Foglie, And ea Delle; Pola, Gianni
A icle
Make he bes om compa ing con en ional and Islamic
asse classes: A design o an all-seasons combined
po olio
Jou nal o Risk and Financial Managemen
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Foglie, And ea Delle; Pola, Gianni (2021) : Make he bes om compa ing
con en ional and Islamic asse classes: A design o an all-seasons combined po olio, Jou nal o
Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol. 14, Iss. 10, pp. 1-17,
h ps://doi.o g/10.3390/j m14100484
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/258588
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
Jou nal o
Risk and Financial
Managemen
A icle
Make he Bes om Compa ing Con en ional and Islamic Asse
Classes: A Design o an All-Seasons Combined Po olio
And ea Delle Foglie 1,* and Gianni Pola 2


Ci a ion: Delle Foglie, And ea, and
Gianni Pola. 2021. Make he Bes
om Compa ing Con en ional and
Islamic Asse Classes: A Design o an
All-Seasons Combined Po olio.Jou nal
o Risk and Financial Managemen 14:
484. h ps://doi.o g/10.3390/
j m14100484
Academic Edi o : Muhammad
Ishaq Bha i
Recei ed: 20 Augus 2021
Accep ed: 7 Oc obe 2021
Published: 13 Oc obe 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Depa men o Managemen , Facul y o Economy, Sapienza Uni e si y o Rome, 00161 Rome, I aly
2Anima SGR S.p.A., Anima Holding, 0121 Milan, I aly; gianni.pola@animasg .i
*Co espondence: and ea.delle oglie@uni oma1.i
Abs ac :
This pape aims o con ibu e o he exis ing li e a u e in po olio managemen and s a egy
by in es iga ing he pe o mance, di e si ica ion, and hedging bene i s a ising om in eg a ing
Sha ia-complian s ocks in o a con en ional po olio. Thus, his pape es s he pe o mance o a
Combined Po olio, esul ing om he combina ion o con en ional Islamic ins umen s, co e ing
di e en mac oeconomic scena ios in he las decade (2010–2020). The s a egic asse alloca ion was
designed ollowing he Global Mac o Anima (GMA) s a egy, sol ing a isk-pa i y op imisa ion
p oblem using a speci ically de eloped MATLAB
™
algo i hm. The indings will con ibu e o
answe ing he ques ion ela ed o he possibili y o including al e na i e ins umen s o inc ease
di e si ica ion wi h hedging bene i s by building asse alloca ions ha pe o m well ac oss di e en
mac oeconomic scena ios.
Keywo ds:
asse alloca ion; po olio managemen ; isk pa i y; Islamic equi ies; mac oeconomic
condi ions
JEL Classi ica ion: G11; G15; G17; G19
“The COVID-19 is no a Black Swan. I was mo e p edic able han people ealise.
The Black Swan was mean o explain why, in a ne wo ked wo ld, we need o change
business p ac ices and social no ms no o p o ide a cliché o any bad hing ha su -
p ises us.”
Nassim Taleb
1. In oduc ion
COVID-19 has s ongly s essed and es ed he global inancial ma ke s, ep esen ing
a ough challenge o asse managemen . The pandemic ep esen s an exogenous economic
shock, al hough his depends on unp edic able noneconomic ac o s, di e en om he
global inancial c isis in 2008 (GFC) o he Eu opean so e eign deb c isis in 2010–2013,
which we e endogenous shocks due o inancial easons (Bo io 2020). I a ec s he global
economy, igge ing se e al sec o s, such as labou ma ke s, global supply chains, and
consump ion beha iou , since na ional au ho i ies ha e decla ed lockdowns o de ing
he shu down o mos nonco e business ac i i ies. These s ic ac ions gene a ed se e e
demand and supply-chain sh inking in he inancial ma ke ollowing he ou b eak. The
eme ging and de eloped s ock ma ke s ha e g adually allen, decla ing he hun ’s open
season o sa e-ha en asse s o limi he con agion e ec s. Fo his pu pose, schola s a e
also keeping an eye on he Islamic Financial Sys em (IFS), which demons a ed mo e
s abili y and esilience due o he in insic unde pinnings o e hicali y and sus ainabil-
i y o he Sha iah-complian p inciples (A ou i e al. 2013;Ash a e al. 2020;Hengchao
and Hamid 2015;Jawadi e al. 2014;Pal inie i e al. 2019). Ash a e al. (2020) es he
pe o mance o he S&P Dow Jones, con i ming ha i unde pe o med compa ed o i s
Islamic coun e pa s du ing Q1 o 2020, and suppo ing he hypo hesis ha Islamic equi ies
J. Risk Financial Manag. 2021,14, 484. h ps://doi.o g/10.3390/j m14100484 h ps://www.mdpi.com/jou nal/j m
J. Risk Financial Manag. 2021,14, 484 2 o 17
p o ide po olio-hedging bene i s du ing se e e ma ke down alls.
Ha oon e al. (2021)
also epo ed Islamic equi y di e si ica ion oppo uni ies due o hei lowe sys ema ic
isk. Howe e , Hasan e al. (2021) con es ed he decoupling hypo hesis o he Islamic s ock
ma ke om he con en ional one, since he Sha iah sc eening p ocess ails o p o ide
immuni y. Fu he mo e, he majo Islamic s ock indexes ou pe o med hei coun e -
pa s du ing he pe iod 2019–2020, especially a e he s ock-ma ke c ash in Ma ch 2020
(She i 2020)
. The li e a u e also ocuses on he pe o mance o Islamic mu ual unds,
compa ed wi h he con en ional ones, depending on he manage ial and ma ke iming
skills (Manso e al. 2020). In addi ion, du ing he COVID-19 c ash, which was se e e and
quick, e e y ma ke c ash awakened and ein o ced some longs anding ending opics o
academics and p ac i ione s, such as:
•
The enewed a en ion o asse s unco ela ed o nega i ely co ela ed wi h o he
adi ional asse s, such as gold, p ecious me als, commodi ies, o easu ies, p o iding
po olio-di e si ica ion bene i s in e ms o ola ili y, downside isk, and maximum
d awdown mi iga ion powe , pa icula ly du ing inancial down u ns (Bau and
Lucey 2010;Bau and McDe mo 2010;Bou i e al. 2020;Ji e al. 2020;K is ou ek 2020;
Rebo edo 2013).
