Does the Assumption on Innovation Process Play an Important Role for Filtered Historical ...

13 Pages·2017·0.34 MB·English

Journal of
Risk and Financial
Management
Article
Does the Assumption on Innovation Process
Play an Important Role for Filtered Historical
Simulation Model?
EmrahAltun1,*,HuseyinTatlidil1,GamzeOzel1andSaraleesNadarajah2
1 DepartmentofStatistics,HacettepeUniversity,06800Ankara,Turkey;[email protected](H.T.);
[email protected](G.O.)
2 SchoolofMathematics,UniversityofManchester,ManchesterM139PL,UK;
[email protected]
* Correspondence:[email protected]
Received:30November2017;Accepted:22January2018;Published:23January2018
Abstract: Most of the financial institutions compute the Value-at-Risk (VaR) of their trading
portfoliosusinghistoricalsimulation-basedmethods.Inthispaper,weexaminetheFilteredHistorical
Simulation (FHS) model introduced by Barone-Adesi et al. (1999) theoretically and empirically.
Themaingoalofthisstudyistofindananswerforthefollowingquestion: “Doestheassumptionon
innovationprocessplayanimportantrolefortheFilteredHistoricalSimulationmodel?”.Forthisgoal,
weinvestigatetheperformanceofFHSmodelwithskewedandfat-tailedinnovationsdistributions
such as normal, skew normal, Student’s-t, skew-T, generalized error, and skewed generalized
errordistributions.TheperformancesofFHSmodelsareevaluatedbymeansofunconditionaland
conditionallikelihoodratiotestsandlossfunctions.Basedontheempiricalresults,weconcludethat
theFHSmodelswithgeneralizederrorandskew-TdistributionsproducemoreaccurateVaRforecasts.
Keywords: FilteredHistoricalSimulationModel;Value-at-Risk;volatility;backtesting
1. Introduction
Themostwellknownriskmeasure, Value-at-Risk(VaR),isusedtomeasureandquantifythe
leveloffinancialriskwithinafirmorinvestmentportfoliooveraspecificholdingperiod. TheVaR
measuresthepotentiallossofriskyassetorportfoliooveradefinedperiodandforagivenconfidence
level. TheVaRisdefinedas
VaR = F−1(1−p), (1)
p
where Fisthecumulativedistributionfunction(cdf)offinanciallosses, F−1 denotestheinverseof
Fand pisthequantileatwhichVaRiscalculated. TheapproachestoVaRcouldbeinvestigatedin
threecategories: (i)fullyparametricmodelsapproachbasedonavolatilitymodels;(ii)non-parametric
approachesbasedontheHistoricalSimulation(HS)methodsand(iii)ExtremeValueTheoryapproach
basedonmodelingthetailsofthereturndistribution.
In this paper, we focus on the non-parametric HS models. The HS model is based on the
assumptionthathistoricaldistributionofreturnswillremainthesameoverthenextperiods. TheHS
modelassumesthatpricechangebehaviourrepeatsitselfoverthetime. Thus,futuredistributionof
assetreturnscouldbedescribedbytheempiricalone. Theone-day-aheadVaRRforecastforHSmodel
isgivenby
VaRt+1 = Quantile(cid:8){Xt}nt=1,p(cid:9), (2)
J.RiskFinancialManag.2018,11,7;doi:10.3390/jrfm11010007 www.mdpi.com/journal/jrfm
J.RiskFinancialManag.2018,11,7 2of13
where pisthequantileatwhichVaRiscalculated. MögelandAuer(2017)comparedtheperformance
ofHSmodelwithseveralcompetitiveVaRmodelsandstatedthatHSmodelproducesthesimilarVaR
forecastswithunconditionalgeneralizedParetodistribution.
