Jelesko ic, Vahidin; La ini, Claudio; Younas, Zahid I.; Al‐Fa yan, Mamdouh A. S.
A icle — Published Ve sion
C yp ocu ency po olio op imiza ion: U ilizing a GARCH‐
copula model wi hin he Ma kowi z amewo k
Jou nal o Co po a e Accoun ing & Finance
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John Wiley & Sons
Sugges ed Ci a ion: Jelesko ic, Vahidin; La ini, Claudio; Younas, Zahid I.; Al‐Fa yan, Mamdouh A.
S. (2024) : C yp ocu ency po olio op imiza ion: U ilizing a GARCH‐copula model wi hin he
Ma kowi z amewo k, Jou nal o Co po a e Accoun ing & Finance, ISSN 1097-0053, Wiley, Hoboken,
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DOI: 10.1002/jca .22721
RESEARCH ARTICLE
C yp ocu ency po olio op imiza ion: U ilizing a
GARCH-copula model wi hin he Ma kowi z amewo k
Vahidin Jelesko ic1Claudio La ini2Zahid I. Younas3
Mamdouh A. S. Al-Fa yan4
1Depa men o econmics,
Humbold -Uni e si ä o Be lin, Be lin,
Ge many
2Independen Resea che , Vi e bo, I aly
3Be lin School o Business and
Inno a ion, Be lin, Ge many
4Uni e si y o Po smou h, Po smou h,
UK
Co espondence
Vahidin Jelesko ic, Humbold -Uni e si ä
o Be lin, Be lin, Ge many.
Email: ahidin.jelesk[email p o ec ed]
Abs ac
The g owing in e es in c yp ocu encies has b ough his new means o
exchange o he a en ion o he inancial wo ld. This s udy aims o in es iga e
he e ec s ha a c yp ocu ency can ha e when i is conside ed as a inancial
asse . The analysis is ca ied ou om an ex-pos pe spec i e, e alua ing he
pe o mance achie ed in a ce ain pe iod by h ee di e en po olios. These
a e he one composed only o equi ies, bonds and commodi ies, he second
one only o c yp ocu encies, and he hi d one is a combina ion o hese bo h
ones and hus made up o all conside ed “ adi ional” asse s and he mos
pe o ming c yp ocu ency o he second po olio. Fo hese pu poses, he
classic a iance-co a iance app oach is applied whe e he calcula ion o he isk
s uc u e is done ia he GARCH-Copula and GARCH-Vine Copula app oaches.
The op imal weigh s o he asse s in he op imized po olios a e de e mined
h ough Ma kowi z op imiza ion p oblem. The analysis mainly showed ha
he po olio composed o c yp ocu ency and adi ional asse s has a highe
Sha pe index, om an ex-pos pe spec i e, and mo e s able pe o mances, om
an ex-an e pe spec i e. We jus i y ou selec ion o he Ma kowi z app oach o e
condi ional VaR and expec ed sho all due o hei heigh ened sensi i i y o
unsys ema ic ex eme e en s in c yp o ma ke s.
KEYWORDS
Copula, c yp ocu encies, GARCH, Ma kowi z op imiza ion, Vine Copula
JEL CLASSIFICATION
C10, G11, G17
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s 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.
© 2024 The Au ho s. Jou nal o Co po a e Accoun ing & Finance published by Wiley Pe iodicals LLC.
J Co p Accoun Finance. 2024;35:139–155. wileyonlinelib a y.com/jou nal/jca 139
140 JELESKOVIC e al.
1 INTRODUCTION
In oduc ion o c yp ocu encies in inancial sys em
becomes a ho issue o in es men among wo ldwide
in es o s. Like adi ional cu encies, he c yp ocu encies
also o e he holde o cu ency he pu chase o goods
and se ices. Howe e , con a y o legal ende paymen
me hods, he c yp ocu ency sys em is based exclusi ely
on he us o he pa ies in c yp ocu encies and c yp o
okens since he e is no law ye ha obliges o accep c yp-
ocu encies as a means o paymen (Gogo, 2019). A public
egis e o iling o he exchanges (called blockchain) o
which e e yone can access, is he base o he iducia y sys-
em; all he ope a ions occu ing be ween he holde s o
c yp ocu encies a e he e o e alida ed by a mine , whose
ask is o gua an ee and enc yp he ansac ions ha
ha e aken place by en e ing hem in he public egis e
(Tuwine , 2019).
Despi e he absence o egula o y equi emen s, he
accep abili y o c yp ocu ency as a medium o exchange
go emendous a en ion among in es o s and i s olume
o ade has also gone up by leaps and bounds in a e y
small pe iod o ime. The eason behind his success is he
inancial c isis o 2008 ha no only esul ed in he loss
o ha d-ea ned in es men s o people ac oss he globe, bu
also o ced hem o look o p o ec ion agains such c i-
sis and al e na i es o sa e in es men (Pinudom e al.,
2018). Fu he , c yp ocu encies may be un ela ed o he
p e ailing economic si ua ion o any coun y, and as an
al e na i e in es men also p o ide he oppo uni y o
inancial di e si ica ion (Bodie e al., 2014; K ueckebe g &
Scholz, 2018; T imbo n e al., 2017). Mo eo e , adding he
c yp ocu ency o he po olio also bene i s he in es o
agains he a ious mone a y policy- ela ed isks a ached
wi h adi ional cu ency and hus inally imp o e he
pe o mance o po olio (Any an aki & Topaloglou, 2018;
B ie e e al., Sza a z, 2015; Elendne e al., 2018;Maye ,
2018). Financial di e si ica ion, in addi ion o being a isk-
educing ins umen , is also a mo i e o achie ing g ea e
p o i . The in es o s wi h such mo i es a e called “ isk-
seeke s,” ha is, hose who manage o make a p o i om
he high ola ili y o he ma ke s.
Since in es o has s a ed op ing he c yp ocu ency as
an in eg al pa o hei po olio and capi al alloca ion,
hus he e a ques ion a ises ha , whe he c yp ocu en-
cies esul in he mo e bene icial op imiza ion o hei
po olio o no ? Po olio op imiza ion can be de ined
as he maximum bene i ha can de i e om an alloca-
ion o inancial esou ces in ela ion o he isk- e u n
p o ile o he in es o . Ma kowi z (1952), used a e age,
a iance, co a iance and Pea son’s linea co ela ion o
cons uc ion o op imal po olio. Howe e , cons uc ion
o Ma kowi z heo y o op imal po olio selec ion does
no conside ha dis ibu ion o his o ical e u ns o inan-
cial asse s may no be Gaussian. While, his poin o iew
is widely ejec ed in inancial li e a u e wi h empi ical
e idence ha inancial his o ical se ies a e e y o en cha -
ac e ized by phenomena o asymme y and lep oku osis
(Sheikh & Qiao, 2009) o e en mo e d ama ic s ylized ac s
(Con , 2001). Assuming ha inancial e u ns a e no mally
dis ibu ed lead o hei unde es ima ion and inaccu a e
quan i ica ion o isk (Pinudom e al., 2018). Simila ly,
unde he Ma kowi z po olio selec ion, measu ing isk
only by a iance o his o ical e u ns is e y gene al o
speci ic asse and also inco ec . Mo eo e , Ma kowi z
used Pea son´s linea co ela ion as a ool o measu e
he associa ion among asse s o cons uc ion o po o-
lio, howe e , Pea son’s co ela ion igno es he non-linea
associa ion among he asse s and hus conceal hei depen-
dency s uc u e (Rache e al., 2009). To o e come abo e
men ioned issues condi ional au o eg essi e and gene -
alized he e oscedas ic models (GARCH) a e also widely
used o p edic he u u e e olu ion o a iance in esea ch
ela ed o i ual cu encies (A dia e al. 2018b; Chen e al.,
2016; Chu e al., 2017; Ka siampa, 2017; Saha, 2018). While,
he s udy o Capo ale and Zekokh (2019) has ex ended
he analysis o A dia e al. (2018b) on Bi coin, E he eum,
Ripple and Li ecoin, he conside ed a Model Con idence
Se (MCS) p ocedu e o selec he GARCH models o he
Ma ko egime-swi ching o each c yp ocu ency.
