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 tradeoff 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 bestfit model is
as follows:
GE(t) = 1.536PETS(t2) – 0.631TS(t6) + 11.805(t2000) + 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 R^{2}=0.93 is not biased.
Table
1. Statistical estimates for the model coefficients

Coefficients

Standard Error

t Stat

Pvalue

Lower 95%

Upper 95%

d

291.08

10.5133

27.670

6.72E55

270.095

311.7124

b1

1.536

0.0439

34.972

4.42E66

1.62381

1.44985

b2

0.632

0.0512

12.325

2.79E23

0.73194

0.52938

c

11.805

0.4211

28.024

1.73E55

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 nonstationary 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

Feb14

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

Dec13

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

Nov12

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

Dec11

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

Dec10

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(t_{j}) = Î£b_{i}∙CPI_{i}(t_{j}t_{i}) + c∙(t_{j}2000 ) + d + e_{j} (1)
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 ith
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 betsfit 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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