We have been trying to build a pricing model for Goldman Sachs Group (GS) since 2008. This company was included in our study of bankruptcy cases in the USA. 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 behavior of the two defining CPI components. Figure 2 also depicts the high and low monthly prices for the same period, which illustrate the inter-month 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.
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.
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(tj) = Σbi∙CPIi(tj-ti) + c∙(tj-2000 ) + d + ej (1)
where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-ti) is the i-th component of the CPI with the time lag ti, i=1,..,I (I=2 in all our models); bi, c and d are empirical coefficients of the linear and constant term; ej 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.
Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.