Fu he , he linea co ela ion coe icien may also be
inadequa e in iden i ying he s uc u e o in e dependence
o he asse s (Emb ech s e al., 1999; Szegö, 2005). Thus
ela ed li e a u e sugges s ha inancial associa ion among
he inancial asse s should be de e mined by Copula mod-
els as i acili a es o model o each asse , he ma ginal
p obabili ydis ibu ionand oe alua esepa a ely, andsuc-
cessi ely, he mul i a ia e dependence o all asse s in a
po olio (Che ubini e al., 2004; Nelsen, 2007; Skla , 1959).
The Copulas he e o e ocus on de e mining a co ela ion
s uc u e be ween he uni a ia e dis ibu ions o he asse s
and a e mo e lexible han he s anda d mul i a ia e dis i-
bu ions (Kakou is & Rus em, 2014). This in u n makes i
possible o b eak he link wi h he Gaussian dis ibu ion
(Bai & Sun, 2007).
As GARCH-Copula models p o ide a be e o ecas
some mu ual p ope ies u u e e u ns han boo s apping
me hods, hey a e equen ly applied in he ield o po -
olio di e si ica ion (Bouoiyou e al., 2017;K es a,2015;
Os e iede e al., 2016). Howe e , he Copula models a e
o en applied only in bi a ia e con ex s due o he ac
ha he mul i a ia e Copula has a kind o complexi y
ha inc eases as he size inc eases, ending o lose p eci-
sion o i ing on he ails (Deng e al., 2011). Conside able
e o s ha e been made o inc ease he lexibili y o he
mul i a ia e Copula models. The Vine-Copulas models
JELESKOVIC e al. 141
a e he esul o his endea o and i was u he mod-
i ied in o i s subclasses o C-Vine and D-Vine (Bed o d
& Cooke, 2001; Joe, 1996). La e on, many s udies ha e
no only used hese Copulas and hei subclasses in o
hei esea ch on inancial e u ns, po olio managemen ,
exchange a e managemen , bu also p o ed empi ically
ha Vine Copula app oach ou pe o ms he mul i a ia e
-Copula, especially when e u ns ha e asymme y and a
di e en dependency s uc u e be ween pai s o inancial
asse s (Aas e al., 2009; Fische e al. 2007; Mendes e al.,
2010; Saha, 2018; Schi mache & Schi mache , 2008).
In he ligh o ela ionship o isk and e u n, his
s udy aims o explo e he impac on isk and e u n s uc-
u e o po olio i c yp ocu encies as inancial asse s a e
conside ed as a pa o op imized po olios. The s udy
uses GARCH models o quan i y he a iance o e u ns
whe eas among Copula Models Vine Copula models a e
applied o cap u e he co ela ion s uc u e be ween he
di e en asse s. The use o a GARCH p ocess combined
wi h a Copula model allows o di ide he isk due o se -
e al ac o s (s anda d de ia ion, skewness and ku osis
and co-mo emen s be ween asse s), in di e en ma he-
ma ical s eps ying o speci y he isk o po olio in he
bes possible way. In his pape , he analysis is based on
a compa a i e pe spec i e. Th ee po olios a e e alua ed
on he basis o he pe o mance achie ed: he i s one
composed only by adi ional asse s, we call i “T adi ional
po olio,” he second one composed by he same asse s
wi h he inclusion o a c yp ocu ency, hus “T adi ional
C yp o po olio,” and he hi d one makes up by only
c yp ocu encies—“C yp os po olio”. The h ee po o-
lios a e cons uc ed unde conside a ion o h ee di e en
speci ica ions ega ding e u ns’ dis ibu ion: he i s one
assumes a no mal dis ibu ion o he his o ical e u ns;
he second one and hi d one combine bo h mul i a ia e
Copula model and Vine Copula model, espec i ely, wi h
he GARCH modeling o he a iance. Ou esul s show
ha combining he adi ional asse s wi h he c yp oas-
se s may lead o highe gains in e ms o ewa d- isk a io,
and hus should be conside ed in he mo e in ensi e u u e
wo k. The inal con ibu ion o he s udy is also o e alua e
ha , which o he model among p oposed models p o ide
he be e pe o mance and s abili y o po olio o e ime
conside ing di e en op imali y c i e ia.
In his con ex , we wish o emphasize ou delibe a e
choice o he Ma kowi z app oach o e me hods based
on Condi ional Value a Risk—CVaR and i s coun e pa
Expec ed Sho all—ES (see, e.g., K okhmal e al., 2002;
Rocka ella & U yase , 2002;S oyano e al.,2007;Ziemba
&Vickson, 2006).In hissense, he ela edpape s a egi en
by Ben Osman e al. (2023) Howe e , he e a e se e al
compelling easons behind his decision p o-Ma kowi z.
The CVaR and ES me hods ake in o accoun he so-called
a ails in he e u n dis ibu ion o he unde lying asse ,
e ec i ely measu ing he isk o signi ican losses when
op imizing po olios. Howe e , i ’s impo an o no e ha
e u ns in egula ed ma ke s s em p ima ily om he un-
damen al alue and he ading p ocess o he asse . In
con as , he p ices o cybe asse s can be hea ily in lu-
enced by a mul i ude o nega i e exogenous e en s ha
a e no di ec ly linked o hei undamen al alues o ad-
ing p ocesses. These e en s can include cybe -a acks and
hei media co e age, he impac o in luen ial igu es on
social media (such as Elon Musk’s e ec on Dogecoin),
ailu es in implemen ing mainne and signi ican p o o-
col upg ades, and mo e. Resea che s o en o e look his
dis inc ion and ea hese e en s as i hey we e endoge-
nous, which can lead o misleading in es men s a egies
o c yp oasse s. To summa ize, while ex eme e en s in
adi ional inancial ma ke s a e a e bu sys ema ic and
end o ecu , he wo ld o c yp o ma ke s wi nesses a as
numbe o ex eme e en s ha a e o en unique and do no
epea . Fu he mo e, wi hin he c yp o in es ing commu-
ni y, a common man a p e ails: “ end is you iend,” and
in es o s end o ollow he p inciple o “buying on he dip
and selling on he peak.” This implies ha in es o s a e
mo e ocused on iden i ying and capi alizing on posi i e
ends, whe e he middle o he e u n dis ibu ion holds
g ea e impo ance han isola ed ex eme e en s a he
ails. Consequen ly, ou choice o he Ma kowi z app oach
aligns wi h he needs o medium and long- e m in es o s
a he han hose wi h a e y sho - e m pe spec i e.
In conclusion, ou decision has been o u ilize da a p e-
da ing he eme gence o he COVID-19 pandemic and he
Uk aine con lic . These e en s ha e gi en ise o ecen
economic and ene gy c ises, as well as excep ional ma ke
condi ions. Ou a ionale is oo ed in he belie ha using
da a om pe iods p io o hese c ises p o ides a mo e
ai h ul ep esen a ion o he o dina y and s able economic
en i onmen , which we deem o be he p e ailing s a e o
he majo i y o he ime.
The emainingsec ionso hepape a es uc u edas ol-
lows. In Sec ion II, we in oduce Copula models and hei
ele ance o ou analysis. Sec ion III p o ides a de ailed
explana iono heme hodology employed o da aanalysis
andop imiza ion. Mo ing o wa d o Sec ionIV wep esen
he examined da a and epo he esul s ob ained om
each po olio. Sec ion V p esen s esul s. Sec ion IV is ded-
ica ed o a comp ehensi e discussion o he main indings
ollowedbysec ion VII o conclusionan ou look o u u e
esea ch and implica ions.
2METHODOLOGY
2.1 Loca ion-scale model
The Loca ion-Scale model (o Loca ion-Scale Family) is
he heo e ical basis on which he en i e ma hema ical
142 JELESKOVIC e al.
sys em o he GARCH-Copula model is based. A andom
a iable 𝑋is said o belong o he loca ion-scale amily
when i s cumula i e dis ibu ion (CDF).