TheHSmodelhasseveraladvantages. Forinstance,itiseasytounderstandandimplement. Itis
anonparametricmodelanddoesnotrequireanydistributionalassumption. However,theHSmodel
hasalsoseveralshortcomings. TheHSmodelignoresthetime-varyingvolatilitydynamics. Inorder
toremovelackofHSmodel, HullandWhite(1998)andBarone-Adesietal.(1999)introducedthe
FHSmodel. ThisapproachcanbeviewedasmixtureoftheHSandtheGeneralizedAutoregressive
ConditionalHeteroskedasticity(GARCH)models. Specifically,itdoesnotmakeanydistributional
assumption about the standardized returns, while it forecasts the variance through a volatility
model. Hence,itismixtureofparametricandnon-parametricstatisticalprocedures. Barone-Adesi
and Giannopoulos (2001) demonstrated the usefulness of the FHS model over the historical one.
Kuester et al. (2006) compared the forecasting performance of several advanced VaR models.
Kuesteretal.(2006)concludedthatGARCH-Skew-T,ExtremeValueTheory(EVT)approachwith
normal and Skew-T innovations and FHS model with normal and Skew-T innovations perform
the best. Angelidis et al. (2007) compared the FHS model with GARCH models specified under
differentinnovationdistributionssuchasnormal,Student’s-tandSkewed-T.Roy(2011)estimated
the VaR of the daily return of Indian capital market using FHS model. Omari (2017) compared
FHS, Exponentially Weighted Moving Average (EWMA), GARCH-normal, GARCH-Student’s-t,
GJR-GARCH-normal and GJR-GARCH-Student’s-t models in terms of accuracy of VAR forecasts.
Omari(2017)demonstratedthatGJR-GARCH-Stundet’s-tapproachandFilteredHistoricalSimulation
method with GARCH volatility specification perform competitively accurate in estimating VaR
forecasts for both standard and more extreme quantiles thereby generally out-performing all the
othermodelsunderconsideration.
ThegoalofthispaperistoinvestigatetheVaRforecastingperformanceoftheFHSmodelspecified
underskewedandfat-tailedinnovationsdistributions. Forthisgoal,thecomprehensiveintroduction
to the FHS and GARCH models is given. The FHS model under six innovation distributions are
introduced. Rolling window estimation produce is used to obtain both unknown parameters of
GARCHmodelsandVaRforecasts. TheperformanceoftheFHSmodels,intermsofaccuracyofVaR
forecasts,areevaluatedbymeansofbacktestingmethodsandlossfunctions.
Therestofthepaperisorganizedasfollows: Section2isdevotedtotheoreticalpropertiesofthe
FHSandGARCHmodelsundernormal, Student’s-t, skew-normal, skew-T,generalizederrorand
skewedgeneralizederrorinnovationdistributions. BacktestingmethodologyisgiveninSection3.
EmpiricalfindingsandmodelcomparisonsarepresentedinSection4. Concludingremarksaregiven
inSection5.
2. FilteredHistoricalSimulationModels
Inthissection,theFHSmodelisdefined. Then,thelog-likelihoodfunctionsofGARCHmodel
specifiedundernormal,skew-normal,Student’s-t,skew-T,generalizederrorandskewedgenealized
errorinnovationdistributionsarepresented.
FHSmodelcanbesummarizedasfollows:
√
Let R denotes the daily log-returns. The benchmark GARCH(1,1) model, introduced by
t
Bollerslev(1986),isdefinedby
R = µ+e ,
t t
e = ε h , ε ∼i.i.d. (3)
t t t t
h2 = ω+γ e2 +γ h2 ,
t 1 t−1 2 t−1
J.RiskFinancialManag.2018,11,7 3of13
where ω > 0, γ > 0,γ > 0, µ and h2 are the conditional mean and variance, respectively,
1 2 t t
andε istheinnovationdistributionwithzeromeanandunitvariance. MaximumLikelihood
t
Estimation(MLE)methodiswidelyusedtoestimateparametersofGARCHmodels. Underthe
assumptionofindependentlyandidenticallydistributed(iid)innovationswith f (ε ;τ)density
t
function,thelog-likelihoodfunctionofr forasampleofTobservationsisgivenby
t
(cid:96)(ψ) = ∑T (cid:20)ln(f (ε ;τ))− 1ln(cid:16)h2(cid:17)(cid:21) (4)
t 2 t
t=1
whereψ = (µ,ω,γ ,γ ,τ)istheparametervectorofGARCHmodel,τistheshapeparameter(s)
1 2
of f (ε ;τ)andε = et.
