Here, we revisit our stock price model for Advanced Micro Devices (AMD) first presented in 2012. The original model was obtained using our concept of share pricing. The intuition behind this concept is simple; a faster growth in the CPI directly related to the share price (e.g. energy consumer price for energy companies) relative to some independent and dynamic reference (e.g. some goods and services which price does not depend on energy) should be manifested in a higher pricing power for the company. Our model selects (using the LSQ method) a defining CPI and the best reference index from a set of 92 CPI with estimates started before 2000. This set is fixed - it is important for model stability. Both CPIs for a given model must define the studied price for at least 8 months in a row, i.e. the model has to be the same for a relatively long time: the longer – the better. Mathematical details are presented in Appendix.
We have borrowed the time series of monthly closing prices of AMD from Yahoo.com and the relevant (seasonally not adjusted) CPI estimates through February 2014 are published by the BLS. As in the original model, the evolution of AMD share price is defined by the consumer price index of rent of primary residence (RPR) and that hospital and related services (HOSP); both indices are shown in Figure 1. The defining time lags are as follows: the RPR index leads the share price by 1 month and the HOSP by 5 months. The relevant best-fit models (for 2011 and 2014) for AMD(t) are as follows:
AMD(t) = -2.21RPR(t-1) – 0.82HOSP(t-5) + 37.87(t-1990) + 267.01, December 2011
AMD(t) = -2.16RPR(t-1) – 0.77HOSP(t-5) + 36.09(t-2000) + 625.65, March 2014
where AMD(t) is the AMD share price in U.S. dollars, t is calendar time. Figure 2 depicts the high and low monthly prices for an AMD share together with the predicted and measured monthly closing prices (adjusted for dividends and splits). As a rule, the predicted prices are well within the bounds of the share price uncertainty. The model residual error is shown in Figure 3; the standard model error is $3.29 for the period between July 2003 and March 2014.
All in all, with the current trends in both defining CPIs retained over a longer period AMD has no chance to recover to the 2006 level. It is hard to imagine that the index of rent of primary residence will not be growing in the future. All CPIs related to medical care, including HOSP, have stable linear time trends since the very beginning.
Figure 1. Evolution of defining consumer price indices
Figure 2. Observed and predicted AMD share prices.
Figure 3. The model residual error.
We introduced a simple deterministic pricing model in 2009 . 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 :
sp(tj) = Σbi∙CPIi(tj-hi) + c∙(tj-2000 ) + d + ej (1)
where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-hi) is the i-th component of the CPI with the time lag hi, 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 eleven 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. We fix 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 (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). 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 (2014).
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.