CHK, COP, CVX, DVN, ESV, HAL, and XOM. Improved models

Two days ago we revisited share price models for five energy companies from the S&P 500 list: ConocoPhillips (COP), Chevron (CVX), Devon Energy Corporation (DVN), Halliburton (HAL), and Exxon Mobil (XOM). Our pricing model links shares and CPI components (Kitov, 2009). It was demonstrated that all five original models, as had been obtained in 2008, predicted the time history of these prices with a good accuracy. In addition, since the difference between core CPI and headline CPI in the United States is well approximated by a linear function of time this difference can be used to predict share prices in the energy subcategory of the S&P 500 at time horizons of several years.

In this post we are going to improve these models for energy companies using an extended set of 17 individual CPIs, which includes all major subcategories of the headline CPI:

C headline CPI

F food and beverages

H housing

A apparel

T transportation

M medical care

R recreation

ED education

CO communication

O other goods and services

CE CPI less energy

CF CPI less food

CC core CPI

COMM commodities

E energy

NDUR nondurables

CFSHE CPI less food, shelter and energy

This step is in line with our general approach to price modeling, as described by Kitov (2010). We also introduce a quantitative measured of the prediction accuracy – RMSE (root-mean-square error) and check the predictive power of the underlying model.

The pricing model is common for all companies. We assume the presence of a linear link between a share price and the difference between the core (or headline) CPI and some other subset of goods and services comprising the headline CPI. The intuition behind the model is simple; a higher pricing power for a given subcategory of goods and services, and thus related companies, is expressed in a faster increase in corresponding stock prices. In the first approximation, the deviation between relevant price indices is proportional to the ratio of the pricing powers.

The pricing model states that a share price, SP(t), can be approximated by a linear function of the difference between two CPI components, dCPI:

SP(t) = A + B1CPI1(t + t1) + B2CPI2(t + t2) + C(t-t0) (1)

where A, Bi, and C are empirical constants for the studied period (between July 2003 (t0) and November 2010); t is the elapsed time; t1 and t2 are the time delays between the share and the CPIs, both to be determined. Here we introduce a linear trend element, C(t-t0), which has to compensate the trend component in the CPI difference (Kitov&Kitov, 2008). Without loss of generality we minimize the model error by standard LSQ method to find all 6 coefficients (A,Bi,C,ti) for those 2 CPI components among the set of 17, which provide the lowermost RMSE.

Figures 1 and 2 compare two updated predictions for COP and XOM obtained in the previous post with new model based on 17 components. The new models are as follows:

COP(t)= 2.69*H(t+8) – 0.76*T(t+1) – 12.47(t-2000) – 422.1; RMSE=$3.34;

XOM(t)= -2.44*F(t-12) - 1.13*O(t-13) + 31.84(t-2000) + 350.1; RMSE=$4.41

Figure 1. Upper panel: original observed and predicted COP price. A2=72, B2=-5.5 (1999-2009). Lower panel: the improved model described above. The prediction lags by 7 months behind the price.

Figure 2. Same as in Figure 1 for XOM. Original model is characterized by A2=90, B2=-6 (1999-2009).

Figures 3 through 5 compare the original and improved predictions for Chevron, Devon Energy, and Halliburton:

CVX = -5.5*(cCPI - CPI) +85; 1999-2009

CVX(t)= 3.16*H(t) – 10.51*R(t) – 0.90(t-2000) – 565; RMSE=$4.90;

DVN = -7.7*(cCPI - CPI) +97; 1999-2009

DVN(t)= 2.15*CF(t+1) – 0.50*E(t) – 9.72(t-2000) – 304.8; RMSE=$5.88;

HAL = -3.5*(cCPI - CPI) + 43; 1999-2009

HAL(t)= 0.40*T(t+1) – 0.90*O(t+6) + 11.64(t-2000) + 44; RMSE=$2.44

Figure 3. The original and improved share price prediction for CVX

Figure 4. The original and improved share price prediction for DVN

Figure 5. The original and improved share price prediction for HAL

All in all, the extension of the CPI set has significantly improved the predictive power of the model. We have also calculated models for other energy companies. Figure 6 displays two of them:

CHK(t)= -1.47*ED(t-7) + 0.41*E(t+1) +11.4(t-2000) – 12.1; RMSE=$2.68;

ESV(t)= 2.94*C(t) – 2.41*F(t-5) + 2.17(t-2000) – 113; RMSE=$4.14;

Figure 6. Models for CHK and ESV.

