Mappadang, Agoes ina; Nug oho, Bayu Adi; Les a i, Se yani Dwi; Elizabe h; Les a i,
Ti i Kan i
A icle
Measu ing alue-a - isk and expec ed sho all o newe
c yp ocu encies: new insigh s
Cogen Business & Managemen
P o ided in Coope a ion wi h:
Taylo & F ancis G oup
Sugges ed Ci a ion: Mappadang, Agoes ina; Nug oho, Bayu Adi; Les a i, Se yani Dwi; Elizabe h;
Les a i, Ti i Kan i (2024) : Measu ing alue-a - isk and expec ed sho all o newe c yp ocu encies:
new insigh s, Cogen Business & Managemen , ISSN 2331-1975, Taylo & F ancis, Abingdon, Vol. 11,
Iss. 1, pp. 1-29,
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2024, VoL. 11, no. 1, 2416096
Measu ing alue-a - isk and expec ed sho all o newe
c yp ocu encies: new insigh s
agoes ina Mappadanga , Bayu adi nug ohob , se yani Dwi les a ia, elizabe ha and
i i kan i les a ic
aFacul y o economics and Business, uni e si as Budi Luhu , sou h Jaka a, indonesia; bYKPn school o Business, Yogyaka a,
indonesia; cPos g adua e P og am Magis e o applied economics, uni e si as a majaya, sou h Jaka a, indonesia
ABSTRACT
a signi ican amoun o his o ical e u ns is needed o he gene alized au o eg essi e
condi ional he e oscedas ici y (gaRch) models o be calib a ed. newe c yp ocu encies,
such as non- ungible okens (nF s), ha e ela i ely limi ed da a o c ea e obus
pa ame e es ima es. his s udy uses a newly de eloped me hod, he exponen ially
weigh ed mo ing a e age (eWMa) model, ha akes in o accoun he a - ailed
dis ibu ions o e u ns and ola ili y esponse o o ecas Value-a -Risk (VaR) and
expec ed sho all (es). We employ ho ough back es s o daily VaR and es o ecas s,
which a e widely u ilized o egula o y app o al and a e conside ed o be indus y
s anda ds. We also use loss unc ion a ios o selec he bes model. Ou esul s indica e
ha simple models a e jus as good as he complica ed ones, p o ided he simple
models cap u e a - ailed dis ibu ions o e u ns. he p ima y indings hold up h ough
se e al es s.
1. In oduc ion
non- ungible okens (nF s) a e essen ially Blockchain-based i ual asse igh s wi h dis inc i e iden i ies
and da a (e.g. collec ibles, ideo, audio, a , in-game i ems) (an e, 2022). nF s a e di e en om adi-
ional c yp ocu encies in ha each one has a unique iden i y and collec ion o da a, making hem
non-in e changeable. hei abili y o p o ide p oo o owne ship and au hen ici y o digi al goods is
wha has caused hem o become inc easingly popula , p o iding new oppo uni ies o musicians, a -
is s, and o he make s o make money o o hei wo k (Belgui h e al., 2024). e en hough he sec o
is s ill in i s in ancy, ansac ion olumes o nF asse s ha e inc eased signi ican ly in ecen yea s (ka im
e al., 2022). hus, i is pa icula ly impo an o unde s and he isk cha ac e is ics o nF s.
he modeling o quan ile isk le els o c yp ocu encies is a signi ican esea ch opic. his b anch o
s udy has become inc easingly in ica e, assessing nume ous e sions o he gaRch (gene alized au o e-
g essi e condi ional he e oscedas ici y) model se ies ha (Bolle sle , 1986) i s in oduced. howe e , he
complexi y o modeling op ions o c yp ocu ency isk modeling in he academic li e a u e con as s
sha ply wi h he cu en indus y p ac ice. Value-a - isk (VaR) and expec ed sho all (es) a e used by
se e al online si es ha analyze, u ilize, o e ola ili y o ecas s, and use an equally weigh ed me hodol-
ogy. Fo ins ance, he c yp ocu ency exchange Okex p o ides VaR p edic ion o Bi coin on i s blog. he
assump ion used o de ine his o ecas is ha Bi coin e u ns ollow a no mal dis ibu ion.
he accu acy o gaRch models equi es a signi ican numbe o his o ical e u ns (chu e al., 2017;
a dia e al., 2019; guesmi e al., 2019; sosa e al., 2019; iwa i e al., 2019; alexande and Dakos, 2020;
ha o i, 2020; segnon and Beki os, 2020; Maciel, 2021; agga wal, 2022; nug oho, 2022). While some c yp-
ocu encies, such as Bi coin and e he eum, ha e been in ci cula ion o qui e some ime, he cons an
a i al o new c yp ocu encies ha a ac in es o in e es indica es ha he e is o en inadequa e da a
© 2024 he au ho (s). Published by in o ma uK Limi ed, ading as aylo & F ancis g oup
CONTACT agoes ina Mappadang agus ina.mappadang@budiluhu .ac.id uni e si as Budi Luhu , sou h Jaka a, indonesia
h ps://doi.o g/10.1080/23311975.2024.2416096
his is an open access a icle dis ibu ed unde he e ms o he C ea i e Commons a ibu ion License (h p://c ea i ecommons.o g/licenses/by/4.0/), which
pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. he e ms on which his a icle has been
published allow he pos ing o he accep ed Manusc ip in a eposi o y by he au ho (s) o wi h hei consen .
ARTICLE HISTORY
Recei ed 13 Feb ua y
2024
Re ised 21 augus 2024
accep ed 8 Oc obe 2024
KEYWORDS
expec ed sho all;
exponen ially-weigh ed
mo ing a e age; eV ;
gaRch; alue-a - isk
JEL CLASSIFICATION
c46; c58; F31
SUBJECTS
Finance; business,
managemen and
accoun ing; inancial
accoun ing
2 a. MaPPaDang el al.
o gene a e iable es ima ions o newe c yp ocu encies. alexande and Dakos (2023) we e he i s o
in oduce an asymme ic eWMa ailo ed o a ola ili y esponse o add ess he p oblem encoun e ed
when employing equally weigh ed and gaRch models.
howe e , no esea ch has been conduc ed o assess he pe o mance o he asymme ic eWMa model
o downside isk measu es o newe c yp ocu encies such as non- ungible okens (nF ). simila ly, a
signi ican limi a ion o he exis ing li e a u e is he lack o discussion on simple models despi e p ac i-
ione s mos popula ly using such models. in addi ion, he e a e a ious gaps in he li e a u e on c yp-
ocu ency isk me ics. Fo example, he li e a u e on a ic ligh s o VaR and es back es ing is sca ce,
which is a common indus y p ac ice. simila ly, ew s udies ha e in es iga ed he VaR and es o sho
posi ions on c yp ocu encies, al hough hey a e aded as con enien ly as long posi ions. addi ionally,
a ew o he s udies ha e in es iga ed he p edic ion accu acy o mul i a ia e models. his s udy has
applied he mul i a ia e model as a obus ness check. he main objec i e o his esea ch is o ill all o
hese gaps in he li e a u e.
he sec ions ha ensue comp ise he li e a u e e iew and discussions o he compu a ion o VaR and
es based on uni a ia e and mul i a ia e se ings, he empi ical esul s, he discussion, and he conclu-
sion, espec i ely.
