Wal-Mart share in 2012

We estimated our price model for Wal-Mart Stores (NYSE: WMT) nine months ago. The model is based on the decomposition of a share price into a sum of two selected consumer price indices. This is a new model 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 10 months and the latter one evolves in sync with the price. Figure 1 depicts the overall evolution of both involved indices through December 2011. A very specific feature of both indices is their linearity over time: they are close to straight lines. 

In this post, we re-estimate the WMT share price using new data through December 2011. This 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 and March 2010 with only one month change in the lag for the HOST index.  All coefficients in (1) are only 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)  (1)

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)

where t is calendar time. The predicted curve in Figure 2 evolves in sync with the observed price. The residual error is $2.13 for the period between June 2003 and December 2011. With both indices growing along their respective trends one can expect a slight increase to the level of $60 to $65 per share in 2012Q1. Figure 3 presents the residual model error.  

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.

Quarterly report: Loews share price model

This is a quarterly report on the performance of our share price model for Loews Corporation (NYSE: L). The model is based on the decomposition into a weighted sum of two consumer price indices (selected from a larger number of CPIs), linear trend and constant, all coefficients and time lags to be estimated by a LSQ procedure. Here we test the previous model and make a regular update using new data. All in all, the original model is valid since October 2008 and does not show any sign of future changes. This is a reliable model valid during the past 50 months!  

A preliminary model for Loews Corp. was obtained in September 2009 and covered the period from October 2008. This old model included the index of food without beverages (FB) and the index of transportation service (TS). The most recent model also used the monthly closing prices as of April 2011 and the CPI estimates published on April 14, 2011. The defining indices were almost the same: the index of food (F) and the TS index. Figure 1 depicts the evolution of the indices which provide the best fit model, i.e. the lowermost RMS residual error, between July 2003 and December 2011.  The F index leads by 5 months and the TS index by 4 months.  When new data through December 2011 are used, the model does not show any tangible change - only coefficients have been slightly drifting:  

L(t) = -2.03F(t-5) – 2.12TS(t-4) +28.23(t-1990) + 448.98, March 2011

L(t) = -2.01F(t-5) – 2.09TS(t-4) +27.96(t-1990) +440.65, September 2011

L(t) = -2.03F(t-5) – 2.02TS(t-4) +27.65(t-1990) +431.99, December 2011      

where L(t) is the share price in US dollars, t is calendar time. The new model is depicted in Figure 2 together with high and low monthly prices as a proxy to the uncertainty bound of the share price. The predicted curve leads the observed one by 4 months. The residual error is of $2.42 for the period between July 2003 and December 2011.  In the first quarter of 2012, the model foresees essentially no change. It is worth noting that the model obtained in March 2011, accurately predicted the small fall observed in the second and third quarters of 2011.   

Figure 1. Evolution of the price indices F and TS.

Figure 2. Observed and predicted share prices.

A model for Harley-Davidson share price: three years of success

This is a quarterly report on the performance of our pricing model. Harley-Davidson (HOG) is one of the best illustrations of our concept (see a brief description of the concept in Appendix) linking stock prices to CPI components. For HOG, the model is stable for many years. The first model was obtained in September 2009 and covered the period from October 2008. Here we revisit the HOG model using the monthly closing price for December 2001 and the CPI estimates published in January 2012.   

For HOG, the defining indices are as follows: the index of rent of primary residence (RPR) and the index of owners' equivalent rent of residence (ORPR). Both CPI components are leading the share price. Figure 1 depicts the evolution of the indices which provide the best fit model, i.e. the lowermost RMS residual error, between July 2008 and December 2011.  The models are as follows: 

HOG(t) = -13.82RPR(t-3) +12.77ORPR(t-4) +17.82(t-1990) – 163.94, before September 2009

HOG(t) = -11.30RPR(t-3) + 9.83ORPR(t-3) +17.53(t-1990) – 36.34, after September 2009

HOG(t) = -11.27RPR(t-3) + 9.55ORPR(t-3) +19.35(t-1990) – 8.57, September 2011

HOG(t) = -11.09RPR(t-4) + 9.40ORPR(t-4) +18.93(t-1990) – 7.53, December 2011

where HOG(t) is the share price in US dollars, t is calendar time. The model is characterised by standard deviation of $4.28 for the period between July 2003 and December 2011.   

Two recent models are depicted in Figure 2. The predicted curve in December 2011 leads the observed ones by 4 months. We do foresee a further fall in the stock price to $26 per share in the first quarter of 2012.  Figure 3 displays the residual error.

Figure 1. Evolution of the price indices ORPR and RPR.

Figure 2. Observed and predicted HOG share prices. Model for March, September, and December  2011.

Figure 3. The model residual error.  

Appendix

In its general form, our pricing model is as follows: 

sp(t_j) = Σb_i∙CPI_i(t_j-t_i) + c∙(t_j-1990 ) + 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-t_i) is the i-th component of the CPI with the time lag t_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.