Is a unc ion only o 𝑥−𝑎
𝑏:
𝐹𝑋(𝑥 |𝑎,𝑏)=𝐹
(𝑥−𝑎
𝑏);𝑎∈𝑅, 𝑏 >0; (1)
The wo pa ame e s 𝑎, 𝑏, a e espec i ely he loca ion
and he scale pa ame e s.1The a iable
𝑍∶𝑋−𝑎
𝑏(2)
is called he educed o s anda dized a iable, whe e 𝑧is
a ealiza ion o 𝑍. The educed a iable 𝑍has 𝑎=0and
𝑏=1.I 𝑌has a cumula i e dis ibu ion unc ion 𝐹𝑍(𝑧) =
𝑃(𝑍 ≤𝑧), hen 𝑋=𝑎+𝑏𝑍has a cumula i e dis ibu ion
unc ion 𝐹𝑋(𝑥) = 𝐹𝑧(𝑥−𝑎
𝑏). In o he s wo lds, “Fo any an-
dom a iable 𝑍whose CDF belongs o such a amily, he
CDF o 𝑋=𝑎+𝑏𝑍also belongs o he amily.”2The
use o his model in inance can ce ainly be aced back
o he s udies o Azzalini (1985), wi h he in oduc ion o a
hi d pa ame e 𝑐, ha is a pa ame e o “shape,” so:
Lemma 1. I 𝑓0is a one-dimensional p obabili y densi y
unc ion symme ic abou 0, and 𝐺is a one-dimensional
dis ibu ion unc ion such ha 𝐺′exis s and is a densi y
symme ic abou 0, hen
𝑓(𝑧)=2𝑓
0(𝑧)𝐺{𝑤(𝑧)} (−∞ < 𝑧 < ∞) (3)
Is a densi y unc ion o any odd unc ion 𝑤(⋅).
On he basis o he Lemma 1and he espec i e demon-
s a ion (see Azzalini 1985), assuming 𝑓0=𝜙,𝐺= Φ
( espec i ely he PDF, and he CDF o he no mal S an-
da d dis ibu ion) and 𝑤(𝑧) = 𝑐 ⋅ 𝑧 (𝑐is a cons an ), he
au ho is able o a i m ha : i Z is Skewed no mal dis-
ibu ed (𝑍∼𝑆𝑁(𝑐))and𝑋=𝑎+𝑏𝑍, whe e 𝑎∈𝑅,𝑏>
0, hen i is possible s a e ha 𝑋∼𝑆𝑁(𝑎,𝑏
2,𝑐). He o -
malizes also he case o Skewed -dis ibu ion (Azzalini
2005).
Thanks o his con ibu e, o he au ho s ha e been
able o gi e impo an e lec ions: A ellano-Valle e al.
(2004) ex ended he numbe o pa ame e s h ough which
i is possible o ep esen he dis ibu ion ha emains
om a p ocess o “s anda diza ion”: he loca ion-scale
skew-gene alized no mal dis ibu ion (SGN) is de ined
as ha 𝑋=𝑎+𝑏𝑍, whe e 𝑍∼𝑆𝐺𝑁(𝑐,𝑑)and 𝑋∼
𝑆𝐺𝑁(𝑎,𝑏2,𝑐,𝑑). Only in mo e ecen s udies Kuma e al.
(2018) ha e demons a ed he possibili y o ep esen ing
a educed a iable h ough a 3- ac o pa ame e iza ion o
he shape, keeping he connec ion wi h he Loca ion-Scale
model.
2.2 Au o eg essi e model
A necessa y bu no su icien condi ion is o accep he
assump ion ha he s udies unde lying he Loca ion-Scale
model a e alid, conside ing ha e u ns a e he a i-
able unde he mic oscope o his analysis. Empi ical
e idences (Engle, 1982) show ha i is possible o conside
he a iance a ian o e ime. This logicassump ion, com-
bined wi h he Loca ion-Scale model, allows o desc ibe
he e olu iona y p ocess o e u ns as:
𝑅𝑡=𝑎
𝑡+𝑏
𝑡𝑍𝑡(4)
whe e 𝑎𝑡,𝑏2
𝑡,𝑍𝑡a e espec i ely he condi ional mean, con-
di ional a iance and he inno a ion o 𝑅𝑡p ocess. This
ype o o mula ion allows he e o e o model he loca-
ion p ocess, as an au o eg essi e mo ing a e age p ocess
ARMA (m, n):
𝒂𝒕=𝒂+
𝒎
∑
𝒋=1
𝜙𝒋(𝑹𝒕−𝒋 −𝒂
)+
𝒏
∑
𝒋=1
𝜽𝒋(𝑹𝒕−𝒋 −𝒂
𝒕−𝒋)(5)
and pa ame e o scale p ocess, using gene alized au o e-
g essi e condi ional he e oscedas ic p ocess GARCH(q,p):
𝑏2
𝑡=𝜔+
𝑞
∑
𝑗=1
𝜑𝑗(𝑅𝑡−𝑗 −𝑎
𝑡−𝑗)2+
𝑝
∑
𝑗=1
𝜓𝑗𝑏2
𝑡−𝑗 (6)
Equa ion (7) shows he o mula ion p o ided by Bolle -
sle (1986), o he i s GARCH model (simple GARCH).
A e his i s model he esea ch has p oposed un il oday
many GARCH models capable in di e en ways o cap u -
ing e ec s o asymme y and in ensi y o he condi ional
a iance.
The in eg a ed GARCH model (Engle & Bolle sle 1986),
deno ed by IGARCH (p,q), is a pa icula case o he simple
GARCH, because
𝑞
∑
𝑗=1
𝜙𝑖+
𝑝
∑
𝑗=1
𝛽𝑖=1. The condi ion makes
he model s ic ly s a iona y.
The exponen ial GARCH model (Nelson, 1991), deno ed
by EGARCH (q, p), has:
ln (𝑏2
𝑡)=𝜔 +
𝑞
∑
𝑗=1 (𝜑𝑗𝑍𝑡−𝑗 +𝛾
𝑗(|||𝑍𝑡−𝑗|||−𝐸|||𝑍𝑡−𝑗|||))
+
𝑝
∑
𝑗=1
𝜓𝑗ln (𝑏2
𝑡−𝑗)(7)
o 𝜑𝑗>0,𝜓
𝑗>0,𝛾
𝑗>0and 𝜔> 0.𝜑𝑗cap u es he sign
e ec , and 𝛾𝑗cap u es he size e ec o he p e ious s an-
da dized inno a ion. The pe sis ence pa ame e o his
model is 𝜓𝑗.
JELESKOVIC e al. 143
The GJR-GARCH (q, p) model due o Glos en e al.
(1993)has:
𝜎2
𝑡=𝜔+
𝑞
∑
𝑗=1 (𝜑𝑗(𝑅𝑡−𝑗 −𝑎
𝑡−𝑗)2+𝛾
𝑗𝐼𝑡−𝑗(𝑅𝑡−𝑗 −𝑎
𝑡−𝑗)2)
+
𝑝
∑
𝑗=1
𝜓𝑗𝜎2
𝑡−𝑗 (8)
o 𝜑𝑗>0,𝜓
𝑗>0,𝛾
𝑗>0and 𝜔> 0, whe e 𝐼𝑡−𝑗 =
1i (𝑅𝑡−𝑗 −𝑎
𝑡−𝑗)≤0and 𝐼𝑡−𝑗 =0i (𝑅𝑡−𝑗 −𝑎
𝑡−𝑗)>0.𝛾𝑗
ep esen s an asymme y pa ame e . A posi i e shock will
inc ease ola ili y by 𝜑𝑗;anega i eshockwillinc ease
ola ili y by 𝜑𝑗+𝛾
𝑗.
Simple GARCH, IGARCH, GJRGARCH and EGARCH
combined wi h he ARMA p ocess a e used o i he con-
di ional ola ili y and condi ional mean o he inancial
asse s. Fo he analysis a e conside ed a maximum o de
o one o he ARMA pa , and a maximum o o de h ee
o he GARCH models. The condi ional dis ibu ion o
he inno a ion is supposed be no mal o Skewed no mal.
