ConocoPhillips’ share price model revisited

Following my recent post on energy related companies and specifically on ConocoPhillips I have compiled a paper revisiting all COP models. Please follow the link to download the full text.

ConocoPhillips’ share price model revisited

Abstract

Three years ago we found a statistically reliable link between ConocoPhillips’ (NYSE: COP) stock price and the difference between the core and headline CPI in the United States. In this article, the original relationship is revisited with new data available since 2009. The agreement between the observed monthly closing price (adjusted for dividends and splits) and that predicted from the CPI difference is confirmed. The original quantitative link is validated. In order to improve the accuracy of the COP price prediction a series of advanced models is developed. The original set of two major CPIs is extended by smaller components of the headline CPIs (e.g. the CPIs of motor fuel and housing energy) and several PPIs (e.g. the PPIs of crude oil and coal) which may be inherently related to ConocoPhillips and other energy companies. These advanced models have demonstrated much lower modeling errors with better statistical properties. The earlier reported quasi-linear trend in the CPI difference is also revisited. This trend allows for an accurate prediction of the COP prices at a five to ten year horizon.

Key words: stock price, ConocoPhillips, predicti

Baker Hughes is likely undervalued

The evolution of Baker Hughes (NYSE: BHI) share price can be predicted quantitatively. Energy category of the S&P 500 list contains many companies linked to various types of energy production and services. BHI supplies oilfield services, products, and technology services and systems to the oil and natural gas industry worldwide.  As for other energy related companies, we assume that BHI share price is driven by the change in some energy-related prices. There are two opportunities: consumer prices and producer prices. In our previous article, we found that Noble Energy (NBL) is rather driven by the PPI of natural gas. On the other hand, many companies are deeply involved in consumer markets and might likely depend on consumer prices.  

Our concept assumes that a BHI share price can be represented as a weighted sum of two individual producer or consumer price indices selected from a predefined set. We split the overall set into two pieces. There are five producer price indices (all borrowed from the Bureau of Labor Statistics): the overall PPI; the PPI of electric power, EL; of natural gas, GAS; of coal, COAL; and the PPI of oil, OIL.  There are nine  CPIs: the headline CPI, C; the core CPI, CC; the CPI less energy, CE; the CPI of energy, CC; the CPI of motor fuel, MF; the CPI of household energy, HHE; the CPI of fuels and utilities, FU;  the CPI of food and beverages, F; and the CPI of housing, H.

All PPIs, CPIs and the monthly closing price are available now through March 2012. Our model seeks for the best pair of PPIs (CPIs) which minimizes the RMS error since 2003. We also allow both defining indices to lead or lag behind the modeled share price. Additionally, we introduced a linear time trend and an intercept term. The best fit model is obtained with the pair C and F:

BHI(t)= 4.409C(t-0) – 4.174F(t-6) 4.797(t-2000) – 50.84; sterr=$6.73

where BHI(t) is the (monthly closing) share price in U.S. dollars. We allowed both time leads to vary between 0 and 12 months. In the best model, the CPI of food leads the share price by 6 months and the headline CPI evolves in sync with the share price.  The model standard error for the period from July 2003 to March 2012 is $4.75. We report only on reliable models which do not change over eight to twelve months in a row. Thus, the above model is valid and reliable since the middle of 2011.  

Figure 1 shows how the model predicts the current BHI price.  As for many energy companies, there were two major fluctuations in the first half of 2010 and in the fourth quarter of 2011 (see Figure 2 for the model residual error). Both ended on the fundamental price curve.  In November 2011, we would estimate the BHI price as an undervalued one. The closing price of March 2012 is undervalued by $15.  We expect the price to return to the fundamental level as defined by the model, i.e. by the CPIs.

Figure 1. The observed and predicted monthly closing prices for BHI between July 2003 and March 2012.

Figure 2. The model residual error.

Noble Energy is likely driven by natural gas

In this article, we quantitatively predict the evolution of Noble Energy (NYSE: NBL) share price. As for other companies from Energy category of the S&P 500 list, one may assume that its share price is driven by the change in some energy-related prices. It might be a commonplace that oil companies depend on oil price as they have to depend on the price and amount of goods and services they produce/sell. When oil has a higher pricing power these companies are expected to show rising profits also reflected in their share prices.

