Hedge and safe haven properties during COVID-19

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Hedge and safe haven properties during COVID-19: Evidence from Bitcoin and gold

Item Type Article

Authors BenSaïda, Ahmed; Tayachi, Tahar; Chemkha, Rahma; ghorbel, ahmed

Citation Rahma Chemkha, Ahmed BenSaïda, Ahmed Ghorbel, Tahar Tayachi, Hedge and safe haven properties during COVID-19:

Evidence from Bitcoin and gold, The Quarterly Review of Economics and Finance, Volume 82, 2021, Pages 71-85, ISSN 1062-9769, https://doi.org/10.1016/j.qref.2021.07.006.

DOI 10.1016/j.qref.2021.07.006

Publisher Elsevier B.V.

Download date 21/06/2023 04:51:21

Item License https://creativecommons.org/licenses/by/4.0/

Link to Item http://hdl.handle.net/20.500.14131/248

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TheQuarterlyReviewofEconomicsandFinance82(2021)71–85

ContentslistsavailableatScienceDirect

The Quarterly Review of Economics and Finance

j o ur na l h o me pa g e :w w w . e l s e v i e r . c o m / l o c a t e / q r e f

Hedge and safe haven properties during COVID-19: Evidence from Bitcoin and gold

Rahma Chemkha

^a

, Ahmed BenSaïda

^b,∗

, Ahmed Ghorbel

^c

, Tahar Tayachi

^b,d

aGFCLaboratory,FSEGSfax,UniversityofSfax,Sfax,Tunisia

bCollegeofBusiness,EffatUniversity,Jeddah,SaudiArabia

cCODECIlaboratory,FSEGSfax,UniversityofSfax,Sfax,Tunisia

dFSEGMahdia,UniversityofMonastir,Monastir,Tunisia

a r t i c l e i n f o

Articlehistory:

Received18March2021

Receivedinrevisedform12July2021 Accepted14July2021

Availableonline5August2021

JELclassication:

G11 G15 G23 Keywords:

Bitcoin DCC Hedge Gold Safehaven

a b s t ra c t

TheCOVID-19pandemichascausedanunprecedentedhumanandhealthcrisis.Themeasurestaken tocontainthedamagecausedaglobaleconomicslowdown.Investorsfaceliquiditypressuresresulting fromthegeneraldownturninthenancialmarkets,andmightchangetheirriskappetite.Thispaper reassessesthesafehavenpropertyofgoldasatraditionalasset,andBitcoinwhichisgraduallyimposing itselfasanewclassofassetwithuniquecharacteristics.Theempiricalresults,appliedonmajorworld stockmarketindicesandcurrencies,andbasedonthemultivariateasymmetricdynamicconditional correlationmodel,showtheeffectivenessofBitcoinandgoldashedgingassetsinreducingtheriskof international portfolios. Moreover, the analysis provides evidence that during the COVID-19 pandemic, goldisaweaksafehavenfortheconsideredassets,whileBitcoincannotprovideshelterduetoits increasedvariability.

1. Introduction

TheCOVID-19pandemichassucceededinlootinganddesta- bilizingtheentireworld.In justfewmonths,itendangerednot onlylives,butalsoeconomicborders,globalbusinessesandthe veryessenceoftheworld’snancialequilibrium.Thiscrisiscaused theclosureofborders,asharpslowdownintheglobaleconomy andastockmarketcrashonMarch11,2020.Consequently,many investorsturnedtouncorrelatedassetstohedgetheirportfoliosin ordertocopewiththecrisisandsupporttherecovery.

Throughoutitshistory,goldhasbeenviewedasavaluestore, portfoliostabilizerandsourceofliquidityintimesofnancialtur- bulence.Goldisknownasahedgeagainstinationarypressuresin theU.S.andU.K.(Hoang,Lahiani,&Heller,2016).Itreactscounter- cyclicallytomacroeconomic news (Elder,Miao, &Ramchander, 2012)andworksdifferentlyfromotherassets,especiallystocks.

∗Correspondingauthor.

E-mailaddresses:[email protected](R.Chemkha), [email protected](A.BenSaïda),[email protected] (A.Ghorbel),[email protected](T.Tayachi).

When prices of stockmarket indices plummet,gold retains its value.Itseffectivenessasahedgeandsafehavenagainstequitiesin differentmarketsiswellconrmedbynumerousstudies(e.g.,Baur

&Lucey,2010;Beckmann,Berger,&Czudaj,2015,amongothers).

However,BaurandGlover(2012)showedthatinvestorbehavior candestroythehedgingpotentialofgoldduetoincreasedinvest- mentinthepreciousmetalforhedgingandspeculationpurposes.

For example,Klein(2017)usedadynamiccorrelationmodelto showthat preciousmetalshad ahedging rolefor stockmarket indicesinAmericanandEuropeancountries;nevertheless,thisrole havedissipatedafter2013.

