4/4/14

Predicting stock prices: General Electric Company to hold


It’s 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 services. This is important for understanding of 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 (CPIs). The background idea is a simplistic one: there is a potential trade-off between a given share price and goods and services the company produces and/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 consumer price index, CPI) drives the company stock price, i.e. the change in the consumer prices is directly transferred into the relevant stock share price. The net effect of the CPI change can be positive or negative. The latter case implies that the rising consumer prices suppress the stock price.  
 
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 on the company's stock price should also depend on all other CPIs. To take into account the net change in the whole variety of market prices, we introduce just one reference CPI best representing the overall dynamics of the changing price environment. Hence, the pricing model has to include at least 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 by a few months. 

We have borrowed the time series of monthly closing prices (through March 2014) of GE from Yahoo.com and the relevant (seasonally not adjusted) CPI estimates through February 2014 are published by the BLS.  The evolution of GE share price is defined by the index of transportation services (TS) and the index of pets, pet products and services (PETS). These indices are selected by LSQ procedure (see Appendix) from a large set of 92 CPIs covering all categories. All possible pairs of CPIs with all possible time lags and leads (but less than 12 months) were tested one by one and the set minimizing the model error is considered as the defining one. For GE, the defining 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 

where GE(t) is the GE share price in U.S. dollars,  t is calendar time. All coefficients were estimated using linear regression together with their unceratinties. Table 1 confirms that all coefficients are statistically significant. In addition, the predicted and observed time series are cointegrated, i.e. the estimate of the coefficient of determination R2=0.93 is not biased. 

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). 

Why does the index of pets, pet products and service in the above model “define” the evolution of GE price? Actually, the model implies that PETS index does NOT affect the share price. This index provides a dynamic 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 (river) stream speed and its 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. Same is with stock prices - knowing the driving CPI is not enough to calculate the price change, one needs to know "the stream speed" or the market movements. The CPI representing the dynamic reference for GE (PETS) is selected from the full set of 90+ CPIs to minimize the LSQ model residual. There is no other interpretation of this reference CPI except the statistical one.   

The model is stable over time. Table 2 lists the best fit models, i.e. coefficients, b1 and b2, defining CPIs, time lags, the slope of time trend, c, and the free term, d, for 7 months. The same model was obtained in 2012, 2011, and 2010 as listed in Table 2. Therefore, the estimated GE model is reliable over 50+ months.  The model residual is shown in Figure 3. The standard deviation between July 2003 and February 2014 is $1.51.  

Overall, the model does not foresee any big change in GE price any time soon, except 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: stdev=$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 (CPI) 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. 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 a linear time trend and constant term. In its general form, the 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 (I=2 in all our models); bi, c and d  are empirical coefficients of the linear and constant term; ej 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 (1) contains J equations for I+2 coefficients. We start our model in July 2003 and the share price time series has more than 100 points. 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 eight months. This number and the diversity of CPI subcategories are both crucial parameter. 

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