4/7/14

Predicting stock prices: ACE Limited may continue growth


In this post, we model the evolution of ACE Limited (NYSE: ACE) stock price. ACE is a company from Financial sector which “provides a range of insurance and reinsurance products to insureds worldwide”. Lately, we presented similar models for the following financial companies: Bank of America (BAC), Franklin Resources (BEN), Morgan Stanley (MS), Lincoln National (LNC), Invesco Ltd. (IVZ), and Goldman Sachs (GS). 

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 (CPIs). The background idea is simple: there is a potential trade-off between a given share price and goods and services the company produces/provides. For example, the energy consumer price should influence the price of energy companies. Here, we assume that some set of consumer prices (as expressed by consumer price index, CPI) drives ACE stock price.  The change in these consumer prices is directly transferred into the share price. Our first task is to find the driving CPI.

Each company has to compete with all other companies on the market. Therefore, the influence of the driving CPI depends on the overall market evolution. We express the net change in the whole variety of market prices by one reference CPI, which should best represent the overall dynamics of the changing price environment. As a result, our pricing model includes two defining CPIs: the driver and the reference. A company can be a price taker or price setter.  Then, the company share should follow the changes in prices of goods and services related to the company or vice versa. Time delays are possible between action and reaction - some time is needed for any price changes to pass through. In our model, 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 of ACE from Yahoo.com and the relevant (seasonally not adjusted) CPI estimates through February 2014 are published by the BLS.  We have found that the evolution of ACE share price is defined by the consumer price index of medical professional services (MPRS) and the index of pets, pet products and services (PETS) from Recreation CPI category. We assume that the index of medical professional services is the price driver (accident and health insurance is likely related.  The defining time lags are as follows: the MPRS index is contemporary to the price and the PETS index has a 4 months lead. The relevant best-fit model for ACE(t) is as follows:  

ACE(t) =  -1.96PETS(t-4) - 1.46MPRS(t-0)  + 28.62(t-2000) + 531.20,  February 2014    (1) 

where ACE(t) is the ACE share price in U.S. dollars,  t is calendar time. Figure 1 displays the evolution of both defining indices since 2002.  

It is important to understand why the index of pets, pet products and service in the above model “define” the evolution of ACE price. Actually, the model implies that PETS index does not affect the share price. Instead, PETS provides a dynamic reference rather than the driving force. To illustrate the overall market evolution we use the S&P 500 index. Figure 2 displays the evolution of dPETS (the first difference of PETS) and dSP500, both normalized to their respective absolute maximums between July 2003 and February 2014. The similarity between the dPETS/dPETSmax and dSP500/dSP500max (term 0.22 is needed to equalize the peaks) is best visible in 12-month moving averages. In that sense, the PETS index is able to represent the market in statistical terms. There is no other interpretation of this reference CPI except the statistical one.  

Figure 3 depicts the high and low monthly prices for ACE 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. slopes, b1 and b2, defining CPIs, time lags, the slope of time trend, c, and free term, d, for a few models for the period between November 2011 and February 2014. These models all have the same defining CPIs, similar coefficients and time lags – they are practically identical. Therefore, the estimated ACE model is reliable over time. The model residual error is shown in Figure 4. The standard deviation between July 2003 and February 2014 is $2.90. 

The model cannot predict the future of ACE price from the defining CPIs since MPRS has no lead. However, there are some medium-term trends in the defining CPIs. Figure 5 depicts the dMPRS and dPETS curves together with their 12-month moving averages. The rate of dPETS growth decreases (together with the S&P 500 return) and its influence on ACE price is getting negligible. In case the fall in dMPRS extends into 2014 the price of ACE will be growing along the same linear trend. 

Table 1. Selected best fit models for the period between November 2011 and February 2014

Month
b1
CPI1
lag1
b2
CPI2
lag2
c
d
Feb-14
-1.460
MPRS
0
-1.964
PETS
4
28.621
531.202
Jan
-1.457
MPRS
0
-1.964
PETS
4
28.596
530.614
Dec-13
-1.482
MPRS
0
-1.972
PETS
4
28.880
537.091
Nov
-1.428
MPRS
0
-1.956
PETS
4
28.276
523.139
Oct
-1.400
MPRS
0
-1.911
PETS
4
27.781
511.361
Sep
-1.375
MPRS
0
-1.899
PETS
4
27.469
504.530
Aug
-1.338
MPRS
0
-1.888
PETS
4
27.059
495.294
Jul
-1.325
MPRS
0
-1.886
PETS
4
26.926
492.172
Nov-12
-1.013
MPRS
0
-1.831
PETS
4
23.674
417.587
Oct
-0.991
MPRS
0
-1.838
PETS
4
23.479
414.261
Sep
-0.954
MPRS
0
-1.833
PETS
4
23.089
405.356
Aug
-0.941
MPRS
0
-1.829
PETS
4
22.940
402.066
Jul
-0.943
MPRS
0
-1.828
PETS
4
22.947
402.404
Jun
-0.933
MPRS
0
-1.825
PETS
4
22.835
399.977
May
-0.916
MPRS
0
-1.822
PETS
4
22.651
395.843
Apr
-0.915
MPRS
0
-1.822
PETS
4
22.647
395.605
Dec-11
-0.886
MPRS
0
-1.856
PETS
4
22.631
166.439
Nov
-0.851
MPRS
0
-1.854
PETS
4
22.292
159.911

 

Figure 1. The evolution of PETS and MPRS indices  


Figure 2. First differences dPETS and dSP500 normalized to their respective absolute maximum values between July 2003 and February 2014. 12-month moving averages of these differences are similar, i.e. dPETS is a good approximation of dSP500 at a medium-term horizon.
 

Figure 3. Observed and predicted ACE share prices. 

Figure 4. The model residual error: stdev=$2.90. 


Figure 5. First differences of monthly estimates: dPETS and dMPRS. Moving averages of these differences show their medium-term trends.

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