Modeling share prices of banks and bankrupts

Abstract

Exxon Mobil and ConocoPhillips stock price has been predicted using the difference between core and headline CPI in the United States. Linear trends in the CPI difference allow accurate prediction of the prices at a five to ten-year horizon.

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Modeling share prices of banks and bankrupts
Ivan O. Kitov, IDG RAS

Abstract
Share prices of financial companies from the S&P 500 list have been modeled by a linear function of consumer price
indices in the USA. The Johansen and Engle-Granger tests for cointegration both demonstrated the presence of an
equilibrium long-term relation between observed and predicted time series. Econometrically, the pricing concept is
valid. For several companies, share prices are defined only by CPI readings in the past. Therefore, our empirical pricing
model is a deterministic one. For a few companies, including Lehman Brothers, AIG, Freddie Mac and Fannie Mae,
negative share prices could be foreseen in May-September 2008. One might interpret the negative share prices as a sign
of approaching bankruptcies.

Key words: share price, modeling, CPI, prediction, the USA, bankruptcy
JEL classification: E4, G1, G2, G3

Introduction
Recently, we have developed and tested statistically and econometrically a deterministic model
predicting share prices of selected S&P 500 companies (Kitov, 2010). We have found that there
exists a linear link between various subcategories of consumer price index (CPI) and some share
prices, with the latter lagging by several months. In order to build a reliable quantitative model from
this link one needs to use standard and simple statistical procedures.
Following the general concept and principal results of the previous study, here we are
predicting stock prices of financial companies from the S&P 500 list. In several cases, robust
predictions are obtained at a time horizon of several months. In close relation to these financial
companies we have also investigated several cases of bankruptcy and bailout. These cases include
Lehman Brothers (LH), American International Group (AIG), Fannie Mae (FNM) and Freddie Mac
(FRE). Regarding these bankruptcies, we have tested our model against its predictive power in
May and September 2008. The main question was: Could the bankruptcies be foreseen? If yes,
which companies should or should not be bailed out as related to the size of their debt?
In the mainstream economics and finances stock prices are treated as not predictable beyond
their stochastic properties. The existence of a deterministic model would undermine the
fundamental assumption of the stock market. If the prices are predictable, the participants would
have not been actively defining new prices in myriads of tries, but blindly followed the driving
force behind the market. It is more comfortable to presume that all available information is already
counted in. However, our study has demonstrated that the stochastic market does not mean an
unpredictable one.
In this paper, we analyze sixty six financial companies from the S&P 500 lists as of January
2010 as well as a few bankrupts from the financials. Some of the companies have been accurately
described by models including two CPI subcategories leading relevant share prices by several

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months. Other companies are characterized by models with at least one of defining CPI components
lagging behind related stock prices. We have intentionally constrained our investigation to S&P 500
- we expect other companies to be described by similar models.
Our deterministic model for the evolution of stock prices is based on a “mechanical”
dependence on the CPI. Under our framework, the term “mechanical” has multiple meanings.
Firstly, it expresses mechanistic character of the link when any change in the CPI is one-to-one
converted into the change in related stock prices, as one would expect with blocks or leverages.
Secondly, the link does not depend on human beings in sense of their rational or irrational behavior
or expectations. In its ultimate form, the macroeconomic concept behind the stock price model
relates the market prices to populations or the numbers of people in various age groups irrelevant to
their skills. Accordingly, the populations consist of the simplest possible objects; only their
numbers matter. Thirdly, the link is a linear one, i.e. the one often met in classical mechanics. In all
these regards, we consider the model as a mechanical one and thus a physical one rather than an
economic or financial one. Essentially, we work with measured numbers not with the piles of
information behind any stock.
For the selected stocks, the model quantitatively foresees at a several month horizon.
Therefore, there exist two or more CPI components unambiguously defining share prices several
months ahead. It is worth noting that the evolution of all CPI components is likely to be defined, in
part, by stochastic forces. According to the mechanical dependence between the share prices and the
CPI, all stochastic features are one-to-one converted into stochastic behavior of share prices. Since
the prices lag behind the CPI, this stochastic behavior is fully predetermined. The predictability of a
measured variable using independent measured variables, as described by mathematical
relationships, is one of the principal requirements for a science to join the club of hard sciences.
Therefore, our stock pricing model indicates that the stock market is likely an object of a hard
science.
A model predicting stock prices in a deterministic way is a sensitive issue. It seems unfair to
give advantages to randomly selected market participants. As thoroughly discussed in (Kitov,
2009b; Kitov and Kitov, 2008; 2009ab) the models are piecewise ones. A given set of empirical
coefficients holds until the trend in the difference between defining CPI is sustained. Such
sustainable trends are observed in a majority of CPI differences and usually last between 5 and 20
years (Kitov and Kitov, 2008). The most recent trend has been reaching its natural end since 2008
and the transition to a new trend in 2009 and 2010 is likely the best time to present our model. As a
result, there is no gain from the empirical models discussed in this paper. Their predictive power
has been fading away since 2008. When the new trend in the CPI is established, one will be able to
estimate new empirical coefficients, all participants having equal chances.