References

Kitov, I. (2009). ConocoPhillips and Exxon Mobil stock price, Journal of Applied Research in Finance, v. 2, pp. 129-134.

Kitov, I. (2010). Modelling Share Prices of Banks and Bankrupts, Theoretical and Practical Research in Economic Fields, Association for Sustainable Education, Research and Science, vol. 0(1), pages 59-85, June

Kitov, I., Kitov, O. (2008). Long-Term Linear Trends In Consumer Price Indices, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(2(4)_Summ), pp. 101-112.

Predicting shares of ConocoPhillips, Chevron, Devon Energy Corporation, Halliburton, and Exxon Mobil. 2010 revision

Approximately two years ago we introduced a concept linking share prices and CPI components (Kitov, 2009). To begin with, we modelled and predicted the evolution of share prices of ConocoPhillips (COP) and Exxon Mobil (XOM), which are classified in the S&P 500 list as related to energy. It was demonstrated that the time history of these prices could be accurately approximated by a linear function of the difference between core CPI and headline CPI in the United States. This difference is found to be the best to predict share prices in the energy subcategory of the S&P 500.

In this posts we are going to revisit those energy companies which shares were modelled in (Kitov, 2009) and (Kitov&Kitov, 2009) and check the predictive power of the underlying model. These companies are as follows: ConocoPhillips (COP), Chevron (CVX), Devon Energy Corporation (DVN), Halliburton (HAL), and Exxon Mobil (XOM).

The pricing model is common for all companies. It is simple. We assume the presence of a linear link between a share price and the difference between the core (or headline) CPI and some other subset of goods and services comprising the headline CPI. The intuition behind the model is simple; a higher pricing power for a given subcategory of goods and services, and thus related companies, is expressed in a faster increase in corresponding stock prices. In the first approximation, the deviation between relevant price indices is proportional to the ratio of the pricing powers. The presence of sustainable (linear or nonlinear) trends in the differences, as found in (Kitov&Kitov, 2008) allows predicting the evolution of the differences, and thus the deviation between prices for corresponding goods and services. The share prices have to follow up.

So, there exist sustainable trends in the differences between various subcategories of consumer (and producer) price indices. We consider the sustainability as an equivalent to the possibility to describe such trends by simple functions of time. Figure 1 shows that the difference between the core CPI, cCPI, and the headline CPI, CPI, can be approximated by a simple time function:

dCPI(t) = a + bt (1)

where dCPI(t) is the difference, a and b are empirical constants, and t is the elapsed time. Between 1981 and 1999, the linear trend has a slope +0.67, and from 2002 to 2008 the slope is (-1.65). Hence, the “distance” between the core CPI and the headline CPI is a linear function of time, with a positive or negative slope b. It might be of fundamental importance that absolute value of the ratio of the slopes is inversely proportional to the ratio of durations: │0.67/(-1.65)│≈7/19. If such a trade-off actually exists, one can predict the duration of the next trend from its slope.

Figure 1. The difference between the core CPI and the headline CPI between 1980 and 2008. There are two distinct periods from 1981 to 1999 and from 2002 to 2008, where the growth in the difference can be accurately approximated by linear functions of time with slopes +0.67 and -1.65, respectively. Notice that absolute value of the ratio of slopes is inversely proportional to the ratio of durations: │0.67/(-1.65)│≈7/19. The lower panel shows the beginning of the new trend in the difference.

Then, the pricing model states that a share price, for example, that of ConocoPhillips, COP(t), can be approximated by a linear function of the difference between the core and headline CPI (Kitov, 2009):

COP(t) = A + BdCPI(t + t1) (2)

where A and B are empirical constants (for COP, A=72 and B=-5.5) for the period between 1998 and 2009); t is the elapsed time; and t1=1/6 year is the time delay between the share and the CPI changes, i.e. the CPI has a lag behind the share price.

Empirical constants in (2) have to be determined for all distinct periods with different trends. This implies the possibility of structural breaks in the link between share price and CPI as caused by the turns to new trends. For example, the set of long-term economic bounds between goods and services, comprising the CPI and defining the linear trend in the dCPI between 1981 and 1999, underwent a three-year-long transition to a new set. In turn, the new set defined the trend observed from 2002 to 2008. So, it is reasonable to assume that the sign of slope in (2) should change to an opposite one after the end of the current transition period.

