All models have been obtained using our concept of stock pricing as a decomposition of a share price into a weighted sum of two consumer price indices. The background idea is a simplistic one: there is a trade-off between a given share price and goods and services the relevant company produces or provides. For example, the energy consumer price should influence the price of energy companies. Let's assume that some set of consumer prices (as expressed by an appropriate consumer price index, CPI) drives the company stock price. The net effect of the relevant CPI change can be positive or negative.

In real world, each company competes not only with those producing similar goods and services, but also with all other companies on the market. Therefore, the influence of the driving CPI should also depend on all other goods and services, and thus, relevant CPIs. To model the net change in the market prices we introduce just one reference CPI. In quantitative terms, it has to best represent the dynamics of changing price environment. Hence, our pricing model includes two defining CPIs: the driver and the reference. Because of possible time delays between action and reaction (the time needed for any price changes to pass through), the defining CPIs may lead the modeled price or lag behind.

We have borrowed the time series of GE monthly closing prices (through March 2014) from Yahoo.com. CPI estimates (not seasonally adjusted) through February 2014 are published by the BLS. According to the procedure described in Appendix, the evolution of GE share price is defined by the index of transportation services (TS) and the index of pets, pet products and services (PETS). These indices were selected from a large set of 92 CPIs covering all consumer price categories. All possible CPI pairs with all possible time lags and leads were tested one by one. The TS/PETS set, which minimizes the model error, is considered as the defining pair. For GE, the time lags are 6 and 2 months, respectively, and the best-fit model is as follows:

*GE(t) = -*1.536*PETS(t-*2*) -* 0.631*TS(t-*6*) +* 11.805*(t-*2000*) +* 291.08, February 2014 (1)

where *GE(t)* is the GE share price in U.S. dollars, *t* is calendar time. All coefficients in the above relationship were estimated together with their uncertainties using the linear regression technique . Table 1 confirms that all coefficients are statistically significant.

Table 1. Statistical estimates for the model coefficients

Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |

d | 291.08 | 10.5133 | 27.670 | 6.72E-55 | 270.095 | 311.7124 |

b1 | -1.536 | 0.0439 | -34.972 | 4.42E-66 | -1.62381 | -1.44985 |

b2 | -0.632 | 0.0512 | -12.325 | 2.79E-23 | -0.73194 | -0.52938 |

c | 11.805 | 0.4211 | 28.024 | 1.73E-55 | 10.96724 | 12.63418 |

Figure 1 displays the evolution of both defining indices since 2003. Figure 2 depicts the high and low monthly prices for GE share together with the predicted and measured monthly closing prices (adjusted for dividends and splits). The model prediction is best described by the coefficient of determination Rsq.=0.93. The predicted and observed time series are cointegrated. Thus the Rsq. estimate is not biased.

From relationship (1), it seems that the index of pets, pet products and services define the evolution of GE price. Actually, the model implies that PETS index does NOT affect the share price. This index provides a dynamic (price) reference rather than the driving force. Here is a simple example how to understand the term "dynamic reference". Imagine that a swimmer needs to swim 20 km along a river. Let's assume that for this experienced swimmer the average speed is 5 km/h. How much time does s/he need? The quick answer 4 hours is wrong. One cannot calculate the time needed without knowing the stream speed and direction. This stream is the dynamic reference (or moving coordinate reference system) for the swimmer. The stream speed can also vary over time producing a non-stationary coordinate reference system. The intuition behind our pricing concept is similar - the driving CPI is not enough to calculate the price change, one needs to know "the stream speed", i.e. the market movements. The CPI representing "dynamic reference" for GE was selected from 91 CPIs.

The model is stable over time: it has the same defining CPIs, coefficients and time lags for longer periods of time. Table 2 lists the best fit models, i.e. slopes, *b1* and *b2*, for two defining CPIs, time lags, slope of time trend, *c*, and free term, *d*, for 7 months between July 2013 and February 2014. It is instructive that the same model was obtained in 2012, 2011, and 2010, also listed in Table 2. Therefore, the estimated GE model is reliable over 50+ months. The model residual for February 2014 is shown in Figure 3. Between July 2003 and February 2014, the model standard error is $1.51.

Overall, the model does not foresee any big change in GE price. One may expect some price fluctuations within the bounds of intermonth changes observed in the past. The predicted value for May 2014 is $27.7 (+-$1.5).

Table 2. The best fit models for the period between May 2010 and February 2014

Month | b1 | CPI1 | lag1 | b2 | CPI2 | lag2 | c | d |

Feb-14 | -1.536 | PETS | 2 | -0.632 | TS | 6 | 11.805 | 291.08 |

Jan | -1.546 | PETS | 2 | -0.633 | TS | 6 | 11.872 | 292.10 |

Dec-13 | -1.547 | PETS | 2 | -0.634 | TS | 6 | 11.883 | 292.39 |

Nov | -1.531 | PETS | 2 | -0.638 | TS | 6 | 11.815 | 291.87 |

Oct | -1.522 | PETS | 2 | -0.641 | TS | 6 | 11.773 | 291.54 |

Sep | -1.513 | PETS | 2 | -0.644 | TS | 6 | 11.732 | 291.24 |

Aug | -1.510 | PETS | 2 | -0.645 | TS | 6 | 11.723 | 291.16 |

Jul | -1.512 | PETS | 2 | -0.644 | TS | 6 | 11.728 | 291.14 |

Nov-12 | -1.549 | PETS | 2 | -0.711 | TS | 6 | 12.275 | 307.94 |

Oct | -1.544 | PETS | 2 | -0.712 | TS | 6 | 12.250 | 307.75 |

Sep | -1.540 | PETS | 2 | -0.714 | TS | 6 | 12.235 | 307.77 |

Aug | -1.530 | PETS | 2 | -0.719 | TS | 6 | 12.198 | 307.95 |

Jul | -1.527 | PETS | 2 | -0.720 | TS | 6 | 12.175 | 307.78 |

Jun | -1.522 | PETS | 2 | -0.723 | TS | 6 | 12.165 | 308.06 |

May | -1.513 | PETS | 2 | -0.732 | TS | 6 | 12.151 | 308.95 |

Apr | -1.507 | PETS | 2 | -0.740 | TS | 6 | 12.152 | 309.86 |

Dec-11 | -1.520 | PETS | 2 | -0.785 | TS | 6 | 12.443 | 196.23 |

Nov | -1.510 | PETS | 2 | -0.802 | TS | 6 | 12.479 | 197.24 |

Oct | -1.506 | PETS | 2 | -0.806 | TS | 6 | 12.475 | 196.70 |

Sep | -1.495 | PETS | 2 | -0.830 | TS | 6 | 12.539 | 198.53 |

Aug | -1.495 | PETS | 2 | -0.828 | TS | 6 | 12.528 | 197.20 |

Jul | -1.499 | PETS | 2 | -0.831 | TS | 6 | 12.566 | 196.60 |

Jun | -1.494 | PETS | 2 | -0.830 | TS | 6 | 12.535 | 195.46 |

May | -1.486 | PETS | 2 | -0.846 | TS | 6 | 12.572 | 196.39 |

Dec-10 | -1.450 | PETS | 2 | -1.006 | TS | 6 | 13.153 | 226.64 |

Nov | -1.427 | PETS | 2 | -1.040 | TS | 6 | 13.200 | 229.69 |

Oct | -1.435 | PETS | 2 | -1.027 | TS | 6 | 13.180 | 227.00 |

Sep | -1.442 | PETS | 2 | -1.016 | TS | 6 | 13.165 | 224.57 |

Aug | -1.447 | PETS | 2 | -1.006 | TS | 6 | 13.147 | 222.22 |

Jul | -1.487 | PETS | 2 | -0.949 | TS | 6 | 13.097 | 214.03 |

Jun | -1.500 | PETS | 2 | -0.925 | TS | 6 | 13.056 | 209.88 |

May | -1.515 | PETS | 2 | -0.899 | TS | 6 | 13.017 | 205.42 |

Figure 1. The evolution of TS and PETS indices

Figure 2. Observed and predicted GE share prices.

Figure 3. The model residual error: sterr=$1.51.

**Appendix**

The concept of share pricing based on the link between consumer and stock prices has been under development since 2008. In the very beginning, we found a statistically reliable relationship between ConocoPhillips' stock price and the difference between the core and headline consumer price index in the United States. Then we extended the pool of defining CPIs to 92 and estimated quantitative models for all companies from the S&P 500 list. The extended model described the evolution of a share price as a weighted sum of two individual consumer price indices selected from this large set of CPIs. We allow only two defining CPIs, which may lead the modeled share price or lag behind it. The intuition behind the lags is that some companies are price setters and some are price takers. The former should influence the relevant CPIs, which include goods and services these companies produce. The latter lag behind the prices of goods and services they are associated with. In order to calibrate the model relative to the starting levels of the involved indices and to compensate sustainable time trends (some indices are subject to secular rise or fall) we introduced two additional terms: linear time trend and constant. In its general form, the pricing model is as follows:

*sp(t _{j}) = Σ*

where

By definition, the bets-fit model minimizes the RMS residual error. It is a fundamental feature of the model that the lags may be both negative and positive. In this study, we limit the largest lag to eleven months. System (2) contains *J* equations for *I+2* coefficients. We start our model in July 2003 and the share price time series has more than 130 readings. To resolve the system, standard methods of matrix inversion are used. A model is considered as a reliable one when the defining CPIs are the same during the previous seven months.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

All models have been obtained using our concept of stock pricing as a decomposition of a share price into a weighted sum of two consumer price indices. The background idea is a simplistic one: there is a trade-off between a given share price and goods and services the relevant company produces or provides. For example, the energy consumer price should influence the price of energy companies. Let's assume that some set of consumer prices (as expressed by an appropriate consumer price index, CPI) drives the company stock price. The net effect of the relevant CPI change can be positive or negative.

