It is time to model the stock price evolution of General Electric Company (NYSE: GE). GE is a company from Industrial Goods sector which "operates as an infrastructure and financial services company worldwide". Among others, the studied company has a Transportation segment covering a wide range of transportation services. This observation is important for understanding the model.
All models have been obtained using our concept of stock pricing as a decomposition of a share price into a weighted sum of two consumer price indices. The background idea is a simplistic one: there is a trade-off between a given share price and goods and services the relevant company produces or provides. For example, the energy consumer price should influence the price of energy companies. Let's assume that some set of consumer prices (as expressed by an appropriate consumer price index, CPI) drives the company stock price. The net effect of the relevant CPI change can be positive or negative.
In real world, each company competes not only with those producing similar goods and services, but also with all other companies on the market. Therefore, the influence of the driving CPI should also depend on all other goods and services, and thus, relevant CPIs. To model the net change in the market prices we introduce just one reference CPI. In quantitative terms, it has to best represent the dynamics of changing price environment. Hence, our pricing model includes two defining CPIs: the driver and the reference. Because of possible time delays between action and reaction (the time needed for any price changes to pass through), the defining CPIs may lead the modeled price or lag behind.
We have borrowed the time series of GE monthly closing prices (through March 2014) from Yahoo.com. CPI estimates (not seasonally adjusted) through February 2014 are published by the BLS. According to the procedure described in Appendix, the evolution of GE share price is defined by the index of transportation services (NYSE:TS) and the index of pets, pet products and services (NASDAQ:PETS). These indices were selected from a large set of 92 CPIs covering all consumer price categories. All possible CPI pairs with all possible time lags and leads were tested one by one. The TS/PETS set, which minimizes the model error, is considered as the defining pair. For GE, the time lags are 6 and 2 months, respectively, and the best-fit model is as follows:
GE(t) = -1.536PETS(t-2) - 0.631TS(t-6) + 11.805(t-2000) + 291.08, February 2014 (1)
where GE(t) is the GE share price in U.S. dollars, t is calendar time. All coefficients in the above relationship were estimated together with their uncertainties using the linear regression technique . Table 1 confirms that all coefficients are statistically significant.
Table 1. Statistical estimates for the model coefficients
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
d | 291.08 | 10.5133 | 27.670 | 6.72E-55 | 270.095 | 311.7124 |
b1 | -1.536 | 0.0439 | -34.972 | 4.42E-66 | -1.62381 | -1.44985 |
b2 | -0.632 | 0.0512 | -12.325 | 2.79E-23 | -0.73194 | -0.52938 |
c | 11.805 | 0.4211 | 28.024 | 1.73E-55 | 10.96724 | 12.63418 |
Figure 1 displays the evolution of both defining indices since 2003. Figure 2 depicts the high and low monthly prices for GE share together with the predicted and measured monthly closing prices (adjusted for dividends and splits). The model prediction is best described by the coefficient of determination Rsq.=0.93. The predicted and observed time series are cointegrated. Thus the Rsq. estimate is not biased.
From relationship (1), it seems that the index of pets, pet products and services define the evolution of GE price. Actually, the model implies that PETS index does NOT affect the share price. This index provides a dynamic (price) reference rather than the driving force. Here is a simple example how to understand the term "dynamic reference". Imagine that a swimmer needs to swim 20 km along a river. Let's assume that for this experienced swimmer the average speed is 5 km/h. How much time does s/he need? The quick answer 4 hours is wrong. One cannot calculate the time needed without knowing the stream speed and direction. This stream is the dynamic reference (or moving coordinate reference system) for the swimmer. The stream speed can also vary over time producing a non-stationary coordinate reference system. The intuition behind our pricing concept is similar - the driving CPI is not enough to calculate the price change, one needs to know "the stream speed", i.e. the market movements. The CPI representing "dynamic reference" for GE was selected from 91 CPIs.
The model is stable over time: it has the same defining CPIs, coefficients and time lags for longer periods of time. Table 2 lists the best fit models, i.e. slopes, b1 and b2, for two defining CPIs, time lags, slope of time trend, c, and free term, d, for 7 months between July 2013 and February 2014. It is instructive that the same model was obtained in 2012, 2011, and 2010, also listed in Table 2. Therefore, the estimated GE model is reliable over 50+ months. The model residual for February 2014 is shown in Figure 3. Between July 2003 and February 2014, the model standard error is $1.51.
Overall, the model does not foresee any big change in GE price. One may expect some price fluctuations within the bounds of intermonth changes observed in the past. The predicted value for May 2014 is $27.7 (+-$1.5).