NBL engages in the acquisition, exploration, development, production, and marketing of crude oil, natural gas, and natural gas liquids.  We model the evolution of the NBL share price as a weighted sum of two individual producer price indices selected from a set of five producer price indices borrowed from the Bureau of Labor Statistics: the overall PPI, the PPI of electric power, EL, of natural gas, GAS, of coal, COAL, and the PPI of oil, OIL. All PPIs and the monthly closing price are available now through March 2012. Our model seeks for the best pair of PPIs which minimizes the error since 2003. We also allow both defining PPIs lead or lag behind the modeled share price. Additionally, we introduced a linear time trend and an intercept term. The best fit model is obtained with the pair GAS and PPI:

NBL(t)= -0.049GAS(t-3) + 1.367PPI(t-0) - 1.22(t-2000) – 158.53; sterr=$4.75

where NBL(t) is the (monthly closing) share price in U.S. dollars. We allowed both time leads to vary between 0 and 12 months. In the best model, the PPI of natural gas leads the share price by 3 months and the PPI lead is zero months. In other words, gas drives the NBL share with a three month delay.

Surprisingly, the slope of GAS is negative, i.e. falling gas prices drive the NBL price in opposite direction.  The PPI slope is positive and the share price rises with the overall producer price. The model standard error for the period from July 2003 to March 2012 is $4.75. We report only on reliable models which do not change over eight to twelve months in a row. Thus, the above model is valid and reliable since the middle of 2011.

Figure 1 shows that the model based on the producer price index of natural gas and the PPI (domestic production) accurately predicts the current NBL price.  As for many energy companies, there were two major fluctuations in the first half of 2010 and in the fourth quarter of 2011 (see Figure 2 for the model residual error). Both ended on the fundamental price curve. This behavior is illustrative – all deviations finally return to the predicted curve. In November 2011, we would estimate the NBL price as a highly undervalued one. Since the actual price gravitates to the predicted one we consider our prediction as the fundamental price level, which is fully defined by the PPI of natural gas and the PPI.    

Figure 1. The observed and predicted monthly closing prices for NBL between July 2003 and March 2012.

Figure 2. The model residual error

EOG Resources is driven by natural gas and oil

Here we model the evolution of EOG Resources (NYSE: EOG) share price. Imagine that you have to predict (describe) the evolution of a share price for a company from Energy sector. A natural first guess is that this share price is driven by the change in some energy-related prices. Apparently, financial health of this company depends on the price and amount goods and services it sells. When these goods and services have a higher pricing power the company is likely healthy and generates some extra profit reflected in its share price.

EOG engages in the exploration, development, production, and marketing of crude oil and natural gas.  We extend our original model and describe the evolution of the EOG share price as a weighted sum of two individual producer price indices selected from a set of five producer price indices borrowed from the Bureau of Labor Statistics: the overall PPI, the PPI of electric power, EL, of natural gas, GAS, of coal, COAL, and the PPI of oil, OIL. (We do not use consumer price indices in for EOG.) Thus, the model seeks for the best pair of PPIs which minimizes the error since 2003. We also allow both defining PPIs lead or lag behind the modeled share price. Additionally, we introduced a linear time trend and an intercept term. Not surprisingly, the best fit model is obtained with the pair GAS and OIL:

EOG(t)= 0.129GAS(t-0) + 0.115OIL(t-0) + 7.85(t-2000) – 45.12; sterr=$7.77  

where EOG(t) is the (monthly closing) share price in U.S. dollars. We allowed both time leads to vary between 0 and 12 months. In the best model, both time leads are zero, i.e. the share price evolves in sync with the PPI of oil and gas. Instructively, both slopes in the model are positive and the share prices rises with that of oil and gas. The time trend coefficient is also positive and provides an annual increase of $7.85.  The model standard error for the period from July 2003 to March 2012 is $7.77. It is important that the above model is valid and reliable since the middle of 2011, i.e. the defining PPIs, lags, and coefficients are the same for all contemporary models since July 2011.   

Figure 1 shows that the model based on the producer price index of natural gas and crude petroleum (domestic production) accurately predicts the current EOG price.  There were two major fluctuations in the first half of 2010 and in the fourth quarter of 2011. Both ended on the fundamental price curve. This behavior is very indicative – all deviations should return to the predicted curve. In November 2011, we would estimate the EOG price as a highly undervalued one.  In that sense, the predicted price might be considered as the fundamental price level, which is fully defined by the producer prices of natural gas and oil.   