•
The hedging bene i s and esilience o Islamic equi ies du ing he las g ea GFC a e
a ibu ed o he limi ed exposu e o high-le e age companies due o he Sha iah
sc eening (Ash a e al. 2020;Jawadi e al. 2014). IFS dis inguishes i sel by p omo ing
a mo e e hical app oach o p o i and isk sha ing, acili a ing ai ness in inancial
ma e s (Al Rahahleh e al. 2019).
•
The academic in e es in he Islamic S ock Ma ke (ISM) compa ed o he con en ional
one has di e gen esul s in e ms o pe o mance (Beloua i e al. 2015;Delle Foglie
and Pane a 2020;Hassan e al. 2019;Masih e al. 2018). Delle Foglie and Pane a (2020)
p oposed a change in he esea ch app oach, sea ching o he possibili y o e alua e
he Sha iah-complian ins umen di e si ica ion, decoupling and hedging bene i s,
and combining he con en ional po olio and no me ely di e en asse classes.
•
The inancial c isis inc eases he need o design a po olio s a egy ha i s all mac oe-
conomic en i onmen s, and aces he cu en pos c ash scena io and u u e economic
and inancial unce ain y. Assuming e e y economic cycle is a se o unp edic able
ch onological e en s a ec ing each speci ic asse class pe o mance, i seems unnec-
essa y o o ecas he nex inancial down u ns, since i is impossible o p edic he
u u e (Economic machine—B idgewa e 2011). This p inciple also co esponds o he
heo e ical backg ound unde lying he ounda ion o Global Mac o Anima (GMA), a
s a egic asse alloca ion based on he di e si ica ion ac oss mac oeconomic scena ios
p oposed by Pola (2013,2021). The GMA app oach o e comes he mean- a iance
amewo k ha domina es po olio s a egies, decla ing ha “asse – e u n dynamics
can be explained mainly by a ia ions o expec a ions a he han he le els o mac oeconomic
a iables”.
Finally, his pape aims o con ibu e o he li e a u e o po olio-managemen and
asse -alloca ion s a egies, conside ing ypical sa e-ha en asse s combined wi h Islamic
s ocks, ollowing a Combined Po olio app oach (Delle Foglie and Pane a 2020). Acco ding
o Delle Foglie and Pane a (2020), he li e a u e has ocused so a on he con aposi ion
o he ISM wi h he con en ional ma ke wi hou in es iga ing he possible posi i e e -
ec s o hei combina ion in a po olio-managemen logic. The Combined Po olio aims
o o e come he me e compa a i e app oach (Islamic s. con en ional), changing he
in es iga ion app oach o he phenomenon h ough he lens o in eg a ion. No ably, his
s udy does no in end o de e mine he bes po olio asse alloca ion and s a egy o bea
he ma ke . Con as ingly, i e u ns o he ques ion ela ed o he possibili y o including
al e na i e s ock ins umen s o inc ease di e si ica ion wi h hedging bene i s, building
asse alloca ions ha pe o m well o e ime. In his ega d, his pape also ollows he li e -
a u e s ands ounded on he GMA and All-Wea he (AW) philosophy o B idgewa e (2012)
in o he s a egic asse -alloca ion choice, combining hem wi h a Risk Pa i y (RP) model
J. Risk Financial Manag. 2021,14, 484 3 o 17
(Qian 2005)
as a good asse alloca ion selec ion c i e ion o hese s a egies. The emainde
o his pape is o ganised as ollows. Sec ion 2p o ides he me hodology, ocusing on he
undamen als o he GMA s a egy, he AW, and he isk-pa i y heu is ic app oach. The
op imisa ion p oblem o he isk pa i y is sol ed using he speci ically designed MATLAB
algo i hm. Sec ion 3p esen s and deba es he da a, desc ip i e s a is ics, and empi ical
esul s o he po olios. Finally, he main conclusions and u he ema ks a e disclosed in
Sec ion 4.
2. Me hodology
2.1. The All-Wea he Philosophy and he GMA S a egy
“Wha kind o in es men po olio would you hold ha would pe o m well ac oss
all en i onmen s, be i a de alua ion o some hing comple ely di e en ?” AW enginee ing
is based on an app oach in which asse e u ns a e b oken down in o building blocks.
This p ocess lies in he Pos -Mode n Po olio Theo y (PMPT) li e a u e. The s and o
li e a u e s udying he Mode n Po olio Theo y (MPT) conside s asse alloca ion as he
asse class combina ion based on expec ed e u ns, isks, and co ela ions. Acco ding o
Lee (2011), he global inancial c isis o 2008 demons a ed he weakness o ypical po olio
s a egies such as he mean- a iance op imisa ion, 60/40, and MPT, which ha e p oblems
wi h di e si ica ion and he unde es ima ion o isk. Ins ead, in he PMPT p oposed by AW
(B idgewa e 2012), any o al in es men e u n can be spli in o i s in insic componen s
and analysed while conside ing he leading componen s o hose indi idual pa s. Thus,
his e u n is a unc ion o he e u n on cash ( he isk- ee posi ion), he excess e u n o
ma ke s abo e he cash a e ( he be as), and he alpha as he manage ial skills in s ock
selec ion. In summa y:
In es men e u n = isk- ee + be a + alpha
The e o e, he AW philosophy is based on h ee o he undamen al keys p o iding
di e en in e p e a ions o he in es o ’s mindse : he ole o ma ke expec a ions, he
en i onmen al biases, and he ole o cash. Fi s , AW conside s ma ke s ha a e b eakable
in o se e al componen s. Ma ke s mo e conside ing hei in insic expec a ions and he
shi in hei p ice condi ions. Thus, he e is a ela ionship be ween ma ke expec a ions and
he de ini ion o su p ise. Clea ly, he g ea e he gap, he la ge he ma ke expec a ions.
Second, all asse classes ha e en i onmen al biases. The idea o en i onmen al bias is
linked o he no ion o asse co ela ion. Some asse s pe o m well in ce ain economic
seasons and poo ly in o he s. Asse -class p icing and pe o mance will depend on ma ke
expec a ions, since hey discoun u u e economic scena ios.
