Wal-Mart Stores is stable in March

We revisited our price model for Wal-Mart Stores (NYSE: WMT) two months ago. Originally, the model was estimated in June 2010 and the same model still worked well in December 2011. This suggests the overall robustness and reliability of the share price model for WTM.

Our concept of share pricing is based on the decomposition into a weighted sum of two selected consumer price indices. The intuition is simple; a faster growth in the CPI related to a given company relative to some independent but dynamic reference should be manifested in a higher pricing power for the studied company. This reference is needed since all consumer prices change over time and those associated with the company should be measured in relative terms. Hence, our stock price model seeks for a defining CPI and the best reference which we both select from a set of 92 different (not seasonally adjusted) CPIs. This set includes the headline and core CPI, all major categories from food to other goods and services, and many minor subcategories with a long enough measurement history (i.e. continuous estimates should be available since 2000).

Here we re-estimate the original model with data available in March 2012, i.e. the closing price for February and CPIs for January 2012. This model is defined by the (seasonally not adjusted) index of hospital and related services (HOSP) and the price index of miscellaneous personal services (MISS), as reported by the US BLS. The former CPI component leads the share price by 9 months and the latter one evolves in sync with the price. Figure 1 depicts the overall evolution of both involved indices through January 2012. A very specific feature of both indices is their linearity over time: they are close to straight lines with small fluctuations.

 

The newly estimated model allows validating the initial model and demonstrating its reliability. The previously obtained defining components are the same and provide the best fit model between June 2010, March and December 2011 with only one month change in the lag for the HOSP index. All coefficients are slightly different for the new model (see below). The slope of the time trend is negative. The best-fit 2-C model for WMT(t) is as follows:

WMT(t) = 0.50HOSP(t-10) + 1.42MISS(t) - 28.39(t-1990) – 158.12 (January 2011)

WMT(t) = 0.46HOSP(t-9) + 1.49MISS(t) - 28.03(t-1990) – 165.50 (March 2011)

WMT(t) = 0.46HOSP(t-9) + 1.30MISS(t) - 26.06(t-1990) – 141.92 (December 2011)

WMT(t) = 0.48HOSP(t-9) + 1.33MISS(t) - 26.75(t-1990) – 145.13 (February 2012)

where t is calendar time. The predicted curve in Figure 2 evolves in sync with the observed price. The residual error is $2.15 for the period between June 2003 and February 2012.

With both indices growing along their respective trends, we foresaw in December 2011 a slight increase to the level of $60 to $65 per share in 2012Q1. It really happened and the price was at the level $62.06 in February. The current model supposes a slight negative correction which also follows from the positive residual error shown in Figure 3. A no change scenario is also possible with the predicted price rising to the level of the measured one in March 2012. In a week, we will re-estimate the model using both CPIs for February.

Figure 1. Evolution of the price of HOSP and MISS.

Figure 2. Observed and predicted WMT share prices.

Figure 3. Residual error of the model; sterr=$2.15.

CarMax - no change in Q1/Q2

This is the first time we present a share price model for CarMax (NYSE: KMX). This company from services category of the S&P 500 index is a retailer of used vehicles in the United States.  Several days ago we published a price model for a similar company - AutoNations (NYSE: AN) which is valid since 2009.  Both models have been obtained by decomposition of the time series of monthly closing share prices (adjusted for splits and dividends) into a weighted sum of two consumer price indices. One might presume that a fast growth in the CPI inherently linked to the KMX share price (e.g. motor vehicle parts and equipment) relative to some independent by dynamic reference (e.g. food) should be manifested in a higher pricing power for the company. We seek for two best (say, in sense of the RMS residual error) defining CPIs. It allows testing of the underlying concept (decomposition into CPIs) and to estimate time lags and coefficients for KMX. 

We have borrowed the time series of monthly closing prices of KMX from Yahoo.com and the relevant (seasonally not adjusted) CPI estimates through January 2012 are published by the BLS.  The evolution of KMX share price is defined by the consumer price of food (F) and the index of motor vehicle parts and equipment (MVP). The defining time lags are as follows: the food index leads the price by 3 months and the MVP index leads by 4 months. The relevant best-fit model for KMX(t) is as follows:  

KMX(t) =  -1.37F(t-3) + 1.20MVP(t-4)  + 10.33(t-1990) + 170.88,  February 2012 

where KMX(t) is the KMX share price in U.S. dollars,  t is calendar time. This model is valid since August 2011 with the same lags and coefficients. Figure 1 displays the evolution of both defining indices since 2002.   

Figure 2 depicts the high and low monthly prices for the share together with the predicted and measured monthly closing prices. The predicted prices are well within the bounds of the share price uncertainty.  The model residual error is shown in Figure 3 with the standard deviation between July 2003 and February 2012 of $2.30.