Figure 1. The observed and predicted monthly closing prices for EOG between July 2003 and March 2012.

Predicting the Oracle of Omaha

It is great pleasure to quantitatively assess the future of Berkshire Hathaway (NYSE: BRK-B). The Oracle of Omaha is considered as one of the greatest investors. Here we would like to introduce a deterministic model for a BRK-B share price. Berkshire Hathaway, Inc. is a publicly owned investment manager. The BRKB structure is highly diversified with interests from GEICO (car insurance) to Dairy Queen.

The model is deterministic since it has been obtained by decomposition of a share price into a weighted sum of two consumer price indices. One may follow up our simple assumption that the growth in the CPIs related to BRK-B, e.g. transportation service, TS, and dairy products, DAIRY, might be seen in the change of the pricing power for the studied company. These two CPIs are revealed as the drivers of the BRK-B price. However, they were actually selected using the LSQ method from a set of 92 different (not seasonally adjusted) CPIs. The best model predicting the monthly closing prices adjusted for dividends and splits has the smallest RMS error from July 2003 to March 2012.

We have borrowed the time series of monthly closing prices of BRK-B from Yahoo.com (the closing price for March 2012 is included) and the CPI estimates through February 2012 are published by the BLS. The best-fit model for BRKB(t) is as follows:

BRKB(t) = -0.802DAIRY(t-11) – 2.24TS(t-7) + 21.13(t-2000) + 584.66, March2012

where BRKB(t) is the BRK-B share price in U.S. dollars, t is calendar time. Figure 1 displays the evolution of the defining CPIs since 2002. Instructively, both indices are relevant to the BRK-B structure and have negative slopes. The DAIRY index likely has a larger impact on the growth of the price becasue of higher amplitude oscillations. The TS index is rather a linear line since 2002 with only a small positive fluctuation (negative impact on the price) in the end of 2008.

Figure 2 depicts the high and low monthly prices for an BRK-B share together with the predicted and measured monthly closing prices. 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 March 2012 of $4.73.

The accelerated growth in the DAIRY index since April 2011 induced a mid-term decline in the share price. This rise in dairy price did not come to end yet and we cannot exclude that the actual BRK-B price will return to the predicted level of $73 per share by the third quarter of 2012. The TS price index does not show any sign of deceleration and food price together with dairy products will likely be rising till 2014. If the model is right, we may observe in 2013 some further decline in BRK-B price.

Figure 1. The evolution of defining indices.

Figure 2. Observed and predicted monthly closing prices for a BRK-B share.

Figure 3. The model residual error: sterr=$4.73.

How deep will the S&P 500 fall?

Several days ago we predicted the current fall in the S&P 500 index. For this reason, we did not enter the stock market and instead invested in a defensive portfolio. We are waiting the level of 1350.  The reason is explained below.

Figure 1 shows the evolution of the S&P 500 index since 1980. After 1995, the index behavior reveals some saw teeth with peaks in 2000 and 2007. The current growth resembles those between 1997 and 2000 and from 2003 and 2007.  There are two deep troughs in 2002 and 2009 which are marked by red and green lines, respectively.  For the current analysis we assume that the repeated shape of the teeth is likely induced by a degree of similarity in the evolution of macroeconomic variables. The intuition behind such an assumption is obvious – in the long run the market depends on the overall economic growth.

Having two peaks and troughs between 1995 and 2009, what can we say about the current growth in the S&P 500? Before making any statistical estimates, in Figure 2 we have shifted forward the original curve in Figure 1 in order to match the 2009 trough (blue line).  When the 2002 and 2009 troughs are matched, one can see that the current growth path closely repeats that after 2002. The first big deviation from the blues curve in Figure 2 started in 2011 and had amplitude of 150 units (from 1210 to 1360).  The black curve returned to the blue one in August/September 2011. A month ago, we observed a middle-size deviation of about 100 units and predicted that the index will have a negative correction down to the level of 1300 any time soon.  If the index will repeat the path of the previous rally one-to-one, one may expect the peak level of 1500 in the end of 2013.  In two to four weeks it might be a good time to invest for a 15% return cumulated to October 2013 (but not more than two months), when the negative correction is over. 