So, an i e a i e p ocedu e es ima es, o all 4 ARMA-
GARCH p ocesses, he bes model o each combina ion
o lags un il he maximum se led, eple e his p ocedu e
wo imes, one o each dis ibu ion hypo hesis o he
inno a ion.3The lowe bi coin (BIC) alue is chosen as
c i e ium o iden i y he mos adap model.
3 COPULA AND VINE COPULA
MODELS
Copulas a e models h ough which i is possible o iso-
la e he dependence s uc u eo amul i a ia edis ibu ion
(Nelsen,2007).Le 𝐻be a n-dimensional dis ibu ion unc-
ion wi h ma ginal dis ibu ion unc ions 𝐹𝑗(𝑍𝑗;𝜃).Then
he e exis s a Copula 𝐶such ha :
𝐻(𝑍1,…, 𝑍
𝑛)=𝐶(𝐹1(𝑍1;𝜃),…, 𝐹
𝑛(𝑍𝑛;𝜃);𝛿)(9)
whe e 𝐶(𝐹1(𝑥1;𝜃),…, 𝐹
𝑛(𝑥𝑛;𝜃);𝛿)is he Copula associ-
a ed o H and 𝛿is he ec o o dependence pa ame e s
o 𝐶. The Copula is unique i he ma ginals 𝐹𝑗(𝑍𝑗;𝜃)a e
con inuous (Skla , 1959). The unici y o he Copula was
demons a ed by Skla (1959). A la ge numbe o Cop-
ulas ha e been p oposed in he li e a u e, and each o
hese imposes a di e en dependence s uc u e on he da a
(T i edi & Zimme 2007). Joe and Xu (1996), T i edi and
Zimme (2007), Balak ishnan and Lai (2009) and Nelsen
(2007) p o ide a de ailed o e iew abou he p ope ies o
Copulas (Elaal 2017). Gaussian (no mal) Copula, S uden ’s
Copula, Gumbal and Clay on Copula a e aken in o con-
side a ion o his analysis. The no mal Copula, p oposed
by Lee (1983), akes he o m:
𝐻(𝑍1,…, 𝑍
𝑛)
=Φ
𝐺(Φ−1 (𝐹1(𝑍1;𝜃)) ,…, Φ
−𝑛 (𝐹𝑛(𝑍𝑛;𝜃)) ;𝛿)(10)
whe e Φis he CDF o he s anda d no mal dis ibu ion,
and Φ𝐺is he s anda d mul i a ia e no mal dis ibu-
ion wi h co ela ion pa ame e es ic ed o he in e al
(−1, 1). A Copula wi h wo dependence pa ame e s is he
mul i a ia e -dis ibu ion wi h 𝜐deg ees o eedom and
co ela ion 𝛿,
𝐻(𝑍1,…, 𝑍
𝑛)
=𝑡
(𝑡−1
𝑚1(𝐹1(𝑍1;𝜃)) ,…, 𝑡−𝑛
𝑚2(𝐹𝑛(𝑍𝑛;𝜃)) ;𝛿,𝜐)(11)
whe e 𝑡−1
𝑚deno es he in e se o he CDF o he s anda d
uni a ia e -dis ibu ion wi h 𝑚deg ees o eedom. The
pa ame e 𝜐con ols he hea iness o he ails. The Clay on
(1978) Copula, p oposed by Kimeldo and Sampson (1975),
akes he o m:
𝐻(𝑍1,…, 𝑍
𝑛)
=(𝐹1(𝑍1;𝜃)−𝛿 +⋯+𝐹
𝑛(𝑍𝑛;𝜃)−𝛿 −1
)−1∕𝛿 (12)
wi h he dependence pa ame e 𝛿 es ic ed on he egion
(0, ∞).As𝛿app oaches ze o, he ma ginal become inde-
penden . The Clay on Copula canno accoun o nega i e
dependence. I has been used o s udy co ela ed isks
because i exhibi s s ong le - ail dependence and el-
a i ely weak igh - ail dependence (T i edi & Zimme
2007).
The Gumbel Copula in oduced by Gumbel (1960) akes
he o m:
𝐻(𝑍1,…, 𝑍
𝑛)
=exp(−(− log 𝐹1(𝑍1;𝜃)𝛿+⋯−log𝐹
𝑛(𝑍𝑛;𝜃)𝛿)1∕𝛿)
(13)
The dependence pa ame e is es ic ed o he in e al
[1, ∞). Values o 1 and ∞co espond o independence.
Simila o Clay onCopula,Gumbel does no allow nega i e
dependence, bu i con as o Clay on, Gumbel exhibi s
s ong igh - ail dependence and ela i ely weak le - ail
dependence (T i edi & Zimme 2007).
The S uden ’s Copula and he Gaussian Copula belong
o he amily o Ellip ical Copulas, he Gumbel and he
Clay on Copulas ins ead a e pa o he amily o he
A chimedeans. The main di e ence be ween wo amilies
144 JELESKOVIC e al.
o Copulas is ha he Ellip ical ones ha e he possibili y
o speci y he le el o co ela ion be ween he ma ginal
dis ibu ions, con a y o he o he . The A chimedean
ones ha e he me i o modeling well he ex eme alues
o he mul i a ia e dis ibu ions. These cha ac e is ics,
he e o e, make one mo e o less sui able han he o he
depending on he case. In a po olio alloca ion p oblem
wi h mo e han wo asse s, i is imp obable ha a single
co ela ion pa ame e (as in he case o Ellip ical Copulas)
can ep esen a co ela ion s uc u e o n-dimensions.
The possibili y o using he Ellip ical unc ions e en in
a con ex ha conce ns mo e han wo dimensions, mak-
ing he Copula app oach e en mo e lexible, led Joe (1996)
o elabo a e he i s Vine Copula model. A Vine Cop-
ula is a ac o iza ion o mul i a ia e Copula densi ies in o
(condi ional) bi a ia e Copula densi ies. Fo example:
Le 𝑐(𝐹1(𝑍1;𝜃),𝐹
2(𝑍2;𝜃), 𝐹
3(𝑍3;𝜃);𝛿) be he densi y
unc ion o a 3-dimen ional Copula, he espec i e Vine
Copula s uc u e is composed by h ee pa s:
𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑝𝑎𝑖𝑟
=𝑐
ℎ𝑔|𝑚(𝐹ℎ|𝑚(𝑍ℎ|𝑍𝑚);𝐹
𝑚|𝑔(𝑍𝑚|𝑍𝑔)) (14)
𝑢𝑛𝑐𝑜𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑝𝑎𝑖𝑟𝑠
=𝑐
𝑚𝑔 (𝐹𝑚(𝑍𝑚);𝐹
𝑔(𝑍𝑔))⋅𝑐
ℎ𝑔 (𝐹ℎ(𝑍ℎ);𝐹
𝑔(𝑍𝑔)) (15)
𝑚𝑎𝑔𝑖𝑛𝑎𝑙 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠
=𝑓
𝑔(𝑍𝑔)⋅𝑓
𝑔(𝑍𝑔)⋅𝑓
ℎ(𝑍ℎ)(16)
𝑐(𝐹1(𝑍1;𝜃),𝐹
2(𝑍2;𝜃),𝐹
3(𝑍3;𝜃);𝛿)
= 𝑐. 𝑑.𝑝𝑎𝑖𝑟 ⋅ 𝑢.𝑑. 𝑝𝑎𝑖𝑟𝑠 ⋅ 𝑚.𝑑.𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 (17)
As i possible o see, he adap abili y o Copula mod-
els makes hem much mo e complex han mul i a ia e
Copulas in e ms o es ima ion. In ac , as shown in
Equa ion (18), hese models a e composed by condi ional
Copula densi ies, join Copula densi ies and he ma ginal
p obabili y unc ions. The ini ial sequence o hese asse s
is no known a p io i: his means ha i is no possible
o know how o couple he inancial asse s om ime o
ime, in o de o o m he a ious Copula bi a ia e. Mo e-
o e , he de e mina ion o he ini ial sequence is e y o en
subo dina e o he choice o he s uc u e o he Vine
Copula. In li e a u e he e a e h ee main s uc u es s ud-
ied: The Regula -Vine Copula, he Canonical-Vine Copula
and he D aw able-Vine Copula; he las wo a e a sub-
ca ego y o he i s one. The C- ine has a s a s uc u e
whe e a unique node is connec ed o all o he ones. The
D- ine has pa h s uc u e whe e each asse has a single
link wi h he nex one. The R- ine is buil conside ing
he 𝑛!
