Modeling share prices of financial companies: AXP

Original paper:

Ivan O. Kitov, 2010. "Modeling share prices of banks and bankrupts," Quantitative Finance Papers 1003.2692, arXiv.org

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 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.

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 dataKitov (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.

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

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; 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 92 from the previous set of 34 components. As demonstrated below, 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. Model for AXP
American Express Company (AXP) has a model predicting at a four month horizon. The defining CPIs are the index for food and beverages leading by 4 months and the index for medical care leading by 10 months. In the previous study (Kitov, 2010) the model was essentially the same. So, the extended CPI set does not make a better model. The model is a robust one and minimizes the standard error for the period between July and November 2009 as well.

The best-fit 2-C model for AXP(t) is as follows:

AXP(t)= -3.71F(t-4) – 2.10M(t-10) + 50.59(t-2000) + 1127.9

where F in the index of food and beverages leading the stock price by 4 months, M is the index of medical care leading by 10 months, (t-2000) is the elapsed time. The predicted curve should lead the observed price by 4 months. In other words, the price of a AXP share is completely defined by the behaviour of the two CPI components. Figure 1 depicts the observed and predicted prices, the latter shifted four months back for synchronization. The model residual error, i.e. standard deviation, is of $2.64 for the period between July 2003 and February 2010.

The model does predict the share price. Therefore, it should be revisited at a monthly basis. In March and April 2010, the price is expected at $47.5 and $50.5, respectively. On the 26th of March, the price was at $41.12.

Figure 1. Observed and predicted AXP share prices.



References
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