IBM share in 2011 and 2012

In July 2010 we presented a share price model for IBM for the period between July 2003 and May 2010:

IBM(t) = 4.93MVR(t-12) – 3.51TS(t-4) - 10.39(t-2000) + 39.39

where MVR is the index of motor vehicle maintenance and repair (CUUR0000SETD) and TS is the index of transportation services ( CUUR0000SAS4). The former CPI component led the share price by 12 months and the latter one led by 4 months.

Here we extend the modeling period in both directions - between January 1995 and December 2010. As before, the model coefficients are obtained by minimizing the RMS residual error. Current IBM model is as follows:

IBM(t) = -4.32*H(t-1) – 1.48*MVI(t-1) + 40.69(t-2000) + 779.0

where H is the index of housing and MVI is the index of motor vehicle insurance. Figure 1 depicts the overall evolution of both involved indices. The index of housing was on rise before 2009. Since December 2008, this index has been slightly decreasing. Since it has negative influences on the share price, one can expect an increase in IBM price. The MVI index has been quickly growing over the entire period, except during some short segments. Thus, did not allow the share to increase to fast since linear trend also has positive influence on the price. All in all, these two defining components provide the best fit model between December 2009 and December 2010.

The predicted curve in Figure 2 leads the observed price by 1 month with the residual error of $9.49 for the period between January 1995 and December 2010. Currently, the price is slightly underestimated by the model, as Figure 3 shows, and one cannot exclude a downward correction in the first quarter of 2011.

In the long run, the index of housing will be decreasing during the next 10 years. This is a helpful background for IBM share. The MVI has a clear rise/plateau structure. The next segment is likely to be a shelf, starting in 2011 of 2012. Hence, the price share looks good at a two-year horizon.

Figure 1. Evolution of the price of H and MVI.

Figure 2. Observed and predicted IBM share prices.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $9.49.

General Electric share between 1995 and 2011

We use the same pricing model as previously. In its general form, this pricing model is as follows:

sp(tj) = Σbi∙CPIi(tj-i) + c∙(tj-2000 ) + d + ej (1)

where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-i) is the i-th component of the CPI with the time lag i, i=1,..,I; bi, c and d are empirical coefficients of the linear and constant term; ej 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 fourteen months. Apparently, this is an artificial limitation and might be changed in a more elaborated model. In any case, a fourteen-month lag seems to be long enough for a price signal to pass through.

System (1) contains J equations for I+2 coefficients. For General Electric (GE) we use a longer time series from January 1995, i.e. 192 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 GE model, we use 73 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics (2011). All involved indices must start before 1993. That’s why we have excluded 19 indices from the previously used set, including such major ones as communication, education, and recreation. They started after 1994.

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. 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.

Currently, General Electric (GE) is the second biggest company in the S&P 500 list just shy from Exxon Mobil. The defining indices are as follows: the index of motor vehicle maintenance and repair (MVR) and the index of motor vehicle insurance (MVI), both are subcategories of the transportation index. Both CPI components are contemporary with the share price. Figure 1 depicts the evolution of both indices which provide the best fit model, i.e. the lowermost RMS residual error, between January and December 2010:

GE(t) = -1.72*MVR(t) – 0.46*MVI(t) +15.67(t-2000) + 292.6

The predicted curve in Figure 2 is in sync with the observed one. The residual error is of $2.84 for the period between January 1995 and December 2010. Both defining components, especially MVI, grew at a slightly higher than usual rate between 2007 and 2009. This effect has pushed down the share price in 2008 and in the beginning of 2009. Since 2009, both indices evolve at a lower rate and the price has been showing a weak increase. The GE price will hardly be growing at a healthy rate in 2011 and looks slightly overestimated, as Figure 3 demonstrates.

Figure 1. Evolution of the price indices MVR and MVI.

Figure 2. Observed and predicted GE share prices.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $2.84.

An estimate of Morgan Stanley share price in 2011

We introduced a simple deterministic pricing model in 2009 [1]. Originally, it was based on an assumption that there exists a linear link between a share price (here only the stock market in the United States is considered) and the differences between various expenditure subcategories of the headline CPI. The intuition behind the model was simple - a higher relative rate of price growth (fall) in a given subcategory of goods and services is likely to result in a faster increase (decrease) in stock prices of related companies. In the first approximation, the deviation between price-defining indices is proportional to the ratio of their pricing powers. The presence of sustainable (linear or nonlinear) trends in the differences allows predicting the evolution of the differences, and thus, the deviation between prices of corresponding goods and services. The trends are the basis of a long-term prediction of share prices. In the short-run, deterministic forecasting is possible only in the case when a given price lags behind defining CPI components.

In its general form, the pricing model is as follows (Kitov, 2010):

sp(tj) = Σbi∙CPIi(tj-ti) + c∙(tj-2000 ) + d + ej     (1)

where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-ti) is the i-th component of the CPI with the time lag ti, i=1,..,I; bi, c and d are empirical coefficients of the linear and constant term; ej 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 fourteen months. Apparently, this is an artificial limitation and might be changed in a more elaborated model. In any case, a fourteen-month lag seems to be long enough for a price signal to pass through.