Mo eo e , asse -class p icing will conside he ole o cash. As p e iously men ioned,
he e u n o cash ep esen s he in es o s’ isk- ee posi ion. The la ge he in es men
isk, he la ge he compensa ion in es o s equi e ( isk p emium). As a esul , in es men
e u n depends on he e u n o any cons i uen asse . I conside s he changes in he isk
p emium and he unp edic able al e a ions in he economic cycle (en i onmen al biases).
The economic cycle ypically depends on he olume o economic ac i i y (g ow h) and
i s p icing (in la ion). The asse -alloca ion mix will depend on he in es o sen imen o
he u u e condi ion o highe /lowe in la ion g ow h (ma ke expec a ions). Figu e 1
below summa ises he undamen als o he AW s a egy. The esul is ha asse alloca ion
will cap u e he ou isk exposu es, mi iga ing he isk h ough he di e ences in en i-
onmen al co ela ion be ween asse classes (Figu e 1). Thus, ollowing he in es men
e u n o mula, in he long e m, in es men choices mus conside asse s ha should
p o ide a e u n abo e cash ( isk- ee). Pola (2013) in oduced he GMA s a egy wi h
he same undamen als, conside ing ha asse e u ns a e mainly a ec ed by changes in
mac oeconomic and s ess ac o expec a ions. “S ocks mo e no due o low o high g ow h bu
mainly due o he ac ha g ow h is abo e o below expec a ions”.
J. Risk Financial Manag. 2021,14, 484 4 o 17
J. Risk Financial Manag. 2021, 14, x FOR PEER REVIEW 4 o 18
e u n o mula, in he long e m, in es men choices mus conside asse s ha should p o-
ide a e u n abo e cash ( isk- ee). Pola (2013) in oduced he GMA s a egy wi h he
same undamen als, conside ing ha asse e u ns a e mainly a ec ed by changes in mac-
oeconomic and s ess ac o expec a ions. “S ocks mo e no due o low o high g ow h bu
mainly due o he ac ha g ow h is abo e o below expec a ions”.
The GMA s a egy suppo s he mean- a iance amewo k c i icisms ela ed o sig-
ni ican inpu pa ame e e o s and he lack o di e si ica ion cha ac e ising he 60/40 (eq-
ui y/bond) po olio. The 60/40 po olio is designed o disin la ion and ising g ow h
scena ios wi h a high concen a ion o equi y isk. On he con a y, he GMA aims o build
po olios ha limi exposu e o unexpec ed mac oeconomic en i onmen s, and conside s
and manages un a ou able in la ion scena ios (Pola 2021). GMA conside s he co e and
hidden d i e s o asse e u n as he olume o economic ac i i y (g ow h), p icing (in la-
ion), and po en ial ma ke s ess. Pola (2013) es ablished he ela ionship be ween asse
sensi i i y and ac o dynamics using a pola isa ion coe icien . Each single asse class
shows di e en beha iou o he in la ion/g ow h scena io. Table 1 summa ises he in la-
ion–g ow h pola isa ion esul s, epo ing he ela ionship be ween he asse classes and
he di e en mac oeconomic condi ions.
Figu e 1. Quad an o he mac oeconomic scena io. Sou ce: au ho s’ elabo a ion.
Table 1. Rela ion be ween he asse classes, ends, and mac oeconomic condi ions.
Mac oeconomic Condi ions
G ow h
In la ion
Ma ke S ess
T end
Rising
Commodi ies
Eme ging deb in local cu -
ency
Equi ies
Commodi ies
Nominal bonds
Co po a e IG bonds
Gold
Eme ging deb in local cu ency
Gold
In la ion-linked bonds
Falling
Nominal bonds
Gold
Nominal bonds
Co po a e IG bonds
Co po a e HY bonds
Commodi ies
Eme ging deb in local cu ency
Equi ies
Sou ce: da a elabo a ion based on Pola (2013).
Figu e 1. Quad an o he mac oeconomic scena io. Sou ce: au ho s’ elabo a ion.
The GMA s a egy suppo s he mean- a iance amewo k c i icisms ela ed o sig-
ni ican inpu pa ame e e o s and he lack o di e si ica ion cha ac e ising he 60/40
(equi y/bond) po olio. The 60/40 po olio is designed o disin la ion and ising g ow h
scena ios wi h a high concen a ion o equi y isk. On he con a y, he GMA aims o
build po olios ha limi exposu e o unexpec ed mac oeconomic en i onmen s, and
conside s and manages un a ou able in la ion scena ios (Pola 2021). GMA conside s he
co e and hidden d i e s o asse e u n as he olume o economic ac i i y (g ow h), p icing
(in la ion), and po en ial ma ke s ess. Pola (2013) es ablished he ela ionship be ween
asse sensi i i y and ac o dynamics using a pola isa ion coe icien . Each single asse
class shows di e en beha iou o he in la ion/g ow h scena io. Table 1summa ises he
in la ion–g ow h pola isa ion esul s, epo ing he ela ionship be ween he asse classes
and he di e en mac oeconomic condi ions.
Table 1. Rela ion be ween he asse classes, ends, and mac oeconomic condi ions.
Mac oeconomic Condi ions
G ow h In la ion Ma ke S ess
T end
Rising
Commodi ies
Eme ging deb in local cu ency
Equi ies
Commodi ies Nominal bonds
Co po a e IG bonds
Gold
Eme ging deb in local cu ency
Gold
In la ion-linked bonds
Falling Nominal bonds
Gold
Nominal bonds
Co po a e IG bonds
Co po a e HY bonds
Commodi ies
Eme ging deb in local cu ency
Equi ies
Sou ce: da a elabo a ion based on Pola (2013).