2. Li e a u e e iew
he RiskMe ics M eWMa me hod is a p ominen model owing o i s cla i y and con enience (longe s aey
and spence , 1996). some esea che s ha e concen a ed on e alua ing i s o ecas ing accu acy using
con en ional asse s and c yp ocu encies. he e is some e idence in c yp ocu ency esea ch ha applies
an in eg a ed gaRch (igaRch). no e ha he eWMa model i s he in eg a ed ola ili y model. Fo
example, köchling e al. (2020) disco e ed ha igaRch bes i s Bi coin and o he c yp ocu encies.
simila ly, Bau e al. (2018) ind ha c yp o ola ili y is in eg a ed.
i also u ns ou ha while e y complex ola ili y models can ob ain p ecise ou -o -sample downside
isk es ima es, simple models may gene a e equally eliable me hods. Fo example, Bonello and suda
(2018) con as ed VaR es ima es o Bi coin by employing bo h single- and dual- egime gaRch, assuming
gaussian and s uden - dis ibu ions. hey disco e ed hese ea u es could achie e p ecise VaR o ecas s
a he 95% con idence le el. Back es ing daily 1% VaR es ima es o Bi coin, os e e al. (2019) disco -
e ed ha a s anda d gaRch model is compa able o he mo e highly complica ed gaRch me hods used
in hei esea ch. ucíos (2019) e iewed VaR es ima ed o Bi coin a ound 2011 and 2017 using six
di e en models and disco e ed ha only he boo s ap VaR me hod gene a es eliable p edic ions a a
99% con idence le el. VaR p edic ions based on simple ola ili y models, such as he s anda d gaRch,
as used by ucíos and aylo (2023), can be conside ed eliable o Bi coin and e he eum. ace eda e al.
(2020) disco e ed ha mo e sophis ica ed con igu a ions o Bi coin VaR do no ou class simpli ied ones,
p o ided ha a a - ailed dis ibu ion is used ins ead o a no mal dis ibu ion. silahli e al. (2021) disco -
e ed ha a simple benchma k model wo ks well in nume ous VaR back es s o a wide ange o
c yp o asse s.
e en when only eWMa models a e conside ed, con adic o y esea ch is e iden . Fo ins ance, silahli
e al. (2021) asse ha a s anda d eWMa me hod gene a es p ecise VaR es ima es o all c yp ocu en-
cies. in con as , liu e al. (2020) ind ha a simila app oach ejec s VaR back es s. (nekhili and sul an
(2020) obse ed he ou -o -sample esul s o a RiskMe ics M eWMa me hod and ound ha i p o ides
p ecise VaR o ecas s only a a 95% con idence le el; howe e , es o ecas s o nea ly all c yp ocu encies
examined, an eWMa p o ides eliable p edic ions based on he eR es .
al hough p e ious s udies ha e in es iga ed some ola ili y me hods, he a ie y o in es iga ed models
has been limi ed. liu e al. (2020) concen a ed on eWMa-based me hodologies and skipped o e o he ,
mo e complica ed sys ems. hei indings did no p o ide any de ini i e e idence o he gene al eliabili y
o eWMa- ype amewo ks in p edic ing c yp ocu ency ola ili y when compa ed wi h o he app oaches
ha we e ei he mo e complica ed o simple . By con as , ca ania and g assi (2022) engaged in highly
complex gas models and ano he model ha used an assump ion o a a - ailed dis ibu ion o e u n.
hey hen es ed hose models wi h an al eady sophis ica ed benchma k, he β-skew- -egaRch amewo k.
hey ound ha hose models o en p oduced simila esul s ega ding hei abili y o p edic isks.
cOgen Business & ManageMen 3
Mo eo e , he g owing inance li e a u e conce ning newe c yp ocu encies, such as non- ungible
okens (nF s), includes po olio managemen , p ice bubbles, hedging p ope ies, in es o s’ a en-
ion, business models, and wash ading. Men ou a e al. (2023), who de eloped a machine-lea ning
ading echnique, disco e ed ha ha ing nF s in a po olio o adi ional and c yp ocu ency asse s
inc eases he sha pe Ra io. simila ly, he ge be s a is ic demons a ed ha nF s ha e a low co ela-
ion wi h ypical asse classes, which may lead o inc eased po olio di e si ica ion (ko e al., 2022).
he esul s o he squa ed wa ele cohe ence me hod demons a e he exis ence o a o able di e -
si ica ion p ope ies (uma e al., 2022). Fu he , se neels (2023) ecen ly p oposed some ac ics ha
can be used o iden i y po en ially audulen wash ading beha io in he nF ma ke s. in addi ion,
he indings om he nonlinea au o eg essi e dis ibu ed lag demons a ed ha nF s ha e he
po en ial o se e as sa e ha ens o he uni ed s a es dolla du ing he cOViD-19 ime ame (Zhang
e al., 2022).
Fu he mo e, geopoli ical isks s ongly p edic c yp opunks and Decen aland e u ns based on he
quan ile eg ession me hodology (u om e al., 2022). acco ding o he Q-join spillo e model, nF s
should no be ega ded as a sepa a e asse class unde ad e se ma ke condi ions (Xia e al., 2022).
addi ionally, an expe imen al echnique was used by Za i is and cheng (2022), who disco e ed ou nF
comme ce s a egies: nF c ea o s, an okens, nF ma ke places, and compu e games. Fu he , he
a en ion index o nF s in oduced by Wang, (2022) adequa ely explains nF e u ns. simila ly, google
sea ch olume o “nF ” posi i ely co ela es wi h signi ican c yp ocu ency e u ns (Pin o-gu ié ez
e al., 2022).
Based on he in o ma ion abo e, he s udy conce ning isk managemen o newe c yp ocu encies
applying a simple bu powe ul model is s ill e y limi ed. hus, he main goal o his esea ch is o ill
his gap in he li e a u e.
3. Me hodology
3.1 Value-a - isk (VaR)
VaR is a loss we a e easonably con iden we canno su pass i he cu en po olio is held o a ce ain
pe iod. VaR has wo undamen al speci ica ions: he signi icance le el, α, and he ime ame, ypically
de e mined on ading days, o which VaR is calcula ed. Fo example, a 5% daily VaR, which ela es o a
95% con idence le el, is a loss scale we expec o encoun e wi h a 5% equency when he exis ing
po olio is main ained o a day. Pu ano he way, i is
VR
F
F
ale ail long posi ion
igh ail
,
,
,
{
α
α
α
=−−
()
−
−
−
()
−
1
1
1ssho posi ion
()
(1)
F
is a one-day ahead o ecas c ea ed a he ime
and i is based on he e u n dis ibu ion assump-
ion. Following alexande and Dakos (2023), he benchma k model e u ns, applying an equally weigh ed
a e age model, is assumed o be no mal dis ibu ion. hus,
XN
∼
()
µσ
,2 (2)
using he s anda d no mal ans o ma ion, we ha e
PX X P
XPZ
<
()
=−<−
=<
−
=
,
σ
µ
σ
µ
σ
µ
σ
α
xx
(3)
i z is a no mal alue, i gene a es:
X
,
σ
µ
σ
θσ
−=
()
−1 (4)
4 a. MaPPaDang el al.
i
θσ θ σ
−−
()
= −
11
1( ) , hen he 100α% 𝒽-day no mal VaR is
VR
a
,
σ
θ σσ µ
=−
()
−
−11 (5)
i a andom a iable T has a s uden - dis ibu ion wi h ν deg ees o eedom, i s densi y unc ion is
V
V
VVV V
()
=
()
+
+
()
−−
−−+
πτ τ
12 22
2
1
2
1
/
1
1
1
(6)
he
α
quin ile o he s anda d S uden - is
−
()
=−
()
−−
VV
11
αα
1 (7)
he s uden - VaR is
S VR V V
VV
uden −a
α
ασ µ
,=−
()
−
()
−
−−11
21 (8)
3.2 Expec ed sho all (ES)
es ep esen s he po en ial losses i he loss su passes VaR. he es in o ms us o how much we can
expec o lose i he VaR is exceeded:
E a x dx
S VR
αα
α
()
=
()
∫
1
0
(9)
le 𝒳 be he discoun ed h-day e u n. he e u ns o he benchma k model we e assumed o mimic
a no mal dis ibu ion:
X
=+ ∼
()
Z ZN
σ
µ
, ,0 1 (10)
i z is a no mal alue, i p oduces
EE
SX S
,
αα
αµ
()
=
()
− (11)
i 𝒳 has a s uden - dis ibu ion wi h mean
µ
ℏ
, s anda d de ia ion
σ
, and ν deg ees o eedom, hen
(12)
x
α
()
deno es he α quan ile o he s anda dized 𝒮 uden - dis ibu ion wi h ν deg ees o eedom.