With the S&P 500 falling down to 1350, the prediction does not seem inappropriate. The next several weeks should decide on the new level. In Figure 2, we have drawn the fall we expect by the end of May 2012. We would wait by the end of April to decide on the following move in the S&P 500. If the current fall will reach 1300, it’s likely a good time to buy. Otherwise, the end of May is the horizon to wait the bottom.

Figure 1. The evolution of the S&P 500 market index between 1980 and 2012.

Figure 2. The curve in Figure 1 peak is shifted forward to match the 2009 trough (blue line). Red line – expected fall in the S&P 500: from 1400 in Mach to 1300 in May.

Anadarko Petroleum will follow oil price

The evolution of Anadarko Petroleum’s (NYSE: APC) share price is of interest since it is different from that demonstrated by Apache (APA) which is presented in our previous article.  Both companies are likely driven by the same forces but their current prices are on opposite sides from their fundamental levels estimated with our pricing model.  According to the previously developed procedure, we also compare the APC pricing model to that for ConocoPhillips (COP), which is a recognized benchmark for energy related companies.

Imagine that you have to predict (describe) the evolution a share price for a company from Energy sector. It would not be a big mistake to assume that this share price is likely to be driven by the change in the overall energy price or some of its components (we use price indices for modeling). Even if the company does not change its production the overall increase in the price of its product should be manifested in the overall profit and thus the share price. On the other hand, when the overall price level (as expressed by the headline CPI) rises faster than the energy price index (say, 10% vs. 1% per year, respectively) one should not expect the energy company to gain extra pricing power. The company would rather suffer a share price decline.  Thus, considering the secular increase in the overall price level, it is not the absolute change in energy prices what affects the stock price but its current deviation from some energy independent price.  We have proposed to use the simplest model as based on the difference between the headline CPI, C, and the core CPI, CC, without any time lag between these indices and the share price. The headline CPI includes all kinds of energy and thus provides the broadest proxy to the energy price index. The core CPI excludes energy (and food) and thus may represent the energy independent and dynamic reference.

First, we present here the model for COP. We use it as a benchmark showing the quality of the concept and its predictive power. Figure 1 depicts the observed and modeled COP prices. Taking into account that both defining CPIs might be not the best proxies to some true defining indices, the accuracy of prediction is very good. We consider the predicted price as a fundamental one, i.e. the price which is defined by two economy-wide or fundamental indices. Quantitatively, we have estimated the following relationships to minimize the model error between 1998 and 2012:

COP(t) = 72.3 – 5.35(CC(t) - C(t))  (1)

where COP(t) is the share price in U.S. dollars at time t.  

Figure 1. Historic (monthly closing) prices for COP (black line) and the scaled difference between the core CPI and the headline CPI (red line).

Now we may suggest that the COP model determines the benchmark behavior for an energy related company, i.e. its share price has to gravitate to the fundamental price defined by the difference between the core and headline CPI.   In our previous article, we estimated an empirical model for APA:  

APA(t) = 110 – 8.1(CC(t) - C(t))

For APC, the best fit relationship is as follows:

APC(t) = 67 – 4.7(CC(t) - C(t))  (2)

Figure 2 depicts both models, i.e. the observed and predicted prices for APA and APC. For APC, the overall agreement is relatively good but the deviation from the predicted price has been much higher since 2009.   From Figure 2, the actual price is currently overvalued by $15. The underlying model is too crude, however, and we have developed several advanced APC models. The best from these advanced models shows that the current APC price is only slightly overvalued. 

Figure 2. Historic (monthly closing) prices for APA (upper panel) and APC and the scaled difference between the core CPI and the headline CPI - relationship (2). 

We have extended the original model and described the evolution of the APC share price as a weighted sum of two individual consumer price indices (or PPIs) selected from a large set of CPIs borrowed from the Bureau of Labor Statistics. We allow both defining CPIs (PPIs) lead the modeled share price. Additionally, we introduced a linear time trend on top of the intercept. As for many already presented companies, we have tested two principal pairs of CPIs: C and CC; CC and the index of energy, E, as well as the pair the PPI and the producer price index of crude oil, OIL. The best fit (as defined by standard error) model is obtained with the pair PPI and OIL:

APC(t)= 3.28C(t) – 5.56CC(t-1) + 12.25(t-2000) + 323.56; sterr=$6.12 (3)

APC(t)= -2.79CC(t-2) + 0.32E(t) + 13.73(t-2000)  + 326.49; sterr=$5.80 (4)

APC(t)= -0.21PPI(t-10) + 0.14OIL(t-9) + 4.31(t-2000) – 19.95; sterr=$5.48 (5)

where APC(t) is the (monthly closing) share price in U.S. dollars. We allowed both time leads in (3) through (5) to vary between 0 and 12 months. Figures 3 through 5 depict the observed and predicted monthly prices from (3) through (5).  For an oil company, it is not excluded that oil controls its price.

