Make the best from comparing conventional and Islamic asset classes: A design of an all-seasons combined portfolio

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. 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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. 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