is densi y. he s anda dized s uden - densi y is
(13)
he es in a s anda dized 𝒮 uden - dis ibu ion wi h ν deg ees o eedom is
(14)
cOgen Business & ManageMen 5
3.3 EWMA and EGARCH
he eWMa o he a iance es ima e a ime o e u ns is
(15)
λ
is he smoo hing cons an . in oduced an asymme ic ola ili y esponse (η) o he o iginal eWMa
model (aeWMa):
(16)
Fu he , he a iance es ima e in he 𝒮 uden - egaRch (1,1) model:
(17)
ϕε ε γ ε ε
()
= + −
()
θE (18)
3.4 Mul i a ia e se ing
in hese se ings, le be he ec o o nF e u ns a ime . he benchma k model is based on he
(19)
Σ
deno es he co a iance ma ix. he co a iance ma ix in he eWMa model:
(20)
he co a iance ma ix o he aeWMa (λ, η, ν) model is gi en by:
(21)
Fo he mul i a ia e gaRch model, he co a iance ma ix is based on:
Σ
=
DCD
(22)
C QQ Q
=
() ()
−−
diag diag
12 12//
(23)
is he condi ional co ela ion, and
is he a iances (diagonal ma ix) o he uni a ia e egaRch.
is based on he aDcc model (cappiello e al., 2006):
(24)
3.5 Measu ing he accu acy and model selec ion
his s udy employs he Basel commi ee’s indus y-s anda d a ic ligh es . addi ionally, his esea ch
u ilizes back es ing o es, ini ia ed by cos anzino and cu an (2018). Following alexande and Dakos
(2023), his esea ch ad ances he es o include le - and igh - ail VaR. he iola ion pa am-
e e XVR
a
is
(25)
6 a. MaPPaDang el al.
he cumula i e alue o VaR iola ions
X
N
VaR
α
()
o e he en i e o ecas ing pe iod = 1, …, N is
compu ed as
XX
N
VR
N
VR
aa
αα
()
=
()
=
∑
1
(26)
i he VaR model is p ecisely i ed, he o al numbe o VaR iola ions is modeled as a binomial
dis ibu ion:
(27)
i
X
VaR is he o al numbe o iola ions du ing he o ecas ing pe iod and z is he s anda d no mal
ans o m, hen he likelihood o ob aining
X
VaR is
θ
(z), whe e z is he s anda d no mal dis ibu ion
unc ion. g een egion i
θ
(z) < 95%, yellow i 95% ⩽
θ
(z) < 99.99%, and ed i
θ
(z)
≥
99.99%. he
h ee-zone echnique is in oduced o econcile he wo e o ypes: ype i e e s o he po en ial o an
accu a e model o be labeled as inaccu a e based on he ou comes o i s back es ing; ype ii e e s o
he po en ial o an e oneous model no o be labeled as such. he back es ing indings a e hough o
be compa ible wi h an accu a e model in he g een zone, whe e he e is li le chance o mis akenly
accep ing an inaccu a e model. he back es ing esul s a e unlikely o come om an accu a e model in
he ed zone. Back es ing indings could be consis en in he yellow zone.
simila ly, his esea ch ad ances he es o include he le - and igh - ail es:
(28)
he a iables exp ess he in ensi y o each VaR exceeding and . g ea e
magni ude e u ns ha su pass bo h he VaR and es domina e XS
E
()
α
, while e u ns ha su pass he
VaR bu no he es a e gi en compa a i ely less weigh . he cumula i e es iola ion is
XX
N
S
N
S
EE
αα
()
=
()
=
∑
1
(29)
as s a ed by cos anzino and cu an (2018),
X
N
SE
α
()
is
N
µσ
EESS
N,2
()
(30)
µ
E
S
is 0.5 (1 –
α
)
N
and
σ
E
S
2 is (1 –
α
)(4 – 3(1 –
α
))/12.
he p obabili y o ecei ing
X
es o less p o ided he o al ac ual es exceedances du ing he o ecas
pe iod o
X
es, is
θ
(z), whe e z is he s anda d no mal dis ibu ion unc ion o
X
es. g een egion i
θ
(z)
< 95%, yellow i 95% ⩽
θ
(z) < 99.99%, and ed i
θ
(z)
≥
99.99%.
Fu he mo e, his s udy also p o ides he condi ional co e age (cc) es , in oduced by ch is o e sen
(1998).
LRCC
expexp
nn
nnnn
10
01 00 11 10
1
11
01 01 11 11
(31)
ψ
01
is he p opo ion o iola ions, p o ided ha he las e u n is no a iola ion, and
ψ
11
is he p o-
po ion o iola ions, gi en ha he las e u n is a iola ion.
Mo eo e , Mcneil and F ey (2000) in oduced a me hodology o back es ing es. i is based on a ime
se ies o s anda dized exceedance esiduals (eR), de ined as
cOgen Business & ManageMen 7
(32)
he eR es s a is ic is
(33)
µ
is based on 1000 boo s ap simula ions.
i nume ous models ob ain he co ec uncondi ional/condi ional co e age, he p ac i ione encoun-
e s he dilemma o being unable o choose be ween di e en op ions. in his case, compa a i e ech-
niques we e applied o selec he model wi h he bes pe o mance. Fissle and Ziegel (2016) demons a e
ha VaR and es a e join ly elici able:
FZL
S
VR VR
S
S
T
α
α
αα
α
α
α
=−
()
+ +−
()
−
1
1
E
d
E
E
a
alog (34)
he ollowing packages om R s a is ical so wa e we e used: mga ch (ghalanos, 2022a), uga ch
(ghalanos, 2022b), cus omized gas (a dia e al., 2019, 2022), cus omized u Risk (Feng e al., 2022) and
o he cus omized R codes.
4. Empi ical esul s
4.1 Da a
he da a include 379 daily log e u ns o aPe, icP, and sanD om Ma ch 18, 2022, o Ma ch 31, 2023.
his s udy selec ed aPe, icP, and sanD because hey we e newe and among he en la ges nF s when
w i ing his s udy. he nF ma ke is open 24 hou s a day, se en days a week; hus, when calcula ing
e u ns, his s udy used he closing p ice a midnigh (u c). hese da a we e ob ained om coinMa ke cap
and es ic ed by he a ailabili y o he aligned e u n se ies. One hund ed wen y-nine obse a ions
we e applied o ob ain he one-day-ahead VaR and es. he emaining da a (n = 250) we e used o back-
es he models. he Basel commi ee p oposes a s aigh o wa d back es o de e mining he s a u o y
isk-managemen capi al equi emen ha co e s 250 days. hus, he olling window me hod was u ilized
o calcula e VaR and es. losses a e ep esen ed as nega i e daily log e u ns.