In real world, each company competes not only with those producing similar goods and services, but also with all other companies on the market. Therefore, the influence of the driving CPI should also depend on all other goods and services, and thus, relevant CPIs. To model the net change in the market prices we introduce just one reference CPI. In quantitative terms, it has to best represent the dynamics of changing price environment. Hence, our pricing model includes two defining CPIs: the driver and the reference. Because of possible time delays between action and reaction (the time needed for any price changes to pass through), the defining CPIs may lead the modeled price or lag behind.

We have borrowed the time series of GE monthly closing prices (through March 2014) from Yahoo.com. CPI estimates (not seasonally adjusted) through February 2014 are published by the BLS. According to the procedure described in Appendix, the evolution of GE share price is defined by the index of transportation services (TS) and the index of pets, pet products and services (PETS). These indices were selected from a large set of 92 CPIs covering all consumer price categories. All possible CPI pairs with all possible time lags and leads were tested one by one. The TS/PETS set, which minimizes the model error, is considered as the defining pair. For GE, the time lags are 6 and 2 months, respectively, and the best-fit model is as follows:

*GE(t) = -*1.536*PETS(t-*2*) -* 0.631*TS(t-*6*) +* 11.805*(t-*2000*) +* 291.08, February 2014 (1)

where *GE(t)* is the GE share price in U.S. dollars, *t* is calendar time. All coefficients in the above relationship were estimated together with their uncertainties using the linear regression technique . Table 1 confirms that all coefficients are statistically significant.

Table 1. Statistical estimates for the model coefficients

Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |

d | 291.08 | 10.5133 | 27.670 | 6.72E-55 | 270.095 | 311.7124 |

b1 | -1.536 | 0.0439 | -34.972 | 4.42E-66 | -1.62381 | -1.44985 |

b2 | -0.632 | 0.0512 | -12.325 | 2.79E-23 | -0.73194 | -0.52938 |

c | 11.805 | 0.4211 | 28.024 | 1.73E-55 | 10.96724 | 12.63418 |

Figure 1 displays the evolution of both defining indices since 2003. Figure 2 depicts the high and low monthly prices for GE share together with the predicted and measured monthly closing prices (adjusted for dividends and splits). The model prediction is best described by the coefficient of determination Rsq.=0.93. The predicted and observed time series are cointegrated. Thus the Rsq. estimate is not biased.

From relationship (1), it seems that the index of pets, pet products and services define the evolution of GE price. Actually, the model implies that PETS index does NOT affect the share price. This index provides a dynamic (price) reference rather than the driving force. Here is a simple example how to understand the term "dynamic reference". Imagine that a swimmer needs to swim 20 km along a river. Let's assume that for this experienced swimmer the average speed is 5 km/h. How much time does s/he need? The quick answer 4 hours is wrong. One cannot calculate the time needed without knowing the stream speed and direction. This stream is the dynamic reference (or moving coordinate reference system) for the swimmer. The stream speed can also vary over time producing a non-stationary coordinate reference system. The intuition behind our pricing concept is similar - the driving CPI is not enough to calculate the price change, one needs to know "the stream speed", i.e. the market movements. The CPI representing "dynamic reference" for GE was selected from 91 CPIs.

The model is stable over time: it has the same defining CPIs, coefficients and time lags for longer periods of time. Table 2 lists the best fit models, i.e. slopes, *b1* and *b2*, for two defining CPIs, time lags, slope of time trend, *c*, and free term, *d*, for 7 months between July 2013 and February 2014. It is instructive that the same model was obtained in 2012, 2011, and 2010, also listed in Table 2. Therefore, the estimated GE model is reliable over 50+ months. The model residual for February 2014 is shown in Figure 3. Between July 2003 and February 2014, the model standard error is $1.51.

Overall, the model does not foresee any big change in GE price. One may expect some price fluctuations within the bounds of intermonth changes observed in the past. The predicted value for May 2014 is $27.7 (+-$1.5).

Table 2. The best fit models for the period between May 2010 and February 2014

Month | b1 | CPI1 | lag1 | b2 | CPI2 | lag2 | c | d |

Feb-14 | -1.536 | PETS | 2 | -0.632 | TS | 6 | 11.805 | 291.08 |

Jan | -1.546 | PETS | 2 | -0.633 | TS | 6 | 11.872 | 292.10 |

Dec-13 | -1.547 | PETS | 2 | -0.634 | TS | 6 | 11.883 | 292.39 |

Nov | -1.531 | PETS | 2 | -0.638 | TS | 6 | 11.815 | 291.87 |

Oct | -1.522 | PETS | 2 | -0.641 | TS | 6 | 11.773 | 291.54 |

Sep | -1.513 | PETS | 2 | -0.644 | TS | 6 | 11.732 | 291.24 |

Aug | -1.510 | PETS | 2 | -0.645 | TS | 6 | 11.723 | 291.16 |

Jul | -1.512 | PETS | 2 | -0.644 | TS | 6 | 11.728 | 291.14 |

Nov-12 | -1.549 | PETS | 2 | -0.711 | TS | 6 | 12.275 | 307.94 |

Oct | -1.544 | PETS | 2 | -0.712 | TS | 6 | 12.250 | 307.75 |

Sep | -1.540 | PETS | 2 | -0.714 | TS | 6 | 12.235 | 307.77 |

Aug | -1.530 | PETS | 2 | -0.719 | TS | 6 | 12.198 | 307.95 |

Jul | -1.527 | PETS | 2 | -0.720 | TS | 6 | 12.175 | 307.78 |

Jun | -1.522 | PETS | 2 | -0.723 | TS | 6 | 12.165 | 308.06 |

May | -1.513 | PETS | 2 | -0.732 | TS | 6 | 12.151 | 308.95 |

Apr | -1.507 | PETS | 2 | -0.740 | TS | 6 | 12.152 | 309.86 |

Dec-11 | -1.520 | PETS | 2 | -0.785 | TS | 6 | 12.443 | 196.23 |

Nov | -1.510 | PETS | 2 | -0.802 | TS | 6 | 12.479 | 197.24 |

Oct | -1.506 | PETS | 2 | -0.806 | TS | 6 | 12.475 | 196.70 |

Sep | -1.495 | PETS | 2 | -0.830 | TS | 6 | 12.539 | 198.53 |

Aug | -1.495 | PETS | 2 | -0.828 | TS | 6 | 12.528 | 197.20 |

Jul | -1.499 | PETS | 2 | -0.831 | TS | 6 | 12.566 | 196.60 |

Jun | -1.494 | PETS | 2 | -0.830 | TS | 6 | 12.535 | 195.46 |

May | -1.486 | PETS | 2 | -0.846 | TS | 6 | 12.572 | 196.39 |

Dec-10 | -1.450 | PETS | 2 | -1.006 | TS | 6 | 13.153 | 226.64 |

Nov | -1.427 | PETS | 2 | -1.040 | TS | 6 | 13.200 | 229.69 |

Oct | -1.435 | PETS | 2 | -1.027 | TS | 6 | 13.180 | 227.00 |

Sep | -1.442 | PETS | 2 | -1.016 | TS | 6 | 13.165 | 224.57 |

Aug | -1.447 | PETS | 2 | -1.006 | TS | 6 | 13.147 | 222.22 |

Jul | -1.487 | PETS | 2 | -0.949 | TS | 6 | 13.097 | 214.03 |

Jun | -1.500 | PETS | 2 | -0.925 | TS | 6 | 13.056 | 209.88 |

May | -1.515 | PETS | 2 | -0.899 | TS | 6 | 13.017 | 205.42 |

Figure 1. The evolution of TS and PETS indices

Figure 2. Observed and predicted GE share prices.

Figure 3. The model residual error: sterr=$1.51.

**Appendix**

The concept of share pricing based on the link between consumer and stock prices has been under development since 2008. In the very beginning, we found a statistically reliable relationship between ConocoPhillips' stock price and the difference between the core and headline consumer price index in the United States. Then we extended the pool of defining CPIs to 92 and estimated quantitative models for all companies from the S&P 500 list. The extended model described the evolution of a share price as a weighted sum of two individual consumer price indices selected from this large set of CPIs. We allow only two defining CPIs, which may lead the modeled share price or lag behind it. The intuition behind the lags is that some companies are price setters and some are price takers. The former should influence the relevant CPIs, which include goods and services these companies produce. The latter lag behind the prices of goods and services they are associated with. In order to calibrate the model relative to the starting levels of the involved indices and to compensate sustainable time trends (some indices are subject to secular rise or fall) we introduced two additional terms: linear time trend and constant. In its general form, the pricing model is as follows:

*sp(t _{j}) = Σ*

where

By definition, the bets-fit model minimizes the RMS residual error. It is a fundamental feature of the model that the lags may be both negative and positive. In this study, we limit the largest lag to eleven months. System (2) contains *J* equations for *I+2* coefficients. We start our model in July 2003 and the share price time series has more than 130 readings. To resolve the system, standard methods of matrix inversion are used. A model is considered as a reliable one when the defining CPIs are the same during the previous seven months.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

According to the reports of the Bureau of Economic Analysis (BEA), the proportion of personal (money) income in the Gross Domestic Product has not been changing since 1947. This is the year when the BEA started to measure personal incomes. We have revealed the source of virtual increase in income inequality - private companies redistribute their income in favor of personal income of their owners. The question is - how do they get extra money to redistribute to their private owners? This post answers this question - the US tax system started to reduce the level of tax for private companies. Primarily, it is made by increasing the rate of depreciation, which enterprises are officially permitted to charge for tax purposes (usually fixed by law). Hence, the tax law in responsible for the increasing income inequality.

We start with a graph showing the growth in GDP, gross personal income (GPI) and compensation of employees (paid) since 1929. Figure 1 demonstrates that the level of GPI has been rising faster than that of the GDP (and the compensation) since 1979. (The share of GPI in the GDP has been rising since 1979!) The difference between the GPI and GDP curves depicted in Figure 2 has a striking kink around 1979. And this is the start of the current rally in the rich families' personal income. In other words, a new political (taxation is a political issue) era started in 1979. We would like to stress again the proportion of the compensation of employees in the GDP has not been changing since 1929, with a small positive deviation in the end of 1990s and a negative deviation since 2009. This observation supports our previous finding that the proportion of personal (money) income in the GDP has not been changing.