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The remainder of the paper is arranged as follows. Section 1 introduces the model and data,
which include stock prices of sixty six S&P 500 financial companies and seventy CPI components.
In Section 2, empirical models are presented both in tabulated and graphical forms. For each model
we have estimated standard deviation, which serves as a proxy to the model accuracy. For a few
companies, the estimated models are robust over the previous 10 months. Section 3 tests these
models statistically and econometrically. The Johansen (1988) and Engle-Granger (Newbold and
Granger, 1967; Hendry and, Juselius, 2001) tests both demonstrate that the null hypothesis of the
existence a cointegrating relation between the observed and predicted time series cannot be rejected
for a majority of companies. Therefore, the model is justified econometrically, and thus, all
statistical inferences are valid. In Section 4, a crucial historical problem is addressed: Could one
predict in May 2008 the evolution of financial stock prices? For some companies, the models
estimated in the beginning of 2008 hold over the next year. Hence, the empirical modeling would
have allowed accurate prediction of the evolution of stock prices, including those related to
companies who filed for bankruptcy in several months. Finally, Section 5 investigates several cases
of bankruptcy and bailout in the United States. It is found that many stock price trajectories would
have been predicted to dive below the zero line.
The results of the presented research open a new field for the future investigations of the stock
market. We do not consider the concept and empirical models as accurate enough or final. There
should be numerous opportunities to amend and elaborate the model. Apparently, one can include
new and improve available estimates of consumer price indices.

1. Model and data
Kitov (2009b) introduced a simple deterministic pricing model. Originally, it was based on an
assumption that there exists a linear link between a share price (here only the stock market in the
United States is considered) and the differences between various expenditure subcategories of the
headline CPI. The intuition behind the model was simple - a higher relative rate of price growth
(fall) in a given subcategory of goods and services is likely to result in a faster increase (decrease)
in stock prices of related companies. In the first approximation, the deviation between price-
defining indices is proportional to the ratio of their pricing powers. The presence of sustainable
(linear or nonlinear) trends in the differences, as described in (Kitov and Kitov, 2008; 2009ab),
allows predicting the evolution of the differences, and thus, the deviation between prices of
corresponding goods and services. The trends are the basis of a long-term prediction of share prices.
In the short-run, deterministic forecasting is possible only in the case when a given price lags
behind defining CPI components.

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In its general form, the pricing model is as follows (Kitov, 2010):

sp(tj) = Σbi·CPIi(tj-τi) + c·(tj-2000 ) + d + ej (1)