Figu e 1 illus a es he exis ence o ola ili y clus e ing. in addi ion, able 1 e eals ha he ku osis
alues we e g ea e han 3, indica ing a lep oku ic dis ibu ion. he indings o he Ja que-
B
e a,
D’agos ino, and
A
nscombe-
G
lynn es s indica e ha he null hypo hesis o no excess ku osis, skew-
ness, and no mali y could no be suppo ed. he assump ion o a ze o mean applies o all nF s, wi h a
mean close o ze o.
selec ing he app op ia e dis ibu ions ha be e cap u e he a - ailed and skewed na u e o c yp o-
cu ency e u ns is essen ial (Yang and Xu, 2021). ins ead o manually selec ing he app op ia e dis ibu-
ions, we use a dynamic e sion o he op imal uni a ia e gaRch selec ion p ocedu e (an onakakis e al.,
2021). in o he wo ds, we use he cus om R code o selec he dis ibu ions au oma ically. he codes a e
in he R package o connec ednessapp oach (Da id gabaue , 2022). hence, we use egaRch wi h he
s uden - dis ibu ion. i is impo an o unde s and ha he main aim o his esea ch is o use he
eWMa me hod wi h ad hoc pa ame e s, and he egaRch model is used as a compa ison.
Fu he , he s uden - egaRch pa ame e s, shown in able 2, show ha he esponse pa ame e
θ
is
small and insigni ican . Mo eo e , he asymme y pa ame e γ is signi ican o all nF s. he ex emely
signi ican γ suppo s he assump ion ha e u ns ollow a a - ailed dis ibu ion. addi ionally, condi ional
ola ili y’s esponse o ma ke shocks is gauged by he gaRch e o pa ame e α. i α is qui e la ge (e.g.
abo e 0.1), hen ola ili y is e y sensi i e o ma ke e en s. he pe sis ence o condi ional ola ili y,
14 a. MaPPaDang el al.
Figu e 4. VaR (do -dash), losses (long dash), and es (solid) es ima es o aPe (long posi ion), he ading days om 25
July 2022 o 31 Ma ch 2023. No es: he ed do s a e he o al iola ions/exceedances.
cOgen Business & ManageMen 15
Table 8. Back es ing ou comes o one-day-ahead 1% VaR and es (Long Posi ion), 25 Feb ua y 2022—31 Ma ch 2023,
mul i a ia e se ings.
Panel a: 1 % VaR long posi ion
a alanche Polygon solana
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 14 0.9999 0.000 12 0.9996 0.000 18 0.9999 0.000
eWMa (94%) 3 0.3942 0.807 3 0.3942 0.807 4 0.5896 0.076
eWMa (92.5%) 2 0.2087 0.481 1 0.0769 0.168 2 0.2087 0.481
aeWMa (94%, 1%) 3 0.3942 0.807 2 0.2087 0.481 3 0.3942 0.032
aeWMa (94%, 2%) 2 0.2087 0.481 1 0.0769 0.168 2 0.2087 0.481
aeWMa (94%, 3%) 1 0.0769 0.168 1 0.0769 0.168 2 0.2087 0.481
aeWMa (92.5%, 1%) 2 0.2087 0.481 1 0.0769 0.168 2 0.2087 0.481
aeWMa (92.5%, 2%) 1 0.0769 0.168 1 0.0146 0.014 1 0.0769 0.168
aeWMa (92.5%, 3%) 1 0.0769 0.168 0 0.0146 0.014 1 0.0769 0.168
aDCC 3 0.3942 0.807 4 0.5896 0.076 3 0.3942 0.032
Panel B: 1 % es long posi ion
a alanche Polygon solana
X
N
S
E
θ
(z) eR
X
N
SE
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 6 0.9999 0.554 4 0.9998 0.068 7 0.9999 0.616
eWMa (94%) 0 0.1123 0.317 0 0.1123 0.817 1 0.2499 0.317
eWMa (92.5%) 0 0.0397 0.317 0 0.0109 1.000 0 0.0397 0.317
aeWMa (94%, 1%) 0 0.1123 1.000 0 0.0397 0.784 1 0.1123 0.317
aeWMa (94%, 2%) 0 0.0397 1.000 0 0.0109 0.869 1 0.0397 0.317
aeWMa (94%, 3%) 0 0.0109 1.000 0 0.0109 0.755 1 0.0397 0.317
aeWMa (92.5%, 1%) 0 0.0397 0.317 0 0.0109 0.848 0 0.0397 0.317
aeWMa (92.5%, 2%) 0 0.0109 0.317 0 0.0023 0.755 0 0.0109 0.317
aeWMa (92.5%, 3%) 0 0.0109 0.317 0 0.0023 1.000 0 0.0109 0.317
aDCC 0 0.1123 0.179 0 0.2499 0.755 1 0.1123 1.000
No es: see able 3.
Table 9. Back es ing ou comes o one-day-ahead 2.5% VaR and es (Long Posi ion), 25 Feb ua y 2022—31 Ma ch 2023,
mul i a ia e se ings.
Panel a: 2.5 % VaR long posi ion
a alanche Polygon solana
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 21 0.9999 0.000 18 0.9999 0.000 23 0.9999 0.000
eWMa (94%) 11 0.9987 0.011 9 0.9892 0.049 10 0.9961 0.026
eWMa (92.5%) 10 0.9961 0.026 7 0.9370 0.114 12 0.9996 0.004
aeWMa (94%, 1%) 9 0.9892 0.049 4 0.5896 0.076 7 0.9370 0.114
aeWMa (94%, 2%) 5 0.7538 0.117 4 0.5896 0.076 7 0.9370 0.114
aeWMa (94%, 3%) 4 0.5896 0.076 4 0.5896 0.076 5 0.7538 0.117
aeWMa (92.5%, 1%) 7 0.9370 0.114 4 0.5896 0.076 8 0.9727 0.082
aeWMa (92.5%, 2%) 5 0.7538 0.117 4 0.5896 0.076 7 0.9370 0.114
aeWMa (92.5%, 3%) 3 0.4962 0.807 3 0.3942 0.032 4 0.5896 0.076
aDCC 9 0.9892 0.049 7 0.9370 0.114 11 0.9987 0.000
Panel B: 2.5 % es long posi ion
a alanche Polkado solana
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 9 1.0000 0.038 7 0.9999 0.038 10 1.0000 0.506
eWMa (94%) 3 0.9990 0.989 1 0.9784 0.786 3 0.9948 0.732
eWMa (92.5%) 2 0.9948 0.709 1 0.8274 0.898 2 0.9998 0.848
aeWMa (94%, 1%) 2 0.9784 0.695 1 0.2499 0.774 3 0.8274 0.651
aeWMa (94%, 2%) 2 0.4463 0.722 1 0.2499 0.779 2 0.8274 0.683
aeWMa (94%, 3%) 1 0.2499 0.391 0 0.2499 0.784 1 0.4463 0.743
aeWMa (92.5%, 1%) 2 0.8274 0.658 0 0.2499 0.867 2 0.9310 0.711
aeWMa (92.5%, 2%) 1 0.4463 0.684 0 0.2499 0.844 1 0.8274 0.647
aeWMa (92.5%, 3%) 0 0.1123 0.646 0 0.1123 0.843 1 0.2499 0.679
aDCC 2 0.9784 0.678 2 0.8274 0.750 3 0.9990 0.729
No es: see able 4.
16 a. MaPPaDang el al.
Table 10. Back es ing ou comes o one-day-ahead 1% VaR and es (sho Posi ion), 25 Feb ua y 2022 – 31 Ma ch 2023,
mul i a ia e se ings.