2.2. The Risk Pa i y Model and he Op imisa ion P oblem
Acco ding o Allen (2010), adi ional po olio s a egies, ounded on he mean-
a iance op imisa ion, a e based on a 60/40 asse alloca ion, in which s ocks explain ap-
p oxima ely 70%/90% o he o al isk con ibu ion, esul ing in an excessi e concen a ion
o a subse o asse s. The po olio esul s a e well di e si ied om he weigh pe spec-
i e, bu no ega ding he ola ili y o s ocks and bonds. In addi ion, he mean- a iance
app oach seems o be oo sensi i e o i s inpu pa ame e s, wi h signi ican di e ences
in small changes (Mailla d e al. 2010). The li e a u e desc ibes his phenomenon as he
concep o all eggs in one baske , since he 60/40 po olio does no o e p ope isk di e si i-
ca ion. The e is much con usion be ween he concep o ola ili y op imisa ion and isk
di e si ica ion. Mos o he con ibu ions apply he same app oach o minimise po olio

J. Risk Financial Manag. 2021,14, 484 5 o 17
ola ili y, and a ge an expec ed e u n. The PMPT a emp s o bypass he op imisa ion me h-
ods, p e e ing heu is ic-solu ion-based isk dis ibu ion (Allen 2010;
Ande son e al. 2012
;
B ude and Roncalli 2012
;
Chouei a y and Coigna d 2008
;Fo es i and Rush 2010;
Le ell 2010
;
Loh e e al. 2012;Mailla d e al. 2010;Meucci 2007,2009). This kind o app oach is con-
cei ed o c ea e (po en ially) highe long- e m p o i s by accep ing he ole ance o highe
isk ( he e o e, i is no only abou minimizing he ola ili y) (B ude and Roncalli 2012;
Qian 2005
). Following he isk pa i y in Equal Risk Con ibu ion (ERC), he asse alloca ion
does no conside any e u ns in he weigh dis ibu ion bu he isk con ibu ion o a single
componen as he ma ginal isk con ibu ion (MRC). I is he sha e o he o al po olio
isk con ibu ion (TRC) associa ed wi h ha speci ic componen . Acco ding o B ude and
Roncalli (2012) and Mailla d e al. (2010), we conside ed a po olio X = (x
1
;x
2
;
. . .
;x
n
) o
n isky asse s, assuming no possibili y o le e age o sho selling. The po olio s anda d
de ia ion is as ollows:
σp(x)=√xTΩx=q∑ix2
iσ2
i+∑i∑i6=jxixjσij (1)
whe e
σij
is he co a iance be ween asse iand j, and
Ω
is he co a iance ma ix. The
MRCi(x)=(Ωx)
√xTΩx
and he
TRCi(x)=xi(Ωx)
√xTΩx
, so i is easy o show ha he po olio
isk can be explained as he sum o he TRCs:
∑n
i=1TRC(x)=∑n
i=1xi
(Ωx)
√xTΩx=√xTΩx=σp(x)(2)
As p e iously men ioned, he ERC undamen als aim o build a isk-balanced po olio
conside ing he asse alloca ion in e ms o isk con ibu ion a he han in e ms o po olio
weigh s ( isk budge ing). Thus, we conside ed a isk budge , b, and he ec o o isk in he
pe cen age o he o al isk, b = (b1, b2,
. . .
,bn), whe e bi = bj = 1/n, he
TRCi(x)=TRCj(x)
,
and he xi(Ωx)
√xTΩx=xj(Ωx)
√xT, so i is easy o show ha :
∑n
i=1TRC(x)=nTRCi(x)(3)
whe e he
TRCi(x)=σ(x)
n
. Finally, he isk pa i y can be o mula ed as he ollowing
op imisa ion p oblem:
X = a g min (x) (4)
whe e
(x)=∑n
i=1∑n
j=1TRCi(x)−TRCj(x)2=∑n
i=1∑n
j=1hxi(Ωx)i−xj(Ωx)ji2,
∑n
i=1xi=1, ∑n
j=1xj, and x≥0(5)
Conside ing he p e ious Eule decomposi ion o he po olio isk measu e, he
p oblem can be sol ed as ollows:
X=a g min ∑n
i=1xi(Ωx)i−σp(x)
n2
(6)
The op imisa ion p oblem can be sol ed using he MATLAB Op imiza ion Toolbox
™
,
which p o ides unc ions o inding pa ame e s ha minimise o maximise objec i es,
while sa is ying cons ain s. No ably, he sequen ial quad a ic p og amming (SQP) demon-
s a es a solu ion o nonlinea p og amming (NLP), gene a ing i e a ions o sol e he
op imisa ion p oblem by se ling a sequence o SQP and app oxima ing he exac solu ion.
The oolbox also includes sol e s o NLP, such as he sol e -based nonlinea op imiza ion,
which inds he minimum cons ained nonlinea mul i a iable unc ion o discuss he
exis ence and uniqueness o he isk pa i y po olio. Cons ained op imisa ion aims o
con e he p oblem in o an easie subp oblem using an i e a i e p ocess ha p o ides
J. Risk Financial Manag. 2021,14, 484 6 o 17
s ochas ic app oxima ion in gene a ing possible solu ions. Speci ically, his algo i hm uses
a heu is ic me hod, since i is used o ind accep able answe s no di e ing oo much om
he exac solu ion. The mincon unc ions o MATLAB p o ide an SQP-based nonlinea p o-
g amming sol e , inding he minimum o a cons ained nonlinea mul i a iable unc ion
o a p oblem speci ied by By d e al. (2000) and Wal z e al. (2006):
minx (x)such ha











c(x)≤0
ceq(x)=0
A·x≤b
Aeq ·x=beq
lb ≤x≤ub
-
band beq a e ec o s, Aand Aeq a e ma ices, c(x) and ceq(x) a e unc ions ha e u n
ec o s, and (x) is a unc ion ha e u ns a scala . (x), c(x), and ceq(x) can be nonlinea
unc ions.
-x, lb, and ub can be passed as ec o s o ma ices.
Mainly, he Op imiza ion Toolbox
™
sol e s accep ec o s o many a gumen s (x0 as
ini ial poin , lowe bounds lb, and uppe bounds ub) and ma ices, whe e he ma ix is an
a ay o any size. Sol e s handle ma ix a gumen s as ollows:
•
In e nally, sol e s con e ma ix a gumen s in o ec o s be o e p ocessing. Fo
example, x0 becomes x0(:);
•Fo ou pu , sol e s eshape he solu ion, x, o he same size as he inpu , x0;
•
When x0 is a ma ix, sol e s pass x as a ma ix o he same size as x0 o bo h he
objec i e unc ion and o any nonlinea cons ain unc ion;
•
Linea cons ain s, howe e , ake xin ec o o m, x(:). In o he wo ds, a linea
cons ain o he o m:
A*x ≤b o Aeq*x = beq, akes xas a ec o , no a ma ix (Ma hWo ks Inc. 2021).
