We built the first deterministic
pricing model for
Morgan Stanley (NYSE: MS) in November 2008 together with
other companies from the S&P 500 list. Morgan Stanley was included in our
study of bankruptcy cases in the USA [1].
The intuition behind our pricing concept was simple 
a higher relative rate of price growth (fall) in a given subcategory of goods
and services (as expressed by the consumer price indices – CPIs) is likely to result
in a faster increase (decrease) in stock prices of related companies. In the
first approximation, the deviation between pricedefining indices is
proportional to the ratio of their pricing powers. The presence of sustainable
(linear or nonlinear) trends in the differences between various CPIs allows
predicting the evolution of these differences, and thus, the deviation between
prices of corresponding goods and services. Such trends are the basis of a
longterm prediction of share prices. In the shortrun, deterministic forecasting
is possible only when a given price lags behind the CPI components defining its
behavior.
Why do we rely on consumer price
indices in our modeling? Many readers may have reasonable doubts that any consumer
price, which is not directly related to goods and services produced by a given
company, may affect its price. We allow the economy to be a more complex system
than described by a number of simple linear relations between share prices and
goods. The connection between a firm and its products may be better expressed
by goods and services which the company does not produce. The demand/supply
balance is not well understood yet and may evolve along many nonlinear paths
with positive and negative feedbacks. It would be too simplistic to directly
define a company price by its products.
Overall, Morgan Stanley (MS) provides
an example of a reliable pricing model. In [1] we
reported that the defining CPIs in 2008 were the index of housing operations
(HO) and the index of food away from home (SEFV). (For a short time in the
second half of 2010, the defining indices were different: the index of food
less beverages (FB) and the index of information technology, hardware and
software (IT)). The model was revised in 2012 and the evolution
of MS price was defined by the index of food away from home (SEFV) and owner’s equivalent rent of
residence (ORPR), which is similar to
HO. Hence, the model obtained in 2008
is the same as estimate in 2012. Figure 1 depicts the evolution of both
indices. Here we revise the previous model with new data available for the
period between November 2012 and March 2014. The defining indices, their
coefficients and time lags are the same as in 2012. This observation validates
the MS pricing model.
The best fit model, i.e. the
lowermost RMS residual error, for February 2014 and October 2012 are as follows:
MS(t) = 7.93SEFV(t0) + 4.42ORPR(t2)
+ 25.23(t2000) + 420.92; October 2012
MS(t) = 7.57SEFV(t0) + 4.26ORPR(t2)
+ 23.66(t2000) + 398.90; February 2014
Table 1 lists the defining CPIs,
model coefficients, lags, and standard errors for 15 models between March 2012
and February 2014. The predicted price curve in Figure 2 is in sync with the observed
one. We also show the high and low monthly prices, which represent the
uncertainty in the modeled price. One can use any price (e.g. high, low, mean,
median) within a given month to model the overall evolution and obtain
different coefficients and residuals. Therefore, the model for the losing price
is a representative of a wider set of pricing models.
The model error for the
period between July 2003 and February 2014 is $3.35. The model accurately predicts the share price
in the past. From the overall behaviour of the defining CPIs and the current
negative residual (Figure 3) one may conclude that MS price is slightly
underestimated, but the possible correction is within the uncertainty bounds. If
the current trends in ORPR and SEFV are retained, MS may continue its
growth.
Table 1. Defining CPIs,
coefficients and standard errors of the models for 2012 and 2013/2014
Month

C_{1}

t_{1}

b_{1}

C_{2}

t_{2}

b_{2}

c

d

sterr,$


2012


October

SEFV

0

7.93

ORPR

2

4.415

25.226

420.919

3.468

September

SEFV

0

7.90

ORPR

2

4.399

25.137

420.060

3.468

August

SEFV

0

7.96

ORPR

2

4.425

25.343

423.817

3.447

July

SEFV

0

7.96

ORPR

2

4.445

25.258

420.687

3.440

June

SEFV

0

8.01

ORPR

2

4.449

25.526

426.655

3.437

May

SEFV

0

8.01

ORPR

2

4.452

25.540

426.579

3.434

April

SEFV

0

7.97

ORPR

2

4.419

25.492

427.246

3.422

March

SEFV

0

8.00

ORPR

2

4.431

25.609

429.254

3.421


2014

and

2013


February

SEFV

0

7.569

ORPR

2

4.2574

23.66

398.90

3.35

January

SEFV

0

7.605

ORPR

2

4.2773

23.81

400.40

3.33

December

SEFV

0

7.735

ORPR

2

4.3249

24.43

411.33

3.28

November

SEFV

0

7.732

ORPR

2

4.3197

24.44

411.77

3.29

October

SEFV

0

7.740

ORPR

2

4.3231

24.47

412.41

3.30

September

SEFV

0

7.754

ORPR

2

4.3227

24.57

414.62

3.31

August

SEFV

0

7.810

ORPR

2

4.3494

24.80

418.21

3.32

Figure 2. Observed and predicted MS
share prices.
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; b_{i}, c and d are empirical coefficients
of the linear and constant term; e_{j}
is the residual error, which statistical properties have to be scrutinized. By
definition, the betsfit model minimizes the RMS residual error. The time lags
are expected because of the delay between the change in one price (stock or
goods and services) and the reaction of related prices. It is a fundamental
feature of the model that the lags in (1) may be both negative and positive. In
this study, we limit the largest lag to fourteen months. Apparently, this is an
artificial limitation and might be changed in a more elaborated model. In any
case, a fourteenmonth lag seems to be long enough for a price signal to pass
through.
System (1) contains J equations for I+2 coefficients. Since the sustainable trends last more than five
years, the share price time series have more than 60 points. For the current
recent trend, the involved series are between 70 and 130 readings. Due to the
negative effects of a larger set of defining CPI components their number for
all models is (I=) 2. To resolve the
system, we use standard methods of matrix inversion.
For the sake of completeness we
always retain all principal subcategories of goods and services. Among them are
the headline CPI (C), the core CPI,
i.e. the headline CPI less food and energy (CC),
the index of food and beverages (F),
housing (H), apparel (A), transportation (T), medical care (M),
recreation (R), education and
communication (EC), and other goods and
services (O). In this model, we use
92 CPI components. They are not seasonally adjusted indices and were retrieved
from the database provided by the Bureau of Labor Statistics (2014).
There are two sources of uncertainty
associated with the difference between observed and predicted prices. First, we
have taken the monthly close prices (adjusted for splits and dividends) from a
large number of recorded prices: monthly and daily open, close, high, and low
prices, their combinations as well as averaged prices. Without loss of
generality, one can randomly select for modeling purposes any of these prices
for a given month. By chance, we have selected the closing price of the last working
day for a given month. The larger is the fluctuation of a given stock price
within and over the months the higher is the uncertainty associated with the
monthly closing price as a representative of the stock price.
Second source of uncertainty is
related to all kinds of measurement errors and intrinsic stochastic properties
of the CPI. One should also bear in mind all uncertainties associated with the
CPI definition based on a fixed basket of goods and services, which prices are
tracked in few selected places. Such
measurement errors are directly mapped into the model residual errors. Both
uncertainties, as related to stocks and CPI, also fluctuate from month to
month.
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