Panel a: 1 % VaR sho posi ion
a alanche Polygon solana
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 6 0.8685 0.015 9 0.9892 0.049 5 0.7538 0.082
eWMa (94%) 1 0.0769 0.649 4 0.5896 0.095 1 0.0769 0.095
eWMa (92.5%) 1 0.0769 0.649 3 0.3942 0.807 1 0.0769 0.095
aeWMa (94%, – 1%) 2 0.2087 0.404 4 0.5896 0.095 1 0.0769 0.095
aeWMa (94%, – 2%) 2 0.2087 0.404 4 0.5896 0.095 1 0.0769 0.095
aeWMa (94%, – 3%) 2 0.2087 0.404 3 0.3942 0.807 1 0.0769 0.095
aeWMa (92.5%, – 1%) 1 0.0769 0.649 3 0.3942 0.807 1 0.0769 0.095
aeWMa (92.5%, – 2%) 2 0.2087 0.404 3 0.3942 0.807 1 0.0769 0.095
aeWMa (92.5%, – 3%) 1 0.0769 0.649 2 0.2087 0.481 1 0.0769 0.095
aDCC 2 0.2087 0.404 3 0.3942 0.807 2 0.2087 0.875
Panel B: 1 % es sho posi ion
a alanche Polygon solana
X
N
SE
θ
(z) eR
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 3 0.6571 0.401 6 0.9784 0.019 3 0.4464 0.378
eWMa (94%) 1 0.0109 0.317 1 0.2499 0.817 0 0.0109 0.317
eWMa (92.5%) 1 0.0109 0.317 0 0.1123 1.000 0 0.0109 0.317
aeWMa (94%, – 1%) 1 0.0397 1.000 1 0.2499 0.784 0 0.0109 0.317
aeWMa (94%, – 2%) 1 0.0397 1.000 1 0.2499 0.869 0 0.0109 0.317
aeWMa (94%, – 3%) 1 0.0397 1.000 1 0.1123 0.755 0 0.0109 0.317
aeWMa (92.5%, – 1%) 1 0.0109 0.317 0 0.1123 0.848 0 0.0109 0.317
aeWMa (92.5%, – 2%) 1 0.0397 1.000 0 0.1123 0.755 0 0.0109 0.317
aeWMa (92.5%, – 3%) 1 0.0109 0.317 0 0.0397 1.000 0 0.0109 0.317
aDCC 1 0.0397 0.179 1 0.1123 0.755 1 0.0397 1.000
No es: see able 5.
Table 11. Back es ing ou comes o one-day-ahead 2.5% VaR and es (sho Posi ion), 25 Feb ua y 2022 – 31 Ma ch
2023, mul i a ia e se ings.
Panel a: 2.5 % VaR sho posi ion
a alanche Polygon solana
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 11 0.9987 0.000 13 0.9998 0.001 9 0.9892 0.004
eWMa (94%) 7 0.9370 0.005 10 0.9961 0.026 8 0.9727 0.001
eWMa (92.5%) 7 0.9370 0.005 11 0.9987 0.011 8 0.9727 0.001
aeWMa (94%, – 1%) 7 0.9370 0.005 10 0.9961 0.026 7 0.9370 0.026
aeWMa (94%, – 2%) 6 0.8685 0.015 9 0.9892 0.049 5 0.7538 0.216
aeWMa (94%, – 3%) 3 0.3942 0.216 7 0.9370 0.404 3 0.3942 0.649
aeWMa (92.5%, – 1%) 6 0.8685 0.015 11 0.9987 0.011 8 0.9727 0.011
aeWMa (92.5%, – 2%) 5 0.7538 0.042 10 0.9961 0.026 7 0.9370 0.042
aeWMa (92.5%, – 3%) 3 0.3942 0.216 7 0.9370 0.404 4 0.5896 0.404
aDCC 5 0.7538 0.042 9 0.9892 0.049 11 0.9987 0.000
Panel B: 2.5 % es sho posi ion
a alanche Polygon solana
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 7 0.9990 0.153 9 0.9999 0.063 6 0.9785 0.125
eWMa (94%) 1 0.8274 0.989 4 0.9948 0.786 1 0.9311 0.732
eWMa (92.5%) 1 0.8274 0.709 1 0.9990 0.898 1 0.9311 0.848
aeWMa (94%, – 1%) 1 0.8274 0.695 4 0.9948 0.774 1 0.8275 0.651
aeWMa (94%, – 2%) 1 0.6571 0.722 3 0.9784 0.779 1 0.4463 0.683
aeWMa (94%, – 3%) 1 0.1123 0.391 2 0.8274 0.784 1 0.1123 0.743
aeWMa (92.5%, – 1%) 1 0.6571 0.658 3 0.9990 0.867 1 0.9311 0.711
aeWMa (92.5%, – 2%) 1 0.4463 0.684 2 0.9948 0.844 1 0.8275 0.647
aeWMa (92.5%, – 3%) 1 0.1123 0.646 2 0.8274 0.843 1 0.2500 0.679
aDCC 2 0.4463 0.678 3 0.9784 0.704 1 0.9990 0.729
No es see able 6.
cOgen Business & ManageMen 17
4.5 ADCC-GARCH Roll pa ame e s
able 13 p esen s he aDcc-gaRch olling pa ame e s. he α pa ame e measu es condi ional ola ili y’s
esponse o ma ke shocks. i α is la ge (e.g. abo e 0.1), hen ola ili y is highly sensi i e o ma ke
e en s. he β pa ame e compu es he pe sis ence o condi ional ola ili y. Vola ili y akes ime o se le
down when β is la ge (e.g., abo e 0.9). he
µ
pa ame e shows he le el o he long- e m a e age ola-
ili y. in addi ion, he asymme ic pa ame e (
ϕ
γ
1
), is posi i e, indica ing ha nega i e esiduals do no
inc ease he condi ional ola ili y signi ican ly mo e han posi i e shocks.
5. GARCH – ine copula simula ion
in his sec ion, we p o ide he indings o Mon e ca lo simula ion-based es s. he sample size makes
his es impo an . none heless, no VaR model p esc ibes he sample size o da a equency— hese a e
pe sonal p e e ences. in addi ion, he selec ion o da a equency and sample size a e closely ela ed. Fo
example, i we use 10-day e u ns, we need a 20-yea sample; i we use weekly e u ns, we need a
10-yea sample; i we use daily e u ns, we need a 2-yea sample; and so on. so a , we ha e compu ed
VaR and es o daily e u ns.
his sec ion also deals wi h 10-day VaR and es. he e a e wo app oaches o compu ing VaR and es
o he long ho izon (i.e. h = 10): analy ic, such as he squa e oo o ime, and nume ical simula ion, such
as Mon e ca lo. he analy ic app oach is based on he assump ion ha daily e u ns a e no mal and i.i.d.
howe e , isk ac o e u ns a he daily o weekly equency a ely ha e no mal dis ibu ions. hence, we
use he Mon e ca lo simula ion app oach. We mainly use copulas.
copulas, mul i a ia e dis ibu ions wi h uni o m ma ginals, can c ea e a wide a ay o isk ac o e u n
dis ibu ions. skla (1959) shows ha a andom ec o .
XX X
d
= …( , ., ˙
1) wi h join dis ibu ion F and ma -
ginals
FF
d
1,.,…, i has a copula unc ion ˘
C
ha gi es
Fx xCFx Fx
ddd
111
,, ,.,
˘
(35)
Table 12. Join VaR and es loss unc ion a ios.
