Avon Products share price model

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

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

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

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

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

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

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

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

Figure 3. Residual error of the model.

Beware of the BEA!

In our previous post, we mentioned a structural break in the Okun’s law around 1978. We explained this break by an artificial change in the definition of the GDP deflator (as defined by the Bureau of Economic Analysis) in the very same time, as also was described in this blog. Here we test quantitatively the change in the estimated Okun’s law coefficients as related to the introduction some new definition of GDP.

Figure 1 displays two time series – the consumer price index, CPI, and the gross domestic purchases price index, dGDP. These time series diverge since 1978 and thus the estimates of real GDP, which is nominal GDP reduced by the GDP deflator, are biased relative to the period before 1978. One should not use the real GDP time series as it is published by the BEA when modelling longer time series including years before and after 1978.

The CPI and dGDP curves coincide when the latter one is corrected by a factor of 1.2, as is also shown in Figure 1. Therefore, one must use the corrected GDP deflator when modelling a time series covering periods before and after 1978.

Figure 1. CPI, dGDP, and 1.2*dGDP.

We made a silly mistake in our study of Okun’s law when tried to use the estimates of real GDP published by the BEA and had to introduce an artificial structural break in order to fit both periods. The best-fit model (Okun’s law) minimizing the RMS error was as follows:

du = -0.406dlnG + 1.113, t<1979

du = -0.465dlnG + 0.866, t>1978

where du is the change in the rate of unemployment and dlnG in the change rate of real GDP per capita. All coefficients we estimated by a LSQ technique minimizing the overall model error. Figure 2 shows the predicted and observed rate of unemployment between 1950 and 2010.

Thus, the structural break expresses itself in a change of slope and intercept around 1978, this year was also estimated in the same LSQ procedure. The ratio of slopes is 1.15, i.e. very close to 1.2 obtained from the CPI and dGDP. Considering the uncertainty in both slopes in the above relationship, which is approximately 0.05, one can conclude that the break in 1978 is entirely artificial and our version of Okun’s law has no structural breaks whatsoever. Essentially, the predicted curve in Figure 2 could be obtained in one piece with a correct dGDP. One has to use the same definition of GDP before and after 1978.

Beware of the BEA! It is not a surprise that economists can not find any clear relationships between macroeconomic variables - they are wrongly measured and misrepresented.

Figure 2. Observed and predicted rate of unemployment in the US.

Some corrections to David Altig's job market charts

David Altig presented some projections of the unemployment rate based on various monthly increments in employment.  It was a crude estimate because it did not include inherent fluctuations in the growth of working age population and labor force participation rate. It is much better to use Okun’s law linking unemployment and the real GDP growth.

Previously in this blog, we presented a version of Okun’s law for the rate of unemployment in the USA since 1955 as defined by real GDP per capita. We have estimated Okun’s law coefficients in two different segments using a standard LSQ technique. The reason behind the split into two segments was the change in realGDP estimation procedure introduced around 1978 - the definition of the GDP deflator was dramatically changed. We discussed this important methodical issuein our blog.

The best-fit (dynamic) model minimizing the RMS error of the cumulative model is as follows:

du = -0.406dlnG + 1.113, t<1979

du = -0.465dlnG + 0.866, t>1978 

This model suggests a smaller shift in the slope and a larger change in the intercept around 1979. This Okun’s law is characterized by a standard error of 0.53% for the period between 1958 and 2010. The average rate of unemployment for the same period is 5.6% with an average annual increment of 1.06%.

            Using the relationship for the period after 1979, one can estimate the evolution of the unemployment rate for various growth rates o real GDP per capita. We have selected three different values: 1% per year, 1.86% per year, and 3% per year. The first value is approximately equal to the mean growth rate between 2000 and 2010 (11 years) which is 0.93% per year. The second value provides a constant rate of unemployment, as defined by the ratio of coefficients 0.866/0.465 and is slightly higher than the mean rate after 1980 (1.63% per year). The third value is very high and just demonstrates the condition to reduce the rate of unemployment to 4% by 2020.  Figure 1 depicts the predicted and observed rate of unemployment after 1980 and these three projections. This is a more accurate projection than that by David Altig.

I do not see any opportunity for the rate of unemployment to fall any time soon. In the long run, unemployment will remain high. For the slow growth scenario expected by the FRB, u may reach 13% by 2020.

 Figure 1. Observed, predicted and projected rate of unemployment in the USA.

Procter and Gamble - stable share price in Q4 2011

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

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

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

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

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

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

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

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

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

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

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

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

Figure 3. The model residual error.

Loews Corporation share price in 2011 Q4

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

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

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

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

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

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

Figure 2. Observed and predicted share prices.

On the evolution of Forest Laboratories stock prices

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

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

Thus, three models mentioned above are as follows: 

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

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

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

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

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

Figure 2. Observed and predicted FRX share prices. 

Figure 3. The model residual.  

BofA, how are you doing?

Twenty months ago we presented a preliminary share price model for Bank of America (NYSE: BAC). It was based on the decomposition of the price into a weighted sum of two consumer price components, a linear time trend and a constant (see details here). All coefficients, time lags and CPIs were estimated by the least squares as applied to the model error.   Here we revisit the model using new data through September 2011. The newly estimated model differs from the preliminary one due to strict constraints on time lags and new data.

The preliminary model was a defined by the index of food without beverages (FB) and that of food away from home (SEFV). The latter CPI component leads by 13 months. From our past experience, the larger is the lag the more unreliable is the model. So it happened to the preliminary model and the best-fit two component model for BAC(t) is as follows:  

BAC(t)= -2.35OHF(t) + 1.19H(t) + 1.97(t-1990) + 165.79

   

where OHF is the consumer price index of other food at home and H(t) is the index of housing, both having notime lag behind the price (see Figure 1 for the evolution of the CPIs). Both CPIs are available only for August 2011, and thus we actually have a one month lead of the CPIs, if they would have been already available for September 2011. One has to reestimate the model when the CPIs for September are published by the BLS . The standard deviation of the modeling error is $2.80 for the period between July 2003 and September 2011.

Figure 2 depicts the observed and predicted prices and indicates that the market overestimates the share above the predicted level. The overestimation is also clear from Figure 3 where the model residual is displayed. All in all, Bank of America might have serious problems, which could bring its share price down to $1 in the near future. Similar problems had Lehman Brothers and other banks several months before bankruptcy. BofA aslo was a bankrupt, but    was bailed out.

Figure 1. The evolution of the defining consumer price indices.

Figure 2. Observed and predicted BAC share prices. 

Figure 3. The model error