Overthepastdecade,attentionhasshiftedfromgoldtoanew asset,Bitcoin. Bitcoinwasrstintroducedin 2009in thewake of thebankruptcy of LehmanBrothersinvestment bank,asthe condenceinnancialinstitutionsdeteriorated.Soonafter,this cryptocurrency succeededin catchingtheattentionofinvestors andinstitutionalbodies.Ithasestablisheditselfthroughitsinno- vativecharacter.TheBitcoinprotocolisbasedonthevoluntary participation oftheparties,it is notsubject toany controland allowseveryonetoaccumulateandtransfervalueina currency thatresistspricemanipulationbycentralbanksandglobalnancial institutions(Chemkha,BenSaïda,&Ghorbel,2021).

https://doi.org/10.1016/j.qref.2021.07.006

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Obviously,thishasledBitcoinandgoldtocompeteforthesta- tusofthemosteffectiveprotectiveasset.Comparedtogold,Bitcoin mustovercomeseveralchallenges,particularlyintermsofhistory, acceptance,consumption,intrinsicvalueandlowvolatility.How- ever,goldandBitcoinshareseveralcharacteristics.(i)First,they bothescapepoliticalinuenceandareregulatedlikecommodities, especiallyintheU.S.whereBitcoinisconsideredacommodityby theCFTCinstitution.(ii)Second,nocentralauthorityintervenes toadjustorcontroltheirminingtransactionsandactivities(Baur, Hong,&Lee,2018);therefore,theyarebothindependentfromina- tion.(iii)Third,bothassetshaveanitesupply,anditisthisscarcity thatmakesthemvaluable.SinceBitcoin’snumberofunitsincircu- lationislimitedto21million,makingitananti-inationaryasset, whichbringsitclosertogold.(iv)Fourth,theasymmetricreaction topositiveandnegativenewscharacterizesbothBitcoin(Bouri, Jalkh,Moln’ar,&Roubaud,2017)andgold(Baur&Glover,2012).

Researchesonthelinkbetweenthepricesofgoldandthoseof otherassets,andthepotentialforhedgeandsafehavencontinue togrowremarkably.Indeed,Bitcoin’slowcorrelationwithtradi- tionalassetsmakesitdesirablefordiversication(Corbet,Meegan, Larkin,Lucey,&Yarovaya,2018;Guesmi,Saadi,Abid,&Ftiti,2019;

Ji,Bouri,Gupta,&Roubaud,2018),andavaluablehedgeagainst stockmarkets(Bauretal.,2018;Dyhrberg,2016)andcommodities (Bourietal.,2017).Furthermore,severalstudiesclaimthatBitcoin wasnotaffectedbytheEuropeandebtcrisisnortheCypriotbanking crisis,yetitprospered(Luther&Salter,2017).

TheeconomicimpactoftheCOVID-19crisisonnancialmarkets hasrecentlybecomeatopicofinterest.Thishasreignitedthedebate overwhetherthecryptocurrencyhastheabilitytomimicorbeat goldhedgingandsafehavenpropertyagainststocks.Moreover, tothebestofourknowledge,alimited numberofstudieshave analyzedBitcoinandgold(Bourietal.,2017;Corbetetal.,2018;

Dyhrberg,2016)withoutacomparablestudybetweentheirroles againstnancialassets.Tollthegapintheliterature,ourstudy drawsnewinsightsintothepotentialofBitcoinandgoldtohedge againstuctuationsinnancialmarketsandtheirroleassafehaven duringtheCOVID-19virusoutbreak.

Theremainderofthispaperisorganizedasfollows.Section2isa briefreviewoftheliterature.Section3explainsthemethodological approach.Section4describesthedatasample.Section5analyzes theempiricalresults.Section6concludes.

2. Literaturereview

Sincetheglobalnancialcrisisof2007-2009,therehasbeena resurgenceofinterestamongacademicsandpractitionerstostudy thehedgeandsafehavenpotentialofvariousassetclassesagainst marketdownturns.Indeed,duringtimesofcrisis,investorsseek todisposeoftheirriskyassetsinfavorofothermoresecureassets (ight-to-safety).Thisispreciselythecaseofgold,whosevaluedoes notdependona company’sperformancenora state’sabilityto repayitsdebt.Thus,whenotherassetscollapse,onecanalways relyongold,sinceitcanbeeasilyresoldwhenneeded.

Lawrence(2003)providesanoverviewoftheevolutionofthe relationshipbetweengoldandstockmarkets.Theauthorconcludes thatgoldappearstobeisolatedfromthebusinesscycle–unlike othercommodities–whichmaymakeitmoreattractiveasadiver- sierandevenasasafehaven.Previousstudies(Fleming,Kirby,&

Ostdiek,1998,amongothers)showevidenceinfavorofgovern- mentbondsprovidingequityinvestorswithasafehavenintimes ofcrisis.Chan,Treepongkaruna,Brooks,andGray(2011)showthat U.S.Treasuriesareasafehavenforequityinvestorsduringtimes of market stress.Flavin, Morley, and Panopoulou (2014)show that 1-yearand10-yearU.S.treasurybonds,inadditiontogold,aresafe havenstoprotectportfolios intheeventofa marketdownturn.

However,evenwithsomeoftheadvantages associatedwithinvest- inginTreasuries,therearestilldrawbackstobeawareof,suchas thelowyield.