Thus, ecalling Equa ion (4), he app op ia e syn ax o he isk-pa i y op imisa ion
p oblem’s solu ion is as ollows:
X = mincon ( un,x0, A,b,Aeq,beq,lb,ub) (7)
which de ines a se o lowe and uppe bounds on he design a iables in x, so ha he
solu ion is always in he ange lb
≤
x
≤
ub (Giuzio 2017;Mussa i and Ismail 2021). By
de aul , he mincon unc ion sol es he in e io -poin algo i hm app oach o cons ained
minimisa ion p oblems. Following, he RP op imisa ion p oblem is sol ed by compu ing
MATLAB mincon. Fi s , Func ion (1) was designed o sol e he op imisa ion p oblem
in Equa ion (4). Func ion (1) ep esen s he MATLAB unc ion compu ed o sol e he
op imisa ion p oblem showed in Equa ion (4):
un = @ {(EW_Sha es) Aeq * (((Va Co a (:,:,i)*(EW_Sha es)/(sq
((EW_Sha es’) * Va Co a (:,:,1) *EW_Sha es))). * EW_Sha es −(sq
((EW_Sha es’) * Va Co a (:,:,1) *EW_Sha es))/nc).ˆ2}
whe e Va Co a is he a iance–co a iance ma ix, and nc is he numbe o he asse classes
composing he po olio. Second, he mincon was applied o Func ion (1) o sol e he
op imisa ion p oblem, w i ing a s ing o ind he RP po olio weigh s. (To imp o e he
compu ing, we se he op imisa ion algo i hm, changing he e mina ion ole ance o he
unc ion alue (se as 1
×
10
−6
) and se ing he maximum numbe o unc ion e alua ions,
a posi i e in ege , as 500,000.)
RPSha es (:,i) = mincon ( un, EW_Sha es, [ ], [ ], Aeq,beq,lb,ub, [ ], op ions)
Finally, o imp o e he op imisa ion p oblem’s solu ion, we se he maximum unc ion
e alua ions o 500,000 and se he op imali y ole ance o 1 ×10−6.
J. Risk Financial Manag. 2021,14, 484 7 o 17
2.3. Da a and Sample Selec ion
As p e iously men ioned, he sample selec ion was designed ollowing he GMA
and he AW s a egy o iden i y an asse alloca ion ha may pe o m ac oss all di e en
economic mac oscena ios. No ably, since majo global indexes a e quo ed in USD, we
also conside ed he geog aphical and cu ency exposu e sui able o Eu opean-based
in es o s, adding a EUR (Eu o)-based bond componen . Thus, ollowing Figu e 1and
Table 1, he asse alloca ion was designed while assuming he equal p obabili y ha each
o he ou scena ios occu s o e ime (25% o he in es o ’s isk p emium). Howe e ,
o acili a e he ope a ion o he isk-pa i y model, we chose no o conside bo h asse s
a ec ed by oo much and oo li le ola ili y, espec i ely, as commodi ies and cash. Thus,
each asse class is included in he po olio designed, conside ing i s speci ic ole o eac
di e en ly when he economic en i onmen a ies. Conside ing he possibili y o in es o s
o use index- acking ins umen s, he con en ional asse alloca ion includes 10 indexes
wi h 506 weekly obse a ions (505 weekly e u ns) om 29 Ap il 2011 o 31 Decembe
2020 (9.7 yea s), ex ac ed om Reu e s Re ini i Eikon. Acco ding o he s a egy, he
da a span co e s di e en mac oeconomic cycles, including c isis and pos c isis pe iods,
de la ion, and economic g ow h, including COVID-19 pandemic ma ke shock.
Figu e 2
summa ises di e en global business cycles co esponding o he di e en le els o in la ion
and annual GDP g ow h. Fo he second s age o he analysis, we added ou s ock indexes
o he Combined Po olio co esponding o he Sha iah-complian coun e pa s p e iously
selec ed. A Bloombe g Te minal was used o ex ac Sha iah s ock index ime se ies.
Table 2
summa ises each mac oasse class and he co esponding index selec ed o he
asse alloca ion.
J. Risk Financial Manag. 2021, 14, x FOR PEER REVIEW 7 o 18
unc ion alue (se as 1 × 10−6) and se ing he maximum numbe o unc ion e alua ions,
a posi i e in ege , as 500,000.)
RPSha es (:,i) = mincon ( un, EW_Sha es, [ ], [ ], Aeq, beq, lb, ub, [ ], op ions)
Finally, o imp o e he op imisa ion p oblem’s solu ion, we se he maximum unc-
ion e alua ions o 500,000 and se he op imali y ole ance o 1 × 10−6.
2.3. Da a and Sample Selec ion
As p e iously men ioned, he sample selec ion was designed ollowing he GMA and
he AW s a egy o iden i y an asse alloca ion ha may pe o m ac oss all di e en eco-
nomic mac oscena ios. No ably, since majo global indexes a e quo ed in USD, we also
conside ed he geog aphical and cu ency exposu e sui able o Eu opean-based in es-
o s, adding a EUR (Eu o)-based bond componen . Thus, ollowing Figu e 1 and Table 1,
he asse alloca ion was designed while assuming he equal p obabili y ha each o he
ou scena ios occu s o e ime (25% o he in es o ’s isk p emium). Howe e , o acili-
a e he ope a ion o he isk-pa i y model, we chose no o conside bo h asse s a ec ed
by oo much and oo li le ola ili y, espec i ely, as commodi ies and cash. Thus, each
asse class is included in he po olio designed, conside ing i s speci ic ole o eac di -
e en ly when he economic en i onmen a ies. Conside ing he possibili y o in es o s
o use index- acking ins umen s, he con en ional asse alloca ion includes 10 indexes
wi h 506 weekly obse a ions (505 weekly e u ns) om 29 Ap il 2011 o 31 Decembe
2020 (9.7 yea s), ex ac ed om Reu e s Re ini i Eikon. Acco ding o he s a egy, he
da a span co e s di e en mac oeconomic cycles, including c isis and pos c isis pe iods,
de la ion, and economic g ow h, including COVID-19 pandemic ma ke shock. Figu e 2
summa ises di e en global business cycles co esponding o he di e en le els o in la-
ion and annual GDP g ow h. Fo he second s age o he analysis, we added ou s ock
indexes o he Combined Po olio co esponding o he Sha iah-complian coun e pa s
p e iously selec ed. A Bloombe g Te minal was used o ex ac Sha iah s ock index ime
se ies. Table 2 summa ises each mac oasse class and he co esponding index selec ed o
he asse alloca ion.
