- We have developed a statistically reliable share price model for Invesco Ltd. based on the consumer price index of appliances and the index of pets, pet products and services.
- The model does not exclude a negative correction to $35 in May 2014.
- The model standard error is $2.05 since July 2003, that corresponds to the level of intermonth price fluctuations.
Here we model the evolution of Invesco Ltd (IVZ) stock price. IVZ is a company from Financial sector which "provides its services to institutional clients including major public entities, corporations, unions, non-profit organizations, endowments, foundations, pension funds, and financial institutions". Lately, we presented models for the following financial companies: Franklin Resources (NYSE:BEN), Morgan Stanley (NYSE:MS), Goldman Sachs (NYSE:GS), and Lincoln National Corporation (NYSE:LNC), which are different from the model for IVZ.
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 potential trade-off between a given share price and goods and services the company produces and/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 consumer price index, CPI) drives the company stock price, i.e. the change in the consumer prices is directly transferred into the relevant stock share price. The net effect of the CPI change can be positive or negative. The latter case implies that the rising consumer prices suppress the stock price.
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 on the company's stock price should also depend on all other CPIs. To take into account the net change in the whole variety of market prices, we introduce just one reference CPI best representing the overall dynamics of the changing price environment. Hence, the pricing model has to include at least 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 by a few months.
We have borrowed the time series of monthly closing prices (through March 2014) of IVZ from Yahoo.com and the relevant (seasonally not adjusted) CPI estimates through February 2014 are published by the BLS. The evolution of IVZ share price is defined by the index of appliances (APL) from the Housing subcategory and the index of pets, pet products and services (PETS). These indices are selected by LSQ procedure (see Appendix) from a large set of 92 CPIs covering all categories. All possible pairs of CPIs with all possible time lags and leads (but less than 12 months) were tested one by one and the set minimizing the model error is considered as the defining one. For IVZ, the defining time lags are 6 and 3 months, respectively, and the best-fit model is as follows:
IVZ(t) = 1.296APL(t-6) - 1.300PETS(t-3) + 8.829(t-2000) + 11.362, February 2014
where IVZ(t) is the IVZ share price in U.S. dollars, t is calendar time. Figure 1 displays the evolution of both defining indices since 2002. Figure 2 depicts the high and low monthly prices for IVZ share together with the predicted and measured monthly closing prices (adjusted for dividends and splits).
The reader may ask here why the index of pets, pet products and service define the evolution of IVZ price? Actually, the model implies that PETS index does NOT affect the share price. This index provides a dynamic reference rather than 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 (professionally high). How much time does s/he need? The answer 4 hours is wrong. One cannot calculate the time needed without knowing the (river) stream speed and its direction. This stream is the dynamic reference (or moving coordinate reference system) for the swimmer. Same is with stock prices - knowing the driving CPI is not enough to calculate the price change, one needs to know "the stream speed". The CPI representing the dynamic reference for IVZ is selected from the full set of 90+ CPIs to minimize the LSQ model residual. There is no other interpretation of this reference CPI (PETS) except the statistical one.
The model is stable over time. Table 1 lists the best fit models, i.e. coefficients, b1 and b2, defining CPIs, time lags, the slope of time trend, c, and the free term, d, for 7 months. In 2012, the same model was obtained, as also listed in Table 1. Therefore, the estimated IVZ model is reliable over longer time. The model residual is shown in Figure 3. The standard deviation between July 2003 and February 2014 is $2.05.
Overall, the model does not foresee any bug change in IVZ price any time soon. The predicted value for May 2014 is lower than the current one (around $38 on April 3). We would not exclude a negative correction to $35 [+-$2.05] in a month.
Table 1. The best fit models for the period between April 2012 and February 2014
Figure 1. The evolution of APL and PETS indices
Figure 2. Observed and predicted IVZ share prices.
Figure 3. The model residual error: stdev=$2.05.
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 (CPI) 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. 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 a linear time trend and constant term. In its general form, the pricing model is as follows:
sp(tj) = Σbi∙CPIi(tj-ti) + c∙(tj-2000 ) + d + ej (1)
where sp(tj) is the share price at discrete (calendar) times tj, j=1,…,J; CPIi(tj-ti) is the i-th component of the CPI with the time lag ti, i=1,..,I (I=2 in all our models); bi, c and d are empirical coefficients of the linear and constant term; ej is the residual error, whose statistical properties have to be scrutinized.
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 (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. A model is considered as a reliable one when the defining CPIs are the same during the previous eight months. This number and the diversity of CPI subcategories are both crucial parameter.