One of the commenters made a good remark that any quantitative model fails to predict for extended periods. In other words, the commenter doubts that our concept is valid and is able to provide an accurate description when the modeling period is extended back into the past. Fortunately, we have already addressed this issue in an article that was published by the Journal of Applied Research in Finances.

In order to explain our approach to stock pricing we have to start with an important observation on the difference between the core and the headline CPI. Four years ago we showed that there exist linear trends in consumer and producer price indices. Basically, it was found that the difference between the core CPI, CC, and the headline CPI, CPI, can be approximated by a linear time function:

dCPI(t) = CC(t) - CPI(t) = A1 + B1t

where dCPI(t) is the difference, A1 and B1 are empirical constants, and t is the elapsed time. Thus, the "distance" between the core CPI and the headline CPI is a linear function of time, with a positive or negative slope B1.

Figure 1 displays this difference from 1960 to 2012. There are three distinct periods of linear dependence on time: from 1960 to 1980, from 1981 to 1998, and from 2002 to 2008. The second period is characterized by a linear trend with slope B1=+0.65, and the third one has a larger negative slope of B1=-1.52. There are also two turning points or short time intervals - between 1980 and 1981, and from 1999 to 2002, where the trends undergo major changes. In 2008, we expected the difference to form a new linear trend, which would repeat the previously observed by duration and slope. In Figure 1, the green solid line represents the expected trend between 2009 and 2015. Currently, there is some deviation from the expected trend. We believe that this deviation is a temporary one and the core CPI will soon regain its pricing power over the prices of energy and food. For this, oil price has to fall to the level of $70 per barrel, which we observed in October 2011.

Figure 1. The difference between the core and headline CPI as a function of time. One can distinguish three periods of quasi-linear behavior with two distinct turning points. For the second and third periods, linear regression lines are characterized by slopes B1=+0.65 and B1=-1.52, respectively. The green line represents the expected trend between 2009 and 2015, which we predicted as a mirror reflection of the previous trend.

This discussion is crucial for our stock pricing concept. It links the change in the difference to the change in pricing power. Apparently, when the difference turns to a trend with an opposite sign, the pricing power of the relevant goods and services has to swap as well. As a result linear coefficients in our pricing model have to change their signs as well.

The pricing model is a simple one. We assume the presence of a linear link between a stock price, say that of ConocoPhillips (NYSE:COP), and the difference between the core and headline CPI,

COP(t) = A2 + B2dCPI(t)

where A2 and B2 are empirical constants, t is the elapsed time, and t2≥0 is the time delay between the stock and the CPI changes, i.e. the CPI may lag behind or lead the price.

We have already reported on the recent ConocoPhillips model and estimated all coefficients which best fit the observed price between July 2003 and February 2012:

COP(t) = -5.5dCPI(t) + 75

Figure 2 depicts the observed and predicted price since 1982. The above model does predict well after 1999, but there is no fit before 1999. Actually, the predicted and observed curves deviate spectacularly. At first glance, one might suggest that the dCPI provides no information about the evolution of the COP price.

The power of our pricing concept easily resolves this conflict. As we mentioned above, Figure 1 shows that the linear trend before 1999 was positive and after 2002 is a negative one. In terms of econometrics, there was a structural break in the dCPI behavior. The set of long-term economic links between goods and services, comprising the CPI and defining the linear trend in the dCPI between 1982 and 1999, underwent a three-year-long transition to a new set of links and constraints. In turn, this new set defines the trend observed from 2002 to 2008. It's likely that the same trend is observed now. A reasonable assumption is that the sign of slope in the equation for COP(t) should also change to an opposite one. Since the positive slope between 1981 and 1999 is only between a half and one third of that between 2002 and 2008, one can expect that the slope observed before 1999 should also be divided by a factor of ~3.

After reversing the sign and calibrating relevant amplitude and level between 1982 and 1998 (we included the transition into the second segment) we have obtained a much better fit as depicted by green line in Figure 2:

COP(t) = 1.7dCPI(t) - 5; t between 1981 and 1998

Finally, a complete prediction of the COP price between 1982 and 2012 is obtained.

There is no special need to describe the price in the early 1980s using the CPI difference. All subcategories of the consumer price index, except the index for energy, are parallel before 1982. Therefore, the difference between any two indices, including the headline and core CPI, is constant, i.e. it contains no information on the changes in stock prices.

Figure 2. Historic (monthly closing) prices for COP (black line) and the scaled difference between the core CPI and the headline CPI (predicted price): green line from 1982 to 1998 and red line since 1998.

We expected that COP stocks will follow the new trend in the dCPI (green line) in Figure 1, as it did between 1985 and 2008, one will be able to predict the "trend price" at any given time before 2015. It did not happen yet and we are waiting for a turn to the new trend when oil price will go down. Meanwhile, any large deviation from the trend which will be compensated at a few year horizons might provide a good hint for short-term trading.

Overall, our pricing concept easily matched the challenge of the commenter. The difference between the core and headline CPI gives a good approximation to the evolution of COP price since 1982. There are short periods of rapid and deep fall in stock price which might be associated with the change in linear trends.

**Disclosure: **I have no positions in any stocks mentioned, but may initiate a long position in COP over the next 72 hours.