We introduced a simple deterministic pricing model in 2009 [1]. 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 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 110 readings. Due to the negative effects of a larger set of defining CPI components 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.

For the sake of completeness we always retain all principal subcategories of goods and services. Among them are the headline CPI (NYSE:C), the core CPI, i.e. the headline CPI less food and energy (NYSE:CC), the index of food and beverages (NYSE:F), housing (NYSE:H), apparel (NYSE:A), transportation (NYSE:T), medical care (NYSE:M), recreation (NYSE:R), education and communication (NYSE:EC), and other goods and services (NYSE:O). In this model, we use 92 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics (2011). 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.

There are two sources of uncertainty associated with the difference between observed and predicted prices. 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.

Morgan Stanley (NYSE:MS) is an example of a changing pricing model. In [3] we reported that the defining CPIs in 2008 were the index of housing operations (HO) and the index of food away from home (SEFV). However, through the second half of 2010 the defining indices are different: the index of food less beverages (NASDAQ:FB) and the index of information technology, hardware and software (NYSE:IT), which is a part of the communication index. The former CPI component is contemporaneous with the share price and the latter one leads by 1 month. Figure 1 depicts the evolution of both indices. As discussed in our previous posts on food, it is likely that the index for food will be slowly growing during the next two years. This growth has a negative influence on the share price – the fall in the share price in 2008/2009 is clearly associated with the spike in the food price index. Same effect was well described by the SEFV index in the previous model. The IT index is characterized by a long term decline and will hardly be growing during the next several years. Both, linear trend and constant term, have positive influence on the price.

These defining components provide the best fit model, i.e. the lowermost RMS residual error, between July 2010 and December 2010. The best-fit 2-C model for MS(t) is as follows:

MS(t) = -3.32*FB(t) – 17.34*IT(t-1) +0.68(t-2000) +904.4

The predicted curve in Figure 2 is in sync with the observed one. The residual error of $3.98 for the period between July 2003 and December 2010. The model accurately predicts the share price in the past. From the overall behaviour of the defining CPIs one may expect that the MS price will be stalled or slightly growing in the first quarter of 2011.

Figure 2. Observed and predicted MS share prices.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $3.98.

**References**

1. Kitov, I. (2009). Predicting ConocoPhillips and Exxon Mobil stock price, Journal of Applied Research in Finance, v., issue 2(2), Winter 2009, pp.129-134.

2. Kitov, I. (2010). Deterministic mechanics of pricing. Saarbrucken, Germany: LAP LAMBERT Academic Publishing.

3. Kitov, I. (2010). Modelling share prices of banks and bankrupts, Theoretical and Practical Research in Economic Fields, ASERS, vol. I(1(1)_Summer) pp. 59-85