System (1) contains J equations for I+2 coefficients. Since the sustainable trends last more than five years, the share price time series have more than 60 points. For the current recent trend, the involved series are between 70 and 110 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.

For the sake of completeness we always retain all principal subcategories of goods and services. Among them are the headline CPI (C), the core CPI, i.e. the headline CPI less food and energy (CC), the index of food and beverages (F), housing (H), apparel (A), transportation (T), medical care (M), recreation (R), education and communication (EC), and other goods and services (O). In this model, we use 92 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics (2011). Many indices were started as late as 1998. It was natural to limit our modeling to the period between 2000 and 2010, i.e. to the current long-term trend.

There are two sources of uncertainty associated with the difference between observed and predicted prices. First, we have taken the monthly close prices (adjusted for splits and dividends) from a large number of recorded prices: monthly and daily open, close, high, and low prices, their combinations as well as averaged prices. Without loss of generality, one can randomly select for modeling purposes any of these prices for a given month. By chance, we have selected the closing price of the last working day for a given month. The larger is the fluctuation of a given stock price within and over the months the higher is the uncertainty associated with the monthly closing price as a representative of the stock price.

Second source of uncertainty is related to all kinds of measurement errors and intrinsic stochastic properties of the CPI. One should also bear in mind all uncertainties associated with the CPI definition based on a fixed basket of goods and services, which prices are tracked in few selected places. Such measurement errors are directly mapped into the model residual errors. Both uncertainties, as related to stocks and CPI, also fluctuate from month to month.

Morgan Stanley (MS) is an example of a changing pricing model. In [3] we reported that the defining CPIs in 2008 were the index of housing operations (HO) and the index of food away from home (SEFV). However, through the second half of 2010 the defining indices are different: the index of food less beverages (FB) and the index of information technology, hardware and software (IT), which is a part of the communication index. The former CPI component is contemporaneous with the share price and the latter one leads by 1 month. Figure 1 depicts the evolution of both indices. As discussed in our previous posts on food, it is likely that the index for food will be slowly growing during the next two years. This growth has a negative influence on the share price – the fall in the share price in 2008/2009 is clearly associated with the spike in the food price index. Same effect was well described by the SEFV index in the previous model. The IT index is characterized by a long term decline and will hardly be growing during the next several years. Both, linear trend and constant term, have positive influence on the price.

These defining components provide the best fit model, i.e. the lowermost RMS residual error, between July 2010 and December 2010. The best-fit 2-C model for MS(t) is as follows:

MS(t) = -3.32*FB(t) – 17.34*IT(t-1) +0.68(t-2000) +904.4

The predicted curve in Figure 2 is in sync with the observed one. The residual error of $3.98 for the period between July 2003 and December 2010. The model accurately predicts the share price in the past. From the overall behaviour of the defining CPIs one may expect that the MS price will be stalled or slightly growing in the first quarter of 2011.

Figure 1. Evolution of the price index of food less beverages (FB) and information technology (IT).

Figure 2. Observed and predicted MS share prices.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $3.98.

References

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

2. Kitov, I. (2010). Deterministic mechanics of pricing. Saarbrucken, Germany: LAP LAMBERT Academic Publishing.

3. Kitov, I. (2010). Modelling share prices of banks and bankrupts, Theoretical and Practical Research in Economic Fields, ASERS, vol. I(1(1)_Summer) pp. 59-85

Legg Mason may spurt

It is a great pleasure to revisit the previous version of the pricing model of Legg Mason (LM). During the last two years it has been stable and is based on the index related to food (food at home, FH) and the index of appliances (APL) from the housing index. Due to the uncertainty in the defining indices and closing prices as the parameter characterizing given shares, the lead of the FH has slightly changed from 4 months in the previous model to 5 months in the current model. The appliance index still leads by 13 months.

Overall, the predicted time series is very close to the observed one in Figure 1, with standard deviation of $7.33 between July 2003 and December 2010. The largest input to the standard deviation comes from a short period in 2006. Otherwise, both curves are very close even during the dramatic fall from $80 per share in the end of 2007 to $10 per share in February 2009 and during the fast recovery in 2009. The best-fit 2-C model [1] for LM(t) is as follows:

LM(t)= -3.61*FH(t-5) – 9.30*APL(t-13) + 16.62(t-2000) + 1317.6

The predicted curve actually leads the observed price by 5 months, as red line in Figure 1 illustrates. Therefore, one can foresee all major changes in the price. Without loss of generality, the LM price is completely defined by the behaviour of the two defining CPI components.

Currently, the model predicts a quick growth in the share price. Therefore, it would be instructive to revisit the prediction at a monthly rate. In January and February 2011, the price is expected at $45 and $56, respectively. Today, the price is at $36.

Figure 1. Observed and predicted LM share prices.

References

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

Xilinx share

This is a good example of a stable pricing model [1]. We already reported on this case in July 2010. The model for Xilinx (XLNX) is defined by the index of communication (CO-CUUR0000SAE2) and that of information and information processing (INF-CUUR0000SAE21). The former CPI component leads the share price by 4 months and the latter one leads by 11 months. From our past experience, the larger is the lag the more unreliable is the model, but this specific model has been valid since 2008.