t t ht
ThestandardizedresidualsofestimatedGARCH(1,1)modelareextractedasfollows:
eˆ
ε = t, (5)
t hˆ
t
whereeˆ istheestimatedresidualandhˆ isthecorrespondingdailyestimatedvolatility.
t t
Now, we can generate the first simulated residual by randomly (with replacement) draw
standardizedresidualsfromthedatasetwithmultiplyingtheone-dayaheadvolatilityforecast:
z∗t+1 = e1∗ht+1. (6)
Thefirstsimulatedreturnforperiodt+1canbeobtainedasfollows:
R∗t+1 = µt+1+z∗t+1, (7)
wherez∗ isthefirstsimulatedresidualforperiodt+1.
t+1
ThisprocedureisrepeatedBtimesoflengthT. Here,Brepresentsthenumberofbootstrapped
samplesand T representstheeachofbootstrappedsamplesize. Then,VaRforperiodt+1canbe
forecastedasfollows:
B (cid:110) (cid:111)
∑ Quantile {R∗}T ,100p
t t=1
VaRt+1 = b=1 . (8)
B
The rest of this section is devoted to present the log-likelihood functions of GARCH model
under normal, skew-normal, Student’s-t, skew-T, generalized error and skewed generalized
errordistributions.
2.1. NormalDistribution
Thelog-likelihoodfunctionoftheGARCHmodelspecifiedundernormalinnovationsisgivenby
(cid:32) (cid:33)
T T
(cid:96)(ψ) = −0.5 Tln2π+ ∑lnh2+ ∑ε2 , (9)
t t
t=1 t=1
whereψ = (µ,ω,γ ,γ )denotestheparametervectoroftheGARCH-normal(GARCH-N)modeland
1 2
h2 = ω+γ e2 +γ h2 .
t 1 t−1 2 t−1
J.RiskFinancialManag.2018,11,7 4of13
2.2. Skew-NormalDistribution
ThefirstskewextensionofnormaldistributionwasproposedbyAzzalini(1985). Theprobability
densityfunction(pdf)ofskew-normal(SN)distributionisgivenby
φ(z;λ) =2φ(z)Φ(zλ),z∈ (cid:60),λ ∈ (cid:60), (10)
whereλisanadditionalparameterthatcontrolstheskewness. Whenλ < 0,theSNdistributionis
leftskewed,otherwise,itisrightskewed. Ifλ = 0,theSNdistributionreducestostandardnormal
distribution. ThekthmomentofSNdistributionisgivenby
(cid:114) (cid:32) (cid:33)
E(cid:16)Z2k+1(cid:17) = 2 (2k+1)! ∑k (−1)i k δ2i+1 . (11)
π 2kk! i 2i+1
i=0
(cid:46)√
here, k = 0,1,2,...,nandδ = λ 1+λ2. NotethattheevenmomentsoftheSNdistributionare
equal to standard normal distribution. The mean and variance of SN distribution is, respectively,
givenby,
µ = bδ
(12)
σ2 =1−(bδ)2
(cid:113)
whereb = 2. ThestandardizedSNdistributionisobtainedusingthetransformedrandomvariable
π
(cid:14)
ε = (z−µ) σwhereE(ε) =0andvar(ε) =1. Therandomvariablezcanbeexpressedasz = εσ+µ
(cid:14)
and∂z ∂ε = σ. Thus,thepdfofthestandardizedSNdistributionisgivenby
f (ε;λ) =2σφ((εσ+µ))Φ((εσ+µ)λ) (13)
Hereafter,usingthestandardizedSNdistribution,thelog-likelihoodfunctionofGARCHmodel
withSNinnovationdistributionisgivenby
(cid:96)(ψ) = ∑T ln[2σφ(ε σ+µ)]+∑T ln[Φ((ε σ+µ)λ)]− 1 ∑T ln(cid:16)h2(cid:17), (14)
t t 2 t
t=1 t=1 t=1
whereψ = (µ,ω,γ ,γ ,λ)istheparametervector.