2⋅2
⎛⎜⎜⎝
𝑛+1
2⎞⎟⎟⎠possible combina ions be ween he asse s
(Mo ales-Nàpoles, 2010). Rega dless o he chosen s uc-
u e, i should be no ed ha he link be ween each pai o
asse s is unique; his allows o conside as a as he C- and
D- Copulas 𝑛!
2possible disposi ions a e conce ned. Mo e-
o e , while o hese las wo he e is a ule ha es ablishes
how o cons uc he pai s ollowing o he s a ing ee,
he s uc u e o he R- ines is always he esul o a com-
pu a ional p ocedu e. Once an ini ial o de o he ma ginal
dis ibu ions has been de e mined and a me hodology able
o p oceed in he cons uc ion o he pai s has been iden i-
ied, i is necessa y o iden i y he ype o Copula ha binds
each pai , es ima ing i s ela i e pa ame e s.
The wo di e en me hodologies (Vine and mul i a i-
a e) a e compa able using he BIC and AIC c i e ia. The
goodness o i ing (GoF) o bo h Copula models s uc u e
will be measu ed h ough he es p oposed by Genes e al.
(2009). This unc ional and ce ainly no exhaus i e excu -
sus on Copula models can be concluded wi h he ollowing
o mal de ini ion:
De ini ion 1. A d-dimensional Copula, 𝐶∶[0;1]
𝑑∶→
[0; 1] is a cumula i e dis ibu ion unc ion (CDF) wi h
uni o m ma ginal.
I is p ecisely his las de ini ion ha c ea es he link
be ween he Loca ion-Scale model, he GARCH and Cop-
ula models. The de ini ion indica es ha he only alues
ha can be used as inpu o c ea e a Copula model
a e be ween 0 and 1. This means ha he s anda dized
esidues, ob ained om he GARCH model, need o be
con e ed in o p obabili y, h ough he iden i ica ion o
a dis ibu i e unc ion. In ac , acco ding o he in eg al
ans o ma ion o p obabili y, i is possible o ans o m
andom a iables belonging o any dis ibu ion in o uni-
o mly dis ibu ed andom a iables; he s a emen is
alid p o ided ha he dis ibu ion used as a means o
ans o ma ion is he eal one. In o de o iden i y he
ue dis ibu ion, he Kolmogo o -Smi no es (K-S es )
(Massey, 1951) and he Uni o mi y es (PIT es ) (Diebold
e al., 1997) ha e been used on a sample o i e di e en dis-
ibu ions (S anda d no mal, -S uden , Skewed -S uden
(Skwd. -S ud.), Gene alized E o dis ibu ion (GED),
Skewed Gene alized E o dis ibu ion (Skwd. GED).
3.1 Po olio op imiza ion
A key s ep owa d a quan i a i e app oach o he issue o
s a egic asse alloca ion was aken by Ha y Ma kowi z, in
his a icle “Po olio Selec ion” published on he Jou nal
o Finance in 1952. I s model assumes essen ially ha he
JELESKOVIC e al. 145
in es o s a e a ional and a e in e es ed in maximizing
he e u ns and minimizing he isk, u he mo e hey
ha e ee access o co ec in o ma ion on he e u ns
and isk; he ma ke s a e e icien and abso b he in o -
ma ion quickly and pe ec ly. Unde hese hypo hesizes,
Ma kowi z conside he isk how he oscilla ion o he
e u ns a ound hei a e age. In hese e ms we can
conside he a iance o he e u ns o a po olio as i s isk
and he a e age o hese e u ns as he expec ed e u n o
he po olio. In s a is ical e ms i is he e o e necessa y
o minimize he a iance (o s anda d de ia ion) and
maximize he a e age, adjus ing he weigh s ha each
asse has inside o he po olio. Fo mally:
min
wi
(𝜎2
𝑝)=min
𝑤𝑖;𝑤𝑗(∑
𝑖∑
𝑗
𝑤𝑖𝑤𝑗𝜎𝑖𝑗)(18)
𝐸(𝑅𝑝)=
𝑛
∑
𝑖=1
𝑤𝑖𝑅𝑖𝑎𝑛𝑑 0 ≤𝑤𝑖≤1(19)
This model is no ee o limi s: i s o all, his way
o op imizing a po olio limi s he numbe o asse s
o be chosen, con a y o he bene i o di e si ica ion,
c ea ing iolen and d as ic shi s in he composi ion o
he po olio o e ime; mo eo e , he pas pe o mance o
a po olio has li le p edic i e powe o i s u u e pe o -
mance (B aga 2016). In he empi ical wo k, o de e mine
he weigh o each asse in po olio, he maximiza ion o
Sha pe a io is se as a ge .4In li e a u e his is known
how Tangency po olio.5
max
𝑤
𝐸(𝑅𝑝)
𝜎𝑝
Subjec o ∑𝑖
𝑤𝑖=1
0≤𝑤𝑖≤1
(20)
The use o his model oge he wi h he abo e limi s
allows a ans e sal analysis o he p oblem: on he one
hand i allows o assess whe he a po olio ha con ains
c yp ocu encies is ac ually mo e pe o ming han hose
composed only o adi ional asse s, on he o he hand i
allows o see i a Loca ion-Scale-GARCH-Copula model
app oach allows o sol e (a leas pa ially) he p oblems
men ioned abo e. Finally, he po olio’s pe o mance is
assessed using he Sha pe index in he e sion p oposed
by Pezie and Whi e (2006).6
4 DATA AND RESULTS
The i s s ep owa d a compa a i e analysis o he h ee
di e en po olios begins wi h he choice o asse s. This
i s phase is conduc ed by aking in o accoun he pas
pe o mance o he e u ns o a se o asse s o e a pe iod
anging om Janua y 01, 2017 o Decembe 31, 2018.7The
i s po olio, called “T adi ional po olio” (o T ad), is
composed exclusi ely o asse s belonging o he in es -
men uni e se ha is ep esen ed by equi ies, go e nmen
bonds and commodi ies. 8As o he s ock ma ke , he
Dow Jones and Eu os oxx 50 indices p o ided a basis o
s ock picking. In ac , ou o he o al 80 equi y asse s, he
h ee mos p e o ming ones o bo h indices we e cho-
sen: Mic oso (MSFT), Boing (BA) and Visa (V) om Dow
Jones, RWE AG (RWEG), Amadeus (AMA) and Ke ing
S.A. (PRTP) o m Eu os oxx 50. Compa ing he go e n-
men bond yieldso F ance, Ge many, I aly,UK andUSAa
10 yea s, i was ound ha US go e nmen bonds we e he
bes in e ms o isk- e u n. In o de o achie e he mos
accu a e possible eplica ion o his ype o e u n, i was
decided o include he ETF (Exchange T aded Fund) IUSM
in he po olio. Finally, o he choice o he ou com-
modi ies, he pe o mance analysis was ca ied ou , aking
in o accoun he mos liquid ma ke s. The mos pe o m-
ing commodi ies we e Palladium (PA), Coppe (HG), Gold
(GC). C ude oil (CL) is o cibly added o gua an ee he
mos di e si ica ion possible. The second po olio, called
“C yp os po olio” (o C yp os), is composed by 10 c yp-
ocu encies, which we e selec ed om he 200 wi h he
highes ma ke capi aliza ion on Decembe 31, 2018; he
c yp ocu encies ha pe o med bes we e PIVX (PIVX),
E he eum (ETH), XRP (XRP), S ella (XLM), NEO (NEO),
Dec ed (DCR), Wa es (WAVES), Ve ge (XVG), Unob a-
nium (UNO) and G oes lcoin (GRS).9The hi d po olio,
called“T adi ional C yp opo olio”(o T ad+C y)is com-
posed o all he asse s o he “T adi ional po olio” wi h
headdi ion o he mos pe o mingc yp ocu encyamong
hose selec ed o he “C yp os po olio”.
