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.

Why the Economic Projections of Federal Reserve Board are inconsistent

The FRB members have recently projected the evolution of key macroeconomic variables including real GDP and the rate of unemployment. In our blog , we have developed a very accurate model linking the rate of unemployment in the US to the rate of real GDP (per capita) growth: (A series of posts has resulted in a working paper.) The following relationship was estimated:

du = -0.62dlnG + 1.09,  (1)

When integrated between t0 and t, equation (1) can be rewritten in the following form:

u(t) = u(t0) + bln[G/G0] +a(t-t0) + c  (2)

Without loss of generality, we assume t0=0. The intercept c≡0, as is clear for t=t0. Instead of integrating (2), we calculate cumulative sums of the annual estimates of du and lnG with appropriate initial conditions. The cumulative sum of du’s is the time series of the unemployment rate. Figure 1 depicts the measured and observed curves for the period between 1958 and 2011. The agreement is excellent and has been obtained by a formal statistical method (LSQR).

 From (1) it follows that higher rates of GDP growth decrease the rate of unemployment. The FRB has projected real GDP with the highest rates of 2.7% in 2012, 3.2% in 2013, and 4% in 2014. We reduce these rates by 1% per year to estimate the per capita rate of growth, i.e. the growth in population is 1% per year. Using (2) we calculate the rate of uneployment which will correspond to the projected real GDP.

Figure 1 also depicts these predicted rates for 2012 to 2014 by open circles. The rates of unemployment projected by the FRB are shown by red circles. There is a significant deviation between the predicted and projected rates, which likely manifests the inconsistency in the FRB member's models of unemployment.

One may check these projections in 2015. 

Figure 1. The observed and predicted rate of unemployment in the USA between 1958 and 2010.The projected rate of unemployment (middle point of the projections) is shown by red circles. 

Who is responsible for income inequality? Blame old men.

I've plotted a series of mean personal income estimates borrowed from the Census Bureau. There are 10-year age bins with data from 1967 for men and women separately. The first plot shows that the male mean income is much higher than that of female. The gap between them has been slightly decreasing since 1974 but not spectacularly. Therefore, a higher rate of income growth for women results in decreasing income inequality, i.e. convergence of mean incomes.
In order to highlight the age and sex groups growing at the the highest rates all mean values in given age groups are normalized to 1967 (except the youngest group normalized to 1974). These plot show the following empirical results:
1. Young women (especially between 25 and 44) have been effectively closing the income gap with men.
2. Younger male groups (from 15 to 34) have suffered absolute decrease in mean income since 1974!
3. For men, the highest rates of income growth belongs to the eldest group. Hence, one has to blame them for increasing inequality. Some of them are blogging on inequality.

Income inequality paradox - family vs. personal median income

One can always find a good graph to illustrate increasing income inequality in the US. Lane Kenworthy and then Paul Krugman have demonstrated a dramatic deviation between the median family income and GDP per capita.  We have already posted on the problems behind the definition of income and GDP. Here we illustrate another paradox of the data published by the US Census Bureau. From the same source we retrieved the following time series: median personal and family income (chained $), GDP per capita (chained $) and Gini ratios for family and personal income distribution. Figure 1 shows the median personal and family income as normalized to 1974, and the GDP per capita time series also normalized to 1974. This is practically the plot from Lane Kenworthy except the personal income median. Instructively,  both median curves are very similar with small deviations likely associated with the estimation procedure than with actual changes.

Figure 1. Median incomes and GDP per capita in the US.

Now we plot another values also published by the Census Bureau. Figure 2 depicts Gini ratios for family and personal income distribution as calculated by the Bureau.

Figure 2. Gini ratios for personal and family income distributions measured by the Census Bureau.

A big surprise - Gini ratio for personal income has been decreasing since 1994 (no estimates before this point). So, the deviation between the GDP per capita and median income does not support increasing inequality.  More likely, it is something wrong with the Census Bureau.

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. 

Quarterly report: the performance of share price model for Hewlett Packard

This is quarterly report of the performance of our share price model. Hewlett Packard (NYSE:HPQ) provides a good example of a successful share price prediction at a several month horizon.  We have already published our predictions at a four month horizon four times (July 2010, January 2011, March 2011, July 2011, and September 2011). All predictions were based on our concept of share pricing as decomposition into a weighted sum of two CPI components.  We calculated the evolution of the monthly closing price (adjusted for dividends and splits). Here we test and update the model using data through December 2011. The model is still accurate and robust. 

Originally, the long term model for HPQ share price was defined by the index of food without beverages (FB) and that of rent of primary residency (RPR). The former CPI component led the share price by 4 months and the latter one led by 5 months. Figure 1 depicts the overall evolution of both involved indices through December 2011 (this is the reason of the time lead increase by 1 month relative to previous models where CPI were not contemporaneous). Below we present four best-fit 2-C models for HPQ(t) obtained at different times:  

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

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

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

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

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

The predicted curves are shown in Figure 2 (March, September, and December 2011). From Figure 2, we predict the price to fall to $20 in the first quarter of 2012 and then rise to $25 in Q2.  

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

Figure 2. Observed and predicted HPQ share prices in March, September, and December 2011. The contemporaneous prediction is shown by solid red line.  High and low prices are shown by dashed lines.