From Figure 2, we expect no large changes in the first half of 2012.  (Same conclusion was made for AN.)

Figure 1. The evolution of the index of rent of food (F) and the index of motor vehicle parts (MVP).  

Figure 2. Observed and predicted KMX share prices. 

Figure 3. The model residual error: sterr=$2.30.

Bank of America: may fall below $5

Four months ago we presented a share price model for Bank of America (NYSE: BAC) and predicted its share falling below zero (an important step to bankruptcy). Seeking Alpha published two years ago our academic model of bankruptcy which foresaw Lehman Brothers, AIG, Freddie and Fanny, Citigroup, CIT, and some others. Some of these financial institutions were bailed out, but this also suggests their negative share prices. Therefore, it’s very important to track the BofA price in order to see early signals of sinking below zero.  

Our model for BofA has a history of three years and is based on the decomposition of the price into a weighted sum of two consumer price components, a linear time trend and a constant. The background idea is a simplistic one: there is a potential trade-off between a given share price and goods and services the company produces and/or provides. For example, the energy consumer price does influence the price of energy companies. It should be taken into account that the defining consumer price (or relevant CPI) has to be related to some independent and dynamic reference, which can also be a consumer price index. A higher relative growth of the defining CPI should be manifested in a higher pricing power for the company.

For the BAC model, all coefficients, time lags and CPIs were estimated by the least squares as applied to the model error.   Here we revisit the model using new data through February 2011. The newly estimated model differs from the preliminary one due to strict constraints on time lags and new data. In October 2011, the best-fit two component model for BAC(t) was as follows:

BAC(t)= -2.35OHF(t) + 1.19H(t) + 1.97(t-1990) + 165.79

where OHF is the consumer price index of other food at home and H(t) is the index of housing, both having notime lag behind the price (see Figure 1 for the evolution of the CPIs between 2002 and 2012). 

In December 2011, the best model changed a bit and was the following: 

BAC(t)= -2.30OHF(t) + 1.12H(t) + 2.17(t-1990) + 167.83    

In February 2012, we also observed a slight change in coeffcients and no change in time lags: 

BAC(t)= -2.27OHF(t) + 1.06H(t) + 2.38(t-1990) + 170.38, sterr=$3.08.     

Both CPIs are available only for January 2012, and thus we actually have a one month lead by both CPIs. We are going to update the model in a week when the BLS estimates for both CPIs are available.

Figure 2 depicts the observed and predicted prices and indicates that the market overvalues the share, i.e. the share is above the predicted level. The deviation from the predicted level is also clear from Figure 3 where the model residual is displayed. Despite a small increase in the predicted price in January/February 20122, we still think that Bank of America is under the pressure of a negative stock price correction. Such a correction could bring its BAC share below $5 per share in the near future. Lehman Brothers, Citigroup, and other banks all had the predicted price below zero several months before bankruptcy.  

Figure 1. The evolution of the defining consumer price indices.

Figure 2. Observed and predicted BAC share prices. 

Figure 3. The model residual.

Hewlett Packard : a slight negative correction

A month ago we presented a quarterly report of the performance of our share price model for Hewlett Packard (NYSE:HPQ). This company provides a good example of a successful share price prediction at a several month horizon.  We have already published our predictions at a four month horizon four times (July 2010, January 2011, March 2011, July 2011, and September 2011). All predictions were based on our concept of share pricing as decomposition into a weighted sum of two CPI components.  The intuition behind our concept is simple; a faster growth in the CPI 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 what 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. Our model for HPQ is stable and shows an excellent predictive power at a four month horizon for more than 30 months without gaps.

Originally, the long term model for HPQ share price was defined by the index of food without beverages (FB) and that of rent of primary residency (RPR). The former CPI component led the share price by 4 months and the latter one led by 5 months. Figure 1 depicts the overall evolution of both involved indices through February 2012. Below we present five best-fit models for HPQ(t) obtained at different times:

HPQ(t) = -3.20FB(t-4) + 2.91RPR(t-5) + 3.64(t-1990) - 50.82, July 2010

HPQ(t) = -3.34FB(t-4) + 3.41RPR(t-5) + 0.51(t-1990) - 85.44, June 2011

HPQ(t) = -3.46FB(t-4) + 3.68RPR(t-5) – 0.72(t-1990) - 99.88, September 2011

HPQ(t) = -3.40FB(t-5) + 3.60RPR(t-6) – 0.57(t-1990) – 97.72, December 2011

HPQ(t) = -3.27FB(t-4) + 3.46RPR(t-5) – 0.39(t-1990) – 95.71, February 2011 

where HPQ(t) is the price in US dollars, t is calendar time. All coefficients have been slightly drifting but very close. This process expresses the trade-off between the linear trend in the difference between  the defining CPIs and the time trend term in the above equtions. 