Table 1shows he pe o mances o he asse s and
selec ed c yp ocu encies.10
A pa icula GARCH p ocess o he a iance is speci-
ied o each asse . Table 2summa ized o each asse he
bes selec ed ARMA-GARCH models,11 he assumed con-
di ional dis ibu ion and he alue o he BIC, es ima ed as
indica ed in he Me hodology Sec ion.12
5 ESTIMATION OF HISTORICAL
PERFORMANCE
By using he s anda dized esiduals, i is possible o p o-
ceed wi h he calculus o he co ela ion among he asse s
in di e en ways and o de e mine he his o ical pe o -
mance o each o he h ee di e en po olios abo e
men ioned. The op imiza ion p ocess is ca ied ou min-
imizing he a iance-co a iance ma ix and maximizing
146 JELESKOVIC e al.
TABLE 1 Pe o mances o c yp ocu encies and adi ional inancial asse s om Janua y 1, 2017 o Decembe 31, 2018.
Asse name PRTP RWEG AMA IUSM BA Va MSFT PA HG GC CL
Expec ed e u n .14% .10% .06% −.02% .12% .09% .08% .10% .02% .01% .00%
S anda d de ia ion .017 .018 .012 .005 .016 .013 .014 .015 .013 .010 .017
Ku osis .073 −.910 −.408 −.026 .023 −.170 .129 −.438 −.049 .222 −.851
Skewness 5.945 9.308 1.414 .765 3.220 3.094 3.476 3.033 1.564 5.412 2.528
Adjus ed sha pe .080 .054 .051 −.042 .076 .068 .060 .072 .017 .011 −.002
Asse name PIVX ETH XRP UNO NEO DCR XLM GRS XVG WAVES
Expec ed e u n .66% .38% .55% .53% .54% .48% .52% .72% .79% .37%
S anda d de ia ion .101 .064 .093 .089 .100 .090 .098 .136 .161 .079
Ku osis .841 .297 2.551 −.615 1.518 .869 1.750 1.943 .989 .286
Skewness 3.830 3.372 27.136 20.788 11.489 2.953 11.543 10.030 7.481 2.321
Adjus ed sha pe .066 .060 .057 .057 .054 .054 .053 .053 .049 .047
TABLE 2 ARMA-GARCH models o c yp ocu encies and inancial asse s.
PRTP RWEG AMA IUSM BA Va MSFT PA HG GC CL
EGARCH model (2.3) (1.3) (2.1) (2.3) (1.2) (2.1) (3.3) (1.3) (2.2) (3.2) (3.3)
ARMA model (.0) (1.0) (.1) (1.0) (.1) (.0) (.0) (.0) (1.0) (.0.0) (1.0)
Cond. dis no m no m no m no m s-no m no m no m no m no m no m s-no m
BIC −5.93 −6.03 −6.60 −8.64 −6.14 −6.67 −6.43 −6.16 −6.55 −7.17 −5.85
ETH XRP XLM NEO DCR WAVES GRS XVG PIVX UNO
EGARCH model (3.3) (1.1) (1.2) (1.2) (3.3) (3.3) (2.3) (3.3) (3.3) (2.3)
ARMA model (.0) (.0) (.0) (.0) (.1) (.0) (.1) (.1) (.1) (.0)
Cond. dis no m s-no m no m s-no m s-no m s-no m no m no m no m no m
BIC −2.75 −2.44 −2.15 −2.08 −2.11 −2.35 −1.57 −1.4 −1.97 −2.21
TABLE 3 S a is ical measu es and inal pe o mance o he simple “Ma kowi z op imiza ion” app oach.
T ad+C y
T ad+C y
(bound) T ad
T ad
(bound) C yp os
C yp os
(bound)
Expec ed e u n .00092 .00090 .00071 .00069 .00584 .00583
S anda d de ia ion .00741 .00732 .00653 .00644 .06032 .06026
Ku osis −.25233 −.26028 −.6567 −.68337 −.28524 −.28772
Skewness 2.18322 2.14843 4.98413 5.01445 1.44485 1.44738
Adjus ed sha pe .12474 .12318 .10667 .10488 .09702 .09690
Sha pe .12486 .12330 .10895 .10715 .09686 .09674
heSha pe a io13 o hepo olio (Bili 2016).Table3shows
he inal pe o mances o he h ee po olios when he
simple “Ma kowi z op imiza ion” (bounded and no )14 is
applied o he his o ical e u ns o each po olio.
The Copula and Vine Copula app oach a e in oduced
in o de o unlink he linea co ela ion om Ma kowi z
op imiza ion. As explained in Sec ion 2, he i s s ep is
o ans o m he s anda dized esiduals in o PITs. Table 4
shows he dis ibu ion ha was indi idua ed o each
asse conside ing he p alues o K-S and Uni o mi y
es s.
The esul ing PITs a e used as inpu o he Copula and
Vine Copula models. Table 5shows he p alues o GoF
es , he BIC and he AIC alues o he ou es ed Copula
amilies.15
As i is possible o no ice, he -Copula p esen s he
bes alues o each ca ego y o pa ame e s so i will be
used o model he co ela ion s uc u e o all h ee po -
olios. Table 6shows he inal pe o mances o he h ee
po olios when he “GARCH- -Copula Ma kowi z op i-
miza ion” (bounded and no ) is applied o he his o ical
e u ns o each po olio.
JELESKOVIC e al. 153
p ocedu es),i is possible o mul iply hem by he condi ioned s an-
da d de ia ion and add he condi ioned a e age (whe e he u u e
es ima es o bo h condi ioned measu es a e calcula ed using he
ARMA-GARCH model selec ed ea lie ), ob aining an es ima e o
he e u ns o each day o he o ecas pe iod. 5000 simula ions o
a ma ix a e ca ied ou : i has as columns he numbe o asse s ha
make up he po olio and as ows he numbe o days o which he
o ecas is made.
18 Th ee in es men ime ho izons o 30, 60 and 90 days a e consid-
e ed, o each o which a p ocess o Ma kowi z op imiza ion akes
place.
19 The ime window is ixed a 30 days and he e o e he po olio will
be ebalanced and op imized h ee imes wi hin he o ecas pe iod.
REFERENCES
Aas, K., Czado, C., F igessi, A., & Bakken, H. (2009). Pai -copula con-
s uc ions o mul iple dependence. Insu ance: Ma hema ics and
economics,44(2), 182–198.
Aepli, M. D., F auendo e , K., Füss, R., & Pa aschi , F. (2015). Mul i-
a ia e dynamic copula models: Pa ame e es ima ion and o ecas
e alua ion. School o Finance, Uni e si y o S . Gallen.
An onakakis, N., Cha zian oniou, I., & Gabaue , D. (2019). C yp-
ocu ency ma ke con agion: Ma ke unce ain y, ma ke com-
plexi y, and dynamic po olios. Science Di ec ,61,37–51.
Any an aki, S., & Topaloglou, N. (2018). Di e si ica ion, in eg a-
ion and c yp ocu ency ma ke . In eg a ion and C yp ocu ency
Ma ke (Ma ch 29, 2018).
A acioĞlu, B., Demi can, F., & Soyue , H. (2011). Mean– a iance–
skewness–ku osis app oach o po olio op imiza ion: An appli-
ca ion in İs anbul S ock Exchange. Ege Akademik Bakış De gisi,
11(5), 9–17.
A dia, D., Blu eau, K., Boud , K., & Ca ania, L. (2018). Fo ecas -
ing isk wi h Ma ko -swi ching GARCH models: A la ge-scale
pe o mance s udy. In e na ional Jou nal o Fo ecas ing,34(4),
733–747.
A ellano-Valle, R. B., Gómez, H. W., & Quin ana, F. A. (2004). A new
class o skew-no mal dis ibu ions. Communica ions in s a is ics-
Theo y and Me hods,33(7), 1465–1480.
Azzalini, A. (1985). A class o dis ibu ions which includes he no mal
ones. Scandina ian jou nal o s a is ics,12,171–178.