So, where the extra money is from? The level of personal income has been actually increasing faster than that of the GDP and it should be the last fool, who lost its share in the GDP. Figure 3 shows two major components of the GPI. The net operating surplus (private) has been changing at the same rate as the GDP since 1929, while the proportion of taxes on production and imports has been growing at lower rate since 1980. We have allocated the source of income for rich families. They take money from the decreasing taxes. But what is the mechanism of money appropriation? Figure 4 demonstrates that the decrease in taxes goes directly into the increasing share of consumption of fixed capital.

This is the force behind the increasing income inequality.

The increasing share of the consumption of fixed capital is successfully converted in private money, not in investments! This is a political problem started in 1979.

It seems that there is no economic problem behind increasing income inequality.

Figure 1. GDP, GPI, and compensation of employees normalized to their respective levels in 1960.

Figure 2. The difference between the GPI and GDP curves in Figure 1.

Figure 3. GDP, net operation surplus (private), and taxes (on production and imports) normalized to their respective levels in 1960.

Figure 4. GDP and consumption of fixed capital normalized to their respective levels in 1960.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

According to the reports of the Bureau of Economic Analysis (BEA), the proportion of personal (money) income in the Gross Domestic Product has not been changing since 1947. This is the year when the BEA started to measure personal incomes. We have revealed the source of virtual increase in income inequality - private companies redistribute their income in favor of personal income of their owners. The question is - how do they get extra money to redistribute to their private owners? This post answers this question - the US tax system started to reduce the level of tax for private companies. Primarily, it is made by increasing the rate of depreciation, which enterprises are officially permitted to charge for tax purposes (usually fixed by law). Hence, the tax law in responsible for the increasing income inequality.

We start with a graph showing the growth in GDP, gross personal income (GPI) and compensation of employees (paid) since 1929. Figure 1 demonstrates that the level of GPI has been rising faster than that of the GDP (and the compensation) since 1979. (The share of GPI in the GDP has been rising since 1979!) The difference between the GPI and GDP curves depicted in Figure 2 has a striking kink around 1979. And this is the start of the current rally in the rich families' personal income. In other words, a new political (taxation is a political issue) era started in 1979. We would like to stress again the proportion of the compensation of employees in the GDP has not been changing since 1929, with a small positive deviation in the end of 1990s and a negative deviation since 2009. This observation supports our previous finding that the proportion of personal (money) income in the GDP has not been changing.

So, where the extra money is from? The level of personal income has been actually increasing faster than that of the GDP and it should be the last fool, who lost its share in the GDP. Figure 3 shows two major components of the GPI. The net operating surplus (private) has been changing at the same rate as the GDP since 1929, while the proportion of taxes on production and imports has been growing at lower rate since 1980. We have allocated the source of income for rich families. They take money from the decreasing taxes. But what is the mechanism of money appropriation? Figure 4 demonstrates that the decrease in taxes goes directly into the increasing share of consumption of fixed capital.

This is the force behind the increasing income inequality.

The increasing share of the consumption of fixed capital is successfully converted in private money, not in investments! This is a political problem started in 1979.

It seems that there is no economic problem behind increasing income inequality.

Figure 1. GDP, GPI, and compensation of employees normalized to their respective levels in 1960.

Figure 2. The difference between the GPI and GDP curves in Figure 1.

Figure 3. GDP, net operation surplus (private), and taxes (on production and imports) normalized to their respective levels in 1960.

Figure 4. GDP and consumption of fixed capital normalized to their respective levels in 1960.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

Right after this fall, we predicted a rally to 1550 according to our simple and straightforward investing tactics - we follow the trajectory of the S&P 500 observed during the rally between 2002 and 2007. In March 2013, the S&P 500 exceeded 1560 and currently we expect it to fall to 1450 at a one month horizon. If to follow up our procedure, as explained in Figures 1 through 6, it's time to sell and wait for a new short-term rally in June-October 2013. Then a tremendous fall down to 750 in 2015 is not excluded. Since 2009, the S&P 500 has been following the previous (2002-2007) trajectory up. This observation is the basis of our best guess that the index should not stay high after October 2013? Currently, I am in bonds and look for 1.5% to 1.7% yield to sell them in June.

In March 2012, we first published a graph which showed that the S&P 500 index would have a local fall in May 2012 at the level 1300. In a few sessions, we bought the S&P 500 index in May/June 2012 at the level 1287 to 1322. The initial idea was to sell by the end of 2013 at 1525 and get a 12% to 14% return. In September 2012, the S&P was at 1450, which was far above the expected level, and we decided to sell and wait a negative correction to 1350 to 1375 to re-enter the index. Selling at ~1460, we obtained an approximately 10% return in September. In October, the S&P 500 fell to 1350 as had been predicted in September and we re-entered in two sessions at 1348 and 1355. By the end of November the S&P 500 regained 4.5% (1416). Two weeks ago, we expected the index to rise to 1430 to 1450 in December 2012. This was considered as the best time to sell before the next negative correction in December 2012/ January 2013. In reality, this correction to 1400 was very short.

2012

Below we present the evolution of the S&P 500 and the step-by-step assumptions illustrating the decisions we have made since March 2012.

Figure 1 shows the evolution of the S&P 500 index since 1980. After 1995, the index behavior reveals some saw teeth with peaks in 2000 and 2007. The current growth resembles those between 1997 and 2000 and from 2003 and 2007. There are two deep troughs in 2002 and 2009 which are marked by red and green lines. For the current analysis we assume that the repeated shape of the teeth is likely induced by a degree of similarity in the evolution of macroeconomic variables. The intuition behind such an assumption is obvious - in the long run the stock market depends on the overall economic growth.

Having two peaks and troughs between 1995 and 2009, what can we say about the current growth in the S&P 500? Before making any statistical estimates, in Figure 2 we have shifted forward the original curve in Figure 1 in order to match the 2009 trough (blue line). When the 2002 and 2009 troughs are matched, one can see that the current growth path closely repeats that after 2002. The first big deviation from the blues curve in Figure 2 started in 2011 and had amplitude of 150 units (from 1210 to 1360). The black curve returned to the blue one in August/September 2011. From December 2011, we observed a middle-size deviation of about 100 units**.**

Figure 1. The evolution of the S&P 500 market index between 1980 and 2012.

In April 2012, we predicted a drop in the S&P 500 to the level of 1300 by the end of May. Figure 2 shows the predicted behavior in April and May 2012, with the predicted segment shown by red line. We expected that the path observed in the previous rally would be repeated with the bottom points coinciding. When this prediction realized, we invested at the average price 1320. In May 2012, the expected exit level was 1500 in October 2013.

Figure 2. The original S&P 500 curve (black line) and that shifted forward to match the 2009 trough (blue line). Red line - expected fall in the S&P 500: from 1400 in March to 1300 in May.

Figure 3 shows the evolution of the S&P 500 monthly closing price between May and August 2012. The S&P 500 closing level for August was 1430 and reached 1469 in the middle of September. This level provided a ten percent return over approximately 4 months. One can see that the observed level was far above the expected level (blue line). The return and the deviation from the expected level both made us think that this was the best time to exit. We sold the index on September 21 (1460) anticipating strong turbulence (economic, financial, and political) and an overall fall to 1375 at a few month horizon.

Figure 3. Same as in Figure 2 with an extension between May and August.

Figure 4 shows the evolution of the S&P 500 monthly closing price in September-November 2012. The October's closing level was 1411. On October 26, we put the November's level down to 1375. One can see that the red line intersects the blue curve. The previous history of the black and red lines intersection with the blue one made us think that the time to enter the market (S&P 500 index) was approaching. We expected to buy at 1350 to 1375.

Figure 4. Same as in Figure 2 with an extension between September and November 2012.

Figure 5 depicts the state of the S&P 500 on December 15, 2012. The index had a local minimum of 1347 in the middle of November and recovered to 1416 on November 30. This was 15 points less than the expected level of 1430 in December 2012. The red line intersects the blue one in January 2013 and a negative correction to 1400 (or less) was expected.

Figure 5. Same as in Figure 4 with an extension into December 2012.

Figure 6 displays the current situation with the S&P 500. As in a few cases after 2010, the actual index (red line) leads the expected one (blue) by a month or so. It has reached the (closing) level 1563 on March 14 and has been falling since then. As in April and November 2012, we expect the index to fall. The level is likely around 1455. Then it might regain its power and have a final spurt to 1560 in October 2013. Do not miss this last opportunity.

Figure 6. Same as in Figure 5 with an extension into March 2013.

]]>Right after this fall, we predicted a rally to 1550 according to our simple and straightforward investing tactics - we follow the trajectory of the S&P 500 observed during the rally between 2002 and 2007. In March 2013, the S&P 500 exceeded 1560 and currently we expect it to fall to 1450 at a one month horizon. If to follow up our procedure, as explained in Figures 1 through 6, it's time to sell and wait for a new short-term rally in June-October 2013. Then a tremendous fall down to 750 in 2015 is not excluded. Since 2009, the S&P 500 has been following the previous (2002-2007) trajectory up. This observation is the basis of our best guess that the index should not stay high after October 2013? Currently, I am in bonds and look for 1.5% to 1.7% yield to sell them in June.

In March 2012, we first published a graph which showed that the S&P 500 index would have a local fall in May 2012 at the level 1300. In a few sessions, we bought the S&P 500 index in May/June 2012 at the level 1287 to 1322. The initial idea was to sell by the end of 2013 at 1525 and get a 12% to 14% return. In September 2012, the S&P was at 1450, which was far above the expected level, and we decided to sell and wait a negative correction to 1350 to 1375 to re-enter the index. Selling at ~1460, we obtained an approximately 10% return in September. In October, the S&P 500 fell to 1350 as had been predicted in September and we re-entered in two sessions at 1348 and 1355. By the end of November the S&P 500 regained 4.5% (1416). Two weeks ago, we expected the index to rise to 1430 to 1450 in December 2012. This was considered as the best time to sell before the next negative correction in December 2012/ January 2013. In reality, this correction to 1400 was very short.