System (1) contains J equations for I+2 coefficients. For POM we use a time series from July 2003 to March 2011, i.e. 94 monthly 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. In the POM 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.

Due to obvious reasons, longer time series guarantee a better resolution between defining CPIs. In general, 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. Second source of uncertainty is related to all kinds of measurement errors and intrinsic stochastic properties of the CPI and its components. 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. 

Will BofA shares fall to $1?

Two months ago we presented a share price model for Bank of America (BAC) and predicted the share falling below zero (a step to bankruptcy). The model 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. 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 December 2011. The newly estimated model differs from the preliminary one due to strict constraints on time lags and new data. The best-fit two component model for BAC(t) is as follows:  

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

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). Both CPIs are available only for November 2011, and thus we actually have a one month lead of the CPIs, if they would have been already available for December 2011 (will be available on January 19, 2012). One has to reestimate the model when the CPIs for September are published by the BLS . The standard deviation of the modeling error is $2.90 for the period between July 2003 and December 2011. Figure 2 depicts the observed and predicted prices and indicates that the market overestimates the share above the predicted level. The overestimation is also clear from Figure 3 where the model residual is displayed.

Despite a small increase in the predicted price in December, we still suggest that Bank of America might have  problems, which could bring its share price down to $1 in the near future. Similar problems had Lehman Brothers and other banks 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.

... and a bit more models

Share price models for selected S&P 500 companies

Avon Products share price model

Here we present a brand new share price model for Avon Products (NYSE: AVP). In April 2011, this model also showed a higher level of reliability and described the price through March 2011. We intentionally skipped it to publish in this blog because the predicted price showed just a small fall. The model was based on our concept linking share pieces and consumer price indices. The share price model for Avon Products was defined by the index of other household equipment and furnishing (OHEF) and that of public transportations (TPU). Figure 1 illustrates the evolution of these indices. In the April model, the former CPI component led the share price by 8 months and the latter one led by 5 months.

Here we revisit the model using the monthly closing prices (adjusted for splits and dividends) and CPIs for the period through September 2011. (The CPIs are available only for August 2011.) The principal result is that the underlying model is practically the same as six months ago with the same time lags but slightly different coefficients. This model predicted the fall in the price observed since June 2011 five months in advance.  The March and September models for AVP(t) are as follows:

AVP(t) =  -2.43OHEF(t-8) – 0.33TPU(t-5)  - 5.22(t-1990) + 392.49, March 2011

AVP(t) =  -2.25OHEF(t-8) – 0.33TPU(t-5)  - 4.58(t-1990) + 366.41, September 2011 

where AVP(t) is a share price in US dolalrs, t is calendar time. Relevant coefficients are both negative. The slope of time trend is also negative.  There is some fluctuation in all coefficients caused by the uncertainty in measurements of both the stock prices and CPIs.  Nevertheless, both models provide an accurate prediction at a five-month horizon as Figure 2 depicts.

Currently, the model shows  that the price has reached a stable level of $20 per share and will not be changing in the fourth quarter of 2011.  The predicted curve in Figure 2 (both versions are depicted) leads the observed price by 5 months with the residual error of $2.90 ($3.08 in April) for the period between July 2003 and September 2011. The model residual for the same period is shown in Figure 3.    

Figure 1. Evolution of the price of OHEF and TPU. 

Figure 2. Observed and predicted AVP share prices. Upper panel – March 2011; lower panel – September 2011.

Figure 3. Residual error of the model.

Procter and Gamble - stable share price in Q4 2011

In this post, we revisit a share price model for Procter and Gamble as based on the decomposition into a weighted sum of two consumer price indices (to be determined), linear time trend and constant. It is shown that the model is valid since September 2009 at least and does not show any sign of possible failure. It predicts the share price at a four month horizon.

A share price model for Procter and Gamble (NYSE: PG) was originally published in this blog in July 2010. According to our concept, it was defined by the index of food away from home (SEFV - CUUS0000SEFV) and that of rent of primary residency (RPR); the evolution of these indices is presented in Figure 1. The former CPI component led the share price by 3 months and the latter one led by 8 months. The upper panel of Figure 2 depicts the original model and the monthly closing prices available in July 2010. This model was stable for the previous 11 months, i.e. for the period from September 2009.

In April 2010, we updated the original model using some new data (closing price for March 2011) and found that the same model was also applicable with a small change in the time lead for the SEFV – it was 4 months instead of 3 months in the original model. New coefficients were also slightly different, but very close to the original ones.

The most recent update uses the monthly closing price for September 2011 and CPIs for August 2011. It validates the model obtained for the previous period but is characterized by the same time lags and a small shift in the coefficients estimated by the LSQ technique. Three best-fit models for PG(t) are as follows:

PG(t) = -5.88SEFV(t-3) + 3.43RPR(t-8) + 17.60(t-1990) + 174.08, July 2010

PG(t) = -5.40SEFV(t-4) + 2.93RPR(t-8) + 18.16(t-1990) + 187.47, March 2011

PG(t) = -4.94SEFV(t-4) + 2.47RPR(t-8) + 18.15(t-1990) + 184.89, September 2011

where PG(t) is the monthly closing price (dividend and split adjusted) in US dollars, t is calendar time.