1 2
2.3. Student’s-tDistribution
Since financial return series has fatter tails than normal distribution, Bollerslev (1986, 1987)
proposedtheGARCHmodelwiththeStudent’s-tinnovations. GARCHmodelwiththeStudent’s-t
innovations enables to model both fat-tail and excess kurtosis observed in financial return series.
Thelog-likelihoodfunctionoftheGARCH-Student’s-t(GARCH-T)modelisgivenby
(cid:96)(ψ) = T(cid:104)lnΓ(cid:16)υ+1(cid:17)−lnΓ(cid:0)υ(cid:1)− 1ln[π(υ−2)](cid:105)
2 2 2
−1 ∑T (cid:104)lnh2+(1+υ)ln(cid:16)1+ ε2t (cid:17)(cid:105) , (15)
2 t υ−2
t=1
where ψ = (µ,ω,γ ,γ ,υ) is the parameter vector, Γ(υ) is the gamma function and parameter υ
1 2
controlsthetailsofthedistribution.
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2.4. Skew-TDistribution
Thepdfofskew-TdistributionobtainedbyAzzaliniandCapitanio(2003)isgivenby
(cid:32)(cid:114) (cid:33)
1+υ
f (x;λ,υ) =2t(x;υ)T λt;υ+1 ,x ∈ (cid:60), (16)
x2+υ
where t(·) and T(·) are pdf and cdf of Student’s-t distribution, respectively, and λ controls the
skewness. When λ = 0, ST distribution reduces to Student’s-t distribution in Equation (16).
ThemomentsofSTdistributionaregivenby
E(cid:16)Xk(cid:17) = (cid:0)υ2(cid:1)2kΓΓ(cid:0)(cid:16)υ((cid:1)υ−2k)(cid:17)E(cid:16)Zk(cid:17), (17)
2
ThemeanandvarianceofSTdistributionare,respectively,givenby
(υ)12Γ(cid:16)(υ−1)(cid:17)
µ = π 2 √ λ ,
Γ(υ) 1+λ2 (18)
σ2 = (cid:0) υ −2 µ2(cid:1).
υ−2
The standardized ST distribution is obtained using the transformed random variable
(cid:14)
ε = (z−µ) σ,whereE(ε) =0andvar(ε) =1. Therandomvariablezcanbeexpressedasz = εσ+µ
(cid:14)
and∂z ∂ε = σ. Thus,thepdfofstandardizedSTdistributionisgivenby
(cid:32)(cid:115) (cid:33)
1+υ
f (ε;λ,υ) =2σt((εσ+µ);υ)T λ(εσ+µ);υ+1 ,υ >2 (19)
(εσ+µ)2+υ
whereµandσaremeanandstandarddeviationofSTdistribution,respectively. Thelog-likelihood
functionofGARCHmodelwiththeSTinnovationdistributionisgivenby
T
(cid:96)(ψ) = Tln(2)+Tln(σ)+ ∑ ln[t((ε σ+µ);υ)]
t
(cid:34) (cid:32) t=1 (cid:33)(cid:35) (20)
+ ∑T ln T (cid:114) 1+υ λ(ε σ+µ);υ+1 − 1 ∑T ln(cid:0)h2(cid:1)
t=1 (εtσ+µ)2+υ t 2t=1 t
whereψ = (µ,ω,γ ,γ ,λ,υ)istheparametervector.
1 2
2.5. GeneralizedErrorDistribution
Nelson(1991)introducedtheGARCHvolatilitymodelofgeneralizederrordistribution(GED).
Thelog-likelihoodfunctionofGARCH-GEDmodelisgivenby
(cid:96)(ψ) = ∑T (cid:20)ln(cid:16)υ(cid:17)− 1(cid:12)(cid:12)εt(cid:12)(cid:12)υ−(1+υ−1)ln(2)−lnΓ(cid:18)1(cid:19)− 1ln(cid:16)h2(cid:17)(cid:21) (21)
2 2(cid:12)λ(cid:12) 2 2 t
t=1
whereψ = (µ,ω,γ ,γ ,υ)istheparametervector,υistail-thicknessparameterand
1 2
 (cid:16) (cid:17) 1
Γ 1 2
υ
λ =   . (22)
2υ2Γ(cid:0)3(cid:1)
υ
NotethatthenormaldistributionisaspecialcaseoftheGEDwhenυ =2. Ifυ <2,theGEDhas
heaviertailsthantheGaussiandistribution.