2012

Below we present the evolution of the S&P 500 and the step-by-step assumptions illustrating the decisions we have made since March 2012.

Figure 1 shows the evolution of the S&P 500 index since 1980. After 1995, the index behavior reveals some saw teeth with peaks in 2000 and 2007. The current growth resembles those between 1997 and 2000 and from 2003 and 2007. There are two deep troughs in 2002 and 2009 which are marked by red and green lines. For the current analysis we assume that the repeated shape of the teeth is likely induced by a degree of similarity in the evolution of macroeconomic variables. The intuition behind such an assumption is obvious - in the long run the stock market depends on the overall economic growth.

Having two peaks and troughs between 1995 and 2009, what can we say about the current growth in the S&P 500? Before making any statistical estimates, in Figure 2 we have shifted forward the original curve in Figure 1 in order to match the 2009 trough (blue line). When the 2002 and 2009 troughs are matched, one can see that the current growth path closely repeats that after 2002. The first big deviation from the blues curve in Figure 2 started in 2011 and had amplitude of 150 units (from 1210 to 1360). The black curve returned to the blue one in August/September 2011. From December 2011, we observed a middle-size deviation of about 100 units**.**

Figure 1. The evolution of the S&P 500 market index between 1980 and 2012.

In April 2012, we predicted a drop in the S&P 500 to the level of 1300 by the end of May. Figure 2 shows the predicted behavior in April and May 2012, with the predicted segment shown by red line. We expected that the path observed in the previous rally would be repeated with the bottom points coinciding. When this prediction realized, we invested at the average price 1320. In May 2012, the expected exit level was 1500 in October 2013.

Figure 2. The original S&P 500 curve (black line) and that shifted forward to match the 2009 trough (blue line). Red line - expected fall in the S&P 500: from 1400 in March to 1300 in May.

Figure 3 shows the evolution of the S&P 500 monthly closing price between May and August 2012. The S&P 500 closing level for August was 1430 and reached 1469 in the middle of September. This level provided a ten percent return over approximately 4 months. One can see that the observed level was far above the expected level (blue line). The return and the deviation from the expected level both made us think that this was the best time to exit. We sold the index on September 21 (1460) anticipating strong turbulence (economic, financial, and political) and an overall fall to 1375 at a few month horizon.

Figure 3. Same as in Figure 2 with an extension between May and August.

Figure 4 shows the evolution of the S&P 500 monthly closing price in September-November 2012. The October's closing level was 1411. On October 26, we put the November's level down to 1375. One can see that the red line intersects the blue curve. The previous history of the black and red lines intersection with the blue one made us think that the time to enter the market (S&P 500 index) was approaching. We expected to buy at 1350 to 1375.

Figure 4. Same as in Figure 2 with an extension between September and November 2012.

Figure 5 depicts the state of the S&P 500 on December 15, 2012. The index had a local minimum of 1347 in the middle of November and recovered to 1416 on November 30. This was 15 points less than the expected level of 1430 in December 2012. The red line intersects the blue one in January 2013 and a negative correction to 1400 (or less) was expected.

Figure 5. Same as in Figure 4 with an extension into December 2012.

Figure 6 displays the current situation with the S&P 500. As in a few cases after 2010, the actual index (red line) leads the expected one (blue) by a month or so. It has reached the (closing) level 1563 on March 14 and has been falling since then. As in April and November 2012, we expect the index to fall. The level is likely around 1455. Then it might regain its power and have a final spurt to 1560 in October 2013. Do not miss this last opportunity.

Figure 6. Same as in Figure 5 with an extension into March 2013.

]]>We have presented share price models based on the evolution consumer price indices (CPI) for three financial companies from the S&P 500 list - Franklin Resources (NYSE: [[BEN]]), JPMorgan Chase (JPM), and Goldman Sachs (GS). Our previous stock price model for BAC (published on March 14, 2012), which predicted a fall down to $5 in 2012Q2 with a standard deviation of $3.08, is updated. We have to admit that BAC fell to its lowest level $6.72 on May 21 from its March high of $9.55. The observed level in Q2 is well within the reported uncertainty. Here we also introduce a new quantitative model for Morgan Stanley (MS).

Having five different models it is instructive to compare them. Our goal is to reveal similarities and differences between the models and thus between the companies. When two companies are driven by similar forces it is always helpful to understand which of the companies provides larger returns. Companies with not correlating price histories driven by different forces may be a natural choice for a defensive portfolio in order to count various possible scenarios in. This is a feasibility study which might be extended to a larger set of banks from and in addition to the S&P 500 list.

To begin with, we characterise five time series statistically. Cross correlation coefficients are estimated for all pairs of stock price series. Then we model all original time series and demonstrate their reliability over time. Since standard unit root tests show that these series are non-stationary, I(1), processes, we (successfully) test the predicted and observed prices for cointegration. Finally, we compare the pricing models and discuss their similarity and difference in terms of investment opportunities and ideas.

**Statistical estimates**

Figure 1 displays the monthly closing (adjusted for splits and dividends) prices for the five studied financial companies. All curves have peaks in 2007 and troughs in 2009. There are significant differences, however. Two companies have recovered to and above their peak pre-crisis levels with the other three companies hovering around lower levels: 0.2 for BAC, 0.25 for MS, and 0.5 for GS, as Figure 2 shows. Table 1 lists the cross correlation coefficients for all pairs of five time series of actual monthly closing prices. All series span the interval between July 2003 and October 2012, which includes 113 readings. There are highly correlated series and not correlating ones. Not surprisingly, the cross correlation coefficient between BAC and MS, which both have been suffering most after 2007, is 0.92. At the same time, the BAC share price series does not correlate with the series from other three banks. Franklin Resources correlate with Goldman Sachs and JPMorgan Chase, with the cross correlation coefficient between the latter two companies of 0.8. Higher cross correlation coefficients suggest that driving forces behind the relevant time series are likely similar. For a quantitative model we discuss in this article, this similarity assumes close defining CPIs.

In Table 1, we also present simple statistical estimates of the model reliability, which will be discussed later on. Diagonal elements (highlighted red) are the coefficients of determination, *R ^{2}*, as estimated from a linear regression of an actual and predicted time series for a given company. All involved series of monthly share prices are likely non-stationary processes. We have carried out several unit root tests (the augmented Dickey-Fuller and Phillips-Perron), which showed that they are all I(1) processes. (We skip technical details, which might be excessive to the broader audience. All results are available by request.) This means that cross correlation coefficients in Table 1 are subject to a positive bias.

Figure 1. The evolution of JPM, MS, GS, BAC, and BEN share prices.

Figure 2. The evolution of JPM, MS, GS, BAC, and BEN share prices, all normalized to their peak values between 2003 and 2009.

Table 1. Cross correlation coefficients for five time series of actual monthly closing prices. Diagonal elements (highlighted red) are the coefficients of determination, R^{2}, as estimated from a linear regression of actual and predicted time series for a given company.

BAC | BEN | GS | JPM | MS | |

BAC | 0.950 | ||||

BEN | -0.194 | 0.925 | |||

GS | 0.313 | 0.657 | 0.859 | ||

JPM | 0.098 | 0.809 | 0.795 | 0.718 | |

MS | 0.921 | -0.010 | 0.547 | 0.259 | 0.935 |

**Quantitative model**

The concept of share pricing based on the link between consumer and stock priceshas been under development since 2008.In the very beginning, we found a statistically reliable relationship between ConocoPhillips' stock price and the difference between the core and headline consumer price index in the United States. In order to increase the accuracy and reliability of the quantitative model we extended the pool of defining CPIs to 92, which includes all major categories like food, housing, transportation etc. and many smaller subcategories. In this set, there are CPIs with similar time series, e.g. the price index of food and beverages, *F*, and the index of food only, *FB*. We tested the model for stability relative to these highly correlated time series.

With the extended set of defining CPIs, we estimated quantitative models for all companies from the S&P 500. A few additional companies with traded stocks were also estimated. Our model describes the evolution of a share price as a weighted sum of two individual consumer price indices selected from the set of CPIs. We allow only two defining CPIs, which may lead the modelled share price or lag behind it by several months. The intuition behind these positive and negative lags is that some companies are price setters and some are price takers. The former should influence the relevant CPIs, which include goods and services these companies produce. The latter companies lag behind the prices of goods and services they are associated with. In order to calibrate the model relative to the starting levels of the involved indices and to compensate sustainable time trends (some indices are subject to secular rise or fall) we introduced a linear time trend and constant term. In its general form, the pricing model is as follows:

*p( t_{j}) = Σb_{i}∙CPI_{i}(t_{j}-τ_{i}) + c∙(t_{j}-2000 ) + d + e_{j} (1)*

where *p(**t _{j}*

By definition, the bets-fit model minimizes the RMS residual error. (One may introduce various metrics to define the best fit.) It is a fundamental feature of the model that the lags may be both negative and positive. In this study, we limit the largest lag (lead) to eleven (eight) months. System (1) contains *J* equations for *I+**2* coefficients. We start our model in July 2003 and the share price time series has more than 100 points. To resolve the system, standard methods of matrix inversion are used.

Since November 17, we have the CPI estimates together with the monthly closing prices for October 2012. We first estimate the model with contemporary (October) readings of stock price and CPIs, with all possible CPI pairs tested with (1). Then we allow both CPIs lead (to be earlier in time) the (October) price by one and more (but less than 12) months and also estimate all possible pairs of CPI with all possible (negative) lags. For October, the best fit model has to have the smallest standard error among all estimated models.

