We have described the evolution of a number of share prices for S&P 500 companies using empirical relationship between the prices and the differences between various CPI components [1-3], [Seeking Alpha posts]: Bank of America (BAC) and Goldman Sachs (GS, both financials), Microsoft (MSFT, information technology), and Alcoa (AA, materials), Boeing (BA, industrials), Exxon mobile (XOM, energy), ConocoPhillips (COP, energy), etc. All these companies were described with a reasonable accuracy considering the uncertainty associated with the monthly estimates of CPI expenditure subcategories and slightly stochastic character of the monthly closing price (adjusted for dividends and splits) as a representative of corresponding share price. All in all, the concept has demonstrated a high level of consistency.
This time our task is a more exciting and challenging one. Can our model describe the approach to bankruptcy? For a regular investor this question is of the top priority. The history of dramatic decline in share prices of financial companies in 2008 and 2009 dictates an inevitable choice – comparison of the Colonial Bank (CNB) and CIT Group (CIT), which both were beyond the edge of bankruptcy but with different outcome. In this study, all share prices are retrieved from YahooFinance.
Previously, we introduced and tested several models with varying number of defining CPI components. It was found that the best model includes two defining CPI components, free term (constant) and linear time term [4-6], which compensates well known the linear trends between the defining CPI components. For historical reasons we call it 3-C model. In this study, we are seeking for those two CPI components from a full set of components, which minimize the (RMS) difference between observed and predicted prices.
The 3-C model for a share price sp(t) is as follows:
sp(t) = A*CPI1(t+t1) + B*CPI2(t+t2) + C*(t-2000) + D (1)
where CPI1 and CPI2 are defining CPI components; t is the calendar time; A, B, C, and D are empirically determined coefficients. Because prices for good and services can lag or lead relevant share prices the time lags t1 and t2 are introduced. Both time lags can be positive or negative. In our model, we test all possible lags between 0 and +13 months. So, we chose the best from 34*33*14*14 ~ 200,000 possible models, with two extra degrees of freedom introduced by linear and free terms.
The full set includes 34 CPI components, which are tested as predictors for each share price. Among these components are major expenditure categories: the headline CPI (C), the core CPI, i.e. the headline CPI less food and energy (CC), food and beverages (F), housing (H), apparel (A), transportation (T), medical care (M), recreation (R), education (ED), communication (CO), and other goods and services (O). All components used in the model are not seasonally adjusted ones and retrieved from the Bureau of Labor Statistics.
In  we have revealed and explained the presence of linear trends in the differences between various components of the CPI. It was also found that the trends exist during long but finite periods and then their slops turn to opposite signs. Accordingly, the evolution of the differences between CPI components since 1960 can be approximated by a piece-wise linear function. When the trends turn, the coefficients in the link between CPI and share price also change. Because of this constraint we limit our modeling to the period between July 2003 and July 2009. The early date is a conservative estimate of the start of the current trend, and the latter one corresponds to the bankruptcy of Colonial Bank.
Here, we would like to emphasise that the usage of the differences instead of CPI components is caused by two crucial factors. First, we presume that the evolution of a share price depends only on the pricing power of goods and services associated with the company relative to other goods and services. Then, the difference in pricing powers of two companies should be directly mapped into the difference in the growth in relevant consumer price indices. Therefore, we assume that there exists a pair of CPI components, which reflect the best relative evolution of the share prices. Such a pair might not be always available among eight major expenditure categories. As a result, one needs two or more differences between CPI components to accurately predict the evolution of a share price.
Second factor consists in the presence of sustainable (linear and nonlinear) trends in the difference. This is the phenomenon we have described in [4-6]. Effectively, one can accurately decompose any share price using several CPI components. However, such decomposition does not allow forecasting without knowledge of the evolution of relevant CPI subcategories. The trends in the differences make it possible to predict at longer time horizons.
Now we apply the 3-C model (1) to the CNB’s and CIT Group’s share prices. The (LSQR) best-fit empirical relationship is as follows:
CNB(t) = 1.58*F(t) – 1.39*O (t-10) + 20.61*(t-2000) + 633 (2)
CIT(t) = -4.14*F(t-6) – 3.10*FB (t+4) + 43.18*(t-2000) + 1776 (3)
Empirical relationship (2) implies that CNB’s share price evolve in sync with the index of food and beverages (F), but lags behind the index for other goods and services (O) 10 months. Therefore, the share price depends on past and contemporary values of the defining CPI components. Effectively, it allows predicting CNB price only in the past. Figure 1 displays the observed and predicted price as well as their residual.
Empirical relationship (3) shows that CIT’s share price leads by six months the index of food and beverages (F) and lags four months behind the index for food only (FB). Therefore, the share price depends on past and future values of the defining CPI components. Effectively, it allows predicting CIT price only in the past.
Comparing the time histories of CNB and CIT one can conclude that the former sank right below the zero price and thus this bank deserved the bankruptcy. CIT has been closing the negative zone but still is slightly above it. The next several readings of FB will answer the question – Was it worth to bail out CIT Group? In other words, will the predicted price sink below the zero line?
Figure 1. Upper panel: Comparison of the observed and predicted CNB’s share price. The latter is obtained using (2). Lower panel: the difference between the observed and predicted price.
Figure 2. Upper panel: Comparison of the observed and predicted CIT’s (adjusted for dividends and splits) share price. The latter is obtained using (3). Lower panel: the difference between the observed and predicted price.
 Kitov, I., (2009). ConocoPhillips and Exxon Mobil stock price, Journal of Applied Research in Finances, v. 1, issue 2 (in press).
 Kitov, I., Kitov, O., (2009). Predicting share price of energy companies: June-September 2009, MPRA Paper 15863, University Library of Munich, Germany.
 Kitov, I., Kitov, O., (2009). Modelling selected S&P 500 share prices, MPRA Paper 15862, University Library of Munich, Germany
 Kitov, I., Kitov, O., (2008). Long-Term Linear Trends In Consumer Price Indices, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(2(4)_Summ), pp. 101-112.
 Kitov, I., Kitov, O., (2009a). A fair price for motor fuel in the United States, MPRA Paper 15039, University Library of Munich, Germany
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