Figu e 2.
Business cycles in 2011–2020. Sou ce: au ho s’ elabo a ion based on Wo ld Bank Da a. No es: The in la ion and
GDP g ow h wo ld da a a e yea -on-yea . The ed do ed lines delimi he end o he business cycle.
J. Risk Financial Manag. 2021,14, 484 8 o 17
Table 2. Sample composi ion.
Mac oasse Class Index Code
Equi ies
S&P 500 Index—CBOE SP500
MSCI Wo ld P ice Index USD MSCIW
MSCI Eme ging Ma ke s P ice Index USD MSCIEM
MSCI AC Asia-Paci ic P ice Index USD MSCIAP
Islamic Equi ies
S&P 500 Sha iah Index SPS500
MSCI Wo ld Islamic Index MSCIWI
MSCI Eme ging Ma ke Islamic Index MSCIEMI
MSCI AC Asia-Paci ic Islamic Index MSCIAPI
Sho Te m Nominal Bonds ICE Bo A US 1–3-Yea US T easu y Index USTREAS
All-Ma u i y Nominal Bonds
Go e nmen Bonds Ma ki IBoxx EUR Eu ozone Index EUROGOV
Co po a e Bonds IBoxx EUR Co po a e Index EUROCORP
In la ion-Linked Bonds IBoxx Eu o In la ion-Linked Index EUROIL
Con e ible Bonds Re ini i Quali ied Global Con e ible Index CONVBOND
Gold COMEX Gold Composi e Commodi y Fu u e Con inua ion 1 GOLD
Sou ce: au ho s’ compila ion.
3. Empi ical Resul s
3.1. Desc ip i e S a is ics and Co ela ion
Table 3summa ises he desc ip i e s a is ics o he weekly asse e u ns. The co e-
la ions be ween he asse e u ns (Table 4) con i m he inancial end o ecen yea s: o
mo e han 20 yea s, s ocks and bonds we e nega i ely co ela ed, and p ecious me als, such
as gold, played he ole o a sa e-ha en asse , being nega i ely co ela ed wi h s ocks and
neu al o bonds. A e he GFC and he so e eign deb c isis in Eu ope, he global cen al
banks began o launch accommoda i e mone a y policies o s imula e he global economy
and achie e long- e m economic g ow h. Howe e , acco ding o he li e a u e in his ield,
he nega i e s ock–bond co ela ion seems o be ela ed o low and s able isk- ee in e es
a es and in la ion, and he como emen be ween economic g ow h and a es, equi y isk
p emiums and bond isk p emiums. Changes in mac oeconomic condi ions may modi y
he s ock–bond co ela ion om nega i e/neu al o posi i e, as occu ed a e he second
wa e o COVID-19 when in la ion and in e es a e g ow h expec a ions began o de elop
(Ande son e al. 2012;Shen and Weisbe ge 2021;Yang e al. 2009).
Simila ly, p ecious me als such as gold ha e always been conside ed by in es o s
as sa e-ha en asse s, since hey a e no co ela ed wi h s ocks and bonds, con ibu ing o
po olio-di e si ica ion bene i s and he downside isk educ ion (
Bau and Lucey 2010
;
Bau and McDe mo 2010
;Rebo edo 2013). Du ing ma ke dis ess and ecession, in-
es o s end o mo e o sa e-ha en asse s such as gold and cash (as he US dolla ). As
o he s ock and bond ma ke , gold e alua ion has also changed undamen als. While
economic s imulus measu es in ecen yea s ha e suppo ed economic g ow h, egional
and global economic and poli ical issues ha e e ained he backg ound o a clima e o
unce ain y, which explains he lack o co ela ion wi h o he asse s.
J. Risk Financial Manag. 2021,14, 484 15 o 17
s a egies ha i all mac oeconomic scena ios acing pos c isis economic and inancial
unce ain y, ecalling he a en ion o global mac os a egies such as GMA. In his con ex ,
he COVID-19 shock has enewed he a en ion, on he one hand, on he di e si ica ion
oppo uni ies and po en ial hedging bene i s o e ed by Islamic equi ies, and on he o he
hand, on he ejec ion o he hypo hesis o he ISM om he con en ional ma ke , especially
du ing high- ola ili y and unce ain pe iods. Al hough some schola s demons a ed ha
he Sha iah sc eening p ocess ailed o p o ide immuni y du ing a sho c ash a ec ed
by an in e nal demand shock, we decided o explo e he possibili y o combining he
con en ional and Sha iah-complian asse s using he GMA s a egy. The pe o mances o
he Combined Po olios demons a ed he posi i e e ec s o combining he GMA s a egy,
he RP, and Islamic equi ies. The esul s also showed ha hese kinds o po olios may
be app op ia e o isk-a e se in es o s who a e no in e es ed in bea ing he ma ke ,
bu in ising s able e u ns wi h low ola ili y, minimising he downside isk and he
maximum po olio d awdown. The RP app oach can also be adap ed o mo e isk- ake
in es o s using he le e age o maximise he e u n. Finally, global mac os a egies such
as GMA eco ded ewe e u ns han o he , mo e agg essi e s a egies, especially in sho
pe iods. In hese s a egies, which conside ed ha asse e u ns we e mainly a ec ed by
mac oeconomic and s ess ac o s, asse alloca ion seemed immune o he sho ma ke
c ash. The Combined Po olios ep esen ed a di e en choice in his con ex , especially
o isk-a e se and isk-inclined in es o s (such as young in es o s). Fu he esea ch
in he po olio-managemen indus y could con inue, sepa a ely and oge he wi h he
beha iou and pe o mance o his kind o s a egy and he Combined Po olios, o s udy
single pa e ns wi h quan i a i e me hods. Keeping in mind ha i is impossible o o ecas
he nex economic o inancial shock, i migh be be e o include each speci ic asse class
wi h a p ecise ole o allow po olios o pe o m ac oss di e en seasons (mac oeconomic
scena ios).