Figures 2 and 3 compare the original and updated predictions for COP and XOM. Coeffcients in (2) are given in the Figure captions.

Figure 2. The observed and predicted COP price. A2=72, B2=-5.5 (1999-2009). Upper panel - original model of 2009. Lower panel - updated for 2009 and 2010.

Figure 3. Same as in Figure 2 for XOM. A2=90, B2=-6 (1999-2009).

Figures 4 through 6 compare the original and updated predictions for Chevron, Devon Energy, and Halliburton. As for other energy-related companies, the difference driving relevant share prices is likely that between the core and headline CPI. The models for CVX, DVN, and HAL share price are the same in both cases:

CVX = -5.5*(cCPI - CPI) +85; 1999-2009

DVN = -7.7*(cCPI - CPI) +97; 1999-2009

HAL = -3.5*(cCPI - CPI) + 43; 1999-2009

The next move in all three shares will be up in line with the increasing oil price (Kitov, 2009). In the long run, the dCPI will likely be growing. The increasing difference will have the same effect on the prices in the future as always before – share price grows at a rate proportional to the slope in the dCPI.

Three companies in Figures 4 through 6 are characterized by different coefficients B between 2000 and 2010: from -3.5 (HAL) to -7.7 (DVN). Previously, we determined the slopes for COP (-5.5) and XOM (-6.0). Now one can conclude that these five energy-related companies demonstrate different levels of effectiveness in converting of the dCPI into share price.

Figure 4. The original and updated share price prediction for CVX

Figure 5. The original and updated share price prediction for DVN

Figure 6. The original and updated share price prediction for HAL

All in all, our concept gave a good prediction two years ago. All models are sound. Despite its striking dissimilarity to the mainstream concepts, our pricing model is deeply rooted in economics as expressed in terms of common sense: a higher pricing power achieved by a given company should be converted into a faster growth in corresponding consumer price index. So, the link between these two measured variables is, effectively, a causal one. If the evolution of the difference between various components of the CPI would have been a random walk, the mainstream stock pricing models would be correct. However, the existence of sustainable trends in the differences makes these models obsolete, at least for some companies from the S&P 500 list.

References

Kitov, I., (2009). ConocoPhillips and Exxon Mobil stock price, Journal of Applied Research in Finance, v. 2, pp. 129-134.

Kitov, I., Kitov, O., (2008). Long-Term Linear Trends In Consumer Price Indices, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. 3(2(4)_Summ), pp. 101-112.

Kitov, I., Kitov, O. (2009). Modelling of selected S&P 500 share prices, MPRA Paper 15862, University Library of Munich, Germany.

Predicting DeVry's share price. December 2010 revision

The model for DeVry (DV) was introduces in July 2010. Originally, the stock price was defined by the index for the rent of primary residency (RPR-CUUS0000SEHA) and that of pets, pet products and services (PETS-CUUR0000SERB). We have revisited the stock and found that the model has not changed, except the time lag of the second defining CPI has increased from 4 to 5 months. The former CPI (RPR) component again leads the share price by 11 months. All coefficients are essentially as in the original model.

Figure 1 depicts the overall evolution of both involved indices. From our past experience, the larger is the lag the more unreliable is the model. However, both defining components provide the best fit model between November 2009 and December 2010. The positive influence of RPR (+7.10) is compensated by the negative input of all other terms. So, the best-fit 2-C model for DV(t) is as follows:

DV(t) =7 .10RPR(t-11) – 2.43PETS(t-5) – 31.90(t-2000) – 686.06

The predicted curve in Figure 2 leads the observed price by 5 months with the residual error of $3.68 for the period between March 2003 and December 2010. In other words, the price of a DV share is completely defined by the behaviour of the two CPI components.

Comparing the evolution of the observed and predicted prices since the start of modelling (2008) we have found that the model does predict the share price in the past and foresees at a four to five month horizon. One may expect a significant fall in the first quarter of 2011.

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

Figure 2. Observed and predicted DV share prices. A contemporary prediction is shown by red line. Black diamonds present the original line shifted 5 months ahead.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $3.68. The largest errors were observed in 2007 and 2008.

HPQ share price. December 2010 revision

Half a year ago we presented a model of the evolution of HPQ share price. We decomposed it into two CPIs, linear time trend and constant. It’s good time to revisit the prediction and the model.