A month ago, we calculated the evolution of the monthly closing price (adjusted for dividends and splits) for February-June 2012. We predicted the price to fall to $20 in the first quarter of 2012 and then rise to $25 in Q2. Currently, the price is on decline and has fallen to $24. We expect a further fall in the near future (March/April) and then an increase to $25. 

Figure 3 depicts the model error. Between July 2003 and February 2012, a standard error is of $2.5 with the current share slightly overvalued.

Figure 1. Evolution of the price of FB and RPR. 

Figure 2. Observed and predicted HPQ share price.

Figure 3. The model residual error; sterr=$2.50.

Alcoa - a slight negative correction is expected

One, three, and nine months ago we reported on a share price model for Alcoa (NYSE: AA), which is company from Materials subcategory of the S&P 500 list specialized in aluminum. Our concept is intuitive and straightforward. A company is what it produces. There are price setters and price takers. Some goods and services drive economic development and some follow up. Let’s imagine a company producing some goods (services) very attractive to people right now. The company may raise the overall price for its goods (services). Accordingly, stocks go up, likely with some time lag. One of the indicators of the overall price is the consumer price index (CPI) for these specific goods and services or some very intimately related G&S. Then it is not excluded that the company’s share depends on this CPI in a statistically reliable way and one can obtain a good link between the price and this CPI. Since the headline CPI evolves under the pressure of a big set of goods and services, one need to find some dynamics reference (another CPI) which would be most independent on the CPI related to the company. Hence, we have to find two CPIs which describe the evolution of the price the best (in the LSQ sense).  When both CPIs lead the price, a deterministic model can be obtained and we are looking for such companies in the S&P 500 list. Alcoa is one of the companies with a deterministic model – both defining CPIs lead by three months at least.  

Here we test and update the AA model using new data, including the monthly closing price in February 2012 and the estimated CPI components for January 2012. The principal result is that the model has the same defining CPI components and time lags with slightly changing coefficients. Therefore, the model is a reliable tool to predict the evolution of Alcoa shares.  

According to our general approach to share price modeling we decompose the observed time history of the monthly closing AA stock price (adjusted for splits and dividends) into a weighted sum of two CPI components, time trend and free term.  Two defining CPI components are selected to minimize the model (RMS) error and may lead or lag behind the share.  

The original and current AA model is defined by the (not seasonally adjusted) index of food away from home (SEFV) and the price index of rent of primary residence (RPR), as reported by the US BLS. The former CPI component leads the share price by 3 months and the latter is 5 months ahead of the share price. Figure 1 depicts the overall evolution of both involved indices through August 2011. It seems these indices have been evolving in sync since 2002 with the only step-like change in the SEFV index in 2008.  We present three empirical models as estimated in April, October, December 2011, and February 2012: 

AA(t) =  -6.71SEFV(t-2) + 3.34RPR(t-4)  + 19.23(t-1990) + 298.87, Aril 2011

AA(t) =  -6.61SEFV(t-2) + 3.22RPR(t-4)  + 19.51(t-1990) + 300.89, October 2011

AA(t) =  -6.47SEFV(t-3) + 3.08RPR(t-5)  + 19.58(t-1990) + 302.45, December 2011

AA(t) =  -6.38SEFV(t-3) + 3.01RPR(t-5)  + 19.53(t-1990) + 301.93, February 2012 

where AA(t)  is a share price in US dollars, t is calendar time. Figure 2 illustrates the observed and predicted models for December 2011. The residual error in Figure 3 is $3.04 ($3.05 in December, $3.04 in October, and $3.12 in April) for the period between July 2003 and February 2011. Figure 2 also shows monthly high and low prices as the uncertainty in the monthly closing price as the best share price estimate. Since the closing price has to characterise the whole month by one value the high and low prices might serve as strict statistical bounds. 

One can expect the share price to fall a bit through March/April 2012 to the level of  $7 to $8.     

Figure 1. Evolution of the price of SEVF and RPR. 

Figure 2. Observed and predicted AA share prices

Figure 3. The model error;  sterr=$3.04.

Exxon Mobil is on a rise

Exxon Mobil (NYSE: XOM) is one of the biggest companies. Its share price influences not only other oil companies but also the S&P 500 index itself. Here we introduce a simple share pricing model for Exxon Mobil, which foresees its price at a three months horizon. In a way this allows to predict the market evolution and to beat the Efficient Market Hypothesis. The model is based on our concept of stock dependence on consumer price index.  The intuition is simple and clear, the evolution of a share price is inherently related some goods and services and thus their relative prices.  For example, one cannot deny that crude oil price has to affect share prices of oil companies. We have proved that such a link exists for ConocoPhillips and formulated an empirical model. For XOM, we use a set of 92 individual consumer price indices to select the best two CPIs to describe the evolution of the share price. 

