A case study on Occidental Petroleum Corporation (NYSE ticker: OXY) using a capital asset pricing model.

1. ECON 3206 - FINANCIAL ECONOMETRICS GROUP ASSIGNMENT
NYSE | OIL AND GAS EXPLORATION | EQUITY PRICE FORECAST
OCCIDENTAL PETROLEUM CORPORATION
NYSE Ticker: OXY
QUESTION 1
Occidental Petroleum Corporation is an
international gas and oil exploration and
production company. The company was
established in 1987 starting as a manufacturer of
basic chemicals and vinyls used in health products.
Occidental is an S&P 500 listed oil&gas exploration
and production company. Its operations span
across the USA, Middle East and Latin America.
It is a large core energy a stock with a market
capitalisation of 55.7 Billion American Dollars and
is currently valued at 73 Dollars and 54 cents
American, per share.
QUESTION 2
LOG RETURNS OVER TIME
The compiled data shows that over the long-term, log-
returns has shown a small positive increase of 0.059 as
demonstrated by its mean. Extreme maximum and
minimum values (16.64323 and -20.44832 respectively)
as well as relatively high standard deviations show a
considerable spread within the data. Hence the data
does not appear to show any discernible trend.
The histogram shows that results are slightly negatively
skewed, which would explain why the mean value is
slightly less than the median. Tests for Skewness and
Kurtosis show that the log returns for Occidental
Petroleum Corp cannot be attributed to a normal
distribution. The relatively high Jarque-Bera value
obtained reinforces this observation.
Log returns over time for Occidental Petroleum Corp.
shows signs of a stationary process as the observations
appear to fluctuate around the mean. The only
exception to this observation is the period 2008-09
where there is very high volatility in returns. The relative
economic uncertainty in this period most likely caused
this, giving basis to the assumption that this period
should be considered an outlier in modelling long term
log returns.

2. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 3
Below is our CAPM estimation for the OXY data:
When markets are in equilibrium, CAPM assumes that
investors are only compensated for systematic risk. OXY is
positively correlated with the market since its beta is positive
(1.09). As beta is greater than one, OXY's systematic risk is
higher than the market proxy S&P500. The R-squared value
is very close to 0, therefore the regression does not explain
the variation of the excess return of OXY's stock price.
As the constant alpha does not equal zero, we reject the null
that the data supports CAPM. Further, as alpha is greater
than zero, this indicates that OXY beats the market. This
excess return indicates the non-linearity between risk and
return, possibly due to the volatile nature of the petroleum
industry.
To overcome this issue we could extend the CAPM model to
include a variable that accounts for unexpected risky events
such as a shock to oil prices, as in the case of the Arbitrage
Pricing Theory.
Shown below is the replication portfolio generated
with the S&P500 index and T-Bill:
The historical expected return of the above portfolio
was calculated to be 2.019 with variance 1.276.
Comparing this to the variance of the OXY share of
4.54, suggests that the bank should invest in the
portfolio given the smaller variance, and hence smaller
risk of loss.

3. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 4
LOG RETURNS MODEL
SQUARED RETURNS
The correlogram for log returns show strong
autocorrelation in returns due to low p values. There
are no obvious lags in the ACF and PACF and this
implies that we should use the constant mean equation
for our return series. The patterns for ACF and PACF for
squared returns appear to be erratic and does not give
a clear indication of what AR,MA or ARMA process
should be used to minimise SIC either.
Since squared returns serve as an imprecise proxy for
modelling volatility in the price of returns we will
instead use our log returns model to present our
forecasts for the stock price of Occidental Petroleum
Corp. SIC is used as our deciding factor in regards to
model choice because SIC is more consistent compared
to AIC across large samples. i.e.: lower probability of
choosing incorrectly, a bigger model.
QUESTION 5
The following are the Schwartz BIC values for our
chosen combinations (Refer Appendix for relevant
Eviews commands):
Process SIC
C 4.334908
MA (1) 4.335085
MA (2) 4.332899
AR (1) 4.334983
AR (2) 4.333851
ARMA (1,1) 4.333768
ARMA (1,2) 4.335041
ARMA (2,1) 4.334738
ARMA (2,2) 4.336612
ARCH (1,0,h) – c 4.244465
ARCH (2,0,h) – c 4.133129
GARCH (1,1) – c 3.979487
GARCH (2,1) – c 3.980113
EGARCH (1,1) – c 3.980538
GJR (1,1) –c 3.977466
GJR (1, 1) – c appears to be the process that achieves
the lowest SIC value. Shown below is the estimate of
our chosen model. A GJR model is a generalisation of
the GARCH process that better addresses volatility
clustering in the innovations process. Shown below are
the results of our estimate of the GJR (1, 1) - c process.

4. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 6
Standardized residuals were obtained using the
Proc/Make Residual Series tab and selecting the
histogram.
To test for a dependence structure in the residuals we
can use the BDS test. This test identifies if the residuals
are independent and identically distributed (iid). It
does so by denoting a given distance, epsilon, between
a pair of points or multiple pairs of points. It tests that
the null hypothesis is that the model is independent
and identically distributed, that the distance is equal to
or less than epsilon. i.e.:
H0: The residuals are iid
H1: The residuals are not iid
We can reject the null hypothesis in favour of the
alternate indicating some structure remains in the
residuals that could include non-linearity and non-
stationarity. The BDS test was calculated using the
epsilon method of standard deviation with a value of
0.5 and dimensions of 5. The eviews output for the
BDS test is as follows:
As the results indicate, the p-values for all dimensions
of the BDS test are above 0.05. At a 95% confidence
level, we would fail to reject the null for all dimensions
and thus conclude that the residuals for our model are
all independent and identically distributed. This
indicates there is not a dependence structure in the
residuals and no reason to suspect non-stationarity or
non-linearity.
QUESTION 7
The conditional mean and conditional variance were
obtained using a dynamic forecast command on
Eviews. The dynamic forecast generated dates
immediately after the estimation period, i.e.
02/09/16. Doing so generated two graphs; the first
graph below featured a forecast of the mean equation,
with two standard deviation bands (2 S.E.). The second
graph below features the forecast of the conditional
variance. These forecasts are depicted below.
Conditional Mean:
Conditional Variance:

5. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 8
Residual distribution
The conditional Value at Risk (VAR) for one day ahead
at 5% significance (95% confidence) is -3.33298.
Standardised Residuals
Conditional Value at Risk using a non-parametric
estimate of quantile =
Value at Risk for portfolio value of $1 million = -
$23590.87
When comparing the residual and standardised
residual distributions normality is rejected in both
cases as demonstrated in the histograms. Both show a
relatively high Jarque-Bera statistic, as well as negative
skew and kurtosis. However the standardised residuals
show considerably improved results in comparison to
residuals. It has a much lower Jarque-Bera statistic of
183, smaller negative skew and kurtosis of 4, only 1 off
a desired 3 for normality. Hence using standardised
residuals will provide for a more precise model.
In the case where the bank has large holdings in the
share and it may take a week to sell the share without
impacting the market a one-day ahead value at risk
may not be sufficient. Rather considering a seven-day
ahead value at risk or following the market capital risk
using ten-day ahead value at risk would be more
suitable.
Value at Risk for portfolio value of $1 million = -
$24515.5

6. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 9
ARMA(1,1)-GARCH(1,1) model
Residual Series from the ARMA(1,1)-GARCH(1,1) model
QUESTION 10
The in sample period is from 1 September 2000 to 20
September 2016 with a total of 4037 OXY log returns.
In order to forecast the conditional mean and
conditional variance for the out of sample period,
dynamic forecasting option is applied as it uses the
chain rule of forecasting to compute the one step
ahead forecast, . Essentially each forecasted value, for
instance, is estimated by inferring the lagged
dependent variable on updated information set that
includes the entire in sample returns as well as the
prior derived one step ahead forecasts. The model we
use to forecast is a simple ARMA(1,1)-GARCH(1,1)
model and we have forecasted for 1 through 500 days
ahead from the end of our sample 21 September 2016
,to 2 November 2018. Our forecasting horizon has
expanded by 400 days more as opposed to the horizon
of 100 days because we believe that a longer horizon
draws a clearer picture on the characteristics of long
term forecast trend.
As we see from graph 1.RF and graph 2. VF in the
following page, they exhibit a nice prediction of the
forecasted values. In graph 1.RF, the conditional mean
is based on ARMA(1,1) model with the assumption that
conditional residual is normally distributed with a zero
residual mean and a time-varying residual variance.
When the forecast horizon increases towards infinity,
the conditional mean will eventually converge to the
constant term of the ARMA(1,1) model but how fast of
such convergence depends on the persistence of OXY
log returns. There is a half-life time method to measure
the stock’s persistence with GARCH(1,1) volatility
model. Half life time for OXY stock is 72 days and it is
supported by the graph 1.RF whereby at approximately
140 days (i.e: July 2017) the conditional mean
stabilises.
Similarly, the conditional variance is derived from
GARCH(1,1) model that shows a mean-reverting
feature, given the sum of the coefficients of lagged
squared residual and lagged conditional variance is less
than one. Our Eviews estimated coefficients for the
GARCH(1,1) model have met the covariance stationary
condition and thus the forecasted conditional variance
will tend towards the unconditional variance in the
long run.

7. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 11
The model used in Question 9 is the ARMA(1,1)-
GARCH(1,1) model.
Conditional mean for one-day-ahead forecast with the
information set up to time T (ie 20 September 2016)
( | ) = ( + | + | + | )
= + +
= 0.005042 + 0.935602*-0.45287 + (-0.955934)*(-
0.68079)
= 0.232122
Conditional variance for one-day-ahead forecast with
the information set up to time T
( | ) = ( + | + | + | )
= ( | ) = ( | ) -
( | )
= ( | )
= + +
= 0.035724 + 0.062245*( 0.68079) +
0.92813 * 2.070825
= 1.986568
We see from our Eviews outputs for the conditional
mean and variance of the ARMA(1,1)-GARCH(1,1)
model (Appendix) that the manual computation
generates the same numbers as the programmatic
outcomes. The conditional mean is precisely 0.232122
and the conditional variance is 1.986568.

8. OCCIDENTAL PETROLEUM CORP EQUITY PRICE FORECAST
QUESTION 12
The model used in Question 9 is the ARMA(1,1)-
GARCH(1,1) model.
According the rule of iterated expectations, the
unconditional mean is given by:
( ) = ( + ( ) + ( ))
= 0.005042 + 0.935602 * 0.057359 + -0.955934 * -
0.02903
= 0.08645
Unconditional variance for model:
( ) = ( + + + )
= + +
= 0.035724 + 0.062245 * 18.93349 + 0.928130 *
0.00329
= 1.21729
APPENDIX
Question 2:
genr Log_Return = log(adj_close/adj_close(-1))*100
plot Log_Return
hist Log_Return
Question 3:
(a) genr tbill_r=100*((1+tbill_annual_r/100)^(1/360)-1)
genr OXY_er = Log_Return - tbill_r
genr Market_r = log(m_adj_close/m_adj_close(-1))*100
genr Market_er = Market_r - tbill_r
*CAPM Estimation
ls oxy_er c market_er
*Portfolio Replication
ls oxy_er market_er tbill_r
Question 4:
genr r=100*(log(oxy_adj_price)-log(oxy_adj_price(-
1)))r.correl(10)
genr r2=r*r
r2.correl(10)
Question 5:
arch(1,1,h) r c
‘change Model to EGARCH and in the Options tab, the
covariance method is Bollerslev_wooldridge
Question 8:
‘Make residual series-After estimating the model, click
ProcMake Residual Series. Click for residual type -
Standardized. Note the default name for the residual
series is resid01. ViewDescriptive StatisticsHistogram
and Stats. To calculate the percentile, click
ViewDescriptive StatisticsStats by Classification. Click
on the Quantile box and type 0.05
Question 9:
arch(1,1,h) r c r(-1) ma(1)
‘Change to Bollerslev-Wooldridge
Date Conditional Variance Conditional Mean Y(t) Residual(t)
9/20/2016 2.07082533 -0.452873687 -0.45287 -0.68078882
9/21/2016 1.986568055 0.232121766 0.232122
9/22/2016 2.003171241 0.222216115 0.222216
9/23/2016 2.01961462 0.212948364 0.212948
9/24/2016 2.035899729 0.204277434 0.204277
9/25/2016 2.052028091 0.196164892 0.196165
9/26/2016 2.068001216 0.188574777 0.188575
9/27/2016 2.083820598 0.181473447 0.181473
9/28/2016 2.099487716 0.174829426 0.174829
9/29/2016 2.115004035 0.168613264 0.168613
9/30/2016 2.130371009 0.162797408 0.162797