Other investors, looking for an alternative togloomy nancialmarkets,prefertheassetbacked securities(ABS), atype of

xed income assets that offer safereturns against market tur- moil.Bernanke,Bertaut,DeMarco,and Kamin(2011)showthat ABShavebecomeanattractivealternativeforinvestors,offering returnsslightly higherthanthoseofTreasuries.However,these typesofABScollateralhaveproventoberisky(Gennaioli,Martin,

&Rossi,2018),involvingriskofassetdevaluation,andcanhaveseri- ousconsequences,suchasanincreaseindefaultsduringtheU.S.

bornsubprimemortgagecrisisin2007(Bertaut,DeMarco,Kamin,

&Tryon,2012).

Unlikerisk-freeassets,goldhasnaturallybeenconsideredasa safehavengivenitshistoricalroleasanaturalcurrencyandavalue store.Itisnegativelycorrelatedwiththenancialcycle,soittends toprovidepositivereturnsduringcrises(Bouri,Lucey,&Roubaud, 2020).Thereisavastliteraturestudyingthepotentialofgoldas ahedgeandasafehaven,buttheresultsaremixed.Forinstance, basedondatafrommajoremerginganddevelopingcountries,Baur andMcDermott(2010)discusswhethergoldistrulysafe.Their resultsconrmthispropertyofgoldforAmericanandEuropean stockindicesbutnotforothermarkets.HoodandMalik(2013)nd thatgoldisahedgefortheU.S.stockmarket,butitssafehavenprop- ertyislowrelativetothevolatilityindex.Beckmannetal.(2015) showthatgoldservesbothasahedgeandasaneffectivesafehaven.

LuceyandLi(2015)studytheroleofpreciousmetalsassafehavens inatimevaryingframeworkandndthatthestrengthofgoldasa safehavenchangesovertime.Foremergingcountries,Chkili(2016) examinestherelationshipbetweengoldandthestockmarketsof BRICScountriesandsuggeststhatgoldcanserveasasafehaven againstextrememovements.Likewise,Akhtaruzzaman,Boubaker, Lucey,andSensoy(2021)examinetheroleofgoldasasafehaven duringtheCOVID-19crisis.TheyfoundthatduringphaseI(before March16,2020)goldwasastrongsafehaven;however,itsproperty hasweakenedduringphaseIIstartingfromMarch17,2020.

Bitcoinisanotherpopularassetthathassucceededingaining theattentionofinvestors asasafehaven.BouoiyourandSelmi (2017) admit that Bitcoin has both the hedge and safe haven propertiesfor theU.S. stockindex. Theyalso demonstratethat preciousmetals have lost their safehaven property over time.

Bouri,Lucey,etal.(2020)ndthatcryptocurrenciescanbeused asahedgeagainstthedownsideriskofequityinvestments.This propertyappliesduringnormalperiodsandtimesofcrisis.Several researcherssuggestthattheroleofBitcoinasasafehavenandhedg- inginstrumentisquitelimited(Eisl,Gasser,&Weinmayer,2015, amongothers).Forexample,basedonanasymmetricdynamiccon- ditionalcorrelation(A-DCC)model,Bourietal.(2017)statethat Bitcoinoffersdiversicationadvantagesovervariousotherassets, suchasstockindices,bonds,oil,gold,commodityindices,andthe U.S.dollar,butitsuseashedgeorsafehavenonlyappearsincertain caseswhichdifferfromonehorizontoanother.Klein,PhamThu, andWalther(2018)suggestthatBitcoinisnotasafehavenand cannothedgeagainstrisk,evenfordevelopedmarkets.Shahzad, Bouri,Roubaud,Kristoufek,andLucey(2019)ndthatgoldhas an“indisputable”safehavenpropertycomparedtothatofBitcoin.

WhilegoldisaneffectivesafehavenforallG7stockindices,Bitcoin onlyoffersasafehavenrolefortheCanadianstockindex.Likewise, ConlonandMcGee(2020)showthatBitcoindoesnotactasasafe haven.ItisevolvingatthesamerateastheS&P500asthehealth crisisdevelops.

Theaforementionedstudiesonthecharacteristicsofgoldand Bitcoin offer mixed results. Therefore, it becomes essential to reassessthehedgeandsafehavenpropertiesofBitcoinintherecent contextofCOVID-19;andtotestwhethergoldretainsitscharacter-

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isticasastablevaluestorethatcanprotectinvestors,policymakers andregulatorsfromtheadverseeffectsofthepandemic.

3. Methodology

ToanalyzewhetherBitcoinandgoldcanserve asahedgeor safehavenforthemaintraditionalassetsofdevelopedmarkets, our methodology is based on the principles presented byBaur and Lucey(2010). Themethods quicklygainedpopularity andhave been appliedin various studies, for example, oncredit default swaps,bonds,metalsand gold(Ciner,Gurdgiev,&Lucey,2013;Li

&Lucey,2017;Lucey&Li,2015).