We have found several years ago that Exxon Mobil provides an example of a company with its share price leading defining CPI components.  Our model is seeking two CPI components from a large number of pre-selected ones, which minimize the difference between observed (monthly closing price adjusted for dividends and splits) and predicted prices for the period between July 2003 and February 2012. Our two-component model also includes a free term (constant) and a linear time term, which compensates well know linear (time) trends between various CPI components. The best-fit model is as follows: 

XOM(t)= -1.70OFH(t-3) – 2.98RRM(t-10) + 22.73(t-1990)  + 581.17, February 2012

where OFH in the index other food at home lagging  the stock price by 3 months, RRM is the index of recreation reading materials leading by 10 months, (t-2000) is the elapsed time. Figure 1 depicts the evolution of both CPIs.  

In the beginning of February 2012, this model predicted the monthly closing price of $89.0, i.e. a $10 increase from January’s closing price.  Figure 2 depicts the observed and predicted prices, the latter shifted three months ahead for synchronization, i.e. the predicted curve leads the observed price by 3 months. The actual monthly closing price in February was $86.5, i.e. $3.5 above that in January.  Therefore, the model has correctly predicted the move in the price. 

The model residual error is shown in Figure 3. It has standard deviation of $3.72 for the period between July 2003 and February 2012.  

The estimated model shows that Exxon Mobil’s share will be growing in 2012Q1 to the level of $90. This is in line with the increasing crude price.

Figure 1. The evolution of defining CPIs.

Figure 2. Observed and predicted XOM share prices. 

Figure 3. The model error; sderr=$3.72. 

Harris Corporation: a negative correction is expected

After Pitney Bowes (PBI), we would like to present a deterministic price model for a technology company from the S&P 500 list.  This time we present a company from technology category – Harris Corporation (NYSE: HRS) which “operates as a communications and information technology company that serves government and commercial markets worldwide”. The model is a deterministic one since we decompose HRS share price into a weighted sum of two consumer price indices, both leading the price by several months. One cannot deny the fact that there exists an inherent trade-off between shares of a given company and goods and services it produces/provides. Therefore, one might assume the growth in a CPI related to HRS (e.g. information technology) relative to some independent but dynamic reference consumer price (e.g. food) should be manifested in a higher pricing power for the studied company. It so happened that both involved CPIs lead the share price. In other words, a HRS share price accommodates all changes in the overall price of information technology with a time lag of several months. Those who know that have a good statistical reason to beat the market.

All in all, our stock price model tries to find one defining CPI and the best reference CPI. Both CPIs are taken from a set of 92 different (not seasonally adjusted) CPIs and the best model has the smallest RMS error between July 2003 and February 2012. This set is not a complete one but includes the headline and core CPI, all major categories from food to other goods and services, and many minor subcategories with long enough history (i.e. continuous estimates should be available since 2000). One can extend this set and might obtain a more robust model. 

We have borrowed the time series of monthly closing prices of HRS from Yahoo.com (February 2012 is included) and the CPI estimates through January 2012 are published by the BLS.  As mentioned above, the evolution of HRS share price is defined by the consumer price of the index of information technology (IT) and the index of food (F). The defining time lags are six and four months, respectively.   The best-fit model for is as follows:  

HRS(t) =  -2.76F(t-4) – 10.39IT(t-6)  + 8.53(t-1990) + 553.00,  February 2012 

where HRS(t) is the HRS share price in U.S. dollars,  t is calendar time. Figure 1 displays the evolution of both defining indices since 2002.  In the empirical model, both indices have negative slopes and the IT index defines the growth of the price.  Higher food prices suppress the level of HRS share price.  

Figure 2 depicts the high and low monthly prices for a HRS share together with the predicted and measured monthly closing prices (adjusted for dividends and splits). The predicted prices are well within the limits of the share price uncertainty.  The model residual error is shown in Figure 3 with the standard deviation between July 2003 and January 2012 of $3.42.  

Due to the sensitive balance between the growth in food price and the fall in the IT index the share price has many peaks and troughs. Currently, this proportion between the growth rates of the F and IT indices has moved the predicted HRS price far from the observed one. Hence, we can not exclude that the actual price will fall by a few dollars by May 2012. A similar correction was observed in 2008.

Figure 1. The evolution of defining indices.

Figure 2. Observed and predicted monthly closing prices for a HRS share. 

Figure 3. The model residual error: sterr=$3.42.