According to [1], the model for Hewlett-Packard (HPQ) is defined by the index of food and beverages (F) and that of rent of primary residency (RPR). The former CPI component leads the share price by 4 months and the latter one leads by 5 months (i.e. one can predict at a four-month horizon). Figure 1 depicts the overall evolution of both involved indices. These two defining components provide the best fit model between November 2009 and December 2010. One coefficients is negative and one is positive together with time trend, with slope of 1.34.

So, the best-fit 2-C model for HPQ(t) share price is as follows:

HPQ(t) = -3.45F(t-4) +3.38RPR(t-5) + 1.34(t-2000) – 69.3

This model is slightly different from that in the previous post on HPQ. The index of food has replaced that of food less beverages (FB). The difference between these indices is practically negligible, however. Both coefficients are practically the same as before. Hence, the model has not changed in practical terms.

The predicted curve in Figure 2 leads the observed price by 4 months with the residual error of $2.39 for the period between March 2003 and December 2010. One can consider the price of a HPQ share as completely defined by the behaviour of the two CPI components.

The model does predict the share price in the past and foresaw the fall in 2010 four months in advance. The HPQ price is not expected to change in the first quarter of 2011.

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

Figure 2. Observed and predicted HPQ share prices. The contemporaneous prediction is shown by red line. Black diamonds present the original line shifted 4 months ahead, i.e. the model.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $2.29. The largest error was observed in July 2010 (-$6.31).

References

1. Kitov, I. (2010). Deterministic mechanics of pricing. Saarbrucken, Germany, LAP Lambert Academic Publishing.

How well can we predict the cumulative inflation ten years ahead?

Our model of inflation [1] allows replacing price index with the cumulative growth of labor force. The biggest world developed economies all fit our model, as Figures 1 through 6 demonstrate (all borrowed from [1]). Moreover, the difference between observed and predicted cumulative inflation is a I(0) process, i.e. all observed/predicted pairs should be cointegrated. This allows linear regression, which is characterized by the goodness of fit above 0.99 for all presented cases: the USA, Austria, Canada, France, Germany, and Japan.

All observed curves have well predicted points of inflection. The most striking case is Japan, where the decline in labor force observed since the late 1990s manifests itself in deflation and cumulative curves (predicted and observed) have synchronized peak points.

We are going to consider each case in details, but it is crystal clear that the relative error of the proposed replacement (i.e. the prediction of cumulative inflation or price index) decreases with time and what one needs is a projection of labor force not all these forecasts of inflation. The former is usually is much more accurate than the latter.

Figure 1. USA

Figure 2. Austria

Figure 3. Canada

Figure 4. France

Figure 5. Germany

Figure 6. Japan

Empirically correct model of real economic growth (France)

We continue our GDP modelling with France. This is one of the biggest developed countries providing information on the population age distribution. As in the previous cases, the model for the GDP growth for France was obtained by the trial-and-error method using a discrete form of equation (1.1). The empirical constant A and the defining age have been varied in order to fit the amplitude and timing of observed peaks and troughs. The best fit values are: $320 and eighteen years of age. In the upper panel of Figure 1, the observed and predicted curves for the period between 1970 and 2009 are presented. Superficial visual inspection allows us to suggest that the agreement between the curves does not contradict our concept. The only principal difference between the US and France is that the defining age for France is eighteen years, as was the case for Japan.

Figure 1. Observed and predicted growth rate of real GDP in France. The predicted curve is obtained from relationship (1.4) with A=$320 (1990 US dollars). Upper panel: Original curves. Lower panel: The original curves smoothed with MA(3).

There are original estimates of the number of 18-year-olds in France, which can be used for the prediction of the past GDP figures. The future GDP can be predicted only by extrapolation of younger age populations. For example, the number of 10-year-olds in 2000 can be used as a proxy to the number of 18-year-olds in 2008. Moreover, it is possible to transform the age pyramid for a given year into the distribution of 18-year-olds, with the accuracy of extrapolation decaying with age. In this study, the number of 5-year-olds in 2001 is the reference distribution. So, using this age we are able to estimate the evolution of GDP up until 2014.

A better prediction could be obtained after censuses, which usually provide a well balanced single-year-of-age distribution. In France, the last general population census with published results was in 1999. By itself, the accuracy of the population estimates is difficult to evaluate, but many features unveil artificial character of the population age pyramid. In any case, one cannot help observing very good correspondence between the slowdowns in both curves in the beginning of 1990s and 2000s in Figure 1.