BaurandLucey(2010)providecleardenitionsanddistinction ofdiversier,hedge,andsafehaven.Anassetisclassiedasdiver- sierifitispositively,butweaklycorrelated,withanotherasset orportfolioonaverage. Ahedgeisanassetthatisuncorrelated (weakhedge)ornegativelycorrelated(stronghedge)withanother assetorportfolioonaverage.Ahedgemaynotreducelossesdur- ingtimesofmarketstressorturbulence,astheassetmayshow positivecorrelationswithotherassetsduringsomeperiods,and negativecorrelationsduringotherperiods,resultinginanegative correlationonaverage.Asafehavenisanassetthatisuncorrelated (weaksafehaven)ornegativelycorrelated(strongsafehaven)with anotherassetorportfoliointimesofturmoil.Thus,asafehaven investmenthasthepotentialtoprotectinvestorsandoffsetlosses intheeventofmarketcrises,suchastheCOVID-19pandemic.

3.1. TheasymmetricDCCmodel

Severalstudiesinnanceusethemultivariatedynamiccon- ditional correlation(DCC) model of Engle (2002) toinvestigate thecorrelationsbetweenassets,andtoconstructreliablehedging strategies(Bourietal.,2017;Chang,McAleer,&Tansuchat,2011;

Cineretal.,2013,amongothers).TheDCCmodelprovidesasimple frameworkforextractingdynamiccorrelationsformultipleassets inasparseparameterconguration.Inthispaper,weextendthe analysisbyemployingtheasymmetricdynamicconditionalcor- relation(A-DCC)modelofCappiello,Engle,andSheppard(2006), whichidentiestheasymmetricresponsesintheconditionalvari- ancesandcorrelationsduringstressperiods.Theasymmetry,also called“leverageeffect”,detectsanoftenobservedstylizedfactof

nancialassets where anunexpected drop(badnews)in asset pricestendstoincreasethevolatilitymorethananunexpectedrise (goodnews)withthesamemagnitude(BenSaïda,2019,2021).

3.1.1. Themodel

Letrt bea(n×1)vectorofreturnsofnassets,suchthatrt=



r_1,t,...,rn,t



_

.Forthebivariatecase,n=2,thevectorrtcontains thereturnsr1,toftheBitcoin(orgold),andr2,tofastockmarket index(orexchangerate).Thereturnsfollowan-variateStudent’s tdistributionwithdegrees-of-freedomasinEq.(1).

r_t= _t+ε_t

rt|I_t−1∼ tn,(t,Ht) (1) where_tstandsfortheconditionalmeanvectorthatmayincludea constantand/orpastobservations.ThemeanequationoftheA-DCC modelisspeciedasanautoregressivemovingaverage(ARMA) process,asdebatedbyKyrtsouandLabys(2007),sinceoverlook- ingthis characteristicmayunderminesomeofthedynamicsof the relationships between the studied variables. The termHtis the conditionalcovariancematrixofrt;andεtdenotesa(n×1)vec- torofresidualsconditionalontheinformationsetI_t₋₁ denedat

timet−1.ThegeneraldynamiccorrelationmodeloftheA-DCCis denedinEq.(2).

H_t=D_tR_tD_t (2)

where Rt represents the time varying conditional correlation matrix,andDtstandsforthe(n×n)diagonalmatrixcontainingthe conditionalstandarddeviationsofunivariateGARCH-typemodels, suchthat:

Dt= diag



h^1/2_1,t,...,h^1/2_n,t



R_t= diag(Q_t)⁻^1/2Q_tdiag(Q_t)⁻^1/2

(3)

ThetermQ_tisapositivedenitematrixthatrepresentstheevo- lutionoftheconditionalcorrelationofthestandardizedresiduals zt.Where,ztdenotesa(n×1)randomvectorofindependentand identicallydistributed(i.i.d.)errors(Engle,2002),asinEq.(4).Note thatz_taretheresidualsstandardizedbytheirconditionalstandard deviation(Engle&Sheppard,2001).

zt =D⁻_t¹(rt−_t) z_t ^i.i.d.∼t_n,(0,R_t)

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It follows from this specication that Et−1



z_tz^_t



= D⁻¹_t Et−1



εtε^_t



D⁻¹_t =D⁻¹_t HtD⁻¹_t =Rt. A typical element of Rt

willhavetheform_ij,t=√_q^q^ij,t

ii,t qjj,t,suchthatQ_t=



q_ij,t



n i,j=1. Eachconditionalvarianceh_i,tofanasseti,fori=



1,...,n



, is estimatedusingtheunivariateasymmetricThreshold GARCH (TARCH)modelofZakoian(1994).TheTARCHmodeloforders(1,1) isrepresentedinEq.(5).

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

r_i,t= c_i+_i,1r_i,t₋₁+ε_i,t ε_i,t= z_i,t



h_i,t,withz_i,t^i.i.d∼t(_i)



h_i,t= ω_i+˛_i,1

 

ε_i,t₋₁

 

₊ˇ_i,1



h_i,t₋₁+_i,11[^εi,t−1<0]|ε_i,t₋₁| (5) where c_i indicatesadriftterm,_i,1 is theautoregressivecoef- cientoforder1inthemeanequation.Alternatively,r_i,t=_i,t+ε_i,t. In thevarianceequation, ω_i isa constantterm,˛_i,1 detectsthe ARCHeffect,ˇ_i,1capturesthepersistenceofthevolatilityprocess, and_i,1representstheasymmetriccoefcient.Theindicatorfunc- tion1[^εi,t<0] equals1ifε_i,t<0and0otherwise.Thecoefcients must satisfy the positivity conditions ω_i>0, ˛_i,1,ˇ_i,1>0 and

˛_i,1+_i,1>0,andthestationaritycondition˛_i,1+ˇ_i,1+¹₂_i,1<

1.Theerrortermsz_i,tarei.i.d.withzeromeanandunitvariance andassumedtofollowastandardStudent’stdistributionwith_i degrees-of-freedom.Forthisspecication,apositivevalueof_i,1 indicatesthatnegativeconditionalresidualstendtoincreasethe volatilitymorethanpositiveshocksofthesamemagnitude.

