Here we model
the evolution of Invesco Ltd (NYSE: IVZ) stock price. IVZ is a company from Financial
sector which “provides
its services to institutional clients including major public entities,
corporations, unions, nonprofit organizations, endowments, foundations,
pension funds, and financial institutions”. Lately, we presented models for
the following financial companies: Franklin Resources (BEN), Morgan Stanley (MS), Goldman Sachs (GS), and Lincoln
National Corporation (LNC), which are different from the model for IVZ.
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
IVZ from Yahoo.com
and the relevant (seasonally not adjusted) CPI estimates through February 2014
are published by the BLS. The evolution of IVZ share price is defined by
the index of appliances (APL) from the Housing subcategory 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 IVZ, the defining time lags are 6 and 3
months, respectively, and the bestfit model is as
follows:
IVZ(t) = 1.296APL(t6) – 1.300PETS(t3) + 8.829(t2000) + 11.362, February 2014
where IVZ(t) is the IVZ share price in U.S.
dollars, t is calendar time. Figure 1 displays the evolution of both
defining indices since 2002. Figure 2
depicts the high and low monthly prices for IVZ share together with the
predicted and measured monthly closing prices (adjusted for dividends and
splits).
The reader may ask here why the index
of pets, pet products and service define the evolution of IVZ price? Actually,
the model implies that PETS index does NOT affect the share price. This index
provides a dynamic reference rather than 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 (professionally high). How much
time does s/he need? The 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.
Same is with stock prices  knowing the driving CPI is not enough to calculate
the price change, one needs to know "the stream speed". The CPI
representing the dynamic reference for IVZ is selected from the full set of 90+
CPIs to minimize the LSQ model residual. There is no other interpretation of
this reference CPI (PETS) except the statistical one.
The
model is stable over time. Table 1 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. In 2012, the same model
was obtained, as also listed in Table 1. Therefore, the estimated IVZ model is
reliable over longer time. The model residual
is shown in Figure 3. The standard deviation between July 2003 and February
2014 is $2.05.
Overall,
the model does not foresee any bug change in IVZ price any time soon. The predicted
value for May 2014 is lower than the current one (around $38 on April 3). We
would not exclude a negative correction to $35 [+$2.05] in a month.
Table
1. The best fit models for the period between April 2012 and February 2014
Month

b1

CPI1

lag1

b2

CPI2

lag2

c

d

Feb14

1.2960

APL

8

1.3004

PETS

5

8.8293

11.3617

Jan

1.2989

APL

9

1.2916

PETS

6

8.7718

10.3373

Dec13

1.3008

APL

10

1.2937

PETS

7

8.7851

10.3574

Nov

1.2944

APL

11

1.2734

PETS

8

8.6568

9.1425

Oct

1.2865

APL

12

1.2599

PETS

9

8.5718

8.6536

Sep

1.2804

APL

13

1.2512

PETS

10

8.5183

8.4184

Aug

1.2785

APL

14

1.2488

PETS

11

8.5023

8.3755

Jul

1.2828

APL

15

1.2541

PETS

12

8.5363

8.4531

Nov12

1.3245

APL

8

1.2824

PETS

4

8.7414

7.5203

Oct

1.3582

APL

9

1.3168

PETS

5

8.9682

7.5219

Sep

1.3971

APL

10

1.3544

PETS

6

9.2178

7.3138

Aug

1.4233

APL

11

1.3779

PETS

7

9.3777

7.0047

Jul

1.4365

APL

12

1.3972

PETS

8

9.5123

7.4526

Jun

1.4291

APL

13

1.3996

PETS

9

9.5371

8.2575

May

1.4272

APL

14

1.4

PETS

10

9.5414

8.4475

Apr

1.4261

APL

15

1.4115

PETS

11

9.6314

9.444

Figure
1. The evolution of APL and PETS indices
Figure 2.
Observed and predicted IVZ share prices.
Figure 3.
The model residual error: stdev=$2.05.
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}) = Σb_{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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