Au ho Con ibu ions:
W i ing, me hodology, in es iga ion, and so wa e, A.D.F.; me hodology,
concep ualiza ion, and supe ision, G.P. All au ho s ha e ead and ag eed o he published e sion
o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Acknowledgmen s:
A.D.F. would like o exp ess his g ea app ecia ion o Fabiomassimo Mango o
his aluable and cons uc i e sugges ions du ing he planning and he de elopmen o he MATLAB
code used in his wo k. Special hanks a e also gi en o his iend Ricca do Ma ini o suppo
du ing he encoding o he model. Thei willingness o gi e hei ime so gene ously was e y
much app ecia ed.
Con lic s o In e es :
The iews exp essed in his wo k a e hose o he au ho s and do no necessa ily
co espond o hose o ANIMA SGR S.p.A. The au ho s decla e ha hey ha e no con lic s o in e es .
Re e ences
Al Rahahleh, Naseem, M. Ishaq Bha i, and Fa idah Najuna Misman. 2019. De elopmen s in Risk Managemen in Islamic Finance: A
Re iew. Jou nal Risk Financial Manag 12: 37. [C ossRe ]
Allen, G ego y. 2010. The Risk Pa i y App oach o Asse Alloca ion. Shankill: Callan In es men s Ins i u e, Callan Associa es.
Ande son, Robe M., S ephen W. Bianchi, and Lisa R. Goldbe g. 2012. Will My Risk Pa i y S a egy Ou pe o m? Financial Analys s
Jou nal 68: 75–93. [C ossRe ]
A ou i, Mohamed El Hedi, Hachmi Ben Ameu , Nabila Jawadi, F edj Jawadi, and Waël Louhichi. 2013. A e Islamic inance inno a ions
enough o in es o s o escape om a inancial down u n? Fu he e idence om po olio simula ions. Applied Economics 45:
3412–20. [C ossRe ]
Ash a , Dawood, Muhammad Suhail Rizwan, and Ghu an Ahmad. 2020. Islamic Equi y In es men s and he COVID-19 Pandemic.
A ailable online: h ps://ss n.com/abs ac =3611898. (accessed on 10 Feb ua y 2021).

J. Risk Financial Manag. 2021,14, 484 16 o 17
Bau , Di k, and B ian Lucey. 2010. Is Gold a Hedge o a Sa e Ha en? An Analysis o S ocks, Bonds and Gold. The Financial Re iew 45:
217–29. [C ossRe ]
Bau , Di k, and Thomas McDe mo . 2010. Is gold a sa e ha en? In e na ional e idence. Jou nal o Banking & Finance 34: 1886–98.
Beloua i, Ahmed, Chaouki Achou Bou akba, and Ka ima Saci. 2015. Islamic Finance and Financial S abili y: A Re iew o he
Li e a u e. Jou nal o King Abdulaziz Uni e si y: Islamic Economics. 28. A ailable online: h ps://ss n.com/abs ac =3063842.
(accessed on 10 Feb ua y 2021).
Bo io, Claudio. 2020. The COVID-19 economic c isis: Dange ously unique. Business Economics 55: 181–90. [C ossRe ]
Bou i, Elie, Rangan Gup a, and Xuan Vinh Vo. 2020. Jumps in Geopoli ical Risk and he C yp ocu ency Ma ke : The Singula i y o
Bi coin. De ence and Peace Economics. [C ossRe ]
B idgewa e , L. P. 2011. The All Wea he S o y. A ailable online: h ps://www.b idgewa e .com/ esea ch-and-insigh s/ he-all-
wea he -s o y (accessed on 10 Feb ua y 2021).
B idgewa e , L. P. 2012. Risk Pa i y Is abou Balance. A ailable online: h ps://www.b idgewa e .com/ esea ch-and-insigh s/ isk-
pa i y-is-abou -balance (accessed on 10 Feb ua y 2021).
B ude , Benjamin, and Thie y Roncalli. 2012. Managing Risk Exposu es Using he Risk Budge ing App oach. A ailable online:
h ps://ss n.com/abs ac =2009778 (accessed on 10 Feb ua y 2021).
By d, Richa d H., Jean Cha les Gilbe , and Jo ge Nocedal. 2000. A us egion me hod based on in e io poin echniques o nonlinea
p og amming. Ma hema ical P og amming 89: 149–85. [C ossRe ]
Chouei a y, Y es, and Y es Coigna d. 2008. Towa d Maximum Di e si ica ion. The Jou nal o Po olio Managemen 35: 40–51. [C ossRe ]
Delle Foglie, And ea, and Ida Claudia Pane a. 2020. Islamic s ock ma ke e sus con en ional: A e islamic in es ing a ‘Sa e Ha en’
o in es o s? A sys ema ic li e a u e e iew. Paci ic-Basin Finance Jou nal 64: 101435. [C ossRe ]
Fo es i, S e en J., and Michael E. Rush. 2010. Risk-Focused Di e si ica ion: U ilising Le e age wi hin Asse Alloca ion. Cali o nia: Wilshi e
Consul ing.
Giuzio, Ma ghe i a. 2017. Gene ic algo i hm e sus classical me hods in spa se index acking. Decisions in Economics and Finance 40:
243–56. [C ossRe ]
Ha oon, Omai , Mohsin Ali, Abdullah Khan, Mudee A. Kha ak, and Syed Aun R. Riz i. 2021. Financial Ma ke Risks du ing he
COVID-19 Pandemic. Eme ging Ma ke s Finance and T ade 57: 2407–14. [C ossRe ]
Hasan, Md Bokh ia , Masnun Mahi, M. Kabi Hassan, and Abul Basha Bhuiyan. 2021. Impac o COVID-19 pandemic on s ock
ma ke s: Con en ional s. Islamic indices using wa ele -based mul i- imescales analysis. The No h Ame ican Jou nal o Economics
and Finance 58: 101504. [C ossRe ]
Hassan, M. Kabi , Si ajo Aliyu, And ea Pal inie i, and Ash a Khan. 2019. A Re iew o Islamic In es men Li e a u e. Economic
Pappe s 38: 345–80. [C ossRe ]
Jawadi, F edj, Nabila Jawadi, and Wael Louhichi. 2014. Con en ional and Islamic s ock p ice pe o mance: An empi ical in es iga ion.