ThedynamicsofQ_tfortheA-DCC(p,q)modelisillustratedin Eq.(6).

Q_t=Q+



p i=1

a_i



z_t−iz^_t−i−Q



+



q j=1

b_j



Q_t₋_j−Q



+



p i=1

g_i



z⁻_t₋_iz⁻_t₋^_i−N



(6)

Whenthelagordersp=1andq=1,themodelcanbereduced toA-DCC(1,1)ofCappielloetal.(2006)inEq.(7),

Q_t=(1−a−b)Q−gN+az_t−1z^_t−1+bQ_t−₁+gz⁻_t₋₁z⁻_t₋^₁ (7) where a and b are non-negative scalars that capture the effects ofpreviousshocksandpreviousconditionalcorrelations,

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respectively,onthecurrentconditionalcorrelation.Anecessary and sufcient condition for Q_t to be positive denite is that a+b+_maxg<1, where _max is the maximum eigenvalue of



Q⁻^1/2NQ⁻^1/2



.ThetermQistheunconditionalcovariancematrix ofthestandardizedresidualsz_i,t=



^ε^i,t

h_i,t suchthatQ=E



ztz^_t



. Theparametergdenotestheasymmetriceffectinthemodel.The quantityz⁻_t =1_[z_t_<0]zt,whereistheHadamardproduct,and 1_[z_t_<0]isa(n×1)indicatorfunctionthattakesonvalue1when theargumentz_i,t<0,i=



1,...,n



and0otherwise. Theterm N=E



z⁻_tz⁻_t^



representstheunconditionalcovariancematrixof z⁻_t.

In practice, the expectations of Q and N are infeasible;

hence,theyarereplacedwiththesampleanalogs ¹_T



T

t=1ztz^_tand

1 T



T

t=1z⁻_tz⁻_t^,respectively.

3.1.2. Estimation

FollowingEngle(2002),theestimationoftheA-DCCmodelis conductedusingatwo-stepmaximumlikelihoodmethod.(1)Ina

rststep,weseparatelyestimatetheconditionalvariancesusinga univariateGARCH-typemodelforeachtimeseriesinEq.(5).(2)In asecondstep,weestimatetheconditionalcorrelationinEq.(7).

Theconditionaljointdistributionofthereturnshasthefollow- ingform,where(·) isthegammafunction:

fn,(rt|It−1)=



_H_t



−¹₂ 



_₊_n

2







_

2



((−2))ⁿ²



1+(rt−t)^H⁻¹_t (rt−t)

−2



−^+n₂

(8)

DenotethevectorofparametersinDt,i.e.,theparametersof allunivariateconditionalvolatilitymodelsinEq.(5);and the vectorofadditionalparametersinRt,i.e.,theparametersofthe conditionalcorrelationmodelinEq.(7).Thelog-likelihoodfunction tomaximizeiswrittenas:

L



,



=



T t=1

lnfn,



rt|,,I_t−1



(9)

=



T t=1



ln



_₊_n

2



−ln



_

2



−n

2((−2))

−1

2ln

 

Ht

 

₋



_₊_n

2



ln



1+(rt−_t)^H⁻_t¹(rt−_t)

−2



=



T t=1



ln



_₊_n

2



−ln



_

2



−n

2((−2))

−ln

 

Dt

 

₋¹

2ln

 

Rt

 

₋



_₊_n

2



ln



1+z^_tR⁻_t¹zt

−2



(9)

ThelikelihoodfunctionisdividedintovolatilitytermL_v







and correlationtermLc



,



inEq.(10).

L



,



=



T t=1



nln



_

+1 2



−nln



 2



−n 2((−2))

−ln



_D_t



₋ ^⁺¹

2 (−2)(rt−t)^D⁻_t²(rt−t)



+



T t=1



ln



+n 2



−nln



_

+1 2



+(n−1) ln



 2



−1 2ln



_R

t



₋



+n

2



ln



1+z^_tR⁻¹t zt

−2



+ +1 2 (−2)z^tzt



= Lv







+Lc



,



(10)

The likelihood function L



,



is maximized by separately maximizingLv







intherststep,then,taketheestimatedcoef-

cients ˆ as input for the second step to maximize Lc



,ˆ 



. FollowingthedevelopmentofXu,Chen,Jiang,andYuan(2018), underthemultivariatet distribution,thevolatilitypartcanbe writtenasthesumofindividuallog-likelihoodsinEq.(11).