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2.6. SkewedGeneralizedErrorDistribution
SkewedGeneralizedErrorDistribution(SGED)providesanopportunitytomodelskewnessand
excesskurtosisobservedinfinancialreturnseries. Leeetal.(2008)introducedtheGARCH-SGED
modelandconcludedthatGARCHmodelwithSGEDinnovationprocessoutperformedtheGARCH-N
modelforallconfidencelevels.
ThepdfofstandardizedSGEDisgivenby,
(cid:18) |ε +δ|κ (cid:19)
f(ε ) =Cexp − t (23)
t [1+sign(ε +δ)λ]κθκ
t
where
C = κ Γ(cid:16)1(cid:17)−1,θ = Γ(cid:16)1(cid:17)0.5Γ(cid:0)3(cid:1)0.5S(λ)−1
2θ √κ κ κ
S(λ) = 1+3λ2−4A2λ2,δ = 2λA (24)
S(λ)
A = Γ(cid:0)2(cid:1)Γ(cid:16)1(cid:17)−0.5Γ(cid:0)3(cid:1)−0.5,
κ κ κ
whereκ >0istheshapeparameter,−1< λ <1isskewnessparameter. TheSGEDturnsouttobethe
standardnormaldistributionwhenκ =2andλ =0. Thelog-likelihoodfunctionofGARCH-SGED
modelisgivenby
(cid:96)(ψ) = −∑T (cid:18) |εt+δ|κ (cid:19)+Tln(c)− ∑T ln(h ) (25)
[1+sign(ε +δ)λ]κθκ t
t=1 t t=1
whereψ = (µ,ω,γ ,γ ,λ,κ)istheparametervector.
1 2
3. EvaluationofVaRForecasts
Now, we introduce backtesting methodology that is used to compare VaR forecast accuracy
of the models. Statistical accuracy of the models is evaluated by backtests of Kupiec (1995),
Christoffersen(1998),EngleandManganelli(2004)andSarmaetal.(2003). Recently,somealternative
backtestingmethodsforVaRforecastswereproposedbyZiggeletal.(2014)andDumitrescuetal.(2012).
Kupiec(1995)proposedalikelihoodratio(LR)testofunconditionalcoverage(LR )toevaluate
uc
themodelaccuracy. Thetestexamineswhetherthefailurerateisequaltotheexpectedvalue. TheLR
teststatisticisgivenby
(cid:20) pn1(1−p)n0 (cid:21)
LR = −2ln ∼ χ2, (26)
πˆn1(1−π)n0 1
whereπˆ = n /(n +n )istheMLEof p,n representsthetotalviolationandn representsthetotal
1 0 1 1 0
non-violationsforecasts. ViolationmeansthatifVaR >r ,violationoccurs,oppositecaseindicates
t t
the non-violation. Under the null hypothesis (H : p = πˆ), the LR statistic follows a chi-square
0
distributionwithonedegreeoffreedom.
The LR test fails to detect if violations are not randomly distributed. Christoffersen (1998)
uc
proposedaLRtestofconditionalcoverage LR toremovethelackofKupiec(1995)test. The LR
cc cc
testinvestigatesbothequalityoffailurerateandexpectedoneandalsoindependentlydistributed
violations. TheLR teststatisticunderthenullhypothesisshowsthatthefailuresareindependent
cc
andequaltotheexpectedone. Itisgivenby
(cid:20) (1−α)n0αn1 (cid:21)
LR = −2ln ∼ χ2, (27)
cc (1−π01)n00π01n01(1−π11)n10π11n11 2
wheren isthenumberofobservationswithvalueifollowedbyjfori,j =0,1andπ = n /∑n is
ij ij ij ij
j
theprobability,fori,j =1. Itdenotesthattheviolationoccurred,otherwiseindicatestheoppositecase.