where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-τi) is the i-th
component of the CPI with the time lag τi, i=1,..,I; bi, c and d are empirical coefficients of the
linear and constant term; ej is the residual error, which statistical properties have to be scrutinized.
By definition, the bets-fit model minimizes the RMS residual error. The time lags are expected
because of the delay between the change in one price (stock or goods and services) and the reaction
of related prices. It is a fundamental feature of the model that the lags in (1) may be both negative
and positive. In this study, we limit the largest lag to fourteen months. Apparently, this is an
artificial limitation and might be changed in a more elaborated model. In any case, a fourteen-
month lag seems to be long enough for a price signal to pass through.
System (1) contains J equations for I+2 coefficients. Since the sustainable trends last more
than five years, the share price time series have more than 60 points. For the current recent trend,
the involved series are between 70 and 90 readings. Due to the negative effects of a larger set of
defining CPI components discussed by Kitov (2010), their number for all models is (I=) 2. To
resolve the system, we use standard methods of matrix inversion. As a rule, solutions of (1) are
stable with all coefficients far from zero.
At the initial stage of our investigation, we do not constraint the set of CPI components in
number or/and content. Kitov (2010) used only 34 components selected from the full set provided
by the US Bureau of Labor Statistics (2010). To some extent, the original choice was random with
many components to be similar. For example, we included the index of food and beverages and the
index for food without beverages. When the model resolution was low, defining CPI components
were swapping between neighbors.
For the sake of completeness we always retain all principal subcategories of goods and
services. Among them are the headline CPI (C), the core CPI, i.e. the headline CPI less food and
energy (CC), the index of food and beverages (F), housing (H), apparel (A), transportation (T),
medical care (M), recreation (R), education and communication (EC), and other goods and services
(O). The involved CPI components are listed in Appendix 1. They are not seasonally adjusted
indices and were retrieved from the database provided by the Bureau of Labor Statistics (2010).
Many indices were started as late as 1998. It was natural to limit our modeling to the period
between 2000 and 2010, i.e. to the current long-term trend.
Since the number and diversity of CPI subcategories is a crucial parameter, we have extended
the set defining components to 70 from the previous set of 34 components. As demonstrated below,

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the extended set has provided a significant improvement in the model resolution and accuracy.
Therefore, we envisage the increase in the number and diversity of defining subcategories as a
powerful tool for obtaining consistent models. In an ideal situation, any stock should find its
genuine pair of CPI components. However, the usage of similar components may have a negative
effect on the model – one may fail to distinguish between very close models.
Every sector in the S&P 500 list might give good examples of companies with defining CPI
components lagging behind relevant stock prices. As of January 2010, there were 66 financial
companies to model, with the freshest readings being the close (adjusted for dividends and splits)
prices taken on December 31, 2009. (All relevant share prices were retrieved from
http://www.finance.yahoo.com.) Some of the modeled companies do present deterministic and
robust share price models. As before, those S&P 500 companies which started after 2004 are not
included. In addition, we have modeled Fannie Mae and Freddie Mac, which are not in the S&P
500 list, and Lehman Brothers and CIT Group (CIT) which are out of the S&P 500 list. Due to the
fact that the latter three companies are both bankrupts, they have been modeled over the period of
their existence. Apparently, there are many more bankrupts to be modeled in the future.
There are two sources of uncertainty associated with the difference between observed and
predicted prices, as discussed by Kitov (2010). First, we have taken the monthly close prices
(adjusted for splits and dividends) from a large number of recorded prices: monthly and daily open,
close, high, and low prices, their combinations as well as averaged prices. Without loss of
generality, one can randomly select for modeling purposes any of these prices for a given month.
By chance, we have selected the closing price of the last working day for a given month. The larger
is the fluctuation of a given stock price within and over the months the higher is the uncertainty
associated with the monthly closing price as a representative of the stock price.
Second source of uncertainty is related to all kinds of measurement errors and intrinsic
stochastic properties of the CPI. One should also bear in mind all uncertainties associated with the
CPI definition based on a fixed basket of goods and services, which prices are tracked in few
selected places. Such measurement errors are directly mapped into the model residual errors. Both
uncertainties, as related to stocks and CPI, also fluctuate from month to month.

2. Modeling financial companies
The results of modeling are presented in Table 1 and Appendix 2: two defining components with
coefficients and lags, linear trend and free terms, and the standard error, σ, expressed in dollars.
Negative lags, which correspond to leading share prices, are shown in bold. Figure 1 and Appendix
3 depict the observed and predicted curves. Five companies will be studied in more detail in Section