to find one defining CPI and the best reference. Both CPIs are taken from a set of 92 different (not seasonally adjusted) CPIs and the best model has the smallest RMS error between July 2003 and February 2012. This set includes the headline and core CPI, all major categories from food to other goods and services, and many minor subcategories with long enough history (i.e. continuous estimates should be available since 2000).
We have borrowed the time series of monthly closing prices of PBI from Yahoo.com (February 2012 is included) and the CPI estimates through January 2012 are published by the BLS. The evolution of PBI share price is defined by the consumer price of the index of information technology (IT) and the index of food away from home (SEVF). The defining time lag is the same for both CPIs – one month. The best-fit model for PBI(t) is as follows:
PBI(t) = -2.09 SEVF(t-1) - 4.87IT(t-1) + 7.66(t-1990) + 381.64, February 2012
where PBI(t) is the PBI share price in U.S. dollars, t is calendar time. Figure 1 displays the evolution of both defining indices since 2002. Both indices have negative slopes and the IT index defines the growth of the price. Higher food prices suppress the level of PBI share price.
Figure 2 depicts the high and low monthly prices for an PBI share together with the predicted and measured monthly closing prices (adjusted for dividends and splits). The predicted prices are well within the limits of the share price uncertainty. The model residual error is shown in Figure 3 with the standard deviation between July 2003 and January 2012 of $1.53.
Since April 2011, the price of food away from home has been growing at a rate much higher than the rate of the IT index growth. This proportion of growth rates has affected the PBI price and it has been falling. We can not exclude that the price will fall below the level of $15 per share by May 2012. The IT price index does not show any sigh of faster growth and food price will likely grow till 2014.
Figure 1. The evolution of defining indices.
Figure 2. Observed and predicted monthly closing prices for a PBI share.
Figure 3. The model residual error: sterr=$1.53.