Azzalini, A. (2005). The skew-no mal dis ibu ion and ela ed mul i-
a ia e amilies. Scandina ian Jou nal o S a is ics,32(2), 159–188.
Bai, M., & Sun, L. (2007). Combina o y, algo i hms, p obabilis ic and
expe imen al me hodologies. The Se ies Lec u e No es in Compu e
Science,46(14), 231–242.
Bai, M., & Sun, L. (2007). Applica ion o Copula and Copula-CVaR
in he Mul i a ia e Po olio Op imiza ion. In: Chen, B., Pa e -
son, M., Zhang, G. (eds.) Combina o ics, Algo i hms, P obabilis ic
and Expe imen al Me hodologies. ESCAPE 2007. Lec u e No es in
Compu e Science, 46(14). Sp inge , Be lin, Heidelbe g. h ps://
doi.o g/10.1007/978-3-540-74450-4_21
Bai, M., & Sun, L. (2007, Ap il). Applica ion o copula and copula-
CVaR in he mul i a ia e po olio op imiza ion. In In e na ional
Symposium on Combina o ics, Algo i hms, P obabilis ic and
Expe imen al Me hodologies (pp. 231–242). Sp inge .
Balak ishnan, N., & Lai, C. D. (2009). Con inuous bi a ia e dis ibu-
ions. Sp inge Science & Business Media.
Basile, I., & Fe a i, P. (Eds.). (2016). Asse managemen and ins i u-
ional in es o s. Sp inge In e na ional Publishing.
Bed o d, T., & Cooke, R. M. (2001). P obabili y densi y decomposi-
ion o condi ionally dependen andom a iables modeled by
ines. Annals o Ma hema ics and A i icial In elligence,32(1-4),
245–268.
Ben Osman, M., Gala io is, E., Guesmi, K., Hamdi, H., & Naoui, K.
(2023). Di e si ica ion in inancial and c yp o ma ke s. In e na-
ional Re iew o Financial Analysis,89,1–13.
Bili , H. (2016). De e mina ion o op imal po olio by using an-
gency po olio and sha pe a io. Resea ch Jou nal o Finance and
Accoun ing,7(5), 2222–2847.
Bodie, Z., Kane, A., & Ma cus, A. J. (2014). In es men s, 10 h global
edi ion. Mc G aw-Hill.
Bolle sle , T. (1986). Gene alized au o eg essi e condi ional he -
e oskedas ici y. Jou nal o econome ics,31(3), 307–327.
Bouchaud, J. P., Ma acz, A., & Po e s, M. (2001). Le e age e ec in
inancial ma ke s: The e a ded ola ili y model. Physical e iew
le e s,87(22), 228701.
Bouoiyou , J., & Selmi, R. (2017). E he : Bi coin’s compe i o o ally?
a Xi p ep in a Xi :1707.07977.
B aga, M. D. (2016). S a egic asse alloca ion wi h mean- a iance
op imisa ion. Asse Managemen and Ins i u ional In es o s,97–
172, Sp inge , Cham.
B echmann, E., & Schepsmeie , U. (2013). Cd ine: Modeling depen-
dence wi h c-and d- ine copulas. Jou nal o S a is ical So wa e,
52(3), 1–27.
B ie e, M., Oos e linck, K., & Sza a z, A. (2015). Vi ual cu ency, an-
gible e u n: Po olio di e si ica ion wi h bi coin. Jou nal o Asse
Managemen ,16(6), 365–373.
Capo ale, G. M., & Zekokh, T. (2019). Modelling ola ili y o c yp-
ocu encies using Ma ko -Swi ching GARCH models. Resea ch
in In e na ional Business and Finance,48,143–155.
Chen, S., Chen, C., Hä dle, W. K., Lee, T. M., & Ong, B. (2016). A
i s econome ic analysis o he CRIX amily. A ailable a SSRN
2832099.
Che ubini, U., Luciano, E., & Vecchia o, W. (2004). Copula me hods
in inance. John Wiley & Sons.
Chu, J., Chan, S., Nada ajah, S., & Os e iede , J. (2017). GARCH
modelling o c yp ocu encies. Jou nal o Risk and Financial
Managemen ,10(4), 17.
Clay on, D. G. (1978). A model o associa ion in bi a ia e li e ables
and i s applica ion in epidemiological s udies o amilial endency
in ch onic disease incidence, Biome ika,65(1), 141–151.
Con , R. (2001). Empi ical P ope ies o Asse Re u ns: S ylized Fac s
and S a is ical Issues. Quan i a i e Finance,1,223–236.h p://dx.
doi.o g/10.1080/713665670
Deng, L., Ma, C., & Yang, W. (2011). Po olio op imiza ion ia pai
copula-GARCH-EVT-CVaR model. Sys ems Enginee ing P ocedia,
2,171–181.
Diebold, F. X., Gun he , T. A., & Tay, A. (1997). E alua ing densi y
o ecas s.
Dissmann, J., B echmann, E. C., Czado, C., & Ku owicka, D. (2013).
Selec ing and es ima ing egula ine copulae and applica ion o
inancial e u ns. Compu a ional S a is ics & Da a Analysis,59,52–
69.
Elaal, M. K., & ALe wi, A. S. (2017). On he es ima ion o he bi a ia e
exponen ia ed pa e o dis ibu ion.
Elendne , H., T imbo n, S., Ong, B., & Lee, T. M. (2018) Chap e
7 – The C oss-Sec ion o C yp o-Cu encies as Financial Asse s:
In es ing in C yp o-Cu encies Beyond Bi coin In Da id Lee
154 JELESKOVIC e al.
Kuo Chuen, Robe Deng (eds). Handbook o Blockchain, Digi al
Finance, and Inclusion Academic P ess, 145–173. h ps://doi.o g/
10.1016/B978-0-12-810441-5.00007-5
Emb ech s, P., Mcneil, E., & S aumann, D. (1999). Co ela ion:
pi alls and al e na i es. In Risk Magazine.
Engle, R. F. (1982). Au o eg essi e condi ional he e oscedas ic-
i y wi h es ima es o he a iance o Uni ed Kingdom in la-
ion. Econome ica: Jou nal o he Econome ic Socie y,50,987–
1007.
Fische , M. J., Köck, C., Schlü e , S., & Weige , F. (2007). Mul-
i a ia e Copula Models a Wo k: Ou pe o ming he dese
island copula? (No. 79/2007). Diskussionspapie e//F ied ich-
Alexande -Uni e si ä E langen-Nü nbe g, Leh s uhl ü S a is ik
und Ökonome ie.
Genes ,C., Rémilla d,B.,& Beaudoin, D. (2009). Goodness-o - i es s
o copulas: A e iew and a powe s udy. Insu ance: Ma hema ics
and economics,44(2), 199–213.
Ghalanos, A. (2019). uga ch: Uni a ia e GARCH models. R package
e sion 1.4-1.
Glos en, L. R., Jaganna han, R., & Runkle, D. E. (1993). On he ela-
ion be ween he expec ed alue and he ola ili y o he nominal
excess e u n on s ocks. The jou nal o inance,48(5), 1779–1801.
Gogo, J. (2019, Janua y 15). Jus Because C yp ocu ency Isn’
‘Legal Tende ’ Doesn’ Make I Illegal. Re ie ed om
h ps://news.bi coin.com/jus -because-c yp ocu ency-isn -
legal- ende -doesn -make-i -illegal/
Gumbel, E. J. (1960). Dis ibu ions des aleu s ex 6mes en plusieu s
dimensions, Publ. Ins . S a is . Uni . Pa is,9,171–173.
Ho e , M., & Mächle , M. (2011). Nes ed A chimedean copulas mee
R: The nacopula package. Jou nal o S a is ical So wa e,39(9), 1–
20.
Ho e , M., Kojadino ic, I., Maechle , M., & Yan, J., (2018). copula:
Mul i a ia e Dependence wi h Copulas. R package e sion 0.999-
19.1.
Joe, H. (1996). Families o m- a ia e dis ibu ions wi h gi en ma gins
and m (m-1)/2 bi a ia e dependence pa ame e s. Lec u e No es-
Monog aph Se ies, 120–141.