A high-amplitude fluctuation in the first derivative is a common feature for most measured macroeconomic variables. This is a direct manifestation of measurement errors associated with numerous limitations in relevant measuring procedures and inappropriately small time step. In the USA, the average annual growth in real GDP per capita during the latter 20 years is around 2% with the average uncertainty of 1 percentage point, i.e. the annual estimates are of the same order of magnitude as the corresponding uncertainty. In the absence of adequate improvements in the measurement methodology per se, better accuracy could be achieved via stretching the time horizon of corresponding GDP readings, i.e. the time step should be larger than one year.

As an intermediate measure one can smooth all time series in order to cancel out measurement noise. There is a variety of smoothing techniques, some of them very complicated, but even a moving average is enough for the original data in Figure 1. In the lower panel, both original curves are smoothed with a three-year moving average, MA(3). After 1985, the curves are very close. That observation supports the assumption that the fluctuations were chiefly induced by high uncertainty of the measurements, and thus, are effectively suppressed by destructive interference. Before 1985, the curves suffer a slight divergence, which can be an indication of the problems with the extrapolation over 20 years back in the past as well as with the reliability of the GDP measurements. According to the predicted curves, France will not suffer a protracted recession in the next four to six years. It is, however, likely that an insignificant GDP decline period will hit France in the near future.

Figure 2. Observed and predicted number of 18-year-olds in France. The former variable is extrapolated from the number of 5-year-olds in 2001 with a 13-year shift, and the latter from the observed real GDP per capita.

With the knowledge of annual GDP estimates it is possible to further double-check our model using the reversed equation (1.6) thus calculating the number of 18-year-olds in France. Figure 2 illustrates the inversion results between 1963 and 2009. In general, the observed and predicted curves are very close after 1985. Before 1985, the curves diverge in minor details, but both show a sharp increase in the 18-year-old population after 1960. This is a major feature which has higher importance for the model than smaller deviations. In the past, annual population estimates in developed countries were highly unreliable.

Did we predict well 2010 inflation in the U.S. five years ago?

In 2005, we published two papers on inflation [1,2] where we first introduced a new concept deterministically linking the rate of price change, π(t), and the change in the level of labor force, dLF/LF. The first model was as follows [2]:

π(t) = 4. dLF(t-2)/LF(t-2) - 0.03

where π(t) is the GDP deflator and LF(t-2) is the level of civilian labor force two years before. Using several labor force projections made by the CBO (2004) and the BLS (2005) we obtained a prediction of inflation at a horizon of 10 years. Figure 9 below is borrowed from [2] and illustrates the original prediction.

Figure 9. Predicted inflation rate for the period between 2006 and 2016 according to the CBO’s (2004) labor force projection. A deflationary period starts in 2012.

We found that the period of the “Great Moderation” was approaching its natural end. The labour force projections made by CBO (2004) undoubtedly indicate a decrease in the participation rate and a decaying growth rate of the working age population. According to these projections, staring from 2010, the annual increase in labour force will be less than 1,200,000 – the value separating inflation and deflation. Hence, the year of 2012 is likely to mark the beginning of the deflationary era in the USA (which hopefully is the global disaster the Mayans talked about) because of the two-year lag between the labour force change and inflation.

Five years later we can compare our prediction with actual observations. (This is a mandatory step to validate any scientific theory. (In this sense, no mainstream macroeconomic theory is a scientific one.) Figure 2.29 is borrowed from our monograph and compares the prediction based on the CBO’s projection of the labour force and observations. After peaked at 3.2% in 2007, the rate of price inflation has been at a gradual decrease in striking agreement with our calculations. Notice that this prediction was actually made “on the back of a napkin” five years ago.

In 2010, we expect the overall price inflation (GDP deflator) at the level slightly below 1%. It will be very instructive to compare our forecast for 2010 with the estimate of the BEA which will be available in couple months.

Figure 2.29. Predicted inflation rate for the period between 2006 and 2016 according to the CBO’s (2004) labour force projection. A deflationary period starts in 2012.

References

1. Kitov, I. (2006). Inflation, unemployment, labor force change in the USA, Working Papers 28, ECINEQ, Society for the Study of Economic Inequality, http://ideas.repec.org/p/inq/inqwps/ecineq2006-28.html

2. Kitov, I., (2006). Exact prediction of inflation in the USA, MPRA Paper 2735, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/2735.html