In the lower panel of Figure 2, the predicted curve leads the observed price by 4 months with the residual error of $2.12 for the period between July 2003 and September 2011 (see Figure 3 for the model residuals). In other words, the price of a PG share is completely defined by the behaviour of these two CPI components.

The model does predict the share price in the past and foresees a period of no growth in the fourth quarter of 2011. In January 2012, the price may fall, but we should revise the model with new data by that time.

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

Figure 2. Observed and predicted PG share prices. In the upper panel, the original prediction published in July 2010 with a three month lead shown by red line. The middle panel – the model published in April 2011. The lower panel – the most recent model with the monthly closing price for September 2011.

Figure 3. The model residual error.

Loews Corporation share price in 2011 Q4

Half a year ago, we presented a share price model for Loews Corporation (NYSE: L) based on the decomposition into a weighted sum of two consumer price indices (selected from a larger number of CPIs), linear trend and constant, all coefficients and time lags to be estimated by a LSQ procedure. Here we test the previous model and make a regular update using new data. All in all, the original model is valid since October 2008 and does not show any sign of future changes. This is a reliable model valid during the past 47 months! 

A preliminary model for Loews Corp. was obtained in September 2009 and covered the period from October 2008. This old model included the index of food without beverages (FB) and the index of transportation service (TS). The most recent model also used the monthly closing prices as of April 2011 and the CPI estimates published on April 14, 2011. The defining indices were almost the same: the index of food (F) and the TS index. Figure 1 depicts the evolution of the indices which provide the best fit model, i.e. the lowermost RMS residual error, between October 2009 and September 2011.  The F index leads by 5 months and the TS index by 4 months.  When new data through September 2011 are used, the model does not show any tangible change - only coefficients have been slightly drifting: 

L(t) = -2.03F(t-5) – 2.12TS(t-4) +28.23(t-1990) + 448.98, March 2011

L(t) = -2.01F(t-5) – 2.09TS(t-4) +27.96(t-1990) +440.65, September 2011      

where L(t) is the share price in US dollars, t is calendar time. The new model is depicted in Figure 2 together with high and low monthly prices as a proxy to the uncertainty bound of the share price. The predicted curve leads the observed one by 4 months. The residual error is of $2.38 for the period between July 2003 and September 2011.  In the fourth quarter of 2011, the model foresees a very slight increase. The model obtained in March 2011, accurately predicted the fall observed in the second and third quarters.   

Figure 1. Evolution of the price indices F and TS.

Figure 2. Observed and predicted share prices.

On the evolution of Forest Laboratories stock prices

During the past two days, we re-estimated share price models for a few companies from the S&P 500 list and found them reliable and accurate for a year and even longer. Forest Laboratories (NYSE: FRX) is one of the first companies with a stable and deterministic share price modelestimated in September 2009 (this early model tracks back into November 2008).  This is a company from Healthcare subcategory specialized in drugs manufacturing. We revisited this model in September 2010, December 2010, and March 2011 and always found the same defining variables with almost the same time lags.  

Our approach to deterministic share pricing is based on the decomposition of a share price into a weighted sum of two selected consumer price indices. For FRX, all models between 2009 and 2011 are defined by the (not seasonally adjusted) index of dairy and related products (DAIRY) and the price index of other household equipment and furnishing (OHEF), as reported by the US BLS. In the September 2011 model, the former CPI component leads the share price by 4 month and the latter is 5 months ahead of the share price. Figure 1 depicts the overall evolution of both involved indices through August 2011.  

Thus, three models mentioned above are as follows: 

FRX(t) =  -0.53DAIRY(t-3) – 4.18OHEF(t-5)  - 13.41(t-1990) + 706.57 (September 2009)   

FRX(t) =  -0.61DAIRY(t-3) – 3.89OHEF(t-4)  - 11.65(t-1990) + 668.75 (March 2011) 

FRX(t) =  -0.66DAIRY(t-4) – 3.45OHEF(t-5)  - 9.78(t-1990) + 613.99 (September 2011)     

where t is calendar time. All coefficients have been slightly drifting due to then trade-off between the linear time trend and the trend in the difference between the CPIs and due to new data. Therefore, the model is effectively the same between November 2008 and September 2011. In other words, we obtained a deterministic (leading by three months) model which was valid during 36(!) months.   Figure 2 depicts the predicted and observed prices together with the relevant monthly highs and lows. The residual error is $4.75 (see Figure 3) for the period between June 2003 and September 2011 ($4.40min March 2011). From the most recent model in Figure 2, we expect a fall to the level of $20 per share in the fourth quarter of 2011.  

Figure 1. Evolution of the price of DAIRY and OHEF. 

Figure 2. Observed and predicted FRX share prices. 

Figure 3. The model residual.