L_v







=



T t=1



nln



_₊₁

2



−nln



_

2



−n

2((−2))

−ln

 

D_t

 

₋ ^⁺¹

2 (−2)(rt−_t)^D⁻_t²(rt−_t)



=



T t=1



nln



_₊₁

2



−nln



_

2



−n

2((−2))

−1 2



n i=1

ln



h_i,t



−+1 2



n i=1



r_i,t−_i,t



2

(−2)h_i,t



≈



T t=1



n i=1



ln



_

i+1 2



−ln



_

i

2



−1

2ln ((i−2))

−



_

i+1 2



ln



1+



r_i,t−_i,t



2

(_i−2)h_i,t



−1 2ln



h_i,t

 

≈



n i=1

L_v,i



_i



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ThemaximizationofLv







isperformedbyseparatelymaximiz- ingthelog-likelihoodofeachassetL_v,i



_i



undertheunivariate Student’s t distribution with _i degrees-of-freedom. The com- mondegrees-of-freedomof themultivariatet_ distributionis anapproximationofeachasset’sdegrees-of-freedom_i.Next,the correlationpartiswrittenas:

Lc



_ˆ

,



=



T t=1



ln



_₊_n

2



−nln



_₊₁

2



+(n−1) ln



_

2



−1

2ln



_R_t



₋



+n 2



ln



1+ˆz^tR⁻¹t zˆt

−2



+ +1 2 (−2)zˆ^tzˆt



⁽¹²⁾

where ˆzt=Dˆ⁻_t¹



rt−ˆ_t



areobtainedfromtheestimationinthe

rststep.¹

The estimated coefcients from thetwo-step procedure are inefcientbutconsistent,sincetheyarelimitedinformationesti- mators,asarguedbyEngle(2002).

3.2. Optimalportfolioweights

Portfoliooptimizationseemstobeanimportantpartofmod- ernquantitativenance thatsolvesmostoftheproblemsposedby investorsthroughseveralalternatives.Inthissection,wecompare therole ofBitcoinandgoldashedgingtoolsfor majorinterna- tionalassets.Inthiscontext,investorsseektominimizetherisk oftheirportfolioswithoutreducingtheexpectedreturns.Forthis purpose,KronerandSultan(1993)introducedamethodbasedon hedgeratios,whichbecamewidelyappliedinnumerousempirical works(Akhtaruzzaman,Boubaker,Lucey,&Sensoy,2021;Chang, McAleer,&Tansuchat,2011;Chkili,2016).Thismethoddetermines

1NumericalestimationsoftheA-DCCmodelareconductedunderRprogramming languageusingtherugarchpackageforunivariatemodelsintherststep,and rmgarchpackageforthecorrelationmodelinthesecondstep.Bothpackagesare availablefromhttps://cran.r-project.org/.

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theoptimalweightofBitcoin(orgold)inaone-dollarwalletofasset attimet.TheoptimalweightisexpressedinEq.(13).

wi/(BTCorGLD)

t = hⁱ_t−hi/(BTCorGLD)

t

hⁱ_t−2hi/(BTCorGLD)

t +h^(BTCorGLD)_t (13)

providedthat,

wi/(BTCorGLD)

t =

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

0 ifwi/(BTCorGLD)

t <0

wi/(BTCorGLD)

t if06wi/(BTCorGLD)

t 61

1 ifwi/(BTCorGLD)

t >1

where hⁱ_t and h^(BTCorGLD)_t are the conditional volatilities of the selected market i and Bitcoin (or gold), respectively; and hi/(BTCorGLD)

t istheconditionalcovariancebetweenBitcoin(orgold) andthereturnoftheassetiattimet.Allthevariancesandcovari- ancesareextractedfromtheA-DCCmodelestimates.Theweight ofassetiintheone-dollarportfolioofBitcoin(orgold)attimet equals



1−wi/(BTCorGLD) t



.

3.3. Hedgeratio

Inadditiontotheoptimalportfolioallocationintheprevious section,investorsandmarketparticipantsseektominimizetherisk ofthehedgedportfolio.Oneofthemostusedstrategy istocompute theoptimalhedgeratio(HR)basedonmultivariateGARCH-class modelsattimetinEq.(14),asdenedbyKronerandSultan(1993).

ˇi/(BTCorGLD)

t = hi/(BTCorGLD)

t

h^(BTCorGLD)_t (14)

Tominimizetheriskofaportfoliooftwoassets,alongpositionof$1takeninanygivenassetshouldbehedgedbyshorting ˇi/(BTCorGLD)

t dollarsintheBitcoin(orgold)market.

Forabetteranalysisoftheperformanceoftheoptimalportfo- lio,wecalculatetheHedgingEffectivenessindex(HE)inEq.(15), inalignmentwithChangetal.(2011).Thisindexevaluatestheper- formanceoftheportfoliobycomparingthevarianceofthehedged portfoliowiththatoftheunhedgedportfolio.