These defining components provide the best fit model, i.e. the lowermost RMS residual error, between August 2009 and December 2010. The best-fit 2-C model for XLNX(t) is as follows:

XLNX(t) = -3.57*CO(t-4) + 4.21*INF(t-4) +1.88(t-2000) – 55.32

The predicted curve in Figure 2 leads the observed price by 4 months with the residual error of $1.89 for the period between July 2003 and December 2010. The model accurately predicts the share price in the past and foresees increasing price in the first quarter of 2011.

Figure 1. Evolution of the price index of communication (CO) and information (INF).

Figure 2. Observed and predicted XLNX share prices.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $1.89. The largest errors were observed in 2004 and 2009.

References

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

Real GDP in New Zealand - a decade of slow growth ahead

Following the post on the German real GDP per capita, we present a model for New Zealand. It was also obtained by the trial-and-error method [1]. Empirical constant A and the specific age, Ns, in the defining equation:

g(t) = dlnG(t)/dt = A/G(t) + 0.5dlnNs(t)/dt (1)

have been varied in order to fit amplitude and major features of the observed curve. The best fit annual increment value is A=$220 (1990 US$), i.e. less than in France and Germany. Surprisingly, the specific age population in New Zealand is 14 years, which is different from that in the US, Japan, France, and Germany. The age pyramid enumerated by the 2006 census was extrapolated in the past and in the future in order to estimate the number of 14-year-olds in (1).

Figure 1 presents the observed and predicted GDP growth rates for New Zealand. As for the other countries, all original readings of GDP were obtained from the Conference Board database. Both curves in the upper panel are characterized by high-amplitude oscillations likely associated with measurement errors. Therefore, in the lower panel of Figure 1, the annual curves are smoothed with MA(5) and MA(3), respectively.

Without prejudice, we have failed to find such a good prediction of real GDP elsewhere and would appreciate any information on a better model. The shape, amplitude and timing of the curves are in an excellent agreement after 1980.

Overall, there is no danger of a deep recession in New Zealand, but the rate of real economic growth will be very low (on average ~0.5% per year) in the years to come. Before 1980, data are likely not reliable due to significant revisions to relevant definitions.

Figure 1. Upper panel: Observed and predicted growth rate of real GDP per capita in New Zealand. Lower panel: The observed curve is smoothed with a 5-year moving average. The predicted rate is smoothed with MA(3). One can observe an outstanding agreement between the smoothed curves.

German GDP and WWII

Several days ago we presented in this blog an empirically correct model of real economic growth. The concept describing the evolution of real Gross Domestic Product (per capita) is very simple and is based solely on the age structure in a given developed country. It was empirically and statistically proved that the growth rate, g(t), of real GDP per capita, G(t), is driven by the attained level of real GDP per capita and the change in a specific age population, Ns. According to this model, the asymptotic growth rate of real GDP in developed countries can be completely characterized by constant annual increment A = const. All fluctuations around this constant increment can be explained by the change in the number of people of the country-specific age:

g(t) = dlnG(t)/dt = A/G(t) + 0.5dlnNs(t)/dt (1)

Equation (1) is the quantitative model that has been constructed empirically and tested statistically.

It is important to stress that the mainstream models such as Solow model and its successors are all based on an assumption that the rate of real economic growth is asymptotically constant. This assumption is empirically wrong and is rejected by statistical tests. We made no theoretical assumptions and our model came directly from data and was formulated at the initial stage of our empirical study. Therefore, we have a model that fits observations best.

We have already presented the cases of US, Japan, and France in this blog. The next country under investigation is Germany. The best fit constant increment is (A=) $260 (1990 US dollars) and the defining age is eighteen years, as in France. The age distribution from 2002 allows a prediction at an 18-year horizon. Figure suggests a slow-down in 2009 and likely a deeper recession in 2011, with a year of growth in 2010. On average, the beginning of 2010s will be characterized by very poor performance of the German economy. It will be even worse than in the 1990s. The population trough in N18 observed the 2010s is likely a remote and decaying echo of the WWII, which was amplified by the reunification turbulence in the early 1990s. The rate of birth was suppressed by the uncertainty of socio-political future of Germany. In the late 2000s and early 2010s, the drop in N18 might be compensated by immigration. In that case, real GDP may grow due to the rising working age population, i.e. due to extensive factors.

Here, we would like to emphasise that the prediction of the 2009 slowdown could be easily obtained in 2002, i.e. seven years before it happened! The estimates of population age structure are slightly noisy, however. Otherwise, the agreement between the observed and predicted curves is excellent after some years of turbulence associated with the reunification.

Figure 1. Observed and predicted rate of real GDP growth in Germany after the reunification. The predicted curve is obtained from relationship (1) with A=$260. Upper panel: Original curves. Lower panel: The original curves smoothed with MA(3). One should not expect a recession period before 2011, but the year of 2009 is very close to recession.