TheLR statisticfollowsachi-squaredistributionwithtwodegreesoffreedom.
cc
J.RiskFinancialManag.2018,11,7 7of13
The Dynamic Quantile (DQ) test, proposed by Engle and Manganelli (2004), examines if the
violations is uncorrelated with any variable that belongs to information set Ωt+1 when the VaR is
calculated. ThemainideaofDQtestistoregressthecurrentviolationsonpastviolationsinorderto
testfordifferentrestrictionsontheparametersofthemodel. Theestimatedlinearregressionmodelis
givenby
p q
∑ ∑
It = β0+ βiIt−i+ µjXj+εt (28)
i=1 i=1
where
(cid:40)
1,r <VaR
I = t t (29)
t 0,r ≥VaR
t t
ThisregressionmodeltestswhethertheprobabilityofviolationdependsontheleveloftheVaR.
Here, pandqareusedas5and1,respectively,forillustrativepurpose.
Inmostinstances,evaluatingtheperformanceofVaRmodelsbymeansofLR ,LR andDCtests
uc cc
maynotbesufficienttodecidethemostadequatemodelamongothers. Forinstance,somemodels
mayhavethesameviolationnumberwithdifferentforecasterrors. Sarmaetal.(2003)definedatest
onthebasisofregulator’slossfunction(RLF)totakeintoaccountdifferencesbetweenrealizedreturns
andVaRforecasts. TheRLFisgivenby
(cid:40)
RLFt+1 = 0(r,t+1−VaRt+1)2,iiff rrtt++11<≥VVaaRRtt++11 (30)
whereVaRt+1representstheone-day-aheadVaRforecastforalongposition.
Theunexpectedloss(UL)isequaltoaveragevalueofdifferencesbetweenrealizedreturnand
VaRforecasts. Theone-day-aheadmagnitudeoftheviolationforlongpositionisgivenby
(cid:40)
ULt+1 = 0(r,t+1−VaRt+1),iiff rrtt++11 <≥VVaaRRtt++11 (31)
TheQLFandULlossfunctionsdonotconsiderthecaseinwhichtherealizedreturnsexceedthe
VaRforecast. Theappropriatelossfunctionshouldtakeintoconsiderationthecostofexcesscapital.
Because,overestimatedVaRforecastsyieldfirmstoholdmuchmorecapitalvaluethanrequiredone.
Themainobjectiveofanyfirmistomaximizethetheirprofits. Forthisreason,Sarmaetal.(2003)is
proposedthenewlossfunction,calledFirm’sLossFunction(FLF).TheFLFisgivenby
(cid:40)
FLFt+1 = −(rtβ+V1a−RtV+a1R, t+1)2,iiff rrtt++11<≥VVaaRRtt++11 (32)
whereβisthecostofexcesscapital.
4. EmpiricalResults
4.1. DataDescription
ToevaluatetheperformanceofFHSmodelsintermsofaccuracyofVaRforecasts,ISE-100index
of Turkey is used. The used time series data contains 1092 daily log-returns from 3 January 2013
to4May2017. Thedescriptivestatisticsofthelog-returnsofISE-100indexaregiveninTable1.
J.RiskFinancialManag.2018,11,7 8of13
Table1.SummarystatisticsfortheISE-100index.
ISE-100
Numberofobservations 1092
Minimum −0.048
Maximum 0.027
Mean 6.6×10−5
Median 2×10−4
Std.Deviation 0.006
Skewness −0.603
Kurtosis 4.957
Jarque-Bera 1190.970(p<0.001)
Table1showsthatthemeanreturnisclosedto0. TheresultsoftheJarque-Beratestprovethatthe
nullhypothesisofnormalityisrejectedatanylevelofsignificance. Itshowsstrongevidenceforhigh
excesskurtosisandnegativeskewness. Thus,itisclearthatlogreturnofISE-100indexhasnon-normal
characteristics,excesskurtosis,andfattails. Figure1displaysthedailylog-returnsofISE-100index.
