On the volatility of daily stock returns of Total …...for why crude oil prices and corresponding stock prices fluctuate heavily and became more volatile during the World War II to
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RESEARCH Open Access
On the volatility of daily stock returns ofTotal Nigeria Plc: evidence from GARCHmodels, value-at-risk and backtestingNgozi G. Emenogu1, Monday Osagie Adenomon2* and Nwaze Obini Nweze2
* Correspondence: [email protected] of Statistics, NasarawaState Univ ersity, Keffi, NasarawaState, NigeriaFull list of author information isavailable at the end of the article
Abstract
This study investigates the volatility in daily stock returns for Total Nigeria Plc usingnine variants of GARCH models: sGARCH, girGARCH, eGARCH, iGARCH, aGARCH,TGARCH, NGARCH, NAGARCH, and AVGARCH along with value at risk estimation andbacktesting. We use daily data for Total Nigeria Plc returns for the period January 2,2001 to May 8, 2017, and conclude that eGARCH and sGARCH perform better fornormal innovations while NGARCH performs better for student t innovations. Thisinvestigation of the volatility, VaR, and backtesting of the daily stock price of TotalNigeria Plc is important as most previous studies covering the Nigerian stock markethave not paid much attention to the application of backtesting as a primaryapproach. We found from the results of the estimations that the persistence of theGARCH models are stable except for few cases for which iGARCH and eGARCH wereunstable. Additionally, for student t innovation, the sGARCH and girGARCH modelsfailed to converge; the mean reverting number of days for returns differed frommodel to model. From the analysis of VaR and its backtesting, this studyrecommends shareholders and investors continue their business with Total NigeriaPlc because possible losses may be overcome in the future by improvements instock prices. Furthermore, risk was reflected by significant up and down movementin the stock price at a 99% confidence level, suggesting that high risk brings a highreturn.
Keywords: Volatility, Returns, Stocks, Total petroleum, Akaike information criterion(AIC), GARCH, Value-at-risk (VaR), Backtesting
IntroductionVolatility is a statistical measure of the dispersion of returns for a given security or
market index. It can be measured using the standard deviation or variance between
returns from the same security or market index. It is often the case that higher levels
of volatility, lead to higher risks associated with a particular security, a leading reason
for why crude oil prices and corresponding stock prices fluctuate heavily and became
more volatile during the World War II to early 1970s (Ulusoy and Ozdurak 2018).
From an economic perspective, world resources are scarce, particularly in developing
countries as Nigeria (Maxwell and Reuvey 2000; International Peace Institute (IPI)
2009). According to Milder et al. (2011), resource scarcity is increasingly perceived as
one of the greatest security risks of the 21st, a characteristic of developing countries
Note: b: the estimated Weibull parameter that, when restricted to the value of 1, results in exponential distribution; uLL:the unrestricted log-likelihood value; rLL: the restricted log-likelihood value; LRp: the likelihood-ratio test statistic
Emenogu et al. Financial Innovation (2020) 6:18 Page 20 of 25
and cleansed returns of Total Nigeria Plc. The persistence of the models was stable except
in few cases where iGARCH and eGARCH were unstable. Additionally, for student t
innovation, the sGARCH and gjrGARCH models failed to converge. The mean-reverting
number of day for the returns of Total Nigeria Plc differed from model to model. The per-
formance of NGARCH was in line with the work of Emenogu and Adenomon (2018). Evi-
dence from the VaR analysis of the selected models revealed that the risk of VaR losses
was high at a 99% confidence level, slightly high at a 95% confidence level and better at a
90% confidence level. Although duration-based tests of independence conducted revealed
that the models were correctly specified, in all cases, the null hypotheses were accepted.
This indicates that the probability of an exception on any day did not depend on the out-
come of the previous day. Finally, both the unconditional (Kupiec) and conditional (Chris-
toffersen) coverage tests for the correct number of exceedances for both Total Nigeria Plc
stock returns and cleansed Total Nigeria Plc returns revealed arejection of the models at a
1% level of significance, which is similar to results obtained for the percentages of viola-
tion rates. This confirms that unconditional (Kupiec) and conditional (Christoffersen)
coverage tests for the correct number of exceedances are reliable compared to the
duration-based tests of independence (Nieppola 2009). This study recommends
shareholders and investors to continue their business with Total Nigeria Plc
because losses may be recouped in the future, based on a long-term view of the
price of the stock. Furthermore, risk was found to be high at a 99% confidence
level, suggesting that high risk brings high return. This is in line with financial
theory, which states that an asset with high expected risk would, on average, pay
higher return (Xekalaki and Degiannakis 2010).
Future studyWe studied Total Nigeria Plc because of its potential in the Nigeria Stock Exchange. In the fu-
ture, we will examine the stock price with GARCH-M models and other more advanced
GARCH models, out-of-sample VaR and Backtesting. We also suggest the need to investigate
Total Nigeria Plc stocks in relation to the interest rate, inflation rate, exchange rate and crude
oil price in the global market during the global financial crisis of 2007 to 2008 using multivari-
ate GARCH (MGARCH) models.