In e na ional Economics 137: 73–87. [C ossRe ]
Ji, Qiang, Dayong Zhang, and Yuqian Zhao. 2020. Sea ching o sa e-ha en asse s du ing he COVID-19 pandemic. In e na ional Re iew
o Financial Analysis 71: 101526. [C ossRe ]
K is ou ek, Ladisla . 2020. G andpa, g andpa, ell me he one abou Bi coin being a sa e ha en: E idence om he COVID-19
pandemics. a Xi a Xi :2004.00047.
Lee, Wai. 2011. Risk-Based Asse Alloca ion: A New Answe o an Old Ques ion? Jou nal o Po olio Managemen 37: 11–28. [C ossRe ]
Le ell, Ch is ophe . 2010. Risk Pa i y: In he Spo ligh a e 50 Yea s. Zag eb: Gene al Resea ch, NEPC.
Loh e, Ha ald, Ul ich Neugebaue , and Ca s en Zimme . 2012. Di e si ied Risk Pa i y S a egies o Equi y Po olio Selec ion. Jou nal
o In es ing 21: 111–28. A ailable online: h ps://ss n.com/abs ac =2049280 (accessed on 10 Feb ua y 2021). [C ossRe ]
Mailla d, Sébas ien, Thie y Roncalli, and Jé ôme Teïle che. 2010. The p ope ies o equally weigh ed isk con ibu ion po olios. The
Jou nal o Po olio Managemen 36: 60–70. [C ossRe ]
Manso , Fadillah, M. Ishaq Bha i, Sha iqu Rahman, and Hung Quang Do. 2020. The In es men Pe o mance o E hical Equi y Funds
in Malaysia. Jou nal o Risk and Financial Managemen 13: 219. [C ossRe ]
Masih, Mansu , Naz ol K. M. Kamil, and Obiya hulla I. Bacha. 2018. Issues in Islamic Equi ies: A Li e a u e Su ey. Eme ging Ma ke s
Finance and T ade 54: 1–26. [C ossRe ]
Ma hWo ks Inc. 2021. Linea P og amming Sol e . A ailable online: h ps://i .ma hwo ks.com/help/op im/ug/linp og.h ml
(accessed on 25 May 2021).
Meucci, A ilio. 2007. Risk Con ibu ions om Gene ic Use -de ined Fac o s. The Risk Magazine, 84–88. A ailable online: h ps:
//ss n.com/abs ac =930034 (accessed on 25 Janua y 2021).
Meucci, A ilio. 2009. Managing Di e si ica ion. Risk 22: 74–79.
Mussa i, Noo Sai Muhammad, and Zuhaimy Ismail. 2021. Op imum Risk-Adjus ed Islamic S ock Po olio Using he Quad a ic
P og amming Model: An Empi ical S udy in Indonesia. Jou nal o Asian Finance, Economics and Business 8: 839–50. [C ossRe ]
Pal inie i, And ea, Mohammad Kabi Hassan, Salman Bahoo, and Ash a Khan. 2019. A bibliome ic e iew o sukuk li e a u e.
In e na ional Re iew o Economics & Finance. [C ossRe ]
Pola, Gianni. 2013. Managing Unce ain y wi h Dams. Asse s’ Segmen a ion in Response o Mac oeconomic Changes. AMUNDI Wo king-
Pape 2013. AMUNDI In es men S a egy Collec ed Resea ch Pape s 2014. Pa is: AMUNDI.
J. Risk Financial Manag. 2021,14, 484 17 o 17
Pola, Gianni. 2021. Un angling Complexi y in Financial Ma ke s. A New Pa adigm o Asse Alloca ion. ANIMA Wo king Pape Se ies,
WPS-2021-04. Milan: ANIMA.
Qian, Edwa d. 2005. Risk Pa i y Po olios: E icien Po olios Th ough T ue Di e si ica ion. Bos on: PanAgo a Resea ch Pape .
Rebo edo, Juan. 2013. Is gold a sa e ha en o a hedge o he US dolla ? Implica ions o isk managemen . Jou nal o Banking and
Finance 37: 2665–76. [C ossRe ]
Shen, Junying, and Noah Weisbe ge . 2021. US S ock-Bond Co ela ion: Wha A e he Mac oeconomic D i e s? PGIM IAS. May.
A ailable online: h ps://ss n.com/abs ac =3855610 (accessed on 25 Janua y 2021).
She i , Mohamed. 2020. The impac o Co ona i us (COVID-19) ou b eak on ai h-based in es men s: An o iginal analysis. Jou nal o
Beha io al and Expe imen al Finance 28: 100403. [C ossRe ]
Wal z, Richa d A., Jose Luis Mo ales, Jo ge Nocedal, and Dominique O ban. 2006. An in e io algo i hm o nonlinea op imisa ion
ha combines line sea ch and us egion s eps. Ma hema ical P og amming 107: 391–408. [C ossRe ]
Yang, Jian, Yinggang Zhou, and Zijun Wang. 2009. The s ock-bond co ela ion and mac oeconomic condi ions: One and a hal cen u ies
o e idence. Jou nal o Banking & Finance 33: 670–80. [C ossRe ]
Hengchao, Zhang, and Za inah Hamid. 2015. The Impac o Subp ime C isis on Asia-Paci ic Islamic S ock Ma ke s. Jou nal o
Asia-Paci ic Business 16: 105–27. [C ossRe ]
Zi o , E ic, and Jiahui Wang. 2006. Rolling Analysis o Time Se ies. In Modeling Financial Time Se ies wi h S-PLUS
®
. New Yo k: Sp inge .
[C ossRe ]