In order to ensure that the same model was the best during a longer period before October we carry out a similar estimate for September 2012 and seven previous months. There is a big difference for these earlier models. Now one has future CPIs estimates (October, etc.) and these CPIs may lag behind the price from one (September's model) to seven (March's model) months. Thus, we have to test the models with the CPIs lagging behind the price. Otherwise, we have the same set of models as for October with all possible CPI pairs and (negative) lags from zero to eleven months. When the best fit model for September is the same as for October, i.e. defined by the same CPIs with similar lags and coefficients in (1), we consider this observation as an indication of the model reliability. For August, the defining CPIs may lag by two months and we have more models to test, both with lagging and leading CPIs. Overall, a model is considered as a reliable one when the defining CPIs are the same during eight months in a row. This number and the diversity of CPI subcategories are both crucial parameters. In further studies, it is important to extend the set of defining CPIs and the length of the model reliability. That's why quarterly revisions to all models are important. They guarantee the reliability.

Why do we rely on consumer price indices in our modelling? Many readers have reasonable doubts that some consumer price, which is not directly related to goods and services produced by a given company, may affect its price. We allow the economy to be a more complex system than described by a number of simple linear relations between share prices and goods. The connection between a firm and its products may be better expressed by goods and services which the company does not produce. The demand/supply balance is not well understood yet and may evolve along many nonlinear paths with positive and negative feedbacks. It would be too simplistic to directly define a company price by its products.

So, the intuition behind our pricing model is likely more insightful - we link a given share to some goods and services (and thus their consumer price indices), which we have to find among various CPIs. In order to provide a dynamic reference we also introduce in the model some relative and independent level of prices (also expressed by CPIs). Hence, one needs two different CPIs to define a share price model. These CPIs we select from a predetermined set of 92 CPIs by minimizing the residual model error. All in all, we assume that any share price can be represented as a weighted sum of two consumer price indices (not seasonally adjusted in our model) which may be leading the share price by several months. Our model also includes a linear time trend and an intercept in order to remove mean and trend components from all involved time series.

**Modelling results**

We have already reported defining parameters for Goldman Sachs for the period between March and October 2012 and repeat them here. Table 2 lists the best fit model for each of eight months. All models are based on the same defining CPIs - the consumer price index of food and beverages, *F*, and the index of owners' rent of primary residence, *ORPR*. In all cases, the lags are the same: three and two months, respectively. Other coefficients and the standard error suffer just slight oscillations or drifts (e.g. *c* and *d*). It is important to stress again that all models the months except October also include those with the future CPIs. Table 2 confirms that no future CPIs drive the share price since March 2012. This company may be considered as a price setter. For the purposes of this study, we use the following best fit model for GS:

*GS*(*t*) = -13.795*F*(*t-*3) + 11.027*ORPR*(*t*-2) + 29.935(*t*-2000) + 33.751, sterr=$14.52 (2)

Table 2. The monthly models for GS.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | F | 3 | -13.795 | ORPR | 2 | 11.027 | 29.935 | 33.751 | 14.521 |

September | F | 3 | -13.791 | ORPR | 2 | 11.013 | 29.992 | 35.827 | 14.584 |

August | F | 3 | -13.787 | ORPR | 2 | 11.003 | 30.023 | 37.106 | 14.649 |

July | F | 3 | -13.759 | ORPR | 2 | 10.978 | 30.018 | 37.647 | 14.707 |

June | F | 3 | -13.731 | ORPR | 2 | 10.933 | 30.124 | 41.985 | 14.758 |

May | F | 3 | -13.704 | ORPR | 2 | 10.876 | 30.342 | 48.755 | 14.770 |

April | F | 3 | -13.661 | ORPR | 2 | 10.819 | 30.449 | 53.171 | 14.805 |

March | F | 3 | -13.787 | ORPR | 2 | 10.943 | 30.440 | 48.639 | 15.055 |

In Tables 3 through 6, we summarize the evolution of models for four banks since March 2012. Taking the defining CPIs and coefficients for October 2012 one obtains the following best fit models:

*BAC*(*t*) = -5.897*SEFV*(*t-*0) + 2.650*RSH*(*t*-2) + 20.609(*t*-2000) + 444.030, sterr=$2.98

*MS*(*t*) = -7.93*SEFV*(*t-*0) + 4.415*ORPR*(*t*-2) + 25.226(*t*-2000) + 420.919, sterr=$3.47

*JPM*(*t*) = -1.856*F*(*t-*4) + 0.993*ORPR*(*t*-2) + 7.037(*t*-2000) + 116.907, sterr=$2.96

*BEN*(*t*) = -7.333*FB*(*t-*4) - 1.519*O*(*t*-9) + 69.578(*t*-2000) + 1536.224, sterr=$7.36

where *SEFV* is the consumer price index of food away from home, *RSH* is the index of rent of shelter, *FB* is the index of food without beverages, and *O* is the index of other goods and services. Therefore, all five models include the index related to food. (We consider April's JPM model as a fluctuation.) Figure 3 shows that *FB* and *F* are practically identical and we might exclude one of them from the full set of CPIs without any significant loss in resolution. On the other hand, the BEN model is stable with *FB* and we retain it in the set.

In four from five models, the second CPI is associated with rent of residence (*ORPR*) or shelter (*RSH*). Figure 3 demonstrates that these indices are also close. Table 7 lists cross correlation coefficients, *CC*, for the six defining CPIs and their first differences. Because of secular growth in prices, these coefficients are extremely high for the original series, but these levels are likely biased up. The first differences characterize the link between the indices in a more reliable way, with *CC*=0.994 for the first differences of *F* and *FB*. The first difference of *SEFV*, *dSEFV*, is well correlated with d*F* and d*FB*. Taking into account all possible time lags between the indices (from 0 to 11 months) in the models one may calculate cross correlation coefficients for the same time series but with various time shifts. Obviously, the highest cross correlation coefficients should not be lower than that for the contemporary time series. In Table 7, the highest *CC*s among all time lags are shown in brackets. For example, the *CC* for d*SEFV* and d*F**/dFB* has increased to 0.49. Interestingly, the first difference of *SEFV, dSEFV,* has the same correlation coefficient with d*ORPR* as d*RSH*, but d*SEFV* and d*RSH*do not correlate. When time lags between the indices are allowed, no big change in the level of correlation of d*SEFV* and d*ORPR* is observed. Overall, it is possible to distinguish three different sets of CPIs: "food", "rent", and "other".

Figure 4 depicts all five models as compared to the relevant actual prices since July 2003. We also plotted the high/low monthly pricing in order to illustrate the level of fluctuations of the intermonth prices. One may model the monthly closing prices as well as the high, low, average, etc. prices and likely obtain slightly different models. As mentioned above, we have estimated R^{2} for five models, as Table 1 lists. All coefficients of determination are larger than 0.7, with three from five models having R^{2}>0.9. In order to prove that these statistical estimates for our quantitative models are not biased we have tested them for cointegration between actual and predicted series. The Johansen tests for cointegration rank have shown cointegration rank 1 in all cases. We have also tested the model residual time series (see Figure 5) for unit roots and found that they are I(0) processes. Therefore the predicted and observed series are cointegrated for all banks and no R^{2} in Table 1 is biased.

Table 3. The monthly models for BAC. The last column lists standard errors.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | SEFV | 0 | -5.897 | RSH | 2 | 2.650 | 20.609 | 444.030 | 2.983 |

September | SEFV | 0 | -5.906 | RSH | 2 | 2.656 | 20.625 | 444.228 | 2.979 |

August | SEFV | 0 | -5.965 | RSH | 2 | 2.679 | 20.868 | 448.932 | 2.962 |

July | SEFV | 0 | -5.953 | RSH | 2 | 2.684 | 20.751 | 446.137 | 2.953 |

June | SEFV | 0 | -5.989 | RSH | 2 | 2.695 | 20.924 | 449.647 | 2.952 |

May | SEFV | 0 | -5.982 | RSH | 2 | 2.699 | 20.850 | 447.823 | 2.949 |

April | SEFV | 0 | -5.960 | RSH | 2 | 2.690 | 20.757 | 446.303 | 2.949 |

March | SEFV | 0 | -5.971 | RSH | 2 | 2.698 | 20.772 | 446.266 | 2.947 |

Table 4. The monthly models for MS.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | SEFV | 0 | -7.93 | ORPR | 2 | 4.415 | 25.226 | 420.919 | 3.468 |

September | SEFV | 0 | -7.90 | ORPR | 2 | 4.399 | 25.137 | 420.060 | 3.468 |

August | SEFV | 0 | -7.96 | ORPR | 2 | 4.425 | 25.343 | 423.817 | 3.447 |

July | SEFV | 0 | -7.96 | ORPR | 2 | 4.445 | 25.258 | 420.687 | 3.440 |

June | SEFV | 0 | -8.01 | ORPR | 2 | 4.449 | 25.526 | 426.655 | 3.437 |

May | SEFV | 0 | -8.01 | ORPR | 2 | 4.452 | 25.540 | 426.579 | 3.434 |

April | SEFV | 0 | -7.97 | ORPR | 2 | 4.419 | 25.492 | 427.246 | 3.422 |

March | SEFV | 0 | -8.00 | ORPR | 2 | 4.431 | 25.609 | 429.254 | 3.421 |

Table 5. The monthly models for JPM.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | F | 4 | -1.856 | ORPR | 2 | 0.993 | 7.037 | 116.907 | 2.955 |

September | F | 4 | -1.859 | ORPR | 2 | 1.006 | 6.965 | 114.846 | 2.932 |

August | F | 4 | -1.861 | ORPR | 2 | 1.018 | 6.898 | 112.917 | 2.914 |

July | F | 4 | -1.863 | ORPR | 2 | 1.024 | 6.873 | 112.112 | 2.912 |

June | F | 4 | -1.865 | ORPR | 2 | 1.024 | 6.883 | 112.342 | 2.912 |

May | F | 4 | -1.863 | ORPR | 2 | 1.024 | 6.877 | 112.182 | 2.912 |

April | FH | 4 | -1.254 | FAB | 2 | 1.770 | 10.219 | -12.460 | 2.905 |

March | F | 4 | -1.878 | ORPR | 2 | 1.051 | 6.791 | 109.260 | 2.839 |

Table 6. The monthly models for BEN.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | FB | 4 | -7.333 | O | 9 | -1.519 | 69.578 | 1536.224 | 7.365 |

September | FB | 4 | -7.319 | O | 9 | -1.515 | 69.428 | 1533.079 | 7.361 |

August | FB | 4 | -7.301 | O | 9 | -1.513 | 69.275 | 1529.960 | 7.353 |

July | FB | 4 | -7.299 | O | 9 | -1.515 | 69.286 | 1530.175 | 7.353 |

June | FB | 4 | -7.311 | O | 9 | -1.515 | 69.375 | 1532.037 | 7.350 |

May | FB | 4 | -7.301 | O | 9 | -1.513 | 69.304 | 1529.705 | 7.343 |

April | FB | 4 | -7.303 | O | 9 | -1.515 | 69.361 | 1530.410 | 7.337 |

March | FB | 4 | -7.309 | O | 9 | -1.513 | 69.312 | 1531.360 | 7.270 |

Table 7. Cross correlation coefficients for six CPI time series and their first differences. Original series include 124 readings, and their first differences - 123 readings.