HE=var_unhedged−var_hedged

var_unhedged (15)

where



var_unhedged



representsthevarianceoftheunhedgedport- folioreturns(i.e.,varianceofthestockindexorcurrencyreturns), and



varhedged



denotesthevarianceofthehedgedportfolioreturns giveninEq.(16).AhigherHEindicatesabetterriskreductionand ahigherhedgingeffectiveness,whichimpliesthattheinvestment methodcanbeconsideredasasuperiorhedgingstrategy.

var



r_h,t|I_t−1



=var



r_j,t|I_t−1



−2ˇi,tcov



r_j,t,r_i,t|I_t−1



+ˇ²_i,tvar



r_i,t|I_t₋₁



(16) wherer_h,tisthehedgedportfolioreturn;r_j,tistheBitcoin(orgold) return;r_i,tisthestockindex(orcurrency)return;andˇ_i,tisthe optimalhedgeratiocalculatedfromEq.(14).

3.4. Safehaven

TheappliedA-DCCmodelestimatesthedynamicrelationship betweenthedifferentvariablesandthepotentialofBitcoinandgold asahedgingassetoverthetotalperiod.However,itdoesnotspec- ifytheusefulnessofBitcoinandgoldduringcrisisperiods.Thus, this section assesses whether the Bitcoin and gold can serve as safe havensduringtheCOVID-19pandemic.FollowingRatnerandChiu (2013)andAkhtaruzzaman,Boubaker,LuceyandSensoy(2021),

theeconometricmodeltotestthesafehavenpropertyispresented asfollows:

ADCC_ij,t=ı₀+ı₁ADCC_ij,t−₁+ı₂D_covid+ϑ_ij,t (17)

where ADCC_ij,t is thepairwise dynamic conditionalcorrelation betweenBitcoin(orgold)jandeachreturnonachosenasseti, andextractedfromtheestimatedQ_t=



q_ij,t



n

i,j=1inEq.(7),such that,

ADCC_ij,t=_ij,t= q_ij,t



_q

ii,t q_jj,t

We furthercreateadummyvariableD_covid fortheCOVID-19 period,thatequalsoneifthereturnsareduringthecrisis,andzero otherwise.AccordingtoBaurandMcDermott(2010)andBourietal.

(2017),iftheconstantı0inEq.(17)isweaklypositive,thenBitcoin (orgold)isaweakdiversieragainstmovementsintheotherasset i.Bitcoin(orgold)isahedgeagainstmovementsintheotherasset ifı0isstatisticallynotdifferentfromzero(weakhedge)ornegative (strong hedge).Finally,Bitcoin(orgold)isaweaksafehavenforthe otherassetduringthecrisisperiodifı₂isnotsignicantlydifferent fromzero,orastrongsafehavenifı2issignicantlynegative.² 4. Dataanddescriptivestatistics

4.1. Data

Intheempiricalpart,wechosetwomajorclassesofnancial assets,namely,stockindicesandcurrencies.Thedataiscollected from Thomson Reuters Datastream database. The daily closing prices of the mainworld stock marketscorrespondto theU.S.

(Standard&Poor’s500,SP500),Eurozone(EuroSTOXX50,ES50), Japan (Nikkei 225, N225) and the U.K. (Financial Times Stock Exchange100,F100).Forcurrencies,weselecttheEuro(EUR),the Japaneseyen(JPY)andtheBritishpound(GBP).Bitcoin(BTC)prices arecollectedfromwww.coinmarketcap.com;andoneounceOR- LBMAofgoldprices(GLD)areobtainedfromwww.lbma.org.uk.All exchangeratesarequotedagainsttheU.S.dollar(USD).Thesample spanstheperiodfromApril29,2013toJanuary5,2021withatotal of2007observations.

ThesampleperiodencompassesastrongspreadoftheCOVID- 19 pandemic. Consequently, we divide the sample into two sub-periodstocomparethebehaviorofBitcoinandgoldinterms of overallperformance. Therstperiod, fromApril29,2013to March 10, 2020,before thecrisis, and thesecond period from March11,2020toJanuary5,2021duringthecrisis.³Theempirical investigationisconductedonthereturnscomputedastheloga- rithmic differencesbetweenprices p_i,t fortheasseti,such that r_i,t=lnp_i,t−lnp_i,t₋₁.

4.2. Preliminaryanalysis

Fig.1exhibitstheevolutionofdailypricesforallassetclasses.

A dramatic pricedropis observedfromthesecond sub-period, especially for stock indices, indicating an increased variability during thepandemic.Indeed, theCOVID-19has ledtoa global recession anda hugeliquidationinthenancialmarkets(Aloui, Goutte, Guesmi,&Hchaichi,2020).Moreover,Bitcoinlostmore

2NotethatRatnerandChiu(2013)improvedthemethodofBaurandMcDermott (2010)toanalyzethesafehavenpropertyofanassetbysubstitutingthetimevarying coefcientbtintheregressionusedbyBaurandMcDermott(2010)withthepairwise conditionalcorrelationADCCij,t.

3OnMarch11,2020,theWorldHealthOrganization(WHO)hasdeclaredthe COVID-19diseaseasaglobalpandemic.Similarstudieshaveusedthisdateas the onsetdateofthecrisis(Corbet,Larkin,&Lucey,2020).