Panel a
α
=
1% long posi ion
α
=
2 5. % long posi ion
a alanche Polygon solana a alanche Polygon solana
Benchma k 1.000 1.000 1.000 1.000 1.000 1.000
eWMa (94%) 0.919 0.916 0.907 0.956 0.955 0.946
eWMa (92.5%) 0.927 0.928 0.918 0.964 0.966 0.956
aeWMa (94%, 1%) 0.903 0.900 0.892 0.941 0.939 0.932
aeWMa (94%, 2%) 0.874 0.869 0.865 0.914 0.911 0.907
aeWMa (94%, 3%) 0.836 0.830 0.830 0.879 0.874 0.875
aeWMa (92.5%, 1%) 0.911 0.910 0.902 0.949 0.949 0.941
aeWMa (92.5%, 2%) 0.880 0.878 0.874 0.920 0.919 0.915
aeWMa (92.5%, 3%) 0.841 0.836 0.837 0.884 0.881 0.882
aDCC 0.837 0.796 0.831 0.912 0.889 0.906
Panel B
α
=1%
sho posi ion
α
=
2 5. % sho posi ion
a alanche Polygon solana a alanche Polygon solana
Benchma k 1.000 1.000 1.000 1.000 1.000 1.000
eWMa (94%) 0.946 0.924 0.942 0.986 0.963 0.984
eWMa (92.5%) 0.955 0.936 0.953 0.994 0.974 0.994
aeWMa (94%, – 1%) 0.930 0.907 0.926 0.971 0.948 0.970
aeWMa (94%, – 2%) 0.900 0.876 0.898 0.943 0.919 0.944
aeWMa (94%, – 3%) 0.860 0.836 0.862 0.906 0.882 0.910
aeWMa (92.5%, – 1%) 0.938 0.918 0.936 0.978 0.958 0.979
aeWMa (92.5%, – 2%) 0.906 0.885 0.907 0.949 0.927 0.952
aeWMa (92.5%, – 3%) 0.866 0.843 0.869 0.911 0.888 0.917
aDCC 0.862 0.806 0.862 0.941 0.884 0.942
No es: Values g ea e han one indica e ou pe o mance o he benchma k model and ice e sa.
18 a. MaPPaDang el al.
Cu uFFu Fu
dd
d
11
1
1
1
,, ,.,
(36)
whe e uFx
iii
.
Mul iple pai -copula cons uc ions exis o mo e han o equal o h ee asse s. Bed o d and cooke
(2002) p oposed a g aphical model o help a ange se e al pai copulas, such as c- and D- ine copulas.
he e a e R- ine copulas o mo e han o equal o i e asse s (aas e al., 2009). see czado e al. (2012)
o u he in o ma ion on pai copula.
in addi ion, we also added ano he model o compu e VaR and es: he ex eme alue heo y (eV ).
Because we ocus on he ail o he e u n dis ibu ion, ex eme alue heo y is an essen ial ins umen
o modeling he ail dis ibu ion wi hou making any assump ions abou he dis ibu ion cen e . in eV ,
he cumula i e dis ibu ion unc ion (CDF) is
(37)
β
and ξ a e he scale and he shape pa ame e s, espec i ely. i ξ > 0, i implies hea y ail dis ibu ion.
hen, VaR and es o ecas s unde eV a e gi en by
(38)
(39)
β
and
ξ
a e es ima ed using maximum likelihood.
Table 13. aDCC-gaRCH oll.
op imal pa ame e s ac oss olls (Fi s 2, Las 2)
Roll-1 Roll-2 Roll-13 Roll-14
µ
A a
0.001 0.001 −0.001 −0.001
ω
A a
0.000 0.004 0.000 0.000
α
A a
0.133 0.138 0.149 0.110
β
A a
0.813 0.812 0.819 0.866
Shape 4.839 4.761 5.203 5.711
µ
Poly
0.000 0.000 −0.003 0.000
ω
Poly
0.000 0.000 0.000 0.000
α
Poly
0.079 0.061 0.159 0.166
β
Poly
0.889 0.904 0.780 0.771
Shape 3.237 3.366 4.219 4.833
µ
Sola
0.002 0.003 −0.002 −0.002
ω
Sola
0.001 0.001 0.001 0.001
α
Sola
0.141 0.132 0.212 0.177
β
Sola
0.712 0.732 0.388 0.476
Shape 4.868 5.007 5.071 4.991
aDCC
ϕα
1
0.035 0.039 0.030 0.012
ϕβ
1
0.904 0.913 0.938 0.936
ϕ
γ
1
0.092 0.089 0.039 0.088
µ
shape 4.209 4.234 4.297 4.628
No es: he alues a e aken om he mga ch package, ixed- olling window app oach, and e i e e y 30 days.
cOgen Business & ManageMen 19
simila o p e ious s udies (gho bel and abelsi, 2014; Mejdoub and gho bel, 2018; nug oho, 2023),
we do he ollowing s eps o gaRch – Vine copula simula ion using R s a is ical packages (alexios
ghalanos, 2022; B echmann and schepsmeie , 2013; nagle e al., 2021):
S ep 1: We ob ain one-s ep-ahead ola ili y based on a hea y- ail dis ibu ion
˘
(,., )
11
, and compu e
one-s ep-ahead s anda dized esiduals, . We do his s ep ecu si ely. We use gaRch wi h
he s uden - dis ibu ion due o au oco ela ed e u ns.
S ep 2: We compu e he pseudo-obse a ions om he s anda dized esiduals compu ed in S ep 1.
S ep 3: Fi a Vine copula o he da a es ima ed in S ep 2. We ha e ound ha C-Vine is he bes i . We also
selec ed he bes copula amily o i ou da a. gumbel copula is he bes choice.
S ep 4: We simula e 10000 pseudo-obse a ions based on pa ame e s selec ed in s ep 3.
S ep 5: We ans o m he pseudo-obse a ions in o e u ns.
S ep 6: We compu e 10-day VaR and es ecu si ely using an equally weigh ed a e age model ( he bench-
ma k), eWMa models, eWMa models adjus ed o a - ailed dis ibu ion, gaRch, and he ex eme
Value heo y (eV ) ecu si ely. We spli he da a 70 – 30. hence, we ha e a ound 300 ou -o -sample
esul s.
able 14 shows he au oco ela ion es s and copulas selec ion. Panel A indica es ha he s anda d-
ized esiduals a e ee om au oco ela ion issues. Panel B shows ha he C-Vine copula and Gumbel
amily a e selec ed o u he simula ion p ocess. Fu he , Figu e 5 shows he densi y ob ained om he
simula ion.
Mo eo e , he back es ing esul s o 10-day 1% VaR and es (long posi ion) a e p esen ed in able 15.
he VaR es ima es e eal ha excep ions gene a ed by all models excep he benchma k (aPe) occu ed
in he g een egions. in addi ion, he cc es was pe o med. i was de e mined ha mos models we e
accu a e o es ima ing VaR. simila ly, he es es ima es show ha he excep ions o exceedances gene -
a ed by all models excep he benchma k (aPe) occu ed in he g een egions.
Fu he mo e, able 16 p esen s he indings om ano he signi icance le el (α = 2.5%). he benchma k
model was no consis en ly accu a e. in e es ingly, he eV model was he only model ha p edic ed VaR
accu a ely o aPe. i is possible ha AEWMA models need mo e han h ee pe cen age adjus men s o
be accu a e. as expec ed, he benchma k model could no consis en ly p o ide accu a e o ecas s. in
addi ion, he back es ing esul s o en-day VaR and es (sho posi ion) a e p esen ed in ables 17
and 18. he esul s show ha models o he han he benchma k p o ided good o ecas s.
in addi ion, able 19 shows he join VaR and es loss unc ion a ios used o ank he accu acy o each
model o o ecas ing he en-day es and VaR. Values g ea e han one indica e he ou pe o mance o
he benchma k model. as expec ed, he gaRch and eV models, which we e mo e complica ed, consis-
en ly ou pe o med he benchma k. in e es ingly, he simple models always opped he benchma k,
p o ided ha he models had he app op ia e pa ame e alues.
Table 14. he goodness o i es s.
