## Summary

- We have developed a statistically reliable share price model for Morgan Stanley as based on consumer price indices of food and rent of residence.
- The model does not exclude further price increase in 2014.
- The model standard error is $3.35 since July 2003 that corresponds to the level of intermonth price fluctuations.

We built the first deterministic pricing model for Morgan Stanley (MS) in November 2008 together with other companies from the S&P 500 list. Morgan Stanley was included in our study of bankruptcy cases in the USA [1]. The intuition behind our pricing concept was simple - a higher relative rate of price growth (fall) in a given subcategory of goods and services (as expressed by the consumer price indices - CPIs) is likely to result in a faster increase (decrease) in stock prices of related companies. In the first approximation, the deviation between price-defining indices is proportional to the ratio of their pricing powers. The presence of sustainable (linear or nonlinear) trends in the differences between various CPIs allows predicting the evolution of these differences, and thus, the deviation between prices of corresponding goods and services. Such trends are the basis of a long-term prediction of share prices. In the short-run, deterministic forecasting is possible only when a given price lags behind the CPI components defining its behavior.

Why do we rely on consumer price indices in our modeling? Many readers may have reasonable doubts that any 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.

Overall, Morgan Stanley provides an example of a reliable pricing model. In [1] we reported that the defining CPIs in 2008 were the index of housing operations (HO) and the index of food away from home (SEFV). [For a short time in the second half of 2010, the defining indices were different: the index of food less beverages (FB) and the index of information technology, hardware and software (IT)]. The model was revised in 2012 and the evolution of MS price was defined by the index of food away from home ((*SEFV)*) and owner's equivalent rent of residence (*(ORPR)*), which is similar to *HO*. Hence, the model obtained in 2008 is the same as estimate in 2012. Figure 1 depicts the evolution of both indices. Here we revise the previous model with new data available for the period between November 2012 and March 2014. The defining indices, their coefficients and time lags are the same as in 2012. This observation validates the MS pricing model.

The best fit model, i.e. the lowermost RMS residual error, for February 2014 and October 2012 are as follows:

MS(*t*) = -7.93*SEFV*(*t-*0) + 4.42*ORPR*(*t*-2) + 25.23(*t*-2000) + 420.92; October 2012

*MS*(*t*) = -7.57*SEFV*(*t-*0) + 4.26*ORPR*(*t*-2) + 23.66(*t*-2000) + 398.90; February 2014

Table 1 lists the defining CPIs, model coefficients, lags, and standard errors for 15 models between March 2012 and February 2014. The predicted price curve in Figure 2 is in sync with the observed one. We also show the high and low monthly prices, which represent the uncertainty in the modeled price. One can use any price (e.g. high, low, mean, median) within a given month to model the overall evolution and obtain different coefficients and residuals. Therefore, the model for the losing price is a representative of a wider set of pricing models.

The model error for the period between July 2003 and February 2014 is $3.35. The model accurately predicts the share price in the past. From the overall behavior of the defining CPIs and the current negative residual (Figure 3) one may conclude that MS price is slightly underestimated, but the possible correction is within the uncertainty bounds. If the current trends in *ORPR* and *SEFV* are retained, MS may continue its growth.

Table 1. Defining CPIs, coefficients and standard errors of the models for 2012 and 2013/2014

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

2012 | |||||||||

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 |

2014 | and | 2013 | |||||||

February | SEFV | 0 | -7.569 | ORPR | 2 | 4.2574 | 23.66 | 398.90 | 3.35 |

January | SEFV | 0 | -7.605 | ORPR | 2 | 4.2773 | 23.81 | 400.40 | 3.33 |

December | SEFV | 0 | -7.735 | ORPR | 2 | 4.3249 | 24.43 | 411.33 | 3.28 |

November | SEFV | 0 | -7.732 | ORPR | 2 | 4.3197 | 24.44 | 411.77 | 3.29 |

October | SEFV | 0 | -7.740 | ORPR | 2 | 4.3231 | 24.47 | 412.41 | 3.30 |

September | SEFV | 0 | -7.754 | ORPR | 2 | 4.3227 | 24.57 | 414.62 | 3.31 |

August | SEFV | 0 | -7.810 | ORPR | 2 | 4.3494 | 24.80 | 418.21 | 3.32 |

Figure 1. Evolution of the price index of food away from home (*(SEFV)*) and owner's equivalent rent of residence (ORPR)

Figure 2. Observed and predicted MS share prices.

Figure 3. Residual error of the model. Mean residual error is 0 with standard deviation of $3.25.

## Appendix

In its general form, the pricing model is as follows:

*sp(t _{j}) = Σb_{i}∙CPI_{i}(t_{j}-*

*t*

_{i}*) + c∙(t*(1)

_{j}-2000 ) + d + e_{j}where *sp(t _{j})* is the share price at discrete (calendar) times

*t*,

_{j}*j*=1,…,

*J*;

*CPI*

_{i}(t_{j}-*t*

_{i}*)*is the

*i*-th component of the CPI with the time lag

*t*

*,*

_{i}*i*=1,..,

*I*;

*b*,

_{i}*c*and

*d*are empirical coefficients of the linear and constant term;

*e*is the residual error, which statistical properties have to be scrutinized. By definition, the bets-fit model minimizes the RMS residual error. The time lags are expected because of the delay between the change in one price (stock or goods and services) and the reaction of related prices. It is a fundamental feature of the model that the lags in (1) may be both negative and positive. In this study, we limit the largest lag to fourteen months. Apparently, this is an artificial limitation and might be changed in a more elaborated model. In any case, a fourteen-month lag seems to be long enough for a price signal to pass through.

_{j}System (1) contains *J* equations for *I+2* coefficients. Since the sustainable trends last more than five years, the share price time series have more than 60 points. For the current recent trend, the involved series are between 70 and 130 readings. Due to the negative effects of a larger set of defining CPI components their number for all models is (*I=*) 2. To resolve the system, we use standard methods of matrix inversion.

For the sake of completeness we always retain all principal subcategories of goods and services. Among them are the headline CPI (*C*), the core CPI, i.e. the headline CPI less food and energy (*CC*), the index of food and beverages (*F*), housing (*H*), apparel (*A*), transportation (*T*), medical care (*M*), recreation (*R*), education and communication (*EC*), and other goods and services (*O*). In this model, we use 92 CPI components. They are not seasonally adjusted indices and were retrieved from the database provided by the Bureau of Labor Statistics (2014).

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

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

Comments()