ISE−100
2
0
0.
0
0
s 0.
n
ur
et
−r
g
Lo 02
0.
−
4
0
0.
−
01/01/2013 01/01/2014 01/01/2015 01/01/2016 01/01/2017
Figure1.Dailylog-returnsoftheISE-100index.
Figure2displaysthetime-varyingskewnessandkurtosisofISE-100. ForFigure2,windowlength
isdeterminedas392andtherollingwindowprocedureisused. BasedonFigure2, itisclearthat
skewnessandkurtosisofISE-100indexexhibitgreatvariabilityacrossthetime.
The benchmark model, GARCH(1,1), is estimated with six different innovation distributions:
Normal, SN,Student’s-t, ST,GEDandSGED.Table2showstheestimatedparametersofGARCH
models. The rugarch package in R software is used to obtain parameter estimation of normal,
Student’s-t, GEDand SGEDmodels.The constrOptim functionin Rsoftware isusedto minimize
negativelog-likelihoodfunctionsofGARCH-STandGARCH-SNmodels.
Based on Table 2, we conclude that GARCH-T and GARCH-SGED models have the lower
log-likelihoodvalueamongothers.SinceGARCH-Tmodelhasthelowestlog-likelihoodvalue,itcould
be chosen as best model for in-sample period. Table 2 also shows that the conditional variance
parametersγ arehighlysignificantforallGARCHmodels.
2
J.RiskFinancialManag.2018,11,7 9of13
Skewness
2
0.
0
0.
2
ess −0.
n
w
Ske 0.4
−
6
0.
−
8
0.
−
01/01/2015 01/01/2016 01/01/2017
Kurtosis
8
7
s
si
urto 6
K
5
4
01/01/2015 01/01/2016 01/01/2017
Figure2.TimevaryingskewnessandkurtosisplotsofISE-100index.
Table2.In-sampleperformanceofGARCHmodelsunderskewedandfat-tailedinnovationdistributions.
Parameters Normal Student-T ST SN GED SGED
µ 5.24×10−4 5.67×10−4 8.56×10−4 4.05×10−4 5.67×10−4 3.51×10−4
3.41×10−4 3.04×10−4 3.60×10−4 3.34×10−4 3.77×10−4 3.26×10−4
ω 3.69×10−6 2.01×10−6 2.27×10−6 3.65×10−6 2.72×10−6 2.63×10−6
1.50×10−6 1.53×10−6 6.71×10−6 1.73×10−6 1.66×10−6 1.57×10−6
γ 0.1194 0.0759 0.1510 0.1460 0.0930 0.0873
1
0.0413 0.0336 0.2940 0.0560 0.0393 0.0353
γ 0.8234 0.8908 0.8240 0.7950 0.8600 0.8650
2
0.0449 0.0485 0.3130 0.0662 0.0518 0.0486
ν - 4.7490 4.8760 - 1.2020 -
- 1.1600 0.5620 - 0.1040 -
λ - - −0.2750 −1.5050 - 0.8860
- - 2.4260 0.2630 - 0.0440
κ - - - - - 1.2200
- - - - - 0.1093
−(cid:96) −1381.1100 −1405.003 −1402.2630 −1388.1800 −1401.0600 −1402.7300
4.2. BacktestingResults
Inthissubsection,rollingwindowestimationprocedureisusedtoestimateparametersofGARCH
models. Then,VaRforecastsofFHSmodelsareobtainedbyusingestimatedparametersofGARCH
J.RiskFinancialManag.2018,11,7 10of13
models, one-day-ahead forecasts of conditional mean and conditional variance and standardized
residualsextractedfromestimatedGARCHmodels. Rollingwindowestimationproduceallowsus
tocapturetime-varyingcharacteristicsofthetimeseriesindifferenttimeperiods. Windowlengthis
determinedas392andnext700dailyreturnsareusedtoevaluatetheoutofsampleperformanceof
VaRmodels.
Table 3 shows the backtesting results for FHS-N, FHS-T, FHS-ST, FHS-SN, FHS-GED and
FHS-SGEDmodels.ThetwostepdecisionmakingprocedureisappliedtodecidethebestVaRmodel.