Figure 5 shows that the advanced model based on the producer price index of crude petroleum (domestic production) accurately predicts the current APC price.  There were two major excursions in the first half of 2010 and in the fourth quarter of 2011. Both ended on the fundamental price curve. This behavior is very instructive – all deviations should return to the predicted curve.

Figure 3. The observed and predicted monthly closing prices for APC between July 2003 and March 2012. The model is based on C and CC.

Figure 4. The observed and predicted monthly closing prices for APC between July 2003 and |March 2012. The model is based on CC and E.

Figure 5. The observed and predicted monthly closing prices for APC between July 2003 and March 2012. The model is based on PPI and OIL.

What is a better investment Apache or ConocoPhillips? An academic model

Here we model the evolution of Apache (NYSE: APA) share price and evaluate its current level relative to that predicted by our pricing model. We also compare the APA model and the ConocoPhillips (COP) model in order to evaluate their relative performance. In other words, we estimate quantitatively which of these two companies provides a better return. Two main findings can be formulated as follows: 1) the current price is slightly lower than that predicted by the model; 2) both companies show the same level of return.


The first finding is not surprising. We have already reported that for many energy related companies (e.g. Newfield Exploration (NFX) and Peabody Energy (BTU)) our empirical models show that their current prices are highly undervalued. In this group, APA is not the worst. However, it is still slightly undervalued despite our model has accurately predicted the rally between 2003 and 2007, the sharp fall in 2008 and the following recovery up to the third quarter of 2011. The second finding is might be an expected one since the stock market seeks for the best return. Therefore, investments are redistributed in a way to provide some constant return. At least this approach should work for all successful companies in the same industry.

We assume that a share price of an energy company is likely to be driven by the change in the overall energy price or some of its components. Considering the secular increase (change) in the overall price level it is not the absolute change in energy prices what affects the stock price but the difference between the energy price and some energy independent price. The simplest model can be based on the difference between the headline CPI, C, and the core CPI, CC, without any time lag between these indices and the share price. The headline CPI includes all kinds of energy and thus provides the broadest proxy to the energy price index. The core CPI excludes energy (and food) and thus represents the energy independent and dynamic reference. Four years ago, these two indices were used in our original models for ConocoPhillips and Exxon Mobil (XOM) and are retained as a benchmark since then.

During these four years, the best model was that for COP. We use it as a benchmark showing the quality of the concept and its predictive power. Figure 1 depicts the observed and modeled COP prices. Taking into account the character of the defining CPIs (they include many irrelevant components which are measurement noise for the model) the agreement between curves is outstanding. One can consider the predicted price as a fundamental one. These are two broadest consumer price indices which define the fundamental price. Quantitatively, we have estimated the following relationships to minimize the model error between 1998 and 2012:

COP(t) = 72.3 – 5.35(CC(t) - C(t)) (1)

where COP(t) is the share price in U.S. dollars at time t.

Figure 1. Historic (monthly closing) prices for COP (black line) and the scaled difference between the core CPI and the headline CPI (red line).

Accordingly, we depict in Figure 2 a similar model for APA. The best fit relationship is as follows:

APA(t) = 110 – 8.1(CC(t) - C(t)) (2)

The overall agreement is also good with the predicted and observed prices very close near the 2008 peak and the 2009 bottom. The recovery since 2009 has been described with a slight underestimation of the measured price, i.e. the actually observed growth was slightly stronger than the predicted one. This undervaluation was quickly compensated by the 2011 fall. Currently, the actual price is undervalued one by a few dollars.

Figure 2. Historic (monthly closing) prices for APA (black line) and the scaled difference between the core CPI and the headline CPI (red line).