Table 2. The best fit models for the period between May 2010 and February 2014
Month | b1 | CPI1 | lag1 | b2 | CPI2 | lag2 | c | d |
Feb-14 | -1.536 | PETS | 2 | -0.632 | TS | 6 | 11.805 | 291.08 |
Jan | -1.546 | PETS | 2 | -0.633 | TS | 6 | 11.872 | 292.10 |
Dec-13 | -1.547 | PETS | 2 | -0.634 | TS | 6 | 11.883 | 292.39 |
Nov | -1.531 | PETS | 2 | -0.638 | TS | 6 | 11.815 | 291.87 |
Oct | -1.522 | PETS | 2 | -0.641 | TS | 6 | 11.773 | 291.54 |
Sep | -1.513 | PETS | 2 | -0.644 | TS | 6 | 11.732 | 291.24 |
Aug | -1.510 | PETS | 2 | -0.645 | TS | 6 | 11.723 | 291.16 |
Jul | -1.512 | PETS | 2 | -0.644 | TS | 6 | 11.728 | 291.14 |
Nov-12 | -1.549 | PETS | 2 | -0.711 | TS | 6 | 12.275 | 307.94 |
Oct | -1.544 | PETS | 2 | -0.712 | TS | 6 | 12.250 | 307.75 |
Sep | -1.540 | PETS | 2 | -0.714 | TS | 6 | 12.235 | 307.77 |
Aug | -1.530 | PETS | 2 | -0.719 | TS | 6 | 12.198 | 307.95 |
Jul | -1.527 | PETS | 2 | -0.720 | TS | 6 | 12.175 | 307.78 |
Jun | -1.522 | PETS | 2 | -0.723 | TS | 6 | 12.165 | 308.06 |
May | -1.513 | PETS | 2 | -0.732 | TS | 6 | 12.151 | 308.95 |
Apr | -1.507 | PETS | 2 | -0.740 | TS | 6 | 12.152 | 309.86 |
Dec-11 | -1.520 | PETS | 2 | -0.785 | TS | 6 | 12.443 | 196.23 |
Nov | -1.510 | PETS | 2 | -0.802 | TS | 6 | 12.479 | 197.24 |
Oct | -1.506 | PETS | 2 | -0.806 | TS | 6 | 12.475 | 196.70 |
Sep | -1.495 | PETS | 2 | -0.830 | TS | 6 | 12.539 | 198.53 |
Aug | -1.495 | PETS | 2 | -0.828 | TS | 6 | 12.528 | 197.20 |
Jul | -1.499 | PETS | 2 | -0.831 | TS | 6 | 12.566 | 196.60 |
Jun | -1.494 | PETS | 2 | -0.830 | TS | 6 | 12.535 | 195.46 |
May | -1.486 | PETS | 2 | -0.846 | TS | 6 | 12.572 | 196.39 |
Dec-10 | -1.450 | PETS | 2 | -1.006 | TS | 6 | 13.153 | 226.64 |
Nov | -1.427 | PETS | 2 | -1.040 | TS | 6 | 13.200 | 229.69 |
Oct | -1.435 | PETS | 2 | -1.027 | TS | 6 | 13.180 | 227.00 |
Sep | -1.442 | PETS | 2 | -1.016 | TS | 6 | 13.165 | 224.57 |
Aug | -1.447 | PETS | 2 | -1.006 | TS | 6 | 13.147 | 222.22 |
Jul | -1.487 | PETS | 2 | -0.949 | TS | 6 | 13.097 | 214.03 |
Jun | -1.500 | PETS | 2 | -0.925 | TS | 6 | 13.056 | 209.88 |
May | -1.515 | PETS | 2 | -0.899 | TS | 6 | 13.017 | 205.42 |
Figure 1. The evolution of TS and PETS indices
Figure 2. Observed and predicted GE share prices.
Figure 3. The model residual error: sterr=$1.51.
Appendix
The concept of share pricing based on the link between consumer and stock prices has been under development since 2008. In the very beginning, we found a statistically reliable relationship between ConocoPhillips' stock price and the difference between the core and headline consumer price index in the United States. Then we extended the pool of defining CPIs to 92 and estimated quantitative models for all companies from the S&P 500 list. The extended model described the evolution of a share price as a weighted sum of two individual consumer price indices selected from this large set of CPIs. We allow only two defining CPIs, which may lead the modeled share price or lag behind it. The intuition behind the lags is that some companies are price setters and some are price takers. The former should influence the relevant CPIs, which include goods and services these companies produce. The latter lag behind the prices of goods and services they are associated with. In order to calibrate the model relative to the starting levels of the involved indices and to compensate sustainable time trends (some indices are subject to secular rise or fall) we introduced two additional terms: linear time trend and constant. In its general form, the pricing model is as follows:
sp(t_{j}) = Σb_{i}∙CPI_{i}(t_{j}-t_{i}) + c∙(t_{j}-2000 ) + d + e_{j} (2)
where sp(t_{j}) is the share price at discrete (calendar) times t_{j}, j=1,…,J; CPI_{i}(t_{j}-t_{i}) is the i-th component of the CPI with the time lag t_{i}, i=1,..,I (I=2 in all our models); b_{i}, c and d are empirical coefficients of the linear and constant term; e_{j} is the residual error, whose statistical properties have to be scrutinized.
By definition, the bets-fit model minimizes the RMS residual error. It is a fundamental feature of the model that the lags may be both negative and positive. In this study, we limit the largest lag to eleven months. System (2) contains J equations for I+2 coefficients. We start our model in July 2003 and the share price time series has more than 130 readings. To resolve the system, standard methods of matrix inversion are used. A model is considered as a reliable one when the defining CPIs are the same during the previous seven months.
Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.