F | FB | SEFV | ORPR | RSH | O | |

F | 1 | |||||

FB | 0.99998 | 1 | ||||

SEFV | 0.99714 | 0.99671 | 1 | |||

ORPR | 0.98356 | 0.98295 | 0.98702 | 1 | ||

RSH | 0.97533 | 0.97478 | 0.97736 | 0.99698 | 1 | |

O | 0.97752 | 0.97661 | 0.98664 | 0.95629 | 0.93924 | 1 |

dF | dFB | dSEFV | dORPR | dRSH | dO | |

dF | 1 | |||||

dFB | 0.994 | 1 | ||||

dSEFV | 0.47 [0.49] | 0.48 [0.49] | 1 | |||

dORPR | 0.12 [0.26] | 0.12 [0.26] | 0.31 [0.35] | 1 | ||

dRSH | 0.13 [0.30] | 0.12 [0.28] | 0.10 [0.29] | 0.31 [0.37] | 1 | |

dO | -0.18 [030] | -0.18 [0.28] | 0.06 [0.29] | 0.002 [-0.21] | 0.04 [-0.29] | 1 |

**Cross comparison**

The stock prices of BAC and MS are well correlated. This observation is supported by the similarity of defining CPIs with equal time lags. It is worth noting that the level of correlation may cease quickly for two models with the same defining CPIs but with increasing difference in time lags. For the same CPIs and lags, the level of correlation depends on the ratio of CPI coefficients. This ratio (*b _{1}/b_{2}*) is -2.23 for BAC and -1.65 for MS. The closeness of the ratios guarantees similar evolution of their prices. It is important to stress, however, that

Goldman Sachs and JPMorgan Chase have the same defining CPIs (*F* and *ORPR*) and practically the same time lags. The ratio of coefficients is -1.23 and -1.87, respectively. According to the ratio of share price to *b _{1}*, JPM is less sensitive to food prices, i.e. one unit change in

The BEN model contains a different defining CPI (*O*) which has a quite specific shape with a high-amplitude step between February and April 2009. Instructively, the first difference of *O* does not correlate with any other involved index. For BEN, the step in *O* series is associated with a sharp fall in the stock price nine months before, as the negative coefficient in Table 6 assumes. One may suggest that despite all companies had the same fall around the same time their further evolution resulted in different models. We interpret this observation as an indication that BEN stocks are driven by some forces different from other companies. Despite its high correlation with MS, the price of BEN may deviate much in the future and corrupt the correlation. For BEN, the best situation is when the defining prices do not grow fast.

Figure 3. The evolution of all defining CPIs. Notice F (blue line) and FB (white line inside the blue line) are practically identical

Figure 4. Observed and predicted share prices together with their high/low monthly prices.

Figure 5. The residual model errors

**Disclosure: **I am long [[SPY]]. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

We have presented share price models based on the evolution consumer price indices (CPI) for three financial companies from the S&P 500 list - Franklin Resources (NYSE: [[BEN]]), JPMorgan Chase (JPM), and Goldman Sachs (GS). Our previous stock price model for BAC (published on March 14, 2012), which predicted a fall down to $5 in 2012Q2 with a standard deviation of $3.08, is updated. We have to admit that BAC fell to its lowest level $6.72 on May 21 from its March high of $9.55. The observed level in Q2 is well within the reported uncertainty. Here we also introduce a new quantitative model for Morgan Stanley (MS).

Having five different models it is instructive to compare them. Our goal is to reveal similarities and differences between the models and thus between the companies. When two companies are driven by similar forces it is always helpful to understand which of the companies provides larger returns. Companies with not correlating price histories driven by different forces may be a natural choice for a defensive portfolio in order to count various possible scenarios in. This is a feasibility study which might be extended to a larger set of banks from and in addition to the S&P 500 list.

To begin with, we characterise five time series statistically. Cross correlation coefficients are estimated for all pairs of stock price series. Then we model all original time series and demonstrate their reliability over time. Since standard unit root tests show that these series are non-stationary, I(1), processes, we (successfully) test the predicted and observed prices for cointegration. Finally, we compare the pricing models and discuss their similarity and difference in terms of investment opportunities and ideas.

**Statistical estimates**

Figure 1 displays the monthly closing (adjusted for splits and dividends) prices for the five studied financial companies. All curves have peaks in 2007 and troughs in 2009. There are significant differences, however. Two companies have recovered to and above their peak pre-crisis levels with the other three companies hovering around lower levels: 0.2 for BAC, 0.25 for MS, and 0.5 for GS, as Figure 2 shows. Table 1 lists the cross correlation coefficients for all pairs of five time series of actual monthly closing prices. All series span the interval between July 2003 and October 2012, which includes 113 readings. There are highly correlated series and not correlating ones. Not surprisingly, the cross correlation coefficient between BAC and MS, which both have been suffering most after 2007, is 0.92. At the same time, the BAC share price series does not correlate with the series from other three banks. Franklin Resources correlate with Goldman Sachs and JPMorgan Chase, with the cross correlation coefficient between the latter two companies of 0.8. Higher cross correlation coefficients suggest that driving forces behind the relevant time series are likely similar. For a quantitative model we discuss in this article, this similarity assumes close defining CPIs.

In Table 1, we also present simple statistical estimates of the model reliability, which will be discussed later on. Diagonal elements (highlighted red) are the coefficients of determination, *R ^{2}*, as estimated from a linear regression of an actual and predicted time series for a given company. All involved series of monthly share prices are likely non-stationary processes. We have carried out several unit root tests (the augmented Dickey-Fuller and Phillips-Perron), which showed that they are all I(1) processes. (We skip technical details, which might be excessive to the broader audience. All results are available by request.) This means that cross correlation coefficients in Table 1 are subject to a positive bias.

Figure 1. The evolution of JPM, MS, GS, BAC, and BEN share prices.

Figure 2. The evolution of JPM, MS, GS, BAC, and BEN share prices, all normalized to their peak values between 2003 and 2009.

Table 1. Cross correlation coefficients for five time series of actual monthly closing prices. Diagonal elements (highlighted red) are the coefficients of determination, R^{2}, as estimated from a linear regression of actual and predicted time series for a given company.

BAC | BEN | GS | JPM | MS | |

BAC | 0.950 | ||||

BEN | -0.194 | 0.925 | |||

GS | 0.313 | 0.657 | 0.859 | ||

JPM | 0.098 | 0.809 | 0.795 | 0.718 | |

MS | 0.921 | -0.010 | 0.547 | 0.259 | 0.935 |

**Quantitative model**

The concept of share pricing based on the link between consumer and stock priceshas been under development since 2008.In the very beginning, we found a statistically reliable relationship between ConocoPhillips' stock price and the difference between the core and headline consumer price index in the United States. In order to increase the accuracy and reliability of the quantitative model we extended the pool of defining CPIs to 92, which includes all major categories like food, housing, transportation etc. and many smaller subcategories. In this set, there are CPIs with similar time series, e.g. the price index of food and beverages, *F*, and the index of food only, *FB*. We tested the model for stability relative to these highly correlated time series.

With the extended set of defining CPIs, we estimated quantitative models for all companies from the S&P 500. A few additional companies with traded stocks were also estimated. Our model describes the evolution of a share price as a weighted sum of two individual consumer price indices selected from the set of CPIs. We allow only two defining CPIs, which may lead the modelled share price or lag behind it by several months. The intuition behind these positive and negative lags is that some companies are price setters and some are price takers. The former should influence the relevant CPIs, which include goods and services these companies produce. The latter companies lag behind the prices of goods and services they are associated with. In order to calibrate the model relative to the starting levels of the involved indices and to compensate sustainable time trends (some indices are subject to secular rise or fall) we introduced a linear time trend and constant term. In its general form, the pricing model is as follows:

*p( t_{j}) = Σb_{i}∙CPI_{i}(t_{j}-τ_{i}) + c∙(t_{j}-2000 ) + d + e_{j} (1)*

where *p(**t _{j}*

By definition, the bets-fit model minimizes the RMS residual error. (One may introduce various metrics to define the best fit.) It is a fundamental feature of the model that the lags may be both negative and positive. In this study, we limit the largest lag (lead) to eleven (eight) months. System (1) contains *J* equations for *I+**2* coefficients. We start our model in July 2003 and the share price time series has more than 100 points. To resolve the system, standard methods of matrix inversion are used.

Since November 17, we have the CPI estimates together with the monthly closing prices for October 2012. We first estimate the model with contemporary (October) readings of stock price and CPIs, with all possible CPI pairs tested with (1). Then we allow both CPIs lead (to be earlier in time) the (October) price by one and more (but less than 12) months and also estimate all possible pairs of CPI with all possible (negative) lags. For October, the best fit model has to have the smallest standard error among all estimated models.