Comparing the slopes in (1) and (2) one can estimate relative performance of APA and COP. These slopes define the price reaction to a given change in the CPI difference. For a one unit change in CC-C, the COP price changes by $5.35 and the APA price by $8.1. Since the price levels are approximately $67 and $100, respectively, the ratio of the slopes (0.66) completely corresponds to the ratio of price levels (0.67). This means that the returns provided by COP and APA are equal if their prices are driven by the difference between CPIs.

The original model is very crude. Both CPIs depend on many other goods and services, what introduces high measurement noise in the model. Also, both CPIs have the same weight (1.0) and cannot lead or lag behind the modeled price or each other. Apparently, it can be some non-zero lag between the change in energy price and in prices of energy companies. Therefore, we extended the model and described the evolution of a share price as a weighted sum of two individual consumer price indices (or PPIs) selected from a large set of CPIs borrowed from the Bureau of Labor Statistics. We allow both defining CPIs (PPIs) lead the modeled share price. Additionally, we introduced a linear time trend on top of the intercept. As for many already presented companies, we have tested two principal pairs of CPIs: C and CC; CC and the index of energy, E, as well as the pair the PPI and the producer price index of crude oil, OIL. The best fit (as defined by standard error) model is obtained with the pair PPI and OIL:

APA(t)= 5.83C(t) – 5.81CC(t-0) + 3.81(t-2000) + 28.53; sterr=$9.13 (3)
APA(t)= 2.69CC(t-9) + 0.57E(t) – 8.25(t-2000) - 447.41; sterr=$8.81 (4)
APA(t)= 1.66PPI(t-0) - 0.059OIL(t-9) – 1.25(t-2000) – 175.90; sterr=$7.68 (5)

where APA(t) is the (monthly closing) share price in U.S. dollars. We allowed both time leads in (3) through (5) to vary between 0 and 12 months. Figures 3 through 5 depict the observed and predicted monthly prices from (3) through (5). For an oil company, it is not excluded that oil controls its price. Interestingly, however, that the index of oil drives the price down, i.e. increasing oil price suppresses the return. From Figure 5, we expect the price to rise to $115 in the near future. Oil price may fall in this case.

Figure 3. The observed and predicted monthly closing prices for APA between July 2003 and March 2012. The model is based on C and CC.

Figure 4. The observed and predicted monthly closing prices for APA between July 2003 and
March 2012. The model is based on CC and E.

Figure 5. The observed and predicted monthly closing prices for APA between July 2003 and March 2012. The model is based on PPI and OIL.

Why Peabody Energy is highly undervalued?

In this article we model the evolution of Peabody Energy (NYSE: BTU) share price since 2003 and evaluate its current level relative to that predicted by the model. The main finding can be formulated as follows: the current price is much lower than that predicted by the model. At the same time, the same model has accurately predicted the rally between 2003 and 2007, the sharp fall in 2008 and the following recovery up to the third quarter of 2011.  The same effect is observed for many energy companies, as we have already reported on Seeking Alpha. For example, Newfield Exploration (NFX) and GeoResources (GEOI) demonstrate high-amplitude deviations from their relevant predicted prices but with opposite signs.  Thus, we have been measuring strikingly abnormal deviations in share prices of many energy related companies since the middle of 2011. This observation needs careful analysis and investors’ attention.

Our pricing concept is very simple and we explain it here step by step. First, we put forward a working hypothesis that a share price of an energy company can be driven by the change in the overall energy price or its components. Obviously, energy prices do not exist in vacuum: they affect and are affected by other goods and services. Therefore, we actually assume that the share price is driven by the difference between the energy price and some energy independent price, both can be presented as indices. 

In its simplest form, the model is based on the difference between the headline CPI, C, and the core CPI, CC, without any time lag between these indices and the share price. The headline CPI includes all kinds of energy and thus provides the broadest proxy to the energy price index. The core CPI excludes energy (and food) and thus represents the energy independent and dynamic reference. Historically, these two indices were used in our original models for ConocoPhillips (COP) and Exxon Mobil (XOM) and are retained as a benchmark since then.

The best ever model was obtained for COP. Figure 1 depicts the observed and modeled COP prices in order to demonstrate the predictive power of the pricing concept. The agreement between curves is excellent. One can see that all deviations of the actual price from the predicted one are only short-term and the predicted curve might be considered as a “fundamental” one. In other words, the actual price gravitates to the predicted one. Quantitatively, we have estimated the following relationships to minimize the model error between 1998 and 2012:

COP(t) = 72.3 – 5.35(CC(t) - C(t))  (1)

where COP(t) is the share price in U.S. dollars at time t.  In any case, both curves are close to each other throughout the whole period and there is deviation growing since 2011. 