Panel a
aPe iCP sanD
Ljung-Box (5) 0.507 (0.956) 3.092 (0.390) 2.507 (0.504)
Ljung-Box (10) 0.233 (0.905) 5.056 (0.497) 3.884 (0.683)
Sign Bias 0.359 (0.719) 0.587 (0.390) 1.209 (0.227)
Panel B
ℓog-ℓikelihood aiC BiC
C-Vine 244.680 −481.360 −466.830
D-Vine 235.180 −464.360 −453.470
Gaussian 204.808 −403.617 −392.724
S uden - 222.882 −433.764 −411.976
Clay on 225.280 −444.561 −433.667
Gumbel 236.488 −466.977 −456.083
F ank 206.756 −407.512 −396.618
Joe 219.765 −433.531 −422.637
No es: Panel a shows s a is ical es s con i ming ha au oco ela ion does no exis . he numbe s in pa en heses a e he p- alues. Panel B
shows ha C-Vine and Gumbel Copula a e p e e ed o e o he s.
20 a. MaPPaDang el al.
Fu he , Figu es 6 and 7 show he exceedances o iola ions. o sa e space, only an analysis o he
aPe is p esen ed. Figu e 6 indica es ha he benchma k model has he wo s b eaches o VaR and es
o he long posi ion o aPe. simila ly, Figu e 7 shows ha he benchma k model also has he wo s
iola ions o VaR and es o he sho posi ion o aPe.
6. Discussion
When he densi y unc ion o a dis ibu ion has a g ea e peak alue and mo e mass in he ails com-
pa ed o he usual densi y unc ion wi h he same a iance, he dis ibu ion is lep oku ic. lep oku osis
is one o he key "s ylized ac s" ha come o ligh when he empi ical dis ibu ions on inancial asse
e u ns a e examined. he skewness o e u n densi ies is also no iceable, as hey equen ly ha e a big-
ge lowe ail and a s ong nega i e skew. When isk ac o e u n dis ibu ions ha e lep oku osis and
nega i e skewness, he VaR based on no mal dis ibu ion will likely unde es ima e he VaR a high con-
idence le els (Figu e 8).
academically speaking, he e is s ill a dea h o nF li e a u e on measu ing nF s’ Value-a -Risk and
expec ed sho all. unde s anding he isk p o iles and inding accu a e and simple models a e essen ial
due o he ma ke dynamics o nF s. he ma ke o nF s saw s ong expansion in 2021–2022 be o e
expe iencing a sha p down u n. la e in 2021 and ea ly 2022, he ma ke exploded, wi h nF s aded o
s agge ing p ices. ne e heless, he ma ke saw a co ec ion by ap il 2022, and nF p ices e e ed o
mo e ai le els. o he bes o ou knowledge, he cu en nF li e a u e has been ocusing on he ding
beha io (De sil a e al., 2024), he di e si ica ion abili ies o nF s (aha on and Demi , 2022; ko e al.,
2022; Zhang e al., 2022), bubbles (Maouchi e al., 2022; Wang e al., 2022), e c.
he issue encoun e ed when employing an equally weigh ed model ( he p ac i ione s’ app oach) is
no caused by equen small jumps in asse p ices bu by in equen la ge jumps (alexande , 2009).
When a leng hy pe iod o a e aging is applied, he signi icance o a single ex ao dina y e en is a e -
aged ac oss a la ge sample o e u ns. consequen ly, an es ima e o he ma ke ’s ola ili y based on a
long-mo ing a e age will no eac o a sudden, sho - e m shock. addi ionally, he model a emp s o
Table 15. Back es ing ou comes o 1% 10-day VaR and es (Long Posi ion), gaRCH – ine copula simula ion.
Panel a: 1 % VaR long posi ion
aPe iCP sanD
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 8 0.996 0.047 5 0.913 0.538 1 0.191 0.391
eWMa (94%) 3 0.638 0.970 0 0.047 0.057 1 0.191 0.391
eWMa (92.5%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (94%, 1%) 3 0.638 0.970 0 0.047 0.057 1 0.191 0.391
aeWMa (94%, 2%) 2 0.413 0.804 0 0.047 0.057 1 0.191 0.391
aeWMa (94%, 3%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (92.5%, 1%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (92.5%, 2%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (92.5%, 3%) 0 0.047 0.057 0 0.047 0.057 1 0.191 0.391
gaRCH 3 0.638 0.970 0 0.047 0.057 2 0.413 0.970
eV 0 0.047 0.057 0 0.047 0.057 1 0.191 0.391
Panel B: 1 % es long posi ion
aPe iCP sanD
X
N
S
E
θ
(z) eR
X
N
SE
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 5 0.996 0.333 5 0.775 0.247 1 0.037 0.397
eWMa (94%) 0 0.305 0.551 0 0.007 0.256 0 0.037 0.513
eWMa (92.5%) 0 0.037 0.723 0 0.007 0.862 0 0.037 0.526
aeWMa (94%, 1%) 0 0.305 0.552 0 0.007 0.463 0 0.037 0.512
aeWMa (94%, 2%) 0 0.126 0.555 0 0.007 0.170 0 0.037 0.513
aeWMa (94%, 3%) 0 0.037 0.559 0 0.007 0.797 0 0.037 0.517
aeWMa (92.5%, 1%) 0 0.037 0.727 0 0.007 0.231 0 0.037 0.524
aeWMa (92.5%, 2%) 0 0.037 0.745 0 0.007 0.431 0 0.037 0.523
aeWMa (92.5%, 3%) 0 0.007 0.785 0 0.007 0.869 0 0.037 0.522
gaRCH 0 0.305 0.488 0 0.007 0.496 0 0.126 0.497
eV 0 0.007 0.946 0 0.007 0.196 1 0.037 0.500
No es: see able 3.
cOgen Business & ManageMen 21
ans o m a o ecas o cons an ola ili y in o an es ima e o ime- a ying ola ili y. Mo eo e , because
i is assumed ha e u ns would ha e cons an ola ili y, p ac i ione s widely employ he benchma k
model, which is compu ed using an equally weigh ed mo ing a e age. ex eme ma ke occu ences can
signi ican ly impac he VaR es ima e, a signi ican issue wi h he equally-weigh ed VaR me hod. he VaR
es ima es won’ accu a ely e lec he s a e o he cu en ma ke .
he eWMa app oach was designed o o e come he limi a ions o he benchma k model. his
app oach p o ides mo e ecen obse a ions, which a e mo e impo an . On a e age, ex eme e en s
lose signi icance when he da a window mo es. alexande and Dakos (2023) u he enhance he eWMa
model. speci ically, his model cap u es he ola ili y esponse. his is an essen ial ea u e because he
Figu e 5. Densi y ob ained om gaRCH – ine copula simula ion.
No es: he densi y is ob ained om he 10000 gaRCH-Vine Copula simula ion.
22 a. MaPPaDang el al.
Table 16. Back es ing ou comes o 2.5% 10-day VaR and es (Long Posi ion), gaRCH – ine copula simula ion.
Panel a: 2.5 % VaR long posi ion
aPe iCP sanD
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 18 0.999 0.000 10 0.999 0.004 5 0.913 0.538
eWMa (94%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
eWMa (92.5%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
aeWMa (94%, 1%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
aeWMa (94%, 2%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
aeWMa (94%, 3%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
aeWMa (92.5%, 1%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
aeWMa (92.5%, 2%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
aeWMa (92.5%, 3%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538
gaRCH 8 0.996 0.047 9 0.998 0.004 5 0.913 0.538
eV 1 0.191 0.391 0 0.047 0.057 2 0.413 0.804
Panel B: 2.5 % es long posi ion
aPe iCP sanD
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 6 1.000 0.084 3 0.999 0.183 1 0.776 0.176
eWMa (94%) 2 0.996 0.174 0 0.550 0.354 1 0.776 0.187
eWMa (92.5%) 0 0.996 0.206 0 0.550 0.405 1 0.776 0.209
aeWMa (94%, 1%) 1 0.996 0.159 0 0.550 0.350 1 0.776 0.177
aeWMa (94%, 2%) 1 0.996 0.146 0 0.550 0.346 1 0.776 0.170
aeWMa (94%, 3%) 0 0.996 0.132 0 0.550 0.342 1 0.776 0.166
aeWMa (92.5%, 1%) 0 0.996 0.193 0 0.550 0.391 1 0.776 0.192
aeWMa (92.5%, 2%) 0 0.996 0.170 0 0.550 0.379 0 0.776 0.180
aeWMa (92.5%, 3%) 0 0.996 0.152 0 0.550 0.370 0 0.776 0.172
gaRCH 3 0.995 0.094 0 0.998 0.228 2 0.776 0.198
eV 0 0.037 0.524 0 0.007 0.499 1 0.126 0.488
No es: see able 4.