Table 10 Implements the VaR Duration Test of Christoffersen and Pelletier on Cleansed TotalNigeria Plc returns
H0: “Duration Between Exceedances have no memory (Weibull b = 1 = Exponential)”
Model VaR alpha b uLL rLL LRp Decision
sGARCH(1,1) with normal 1% 0.9277 −419.8567 − 420.374 0.3091 Accept
5% 1.0387 − 805.6184 −805.884 0.4661 Accept
10% 1.0063 − 1124.353 −1124.366 0.8754 Accept
NGARCH(1,1) With std 1% 0.9974 − 2332.186 −2332.194 0.9030 Accept
5% 0.9941 − 2347.843 −2347.882 0.7806 Accept
10% 0.9936 − 2352.093 −2352.139 0.7620 Accept
Note: b: the estimated Weibull parameter, which when restricted to the value of 1, results in exponential distribution;uLL: the unrestricted log-likelihood value; rLL: the restricted log-likelihood value; LRp: the likelihood-ratio test statistic
Emenogu et al. Financial Innovation (2020) 6:18 Page 21 of 25
Table
11VaRTestforTotalN
igeriaPlcdaily
stockreturns
H0:Correct
Exceed
ances
H0:Correct
Exceed
ances&Inde
pend
ent
Mod
elAlpha
expe
cted
.Exceed
actual.Exceed
uc.LRstat
uc.critical
uc.LRp
Decision
cc.LRstat
cc.critical
cc.LRp
decision
eGARC
Hno
rm1%
4086
39.8476
6.634897
2.745711e-10
Reject
43.61369
9.21034
3.383828e-10
Reject
5%200
216
NaN
3.841459
NaN
NA
NaN
5.991465
NaN
NA
10%
401
340
NaN
2.705543
NaN
NA
NaN
4.60517
NaN
NA
NGARC
H1%
4085
38.31288
6.6348973.
6.026372e-10
Reject
41.99094
9.21034
7.616976e-10
Reject
with
std
5%200
206
NaN
83.841459
NaN
NA
NaN
5.991465
NaN
NA
10%
401
320
NaN
2.705543
NaN
NA
NaN
4.60517
NaN
NA
Note:uc.LRstat:theun
cond
ition
alcoverage
test
likelihoo
d-ratio
statistic;u
c.critical:theun
cond
ition
alcoverage
test
criticalv
alue
;uc.LRp:
theun
cond
ition
alcoverage
test
p-value;
cc.LRstat:thecond
ition
alcoverage
test
likelihoo
d-ratio
statistic;cc.critical:thecond
ition
alcoverage
test
criticalv
alue
;cc.LRp:
thecond
ition
alcoverage
test
p-value;
NAno
tavailable
Emenogu et al. Financial Innovation (2020) 6:18 Page 22 of 25
Table
12VaRTestforcleansed
Totalstock
returns
H0:Correct
Exceed
ances”
H0:Correct
Exceed
ances&Inde
pend
ent
Mod
elAlpha
expe
cted
.exceed
actual.Exceed
uc.LRstat
uc.critical
uc.LRp
Decision
cc.LRstat
cc.critical
cc.LRp
decision
sGARC
Hno
rm1%
4088
42.98808
6.634897
5.507428e-11
Reject
46.9334
9.21034
6.434908e-11
Reject
5%200
203
NaN
3.841459
NaN
NA
NaN
5.991465
NaN
NA
10%
401
319
NaN
2.705543
NaN
NA
NaN
4.60517
NaN
NA
NGARC
Hwith
std
1%40
960
NaN
6.634897
NaN
NA
NaN
9.21034
NaN
NA
5%200
971
NaN
3.841459
NaN
NA
NaN
5.991465
NaN
NA
10%
401
974
NaN
2.705543
NaN
NA
NaN
4.60517
NaN
NA
Note:uc.LRstat:theun
cond
ition
alcoverage
test
likelihoo
d-ratio
statistic;u
c.critical:theun
cond
ition
alcoverage
test
criticalv
alue
;uc.LRp:
theun
cond
ition
alcoverage
test
p-value;
cc.LRstat:thecond
ition
alcoverage
test
likelihoo
d-ratio
statistic;cc.critical:thecond
ition
alcoverage
test
criticalv
alue
;cc.LRp:
thecond
ition
alcoverage
test
p-value;
NAno
tavailable
Emenogu et al. Financial Innovation (2020) 6:18 Page 23 of 25
AcknowledgementsThe authors thank the reviewers of this manuscript. The authors wish to thank the Laboratory for InterdisciplinaryStatistical Analysis (LISA) 2020 Networks for their mentorship.
Authors’ contributionsNgozi G. Emenogu: Initiated the subject, contributed to the methodologies and review of literature, Write and proofread the drafts. Monday Osagie Adenomon: co-Initiated the subject, Analyzed the data in R and interpretation and dis-cussion of results. Nwaze Obini Nweze: Editing of the manuscripts and contributed to the discussion of the results.The author(s) read and approved the final manuscript.
FundingNot Applicable
Availability of data and materialsThe dataset on which the conclusions of the manuscript rely is a secondary data and it will be made available upon request.
Competing interestsThe authors declare that they have no competing interests.
Author details1Department of Statistics, Federal Polytechnic, Bida, Niger State, Nigeria. 2Department of Statistics, Nasarawa State University, Keffi, Nasarawa State, Nigeria.
Received: 20 December 2018 Accepted: 11 February 2020
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