Figure 1. Historic (monthly closing) prices for COP (black line) and the scaled difference between the core CPI and the headline CPI (red line).

In Figure 2 we present a similar model for BTU. The best fit relationship is as follows:

BTU(t) = 59.5 – 5.55(CC(t) - C(t))  (2)

The overall agreement is also excellent with the predicted and observed prices very close near the 2008 peak and the 2009 bottom.   The recovery since 2009 has been also described accurately to the peak value in April 2011. And then the prices started to deviate, which the predicted price falling at a much slower pace.  In this situation one may consider the currently observed price as an highly undervalued one (between $20 and $25). 

Figure 2. Historic (monthly closing) prices for BTU (black line) and the scaled difference between the core CPI and the headline CPI (red line).   

The original model is very crude. Both CPIs depend on many other goods and services, what introduces high measurement noise in the model. Also, both CPIs have the same weight (1.0) and cannot lead or lag behind the modeled price or each other.   Apparently, it can be some non-zero lag between the change in energy price and in prices of energy companies. Therefore, we extended the model and described the evolution of a share price as a weighted sum of two individual consumer price indices (or PPIs) selected from a large set of CPIs borrowed from the Bureau of Labor Statistics. We allow both defining CPIs (PPIs) lead the modeled share price. Additionally, we introduced a linear time trend on top of the intercept.

So, we continue presenting Peabody Energy Corporation which is engaged in the mining of coal. This is not an oil&gas company and we do not expect it directly depend on oil price. As for many already presented companies, we have tested two principal pairs of CPIs: C and CC; CC and the index of energy, E, as well as the pair the PPI and the producer price index of crude oil, OIL. The best fit (as defined by standard error) model is obtained with the pair CC and E:

BTU(t)= 5.21C(t) – 5.09CC(t-0) – 0.45(t-2000) + 40.84; sterr=$7.85 (3)

BTU(t)= 1.34CC(t-9) + 0.52E(t) – 6.57(t-2000)  - 228.58; sterr=$7.02 (4)

BTU(t)= 1.46PPI(t-0) - 0.0346OIL(t-6) – 4.77(t-2000) – 123.01; sterr=$7.86 (5)

where BTU(t) is the share price in U.S. dollars; the core CPI leads the price by 9 months. We allowed both time leads in (3) through (5) to vary between 0 and 12 months. As one can expect, the index of energy drives the price up. Surprisingly, the core CPI also affects the price positively and the long term time trend in both defining CPIs is compensated by the negative time trend. Figures 3 through 5 depict the observed and predicted monthly prices.  

The best model shows even a better overall agreement than the original model with the standard error of $7.02 between July 2003 and February 2012. As we discussed above, the model residual has been growing since the second half of 2011. Currently, the error is -$23. This is an extremely high residual relative to the “fundamental” price. We believe that the current excursion is just a short-term deviation. Therefore, the BTU price is highly undervalued.

The reader may also suggest that the model has failed on BTU. We cannot exclude this explanation but then why the concept works for the biggest energy companies and has also been working relatively well before August 2011?  

Figure 3. The observed and predicted monthly closing prices for BTU between July 2003 and |March 2012. The model is based on C and CC.

Figure 4. The observed and predicted monthly closing prices for BTU between July 2003 and |March 2012. The model is based on CC and E.

Figure 5. The observed and predicted monthly closing prices for BTU between July 2003 and March 2012. The model is based on PPI and OIL.

Safeway is slightly overvalued

Here we model another company from the S&P 500 list.  This is a company from Services category – Safeway Inc. (NYSE: SWY), which operates as a food and drug retailer in North America. SWY share price is approximated by a linear combination of two consumer price indices; with the CPI of food, F, directly related to SWY and the CPI of owner’s equivalent rent of residence, ORPR, representing an independent but dynamic price reference. We suggest that one CPI moves the SWY price but only relative to the overall prices, which evolve freely of SWY.  Thus, our stock price model tries to find one defining CPI and the best reference.  