Joe, H., & Xu, J. J. (1996). The es ima ion me hod o in e -
ence unc ions o ma gins o mul i a ia e models. (h ps://api.
seman icschola .o g/Co pusID:55446837)
Kakou is, I., & Rus em, B. (2014). Robus po olio op imiza ion wi h
copulas. Eu opean Jou nal o Ope a ional Resea ch,235(1), 28–37.
Ka siampa, P. (2017). Vola ili y es ima ion o Bi coin: A compa ison
o GARCH models. Economics Le e s,158,3–6.
Kimeldo i, G., & Sampson, A. (1975). Uni o m ep esen a ions o
bi a ia e dis ibu ions. Communica ions in S a is ics–Theo y and
Me hods,4(7), 617–627.
Kojadino ic, I., & Yan, J. (2010). Modeling mul i a ia e dis ibu ions
wi h con inuous ma gins using he copula R package. Jou nal o
S a is ical So wa e,34(9), 1–20.
K es a, A. (2015). Applica ion o GARCH-copula model in po olio
op imiza ion. Financial Asse s and In es ing,6(2), 7–20.
K okhmal, P., Palmquis , J., & U yase , S. (2002). Po olio op i-
miza ion wi h condi ional alue-a - isk objec i e and cons ain s.
Jou nal o Risk,4(2), 43–68.
K ueckebe g, S., & Scholz, P. (2018). C yp ocu encies as an Asse
Class? A ailable a SSRN 3162800.
Kuma , C. S., & Anila, G. (2018). Asymme ic cu ed no mal
dis ibu ion. Jou nalo S a is icalResea ch,52(2), 173–186.
Lai, K. K., Yu, L., & Wang, S. (2006, June). Mean- a iance-
skewness-ku osis-based po olio op imiza ion. In Fi s In e -
na ional Mul i-Symposiums on Compu e and Compu a ional
Sciences (IMSCCS’06) (Vol. 2, pp. 292–297). IEEE.
Lee, L. F. (1983). Gene alized econome ic models wi h selec i i y.
Econome ica: Jou nal o he Econome ic Socie y,51(2), 507–512.
Ma kowi z, H. (1952). Po olio selec ion. The jou nal o inance,7(1),
77–91.
Massey J , F. J. (1951). The Kolmogo o -Smi no es o goodness o
i . Jou nal o he Ame ican s a is ical Associa ion,46(253), 68–78.
Maye , M. M. L. (2018). C yp ocu enciesas an al e na i e asse class.
Junio Managemen Science,3(4), 1–29.
Mendes, B. V. d. M., Seme a o, M. M., & Leal, R. P. C. (2010).
Pai -copulas modeling in inance. Financial Ma ke s and Po olio
Managemen ,24(2), 193–213.
Mo ales-Nápoles, O. (2010). Coun ing ines. In Dependence model-
ing: Vine copula handbook, 189–218.
Nelsen,R.B.(2007).An in oduc ion o copulas. Sp inge Science &
Business Media.
Nelson, D. B. (1991). Condi ional he e oskedas ici y in asse e u ns:
A new app oach. Econome ica: Jou nal o he Econome ic Socie y,
59,347–370.
Noo , M. N., Yahaya, A. S., Ramli, N. A., & Al Bak i, A. M. M.
(2014). Filling missing da a using in e pola ion me hods: S udy on
he e ec o i ing dis ibu ion (Vol. 594, pp. 889–895). T ans Tech
Publica ions.
Os e iede , J., Lo enz, J., & S ika, M. (2016). Bi coin and
c yp ocu encies-no o he ain -hea ed. A ailable a SSRN
2867671.
Pezie , J., & Whi e, A. (2006). ‘The Rela i e Me i s o In es able
Hedge Fund Indices and o Funds o Hedge Funds in Op i-
mal Passi e Po olios’ ICMA Cen e Discussion Pape s in
Finance icma-dp2006-10, Henley Business School, Uni e si y o
Reading.
Pinudom, B., Tungpisansampun, W., Tansucha , R., & Maneejuk,
P. (2018). Could Bi coin enhance he po olio pe o mance?
In Jou nal o Physics: Con e ence Se ies,1053(1), 012113. IOP
Publishing.
Rache , S. T., Sun, W., & S ein, M. (2009). Copula concep s in
inancial ma ke s. Po olio Ins i u ionell (Ed.), 4,12–15.
Rinne, H. (2010). “P ope ies o loca ion–scale dis ibu ions”. In
Loca ion-scale dis ibu ions—Linea es ima ion and p obabili y
plo ing using MATLAB. Jus us–Liebig–Uni ., Dep . o Economics
and Managemen Science,.
Rocka ella , R. T., & U yase , S. (2002). Condi ional alue-a - isk o
gene al loss dis ibu ions. Jou nal o Banking and Finance,26(7),
1443–1471.
Saha, K. (2018). An in es iga ion in o he dependence s uc u e o
majo c yp ocu encies. A ailable a SSRN 3241216.
Schepsmeie , U., S oebe , J., B echmann, E. C., G aele , B., Nagle ,
T., E ha d , T., & Killiches, M. (2018). Package ‘VineCopula’. R
package e sion, 2.1.8.
Schi mache , D., & Schi mache , E. (2008). Mul i a ia e dependence
modeling using pai -copulas. (pp. 14–16). Technical epo .
Sha pe, W. F. (1994). The sha pe a io. Jou nal o po olio manage-
men ,21(1), 49–58.
Sheikh, A. Z., & Qiao, H. (2009). Non-no mali y o ma ke e u ns:
A amewo k o asse alloca ion decision making. The Jou nal o
Al e na i e In es men s,12(3), 8–35.
JELESKOVIC e al. 155
Skla , M. (1959). Fonc ions de epa i ion an dimensions e leu s
ma ges. Publ. ins . s a is . uni . Pa is,8,229–231.
So, M. K., & Yeung, C. Y. (2014). Vine-copula GARCH model wi h
dynamiccondi ional dependence.Compu a ional S a is ics & Da a
Analysis,76,655–671.
Songa, J. Y., Changa, W., & Songd, J. W. (2019). Clus e analysis on
he s uc u e o he c yp ocu ency ma ke ia Bi coin-E he eum
il e ing. Physica A S a is ical Mechanics and i s Applica ions,527,
121339.
S oyano , S. V., Rache , S. T., & Fabozzi, F. J. (2007). Op imal
inancial po olios, Applied Ma hema ical Finance,14(5), 401–436.
h ps://doi.o g/10.1080/13504860701255292
Sun,L.,&Bai,M.(2007).Applica ion o copula and copula- CVaR in
he mul i a ia e po olio op imiza ion. Sp inge Be lin Heidelbe g,
46(14), 231–242.
Szegö, G. (2005). Measu es o isk. Eu opean Jou nal o Ope a ional
Resea ch,163(1), 5–19.
T imbo n, S., Mingyang, L., & Hä dle, W. K. (2017). In es ing wi h
c yp ocu encies.
T i edi, P. K., & Zimme , D. M. (2007). Copula modeling: An in o-
duc ion o p ac i ione s. Founda ions and T ends R
in Econome -
ics,1(1), 1–111.
Tuwine , J. (2019, Feb ua y 8). Wha is Bi coin Mining and How Does
i Wo k? Re ie ed o m h ps://www.buybi coinwo ldwide.com/
mining/
Yan, J. (2007). Enjoy he joy o copulas: Wi h a package copula.
Jou nal o S a is ical So wa e,21(4), 1–21.
Ziemba, W., & Vickson, R. (2006). S ochas ic op imiza ion models in
inance, 2nd edn. (Wo ld Scien i ic).
How o ci e his a icle: Jelesko ic, V., La ini, C.,
Younas, Z. I., & Al-Fa yan, M. A. S. (2024).
C yp ocu ency po olio op imiza ion: U ilizing a
GARCH-copula model wi hin he Ma kowi z
amewo k. Jou nal o Co po a e Accoun ing &
Finance,35, 139–155.
h ps://doi.o g/10.1002/jca .22721
