We have been trying to build a preliminary pricing
model for Goldman Sachs (NYSE: GS) since 2008. This company was included in our
study of bankruptcy cases in the USA. All in all, the model was not
stable over time and the prediction for 2009 was not fully correct. Originally, the stock price was defined by the
index of housing operations (HO) and that of food away from home (SEFV). In
January 2011, we presented an updated model as based on the CPIs available
till November 2010 and the December monthly closing (adjusted for splits and
dividends) price of GS. In the updated model, the defining CPIs are the index
of other food at home (OFH) and the housing index (H). Thus, the difference
between the preliminary and the updated model was not too large because the pairs
of defining indices are very close. Here we present a revised model, which
includes new data obtained since December 2010. The revised model shows that GS
share may grow to $133 from its current level of $118. This might be a good
long idea together with that for Prudential
Financial.
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. For example, Table 1 lists defining parameters
for GS between March and October 2012. For each month, the best model is based
on the same defining CPIs – the consumer price index of food and beverages, F, and the index of owners’ rent of
primary residence, ORPR. In all cases, the lags are the same: three and
two months, respectively. Other coefficients and the standard error suffer just
slight oscillations or drifts (e.g. c
and d). It is important to stress again that all
models for months except October also include those with future CPIs relative to
the given month. Table 1 confirms that no future CPIs drive the share price in March
2012. The best fir model is always based on the past values of F and ORPR, at least since March 2012.
Figure 1 depicts the overall
evolution of both involved consumer price indices: F and ORPR. It also presents
the evolution of two defining indices from the previous model: the index of
other food at home, OHF, which is a
part of F, and the index of housing, H, with ORPR representing a part of H. The OHF and H indices provided the best fit
model between March 2010 and December 2010. The bestfit models
for GS(t) are as follows:
GS(t) = 11.06OFH(t)
+11.06H(t12)  1.82(t2000) – 99.4, December 2010
GS(t) = 13.79F(t3)
+11.03ORPR(t2) + 29.93(t2000) + 33.75, October 2012
The
predicted curve in Figure 2 leads the observed price by two months. The
residual error is of $14.52 for the period between July 2003 and October 2012. The
price of a GS share is relatively well defined by the behaviour of the two defining
CPI components. Figure 2 also depicts the high and low monthly prices for the
same period, which illustrate the intermonth variation of the share price. These
prices might be considered as natural limits of the monthly price uncertainty
associated with the quantitative model. Since 2009, the predicted price is well
within the high/low band. Figure 3 displays the residual error.
Table 1. The evolution of GS model since March 2012
Month

CPI_{1}

t_{1}

b_{1}

CPI_{2}

t_{2}

b_{2}

c

d

sterr,$

October

F

3

13.795

ORPR

2

11.026

29.934

33.75

14.52

September

F

3

13.791

ORPR

2

11.013

29.992

35.82

14.58

August

F

3

13.786

ORPR

2

11.002

30.023

37.10

14.64

July

F

3

13.759

ORPR

2

10.978

30.018

37.64

14.70

June

F

3

13.730

ORPR

2

10.933

30.124

41.98

14.75

May

F

3

13.703

ORPR

2

10.876

30.342

48.75

14.76

April

F

3

13.661

ORPR

2

10.818

30.449

53.17

14.80

March

F

3

13.786

ORPR

2

10.942

30.439

48.63

14.76

Figure 1. Evolution of F and ORPR. Also shown
are defining CPI of the 2010’s model: OFH
and H.
Figure 2. Observed and predicted GS share prices. The
prediction horizon is two months.
Figure 3. Standard error of the model $14.52.
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