In first step, the performance of VaR models are evaluated according to results of LR c, LR c and
u c
DCtests. Insecondstep,themodels,achievedtopassthesethreebacktest,consideredasaccurate
modelandobtainedtheresultsoflossfunctionsoftheseVaRmodels. Finally,thelowestvaluesofloss
functionsindicatethebestVaRmodels.
Table4showsthatallFHSmodelsperformwellbasedontheresultsofLR ,LR andDCtests
uc cc
results at p = 0.05 ad p = 0.025 levels. However, FHS model with Student’s-t and ST innovation
distributionsprovidebetterVaRforecaststhanothercompetitivemodelsat p =0.01levelbasedonthe
resultofDCtest. Therefore,itcanbeconcludedthatFHSmodelspecifiedunderskewedandfat-tailed
innovationdistributionsprovidesmoreaccurateVaRforecastsespeciallyforhighquantiles.
EvenifFHSmodelshavesimilarresultsinviewofLR ,LR andDCresults,theyhavedifferent
uc cc
failureratesandforecasterrors. LossfunctionsareusefultocompareVaRmodelswiththeirforecast
errors. Based on the ARLF, UL and FLF results, we conclude following results: (i) FHS-SN is the
bestperformedmodelat p =0.05and p =0.025levelsaccordingtoARLFandULcriteria.Basedon
theFLFresults,FHS-GEDmodelhasthelowestexcesscapitalvaluethanothermodelsat p = 0.05
and p = 0.025levels. Therefore,FHS-GEDmodelcouldbechosenasbestmodelfor p = 0.05and
p = 0.025levels;(ii)Basedonthethreebacktestingresults,FHS-TandFHS-STmodelsprovidethe
mostaccurateVaRforecastsamongothersat p =0.01level. Accordingtolossfunctionsresults,itis
easy to see that FHS-ST model has lower values of ARLF, UL and FLF results than FHS-T model.
Therefore,FHS-STmodelcouldbechosenasthebestmodelfor p =0.01model.
Figures3displaystheVaRforecastsofFHSmodelsspecifiedundersixinnovationdistributions.
AsseeninFigure3,theassumptiononinnovationprocessdoesnotaffecttheVaRforecastsofFHS
modelsoulfully. However,theGEDandSTdistributionscouldbepreferabletoreducetheforecast
erroroftheFHSmodel.
Table3.BacktestingresultsofFHSmodelsforlongposition(p=0.05,p=0.025,andp=0.01).
p=0.05
Models MeanVaR(%) N.OfVio. FailureRate LR-uc LR-cc DQ
FSH-N −0.910 29 0.041 1.146(0.284) 1.186(0.552) 4.586(0.710)
FSH-SN −0.911 29 0.041 1.146(0.284) 1.186(0.552) 4.545(0.715)
FSH-T −0.897 32 0.046 0.278(0.597) 0.459(0.794) 2.996(0.885)
FSH-GED −0.899 30 0.043 0.788(0.374) 0.863(0.649) 4.041(0.775)
FSH-SGED −0.904 30 0.043 0.788(0.374) 0.863(0.649) 4.255(0.749)
FSH-ST −0.897 32 0.046 0.278(0.597) 0.459(0.794) 3.187(0.867)
p=0.025
Models MeanVaR(%) N.OfVio. FailureRate LR-uc LR-cc DQ
FSH-N −1.193 20 0.029 0.350(0.554) 0.630(0.729) 6.820(0.448)
FSH-SN −1.196 20 0.029 0.350(0.554) 0.630(0.729) 6.805(0.449)
FSH-T −1.177 20 0.029 0.350(0.554) 0.630(0.729) 4.056(0.773)
FSH-GED −1.179 20 0.029 0.350(0.554) 0.630(0.729) 4.579(0.711)
FSH-SGED −1.187 20 0.029 0.350(0.554) 0.630(0.729) 5.102(0.647)
FSH-ST −1.177 20 0.029 0.350(0.554) 0.630(0.729) 4.051(0.774)