In this
post, we describe a stock pricing model for Cummins Inc. (NYSE: CMI). CMI is a
company from Industrial Goods sector which “designs, manufactures,
distributes, and services diesel and natural gas engines, and engine-related
component products”. The model has
been estimated using our concept of stock
pricing as a decomposition of a share price into a weighted sum of two consumer
price indices (CPIs). The intuition is straightforward: there is a potential
trade-off between a given share price and goods and services the company
produces and/or provides. Cummins might be a good example of stock price
dependence on the goods it produces like some energy related
companies depend on energy price. It is not excluded that some set of consumer
prices (or relevant consumer price index - CPI) drives the company stock price.
Obviously, this company competes with all other companies on the market.
Therefore, the influence of the driving CPI on the company’s stock price also
depends on all other CPIs. To take into account the net change in various
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. 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.
The CMI
monthly closing prices (adjusted for splits and dividends) were borrowed from Yahoo.com
and the relevant (seasonally not adjusted) CPI estimates through February 2014
are published by the BLS. It is worth
noting that it takes approximately two weeks for the BLS to publish its
estimates for the previous month. On April 5, we have the closing price for
March 31, but the CPIs are available only for February. We have to update all
models when obtain new estimates for closing prices or CPIs, i.e. two times a
month.
We have found that the evolution of CMI share price is defined by the consumer price index of household furnishing and operations (HFO) from Housing CPI category and the index of motor fuel (MF) from Transportation category. We assume that the index of motor fuel is the price driver. The defining time lags are both zero. The relevant best-fit model for CMI(t) is as follows:
CMI(t) = -8.40HFO(t-0) + 0.145MF(t-0) + 8.36(t-2000) + 1007.13, February 2014
where CMI(t) is the CMI share price in U.S.
dollars, t is calendar time. Figure 1 displays the evolution of both
defining indices since 2003. Figure 2
depicts the high and low monthly prices for CMI 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 select models for the period
between September 2011 and February 2014. These models all have the same
defining CPIs, time lags, and similar coefficients. All models are practically identical. Therefore,
the estimated CMI model is highly reliable over time. The model residual error
is shown in Figure 3. The model error has standard deviation $7.70 between July
2003 and February 2014.
The
model residual shows that the current price is lower than the closing price on
March 31 - $145.92. The difference between the observed and predicted prices is
approximately $16, i.e. much larger than the standard error. It is not excluded
that CMI price will fall in the next two months.
Table 1.
Selected best fit models for the period between September 2011 and February
2014
|
Month
|
b1
|
CPI1
|
lag1
|
b2
|
CPI2
|
lag2
|
c
|
d
|
|
Feb-14
|
-8.4012
|
HFO
|
0
|
0.1447
|
MF
|
0
|
8.3582
|
1007.132
|
|
Jan
|
-8.2407
|
HFO
|
0
|
0.1479
|
MF
|
0
|
8.2552
|
986.8984
|
|
Dec-13
|
-8.2423
|
HFO
|
0
|
0.1479
|
MF
|
0
|
8.2541
|
987.1116
|
|
Nov
|
-8.1019
|
HFO
|
0
|
0.1511
|
MF
|
0
|
8.1572
|
969.3595
|
|
Oct
|
-8.0189
|
HFO
|
0
|
0.1533
|
MF
|
0
|
8.0896
|
958.8746
|
|
Sep
|
-7.9659
|
HFO
|
0
|
0.1546
|
MF
|
0
|
8.0441
|
952.2164
|
|
Aug
|
-7.8634
|
HFO
|
0
|
0.1559
|
MF
|
0
|
7.9723
|
939.4761
|
|
Jul
|
-7.8298
|
HFO
|
0
|
0.1562
|
MF
|
0
|
7.9494
|
935.3226
|
|
Dec-11
|
-8.1258
|
HFO
|
0
|
0.1607
|
MF
|
0
|
8.0466
|
892.045
|
|
Nov
|
-8.2115
|
HFO
|
0
|
0.16
|
MF
|
0
|
8.1253
|
901.051
|
|
Oct
|
-8.2438
|
HFO
|
0
|
0.16
|
MF
|
0
|
8.1541
|
903.9983
|
|
Sep
|
-8.2357
|
HFO
|
0
|
0.16
|
MF
|
0
|
8.1498
|
902.3621
|
Figure
1. The evolution of HFO and MF indices
Figure 2.
Observed and predicted CMI share prices.
Figure 3.
The model residual error: stdev=$7.70
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