We have borrowed the time series of monthly closing prices of SWY from Yahoo.com (March 2012 is included) and the CPI estimates through February 2012 are published by the BLS.  (The CPI estimates for March 2012 will be reported by the BLS in the middle of April and we will revise all models accordingly).  The best-fit model for SWY(t) is as follows:  

SWY(t) =  -1.59F(t-4) + 1.19ORPR(t-0)  + 3.34(t-2000) + 28.85,  March 2012 

where SWY(t) is the SWY share price in U.S. dollars,  t is calendar time. Figure 1 displays the evolution of both defining indices since 2002.  The ORPR index has a positive influece on the price. Apparently, the negative slope of F implies that higher food prices reduce the SWY return. This is not against common sense.  

Figure 2 depicts the high and low monthly prices for a SWY 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 as defined by the monthly high/low prices.  

The model residual error is shown in Figure 3 with the standard error between July 2003 and January 2012 of $1.86. The model predicts that all deviations are only short-term ones, i.e. the measured curve returns to the predicted one, which may be considered as a “fundamental” price level. Basically, the observed share price gravitates to the predicted curve.   

Considering Figures 2 and 3, one can conclude that they current SWY price is slightly overvalued and a negative correction is not excluded in the beginning of the second quarter of 2012. Such corrections have been observed many times in the past.    

Figure 1. The evolution of defining indices. 

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

Figure 3. The model residual error: sterr=$1.86.

Northeast Utilities will stay at $37 per share in Q2 2012

We have presented quite a few pricing models for energy companies (ConocoPhillips, Exxon Mobil, Devon Energy, Chevron, and Chesapeake among many others) because of their clear relation to consumer and producer indices of energy. For different S&P 500 sectors, our pricing models often include indices not related at first glance. Here we present a new pricing model for a company from Utilities category – Northeast Utilities (NYSE: NU), a public utility holding company, which provides electric and natural gas energy delivery services to residential, commercial, and industrial customers in Connecticut, New Hampshire, and western Massachusetts. In part, NU is also linked to energy. But any company, small or big, has a link and is affected by energy prices. The opposite statement is likely wrong – not every company can affect the overall price of energy.  

Before presenting the best fit deterministic pricing model for NU we would like to brief on the underlying pricing concept.  Our approach consists in decomposition of a share price into a weighted sum of two consumer (or producer) price indices. One may test quantitatively our simple assumption that the growth in some specific CPI related to NU (e.g. housekeeping supplies, the BLS name is CUUR0000SEHN and we abbreviate it to HOS) relative to some independent but dynamic reference (e.g. food away from home, CUUS0000SEF or SEFV) should be seen in a higher pricing power for the studied company. In essence, our stock price model tries to find one defining CPI and the best reference. 

In order to find the best pair of CPIs we calculate the RMS residual errors using all permutations from a set of 92 different (not seasonally adjusted) CPIs. In addition we allow for time lag (fixed between 0 and 12 months) between the studied price and both indices, and introduce a linear trend and intercept terms. The best model has the smallest RMS error between July 2003 and February 2012. The complete set 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 2001). 

We have borrowed the time series of monthly closing prices of NU from Yahoo.com (March 2012 is included) and the CPI estimates through February 2012 are published by the BLS.  (The CPI estimates for March 2012 will be reported by the BLS in the middle of April and we will revise all models accordingly).  The best-fit model for NU(t) is as follows: 

NU(t) =  -2.57SEVF(t-6) + 0.97HOS(t-7)  + 15.18(t-2000) + 267.79,  March 2012 

where NU(t) is the NU share price in U.S. dollars,  t is calendar time. Figure 1 displays the evolution of both defining indices since 2002.  The housekeeping supplies index has a positive influece on the price, i.e. the growth in HOS results in a higher NU price. Apparently, the negative slope of SEFV implies that higher food prices reduce the NU return. This is not against common sense.   

Figure 2 depicts the high and low monthly prices for a NU 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 error between July 2003 and January 2012 of $1.23. All in all, the model is very accurate and all deviations are promptly annihilated, i.e. the measured curve returns to the predicted one, which leads by six months. One can foresee the share price behaviour six months ahead.  

Considering Figures 2 and 3, one can conclude that the price should rise in the second quarter of 2012 to the level of $37 per share. Since the current level is also $37, one may expect no big change in Q2.   

Figure 1. The evolution of defining indices. 

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

Figure 3. The model residual error: sterr=$1.23.