Modeling share prices: JDS Uniphase to hold

Here we model the evolution of JDS Uniphase Corporation (NYSE: JDSU) stock price. JDSU is a company from technology sector which  “provides communications test and measurement solutions, and optical products for telecommunications service providers, wireless operators, cable operators, network-equipment manufacturers, original equipment manufacturers, enterprises, government organizations, distributors, and strategic partners worldwide”. The model has been obtained using our concept of stock pricing as a decomposition of a share price into a weighted sum of two consumer price indices (CPIs). 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 does influence the price of energy companies. It should be taken into account that one defining consumer price (or relevant CPI) has to be related to the share and the other CPI should be an independent one as representing a dynamic reference. We expect a higher relative growth of the defining CPI to manifest itself in a higher pricing power for the company.  Both defining CPIs may lead the price of lag behind by a few months.  

 

We have borrowed the time series of monthly closing prices of JDSU from Yahoo.com and the relevant (seasonally not adjusted) CPI estimates through February 2014 are published by the BLS.  It is instructive that the evolution of JDSU share price is defined by the consumer price index of postage and delivery services (POST) and the index of all items (the headline CPI) less energy index (CE). The defining time lags are as follows: the CE index leads the share price by 4 months and the POST index is contemporaneous with the price. The relevant best-fit model for JDSU(t) is as follows:

 

JDSU(t) =  0.934POST(t-0) – 3.14CE(t-4)  + 7.49(t-2000) + 486.18,  February 2014

 

where JDSU(t) is the JDSU 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 JDSU share together with the predicted and measured monthly closing prices (adjusted for dividends and splits).

 

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 listed in Table 2. Therefore, the estimated JDSU model is reliable over time.  For 2010 and 2011, the model included another reference CPI – the consumer price index of alcohol beverages, AB, which has cross correlation coefficient 0.996 with CE (see Figure 1). Statistically, the model for 2010/2011 was indistinguishable from the current version. The model residual error is shown in Figure 3. The standard deviation between July 2003 and February 2014 is $3.38.

 

Considering the overall evolution of the POST and CE indices we do not expect significant changes in JDSU price. The difference between the headline CPI and the energy index is on a negative trend, i.e. the CPI grows at a higher rate than the consumer energy price.  The index of all items less energy gets some acceleration and this might be a general factor suppressing the JDSU price since CE has a negative coefficient.  The growth in POST index has a positive effect and we observe a few steps affecting the JDSU price in the past. Such a step may raise the price.   

 

Table 1. The best fit models for the period between August 2013 and February 2014

Month	b1	CPI1	lag1	b2	CPI2	lag2	c	d	sterr, $
February 2014	0.9337	POST	0	-3.1359	CE	4	7.4919	486.1826	3.3783
January 2014	0.9732	POST	0	-3.1225	CE	4	7.2528	479.9811	3.3666
December 2013	0.9786	POST	0	-3.1091	CE	4	7.1699	477.1218	3.3797
November 2013	0.9763	POST	0	-3.1464	CE	4	7.3463	483.8175	3.3926
October 2013	0.9855	POST	0	-3.1293	CE	4	7.2357	479.8867	3.4003
September 2012	0.9872	POST	0	-3.1443	CE	4	7.2982	482.2951	3.4122
August 2013	0.9872	POST	0	-3.1284	CE	4	7.227	479.5591	3.4261

 

 

Table 2. The best fit models for 2012

Month	b1	CPI1	lag1	b2	CPI2	lag2	c	d	sterr, $
December	1.163	POST	0	-3.131	CE	4	6.466	463.155	3.33
November	1.163	POST	0	-3.126	CE	4	6.433	462.370	3.33
October	1.164	POST	0	-3.116	CE	4	6.386	460.441	3.33
September	1.158	POST	0	-3.105	CE	4	6.351	459.243	3.32
August	1.154	POST	0	-3.112	CE	4	6.392	460.948	3.32
July	1.156	POST	0	-3.129	CE	4	6.461	463.677	3.32
June	1.190	POST	0	-3.143	CE	4	6.370	461.592	3.31
May	1.214	POST	0	-3.171	CE	4	6.408	463.963	3.29

Figure 1. The evolution of POST and CE indices

 

Figure 2. Observed and predicted JDSU share prices.

 

Figure 3. The model residual error: stdev=$3.38.

Modeling share prices: TECO Energy to hold

In this article, we revise our original pricing model for TECO Energy (NYSE: TE) estimated in November 2012, when we predicted a period of no change. Here we demonstrate that there was no significant change in TE price since 2012 and our prediction was correct.

TE is one of many energy related companies in the S&P 500 list but its model reveals very specific features. Our approach to pricing modeling is based on the link between stocks and consumer price indices.  The intuition is simple and clear, the evolution of a share price is inherently related some goods and services (at least to goods and services the company provides) and thus the evolution of their relative prices.  For example, it is not easy to ignore the intuition that there exists a statistically reliable link between oil price and share prices of oil companies. We have reported that such a link exists for ConocoPhillips and formulated an empirical model. On the other hand, the oil price is not the only changing price and other goods and services should obviously affect share prices of oil companies. Thus one needs also some reference price, which best expresses the overall price evolution.

We are testing various pairs of different CPI components from a large number of pre-selected ones. The best fit pair has to minimize the difference between observed (monthly closing price adjusted for dividends and splits) and predicted prices for the period between July 2003 and February 2014. For TE, we use 92 individual consumer price indices to select the best two (defining) CPIs (i.e. two-component model) in order to describe the evolution of the share price. Our two-component model also includes a free term (constant), which compensates the difference in initial values of the defining CPIs,  and a linear time term, which compensates well know linear (time) trends between various CPI components. We allow the modeled share price to lead and lag behind one or both defining CPIs. When the price is lagged, the model is deterministic one and foresees at the horizon of the smallest lag. When the price leads both CPIs, one cannot predict the future of the price but get some information on the future CPIs. 

In November 2012, both defining CPIs were contemporary to the share price and the best-fit model was as follows: 

TE(t)= -1.43R(t-0) + 0.05E(t-0) + 1.49(t-2000)  + 150.92, October 2012  (1) 

where R(t) in the index recreation at time t, E is the index of energy also contemporary to the price, (t-2000) is the elapsed time. In February 2014, the model was practically identical: 

TE(t)= -1.31(t-0) + 0.047E(t-0) + 1.35 (t-2000)  + 137.81, February 2014   (2) 

Moreover, the same model is defining for each month  between December 2011 and February 2014: the model is valid 27 months in a row. 

       Figure 1 depicts the evolution of both CPIs.  The index of recreation evolves slowly and the energy index has been actually driving the TE price since 2003. Figure 2 depicts the observed and predicted (monthly closing) prices together with the monthly high/low prices, which are natural limits of the intermonth price uncertainty. Our model correctly predicted the price since 2003.  The model residual error is shown in Figure 3. It has standard deviation of $0.78 for the period between July 2003 and February 2014.   

          The energy coefficient in (2) is small but one can see that the evolution of TE is almost fully related to this price index. Our model of oil price evolution implies a significant fall by 2016. This may induce a fall in energy prices in the long run and the TE price will be on a down ward trend as well. On this long term trend, there should be periods of high fluctuations associated with elevated market volatility. When the actual price is far below the predicted price (October 2011), one may consider a profitable (in the short run) share purchase. When the actual price is far above the predicted price (e.g. April 2013), it is the best time to sell. Currently, the predicted and observed prices are almost equal. No action is needed. 

Figure 1. The evolution of defining CPIs.

Figure 2. Observed and predicted TE share prices.  

Figure 3. The model standard error is $0.78.