Predicting share prices: Goldman Sachs Group may rise to $180-$200 per share in April

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 best-fit models for GS(t) are as follows:

GS(t) = -11.06OFH(t) +11.06H(t-12) - 1.82(t-2000) – 99.4, December 2010

GS(t) = -13.79F(t-3) +11.03ORPR(t-2) + 29.93(t-2000) + 33.75, October 2012

GS(t) = -13.038F(t-3) +10.556ORPR(t-2) + 30.44(t-2000) + 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	C1	t1	b1	C2	t2	b2	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) = 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 i-th 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 bets-fit 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.