In order to ensure that the same model was the best during a longer period before October we carry out a similar estimate for September 2012 and seven previous months. There is a big difference for these earlier models. Now one has future CPIs estimates (October, etc.) and these CPIs may lag behind the price from one (September's model) to seven (March's model) months. Thus, we have to test the models with the CPIs lagging behind the price. Otherwise, we have the same set of models as for October with all possible CPI pairs and (negative) lags from zero to eleven months. When the best fit model for September is the same as for October, i.e. defined by the same CPIs with similar lags and coefficients in (1), we consider this observation as an indication of the model reliability. For August, the defining CPIs may lag by two months and we have more models to test, both with lagging and leading CPIs. Overall, a model is considered as a reliable one when the defining CPIs are the same during eight months in a row. This number and the diversity of CPI subcategories are both crucial parameters. In further studies, it is important to extend the set of defining CPIs and the length of the model reliability. That's why quarterly revisions to all models are important. They guarantee the reliability.

Why do we rely on consumer price indices in our modelling? Many readers have reasonable doubts that some consumer price, which is not directly related to goods and services produced by a given company, may affect its price. We allow the economy to be a more complex system than described by a number of simple linear relations between share prices and goods. The connection between a firm and its products may be better expressed by goods and services which the company does not produce. The demand/supply balance is not well understood yet and may evolve along many nonlinear paths with positive and negative feedbacks. It would be too simplistic to directly define a company price by its products.

So, the intuition behind our pricing model is likely more insightful - we link a given share to some goods and services (and thus their consumer price indices), which we have to find among various CPIs. In order to provide a dynamic reference we also introduce in the model some relative and independent level of prices (also expressed by CPIs). Hence, one needs two different CPIs to define a share price model. These CPIs we select from a predetermined set of 92 CPIs by minimizing the residual model error. All in all, we assume that any share price can be represented as a weighted sum of two consumer price indices (not seasonally adjusted in our model) which may be leading the share price by several months. Our model also includes a linear time trend and an intercept in order to remove mean and trend components from all involved time series.

**Modelling results**

We have already reported defining parameters for Goldman Sachs for the period between March and October 2012 and repeat them here. Table 2 lists the best fit model for each of eight months. All models are based on the same defining CPIs - the consumer price index of food and beverages, *F*, and the index of owners' rent of primary residence, *ORPR*. In all cases, the lags are the same: three and two months, respectively. Other coefficients and the standard error suffer just slight oscillations or drifts (e.g. *c* and *d*). It is important to stress again that all models the months except October also include those with the future CPIs. Table 2 confirms that no future CPIs drive the share price since March 2012. This company may be considered as a price setter. For the purposes of this study, we use the following best fit model for GS:

*GS*(*t*) = -13.795*F*(*t-*3) + 11.027*ORPR*(*t*-2) + 29.935(*t*-2000) + 33.751, sterr=$14.52 (2)

Table 2. The monthly models for GS.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | F | 3 | -13.795 | ORPR | 2 | 11.027 | 29.935 | 33.751 | 14.521 |

September | F | 3 | -13.791 | ORPR | 2 | 11.013 | 29.992 | 35.827 | 14.584 |

August | F | 3 | -13.787 | ORPR | 2 | 11.003 | 30.023 | 37.106 | 14.649 |

July | F | 3 | -13.759 | ORPR | 2 | 10.978 | 30.018 | 37.647 | 14.707 |

June | F | 3 | -13.731 | ORPR | 2 | 10.933 | 30.124 | 41.985 | 14.758 |

May | F | 3 | -13.704 | ORPR | 2 | 10.876 | 30.342 | 48.755 | 14.770 |

April | F | 3 | -13.661 | ORPR | 2 | 10.819 | 30.449 | 53.171 | 14.805 |

March | F | 3 | -13.787 | ORPR | 2 | 10.943 | 30.440 | 48.639 | 15.055 |

In Tables 3 through 6, we summarize the evolution of models for four banks since March 2012. Taking the defining CPIs and coefficients for October 2012 one obtains the following best fit models:

*BAC*(*t*) = -5.897*SEFV*(*t-*0) + 2.650*RSH*(*t*-2) + 20.609(*t*-2000) + 444.030, sterr=$2.98

*MS*(*t*) = -7.93*SEFV*(*t-*0) + 4.415*ORPR*(*t*-2) + 25.226(*t*-2000) + 420.919, sterr=$3.47

*JPM*(*t*) = -1.856*F*(*t-*4) + 0.993*ORPR*(*t*-2) + 7.037(*t*-2000) + 116.907, sterr=$2.96

*BEN*(*t*) = -7.333*FB*(*t-*4) - 1.519*O*(*t*-9) + 69.578(*t*-2000) + 1536.224, sterr=$7.36

where *SEFV* is the consumer price index of food away from home, *RSH* is the index of rent of shelter, *FB* is the index of food without beverages, and *O* is the index of other goods and services. Therefore, all five models include the index related to food. (We consider April's JPM model as a fluctuation.) Figure 3 shows that *FB* and *F* are practically identical and we might exclude one of them from the full set of CPIs without any significant loss in resolution. On the other hand, the BEN model is stable with *FB* and we retain it in the set.

In four from five models, the second CPI is associated with rent of residence (*ORPR*) or shelter (*RSH*). Figure 3 demonstrates that these indices are also close. Table 7 lists cross correlation coefficients, *CC*, for the six defining CPIs and their first differences. Because of secular growth in prices, these coefficients are extremely high for the original series, but these levels are likely biased up. The first differences characterize the link between the indices in a more reliable way, with *CC*=0.994 for the first differences of *F* and *FB*. The first difference of *SEFV*, *dSEFV*, is well correlated with d*F* and d*FB*. Taking into account all possible time lags between the indices (from 0 to 11 months) in the models one may calculate cross correlation coefficients for the same time series but with various time shifts. Obviously, the highest cross correlation coefficients should not be lower than that for the contemporary time series. In Table 7, the highest *CC*s among all time lags are shown in brackets. For example, the *CC* for d*SEFV* and d*F**/dFB* has increased to 0.49. Interestingly, the first difference of *SEFV, dSEFV,* has the same correlation coefficient with d*ORPR* as d*RSH*, but d*SEFV* and d*RSH*do not correlate. When time lags between the indices are allowed, no big change in the level of correlation of d*SEFV* and d*ORPR* is observed. Overall, it is possible to distinguish three different sets of CPIs: "food", "rent", and "other".

Figure 4 depicts all five models as compared to the relevant actual prices since July 2003. We also plotted the high/low monthly pricing in order to illustrate the level of fluctuations of the intermonth prices. One may model the monthly closing prices as well as the high, low, average, etc. prices and likely obtain slightly different models. As mentioned above, we have estimated R^{2} for five models, as Table 1 lists. All coefficients of determination are larger than 0.7, with three from five models having R^{2}>0.9. In order to prove that these statistical estimates for our quantitative models are not biased we have tested them for cointegration between actual and predicted series. The Johansen tests for cointegration rank have shown cointegration rank 1 in all cases. We have also tested the model residual time series (see Figure 5) for unit roots and found that they are I(0) processes. Therefore the predicted and observed series are cointegrated for all banks and no R^{2} in Table 1 is biased.

Table 3. The monthly models for BAC. The last column lists standard errors.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | SEFV | 0 | -5.897 | RSH | 2 | 2.650 | 20.609 | 444.030 | 2.983 |

September | SEFV | 0 | -5.906 | RSH | 2 | 2.656 | 20.625 | 444.228 | 2.979 |

August | SEFV | 0 | -5.965 | RSH | 2 | 2.679 | 20.868 | 448.932 | 2.962 |

July | SEFV | 0 | -5.953 | RSH | 2 | 2.684 | 20.751 | 446.137 | 2.953 |

June | SEFV | 0 | -5.989 | RSH | 2 | 2.695 | 20.924 | 449.647 | 2.952 |

May | SEFV | 0 | -5.982 | RSH | 2 | 2.699 | 20.850 | 447.823 | 2.949 |

April | SEFV | 0 | -5.960 | RSH | 2 | 2.690 | 20.757 | 446.303 | 2.949 |

March | SEFV | 0 | -5.971 | RSH | 2 | 2.698 | 20.772 | 446.266 | 2.947 |

Table 4. The monthly models for MS.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | SEFV | 0 | -7.93 | ORPR | 2 | 4.415 | 25.226 | 420.919 | 3.468 |

September | SEFV | 0 | -7.90 | ORPR | 2 | 4.399 | 25.137 | 420.060 | 3.468 |

August | SEFV | 0 | -7.96 | ORPR | 2 | 4.425 | 25.343 | 423.817 | 3.447 |

July | SEFV | 0 | -7.96 | ORPR | 2 | 4.445 | 25.258 | 420.687 | 3.440 |

June | SEFV | 0 | -8.01 | ORPR | 2 | 4.449 | 25.526 | 426.655 | 3.437 |

May | SEFV | 0 | -8.01 | ORPR | 2 | 4.452 | 25.540 | 426.579 | 3.434 |

April | SEFV | 0 | -7.97 | ORPR | 2 | 4.419 | 25.492 | 427.246 | 3.422 |

March | SEFV | 0 | -8.00 | ORPR | 2 | 4.431 | 25.609 | 429.254 | 3.421 |

Table 5. The monthly models for JPM.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | F | 4 | -1.856 | ORPR | 2 | 0.993 | 7.037 | 116.907 | 2.955 |

September | F | 4 | -1.859 | ORPR | 2 | 1.006 | 6.965 | 114.846 | 2.932 |

August | F | 4 | -1.861 | ORPR | 2 | 1.018 | 6.898 | 112.917 | 2.914 |

July | F | 4 | -1.863 | ORPR | 2 | 1.024 | 6.873 | 112.112 | 2.912 |

June | F | 4 | -1.865 | ORPR | 2 | 1.024 | 6.883 | 112.342 | 2.912 |

May | F | 4 | -1.863 | ORPR | 2 | 1.024 | 6.877 | 112.182 | 2.912 |

April | FH | 4 | -1.254 | FAB | 2 | 1.770 | 10.219 | -12.460 | 2.905 |

March | F | 4 | -1.878 | ORPR | 2 | 1.051 | 6.791 | 109.260 | 2.839 |

Table 6. The monthly models for BEN.

