We have been trying to build a pricing model for
Goldman Sachs Group (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 were 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. In December
2012, we published a
paper comparing GS with four financial companies and revised the model,
which includes new data obtained since December 2010. Here we update the model
using new data between December 2012 and March 2014. The 2012 model has not
changed. This validates our approach to stock price modeling (see details in
Appendix).
Table 1
lists defining parameters for GS between March and October 2012, and from
August 2013 to February 2014. 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’ equivalent rent
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).
Figure 1 depicts the overall
evolution of both involved consumer price indices: F and ORPR. 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
GS(t) = 13.038F(t3) +10.556ORPR(t2) + 30.44(t2000) + 12.86,
February 2014
The predicted curve in Figure 2 leads the observed price by two months. The
residual error is of $14.25 for the period between July 2003 and February 2014.
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.
From Figure 2, the modeled price is
approximately $200 in April 2014.
Table 1. The monthly models
for GS for eight months in 2012 and for seven months in 2014/2013.
Month

C_{1}

t_{1}

b_{1}

C_{2}

t_{2}

b_{2}

c

d

sterr,$


2012



October

F

3

13.795

ORPR

2

11.027

29.935

33.751

14.521

September

F

3

13.791

ORPR

2

11.013

29.992

35.827

14.584

August

F

3

13.787

ORPR

2

11.003

30.023

37.106

14.649

July

F

3

13.759

ORPR

2

10.978

30.018

37.647

14.707

June

F

3

13.731

ORPR

2

10.933

30.124

41.985

14.758

May

F

3

13.704

ORPR

2

10.876

30.342

48.755

14.770

April

F

3

13.661

ORPR

2

10.819

30.449

53.171

14.805

March

F

3

13.787

ORPR

2

10.943

30.440

48.639

15.055


2014

and

2013



February

F

3

13.038

ORPR

2

10.556

27.62

12.86

14.25

January

F

3

13.3166

ORPR

2

10.660

28.88

34.69

14.02

December

F

3

13.4606

ORPR

2

10.687

29.71

51.36

13.91

November

F

3

13.4537

ORPR

2

10.676

29.75

52.34

13.96

October

F

3

13.5352

ORPR

2

10.700

30.13

60.04

14.00

September

F

3

13.5638

ORPR

2

10.683

30.44

67.71

14.03

August

F

3

13.6031

ORPR

2

10.691

30.66

72.06

14.07

Figure 1. Evolution of F and ORPR.
Figure 2. Observed and predicted GS
share prices. The prediction horizon is two months.
Figure 3. Model residuals, standard
error of the model $14.25.
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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