Modeling share prices: Advanced Micro Devices

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.

 

Appendix

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 [2]: 

sp(t_j) = Σb_i∙CPI_i(t_j-h_i) + c∙(t_j-2000 ) + d + e_j                                  (1)

 

where sp(t_j) is the share price at discrete (calendar) times t_j, j=1,…,J; CPI_i(t_j-h_i) is the i-th component of the CPI with the time lag h_i, i=1,..,I; b_i, c and d  are empirical coefficients of the linear and constant term; e_j 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.

 

Modeling stock prices: Analog Devices has potential for further growth

 In this blog, we present and track successful quantitative models from the S&P 500 list. They are numerous. We revisit (recalculate) all models using new data and report on successful models. In some cases, a model should hold for a year before we publish it. Our stock pricing concept is very simple as based on deterministic links between share prices and prices of goods and services included in the consumer price index, CPI. Literally, we decompose a share price (monthly closing price adjusted for splits and dividends) into a weighted sum of two individual CPI components, linear time trend component and constant free term. We allow positive and negative time lags between all variables in the relationship and seek to minimize the RMS model error by varying the involved coefficients. The set of CPI components consists of 92 independent price indices of different level: from major (overall and core CPI) to very small (e.g. photo and related materials). When the modeled share lags behind both defining CPI components we have a deterministic model predicting at a horizon of the smallest time lag. This concept gives excellent results in terms of the model error and very stable pricing models which are valid during several years. In 2008, the model successfully predicted bankruptcy of some major banks, including Lehman Brothers. Fannie May and Freddie Mac. We were able to forecast negative share prices several months before the crash [1].  One can also find a formal model description in our monograph.  

 

 In this post, we present a share pricing model for Analog Devices Inc. (NYCE: ADI). It belongs to Information Technology sector and is specialized in analog and digital signal processing integrated circuits.  Here we present a model obtained in March 2014 and covering the period since July 2003.  This model includes the index of motor vehicle maintenance and repair (MVR) and the index of communication (CO).  The latter index makes some sense as directly related to communications. The MVR index does not lead the ADI share price and the CO index leads by 10 months.  Figure 1 depicts the evolution of the indices which provide the best fit model, i.e. the lowermost RMS residual error, between October 2013 and March 2014.  The model is as follows: 

ADI(t) = -1.74MVR(t-0) + 1.76CO(t-10) +14.41(t-2000) + 160.74      

where ADI(t) is a share price in US dollars, t is calendar time.  

The predicted and observed monthly closing prices are depicted in Figure 2 together with the high and low monthly price representing the price uncertainty. The residual error is of $2.55 for the period between July 2003 and March 2014 (see Figure 3 for details).  The dependence on time (linear time term) has been strong enough ($14.4 per year) to overcome negative influence of both indices since 2009: increasing MVR with a negative coefficient and decreasing CO with a positive coefficient both lower ADI price. Figures 1 and 2 suggest that ADI share has a potential for further growth if CO and MVR will retain their long term trends.

 

Figure 1. Evolution of MVR and CO. 

Figure 2. Observed and predicted ADI share prices.  

Figure 3. Model error for ADI. 

Predicting share price: Franklin Resources to grow

Here, we revise our pricing model for Franklin Resources (NYSE: BEN) presented on March 18, 2012.  According to Yahoo.com “The firm provides its services to individuals, institutions, pension plans, trusts, and partnerships. It manages, through its subsidiary, separate client-focused equity, fixed income, and balanced portfolios.” BEN is a financial company and we analyzed it three years ago as a candidate for bankruptcy.  

We presume that any share price can be represented as a weighted sum of two consumer price indices (not seasonally adjusted in our model) which may be leading the share price by several months. Our model also includes a linear time trend and an intercept in order to remove mean and trend components from all involved time series.  The intuition behind our pricing model is obvious – we link a given share to those goods and services which are produced/provided by the company. In order to provide a dynamic reference we also introduce in the model some relative and independent level of prices (also expressed by CPIs). Hence, one needs two different CPIs to define the model. These CPIs we select from a big set of 92 CPIs by minimizing the residual model error. 

 

In March 2012, the tentative model was driven by the consumer price index of food at home, FH, leading the price by five months and the index of other goods and services, O, which led by nine months. In the 2012 revised model, the former index is replaced by the index of food without beverages, FB, which leads the share price by four months. Both indices are shown in Figure. The tentative and revised pricing models are as follows:

BEN(t) = -5.47FH(t-5) – 1.81O(t-9) – 59.55(t-2000) + 1327.36, February 2012  

BEN(t) = -7.33FB(t-4) – 1.52O(t-9) – 69.58(t-2000) + 1536.22, October 2012  

BEN(t) = -2.39FB(t-4) – 0.50O(t-9) + 22.75(t-2000) + 504.08, March 2014    

where t is calendar time.  The standard error between July 2003 and March 2014 is $2.50 ($7.55 in March 2012). Figure 3 shows the model residual.   

Figure 2 depicts the observed and predicted monthly closing prices since 2003 and also provides the high/low monthly prices, which may serve as the estimates of uncertainty in the monthly price. (One can model the monthly high or low price instead of the closing one.) At a four month horizon, the price is expected to grow to the final level of $60.

Figure 1. The evolution of defining CPIs.

Figure 2. Observed and predicted BEN share prices together with the high/low monthly prices.

Figure 3 . The standard model error.