Month | C_{1} | t_{1} | b_{1} | C_{2} | t_{2} | b_{2} | c | d | sterr,$ |

October | FB | 4 | -7.333 | O | 9 | -1.519 | 69.578 | 1536.224 | 7.365 |

September | FB | 4 | -7.319 | O | 9 | -1.515 | 69.428 | 1533.079 | 7.361 |

August | FB | 4 | -7.301 | O | 9 | -1.513 | 69.275 | 1529.960 | 7.353 |

July | FB | 4 | -7.299 | O | 9 | -1.515 | 69.286 | 1530.175 | 7.353 |

June | FB | 4 | -7.311 | O | 9 | -1.515 | 69.375 | 1532.037 | 7.350 |

May | FB | 4 | -7.301 | O | 9 | -1.513 | 69.304 | 1529.705 | 7.343 |

April | FB | 4 | -7.303 | O | 9 | -1.515 | 69.361 | 1530.410 | 7.337 |

March | FB | 4 | -7.309 | O | 9 | -1.513 | 69.312 | 1531.360 | 7.270 |

Table 7. Cross correlation coefficients for six CPI time series and their first differences. Original series include 124 readings, and their first differences - 123 readings.

F | FB | SEFV | ORPR | RSH | O | |

F | 1 | |||||

FB | 0.99998 | 1 | ||||

SEFV | 0.99714 | 0.99671 | 1 | |||

ORPR | 0.98356 | 0.98295 | 0.98702 | 1 | ||

RSH | 0.97533 | 0.97478 | 0.97736 | 0.99698 | 1 | |

O | 0.97752 | 0.97661 | 0.98664 | 0.95629 | 0.93924 | 1 |

dF | dFB | dSEFV | dORPR | dRSH | dO | |

dF | 1 | |||||

dFB | 0.994 | 1 | ||||

dSEFV | 0.47 [0.49] | 0.48 [0.49] | 1 | |||

dORPR | 0.12 [0.26] | 0.12 [0.26] | 0.31 [0.35] | 1 | ||

dRSH | 0.13 [0.30] | 0.12 [0.28] | 0.10 [0.29] | 0.31 [0.37] | 1 | |

dO | -0.18 [030] | -0.18 [0.28] | 0.06 [0.29] | 0.002 [-0.21] | 0.04 [-0.29] | 1 |

**Cross comparison**

The stock prices of BAC and MS are well correlated. This observation is supported by the similarity of defining CPIs with equal time lags. It is worth noting that the level of correlation may cease quickly for two models with the same defining CPIs but with increasing difference in time lags. For the same CPIs and lags, the level of correlation depends on the ratio of CPI coefficients. This ratio (*b _{1}/b_{2}*) is -2.23 for BAC and -1.65 for MS. The closeness of the ratios guarantees similar evolution of their prices. It is important to stress, however, that

Goldman Sachs and JPMorgan Chase have the same defining CPIs (*F* and *ORPR*) and practically the same time lags. The ratio of coefficients is -1.23 and -1.87, respectively. According to the ratio of share price to *b _{1}*, JPM is less sensitive to food prices, i.e. one unit change in

The BEN model contains a different defining CPI (*O*) which has a quite specific shape with a high-amplitude step between February and April 2009. Instructively, the first difference of *O* does not correlate with any other involved index. For BEN, the step in *O* series is associated with a sharp fall in the stock price nine months before, as the negative coefficient in Table 6 assumes. One may suggest that despite all companies had the same fall around the same time their further evolution resulted in different models. We interpret this observation as an indication that BEN stocks are driven by some forces different from other companies. Despite its high correlation with MS, the price of BEN may deviate much in the future and corrupt the correlation. For BEN, the best situation is when the defining prices do not grow fast.

Figure 3. The evolution of all defining CPIs. Notice F (blue line) and FB (white line inside the blue line) are practically identical

Figure 4. Observed and predicted share prices together with their high/low monthly prices.

Figure 5. The residual model errors

**Disclosure: **I am long [[SPY]]. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

In this post, we revisit the trends in the PPI of three commodities related to metals: steel and iron and nonferrous metals. Originally, we reported on these items in 2008 and then revisited in 2010 and February 2012. We expected the index of steel and iron to return to the long term trend, which express a higher rate of growth of the producer price index than that of steel and iron. The index of nonferrous metals had to fluctuate with large amplitude around the PPI and grew at a lower rate than PPI during 2012.

*1.*Figure 1 compares the difference between the PPI and the index for iron and steel (101). The difference is characterized by the presence of a sharp decline between 2001 and 2008. Between 1985 and 2000, the curve fluctuates around the zero line, i.e. there was no linear trend in the absolute difference. Our main assumption was that the negative trend observed before 2008 should start transforming into a positive one after 2008. In Figure 1, the (expected) new trend is shown by green line. This trend suggests that the PPI grows faster than the index of steel and iron by approximately 2 units of index per year. Figure 2 demonstrates the most recent period and confirms that our prediction for 2012 was correct - the difference has been approaching the green line. One may foresee the difference to intersect the green line in the near future as a (pendulum) return motion. The index of steel and iron will likely be falling in absolute terms in the fourth quarter of 2012 and the first quarter of 2013. Accordingly, this is not a commodity to buy in 2012 and 2013.

2. The index for non-ferrous metals (102) shows an example of the absence of sustainable trends in the difference (see Figure 3). The curve is rather a comb with teeth of varying width. Although varying, the distance between consecutive troughs is several years at least. In February 2012, we expected this index to decrease relative to the PPI and the difference in Figure 3 to rise to the level of -10. This prediction is still valid but the difference has gained only 18 points since February (from -62 to -44). Considering the observation that the rate of growth was approximately 3 points per month since February 2012 one may expect the level of -10 in approximately 10 to 12 months, i.e. in September 2013. The price of nonferrous metals will be decreasing in absolute terms as well and does not represent the best investment opportunity.

Figure 1. The difference of the PPI and the index of steel and iron updated for the period between January 2012 and August 2012. As expected, the difference has been increasing during the reported period and closed the new trend (green). The price index for iron and steel will be growing at a lower rate than the overall PPI.

Figure 2. The evolution of the difference between the PPI and the price index of iron and steel between January 2005 and August 2012. Green line predicts the evolution of the difference after 2008. Red circles represent the difference between April 2009 and August 2012.

Figure 3. The evolution of the difference between the PPI and the index of nonferrous metals from 1985 and August 2012. There are no linear trends in the difference, but its behavior demonstrates a clear periodic structure with relatively deep but short troughs.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

In this post, we revisit the trends in the PPI of three commodities related to metals: steel and iron and nonferrous metals. Originally, we reported on these items in 2008 and then revisited in 2010 and February 2012. We expected the index of steel and iron to return to the long term trend, which express a higher rate of growth of the producer price index than that of steel and iron. The index of nonferrous metals had to fluctuate with large amplitude around the PPI and grew at a lower rate than PPI during 2012.

*1.*Figure 1 compares the difference between the PPI and the index for iron and steel (101). The difference is characterized by the presence of a sharp decline between 2001 and 2008. Between 1985 and 2000, the curve fluctuates around the zero line, i.e. there was no linear trend in the absolute difference. Our main assumption was that the negative trend observed before 2008 should start transforming into a positive one after 2008. In Figure 1, the (expected) new trend is shown by green line. This trend suggests that the PPI grows faster than the index of steel and iron by approximately 2 units of index per year. Figure 2 demonstrates the most recent period and confirms that our prediction for 2012 was correct - the difference has been approaching the green line. One may foresee the difference to intersect the green line in the near future as a (pendulum) return motion. The index of steel and iron will likely be falling in absolute terms in the fourth quarter of 2012 and the first quarter of 2013. Accordingly, this is not a commodity to buy in 2012 and 2013.

2. The index for non-ferrous metals (102) shows an example of the absence of sustainable trends in the difference (see Figure 3). The curve is rather a comb with teeth of varying width. Although varying, the distance between consecutive troughs is several years at least. In February 2012, we expected this index to decrease relative to the PPI and the difference in Figure 3 to rise to the level of -10. This prediction is still valid but the difference has gained only 18 points since February (from -62 to -44). Considering the observation that the rate of growth was approximately 3 points per month since February 2012 one may expect the level of -10 in approximately 10 to 12 months, i.e. in September 2013. The price of nonferrous metals will be decreasing in absolute terms as well and does not represent the best investment opportunity.

Figure 1. The difference of the PPI and the index of steel and iron updated for the period between January 2012 and August 2012. As expected, the difference has been increasing during the reported period and closed the new trend (green). The price index for iron and steel will be growing at a lower rate than the overall PPI.

Figure 2. The evolution of the difference between the PPI and the price index of iron and steel between January 2005 and August 2012. Green line predicts the evolution of the difference after 2008. Red circles represent the difference between April 2009 and August 2012.

Figure 3. The evolution of the difference between the PPI and the index of nonferrous metals from 1985 and August 2012. There are no linear trends in the difference, but its behavior demonstrates a clear periodic structure with relatively deep but short troughs.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.

How many people are needed to increase the rate of unemployment by 0.1%? The rate of unemployment is calculated as the ratio of the number of unemployed and labor force. So, 0.1% of unemployment rate with the level of labor force of 155,000,000 corresponds to 155000. Since one person in the CPS represents 1550 people, one needs only 100 people to increase the rate of unemployment by 0.1%. To decrease the rate by 0.3% , only 300 are needed.

I do not say that the result for September 2012 is biased. I say that the Current Population Survey procedure is wide-open for manipulations.

]]>How many people are needed to increase the rate of unemployment by 0.1%? The rate of unemployment is calculated as the ratio of the number of unemployed and labor force. So, 0.1% of unemployment rate with the level of labor force of 155,000,000 corresponds to 155000. Since one person in the CPS represents 1550 people, one needs only 100 people to increase the rate of unemployment by 0.1%. To decrease the rate by 0.3% , only 300 are needed.

I do not say that the result for September 2012 is biased. I say that the Current Population Survey procedure is wide-open for manipulations.

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