Table 17. Back es ing ou comes o 1% 10-day VaR and es (sho Posi ion), gaRCH – ine copula simula ion.
Panel a: 1 % VaR sho posi ion
aPe iCP sanD
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 6 0.965 0.095 4 0.809 0.804 4 0.809 0.538
eWMa (94%) 3 0.638 0.391 0 0.047 0.057 1 0.191 0.391
eWMa (92.5%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (94%, – 1%) 3 0.638 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (94%, – 2%) 2 0.413 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (94%, – 3%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (92.5%, – 1%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (92.5%, – 2%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391
aeWMa (92.5%, – 3%) 0 0.047 0.391 0 0.047 0.057 1 0.191 0.391
gaRCH 3 0.638 0.970 2 0.413 0.804 2 0.413 0.391
eV 3 0.638 0.804 1 0.191 0.391 3 0.638 0.804
Panel B: 1 % es sho posi ion
aPe iCP sanD
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 1 0.918 0.295 0 0.550 0.347 1 0.550 0.163
eWMa (94%) 0 0.305 1.000 0 0.007 0.440 0 0.037 0.933
eWMa (92.5%) 0 0.037 1.000 0 0.007 0.685 0 0.037 1.000
aeWMa (94%, – 1%) 0 0.638 1.000 0 0.007 0.875 0 0.037 0.922
aeWMa (94%, – 2%) 0 0.126 1.000 0 0.007 0.745 0 0.037 0.915
aeWMa (94%, – 3%) 0 0.037 1.000 0 0.007 0.062 0 0.037 0.912
aeWMa (92.5%, – 1%) 0 0.037 1.000 0 0.007 0.811 0 0.037 1.000
aeWMa (92.5%, – 2%) 0 0.037 1.000 0 0.007 0.501 0 0.037 1.000
aeWMa (92.5%, – 3%) 0 0.007 1.000 0 0.007 0.240 0 0.037 1.000
gaRCH 0 0.305 0.559 0 0.126 1.000 1 0.126 0.828
eV 3 0.305 0.505 0 0.037 0.515 3 0.305 0.423
No es: see able 5.
cOgen Business & ManageMen 23
Table 18. Back es ing ou comes o 2.5% en-day VaR and es (sho Posi ion), gaRCH – ine copula simula ion.
Panel a: 2.5 % VaR sho posi ion
aPe iCP sanD
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
X
N
V
aR
θ
(z) CC
Benchma k 10 0.999 0.000 9 0.998 0.005 11 0.999 0.000
eWMa (94%) 8 0.996 0.824 4 0.809 0.804 5 0.913 0.285
eWMa (92.5%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285
aeWMa (94%, – 1%) 8 0.996 0.824 4 0.809 0.804 5 0.913 0.285
aeWMa (94%, – 2%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285
aeWMa (94%, – 3%) 8 0.996 0.538 4 0.809 0.804 5 0.913 0.285
aeWMa (92.5%, – 1%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285
aeWMa (92.5%, – 2%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285
aeWMa (92.5%, – 3%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285
gaRCH 7 0.987 0.824 5 0.913 0.970 6 0.965 0.057
eV 7 0.987 0.538 5 0.913 0.804 3 0.638 0.538
Panel B: 2.5 % es sho posi ion
aPe iCP sanD
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
X
N
S
E
θ
(z) eR
Benchma k 6 0.999 0.005 4 0.999 0.009 1 0.999 0.008
eWMa (94%) 2 0.996 0.239 0 0.550 0.614 1 0.776 0,256
eWMa (92.5%) 0 0.996 0.294 0 0.550 0.865 1 0.776 0.291
aeWMa (94%, – 1%) 1 0.996 0.219 0 0.550 0.590 1 0.776 0.242
aeWMa (94%, – 2%) 1 0.996 0.203 0 0.550 0.570 1 0.776 0.233
aeWMa (94%, – 3%) 0 0.996 0.185 0 0.550 0.554 1 0.776 0.228
aeWMa (92.5%, – 1%) 0 0.996 0.376 0 0.550 0.785 1 0.776 0.679
aeWMa (92.5%, – 2%) 0 0.996 0.350 0 0.550 0.727 0 0.776 0.252
aeWMa (92.5%, – 3%) 0 0.996 0.330 0 0.550 0.683 0 0.776 0.241
gaRCH 1 0.978 0.159 0 0.776 0.434 2 0.916 0.096
eV 6 0.978 0.134 3 0.776 0.310 3 0.305 0.257
No es see able 6.
Table 19. Join VaR and es loss unc ion a ios, gaRCH – ine copula simula ion.
Panel a
α
=1%
long posi ion
α
=
2 5. % long posi ion
aPe iCP sanD aPe iCP sanD
Benchma k 1.000 1.000 1.000 1.000 1.000 1.000
eWMa (94%) 0.653 0.697 0.718 0.802 0.988 0.911
eWMa (92.5%) 0.663 0.707 0.739 0.810 0.999 0.929
aeWMa (94%, 1%) 0.649 0.685 0.714 0.799 0.988 0.908
aeWMa (94%, 2%) 0.643 0.693 0.708 0.793 0.985 0.903
aeWMa (94%, 3%) 0.635 0.697 0.699 0.785 0.978 0.895
aeWMa (92.5%, 1%) 0.659 0.705 0.735 0.806 0.999 0.926
aeWMa (92.5%, 2%) 0.652 0.704 0.729 0.800 0.996 0.920
aeWMa (92.5%,3%) 0.643 0.695 0.720 0.793 0.989 0.912
gaRCH 0.519 0.606 0.468 0.678 0.892 0.774
eV 0.243 0.440 0.422 0.295 0.555 0.453
Panel B
α
=1%
sho posi ion
α
=
2 5. % sho posi ion
aPe iCP sanD aPe iCP sanD
Benchma k 1.000 1.000 1.000 1.000 1.000 1.000
eWMa (94%) 0.752 0.758 0.561 0.899 0.877 0.738
eWMa (92.5%) 0.818 0.749 0.578 0.908 0.887 0.752
aeWMa (94%, – 1%) 0.813 0.737 0.559 0.895 0.877 0.736
aeWMa (94%, – 2%) 0.805 0.734 0.554 0.889 0.874 0.731
aeWMa (94%, – 3%) 0.795 0.727 0.547 0.880 0.868 0.725
aeWMa (92.5%, – 1%) 0.824 0.748 0.575 0.825 0.887 0.750
aeWMa (92.5%, – 2%) 0.816 0.745 0.571 0.819 0.884 0.745
aeWMa (92.5%, – 3%) 0.805 0.738 0.564 0.810 0.875 0.739
gaRCH 0.657 0.669 0.467 0.692 0.665 0.649
eV 0.752 0.804 0.799 0.898 0.865 0.689
No es: Values g ea e han one indica e ou pe o mance o he benchma k model and ice e sa.