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Fig.1. DailydataforBitcoin,gold,indices,andexchangerates.TheshadedareacorrespondstotheCOVID-19pandemicperiod.

thanhalfofitsvalue,unlikegoldwhichbenetedfromamuch more homogeneous and stabilizing distribution. These qualita- tiveresultsprovidearstindicationthatBitcoininvestmentscan increaseportfoliorisk ratherthanactingas asafehaven.How- ever,this dropwasfollowed bya remarkable sharprisefor all assets.Bitcoinisundoubtedlytheassetthathasgrownthemost, afterits rstall-time high inDecember 2017,the priceof Bit- coinexplodestonewrecordswithmorethan$32,000onJanuary 4,2021.

SummarystatisticsofthereturnsarepresentedinTable1.Going fromtheperiodprecedingthecrisis(PanelA)totheperiodfollow- ingtheannouncementoftheCOVID-19pandemic(PanelB),we noticethat,withtheexceptionofBitcoinandtwoatcurrencies Euroand yen,goldandallotherassetsexperiencedanincrease intheiraveragereturnsaccompaniedwithhighvolatility.More- over,Bitcoinexhibitsthehighestreturnandhighestriskduring bothsub-periods.AsindicatedbyBauretal.(2018),thisspectac- ulargrowthis explainedbyastrongdemandfrominstitutional investors.Forinstance,TeslaannouncedonFebruary8,2021that it has bought $1.5 billion worth of Bitcoin, and it started accepting Bitcoinasapaymentmethodforitsproducts.⁴Furthermore,the

4Nevertheless,onMay12,2021,TeslastoppedacceptingBitcoinasapayment methodforitsproducts.Asaresult,Bitcoinpriceplungedfromanearly$58,000to below$38,000injust10days.

Jarque–Beratestrejectsnormalityforalldatareturnsduringboth sub-periods.

The Ljung–Box testrejectsthenullhypothesis ofabsenceof autocorrelation in the squared returns. Additionally, the ARCH teststatisticsforconditionalheteroskedasticityaresignicantfor allreturn series, which suggests the presence of ARCH effects.

TheseresultsconrmourchoiceofGARCH-typemodelstoana- lyzethedynamicrelationshipbetweenBitcoin(andgold)onone hand,andstockandforeignexchange(forex)marketsontheother hand.PanelsBpresentstheresultsoftwounitrootandstationar- itytests,namely,theAugmentedDickeyandFuller(1979)(ADF) andKwiatkowski,Phillips,Schmidt,andShin(1992)(KPSS),which showthatallseriesarestationary.

ThePearsoncorrelationsduringbothsub-periodsarepresented inTable2.ThecorrelationsbetweenBitcoin(andgold)withthe returnsofothermarketsarenegativeandclosetozeroduringthe pre-COVID-19period,which suggeststhatBitcoinand goldcan beusedashedgingassetsduringthestableperiod(Baur&Lucey, 2010).Duringthesecondsub-period,withtheexceptionofBTC/JPY pair, the correlations between Bitcoin (and gold) and othernan- cialassetshavebecomepositive,signalingthattheglobalnancial marketsbecamemoreinterconnected.Thus,Bitcoinandgoldmight nothaveanysafehavenproperty,yettheymayoffersomehedging characteristic.Totestthisassertion,weconductmoreadvanced empiricalmodelsinthefollowingsections.

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Table1

Summarystatisticsandunitroottestsfordailyreturns.

BTC GLD SP500 ES50 N225 F100 EUR JPY GBP

PanelA:BeforetheCOVID-19pandemic Descriptivestatistics

Mean 0.0022 0.0001 0.0003 0.0000 0.0002 −0.0000 −0.0001 −0.0000 −0.0001

Std.dev. 0.0910 0.0083 0.0086 0.0109 0.0127 0.0086 0.0049 0.0056 0.0056

Skewness 0.1637 0.0693 −0.8002 −0.8587 −0.4860 −0.7607 0.0771 0.2674 −1.6691

Kurtosis 445.28 5.5613 11.196 9.6055 8.33187 9.3755 5.4598 7.4248 30.382

J-Btest 8900.0^† 498.02^† 5203.7^† 3476.2^† 2181.5^† 3206.1^† 322.47^† 935.09^† 58009^†

Q²(10) 445.60^c 196.58^c 1050.9*** 206.46*** 202.96*** 349.70*** 169.55*** 139.71*** 85.980***

ARCH(10) 799.19*** 130.92*** 594.24*** 140.97*** 105.92*** 255.12*** 109.51*** 104.12*** 71.732***

Unitrootteststatistics

ADF −10.605*** −12.202*** −11.670*** −12.243*** −12.625*** −11.595*** −12.630*** −11.470*** −12.190***

KPSS 0.0799 0.3516 0.0872 0.1337 0.0929 0.1513 0.1279 0.1666 0.0737

Panel B: During the COVID-19 pandemic Descriptivestatistics

Mean 0.0068 0.0007 0.0012 0.0009 0.0015 0.0005 0.0004 0.0001 0.0002

Std.dev. 0.0498 0.0124 0.0216 0.0203 0.0161 0.0185 0.0047 0.0051 0.0071

Skewness −3.6822 −0.0001 −0.9150 −1.0657 0.3498 −0.8713 −0.5058 −1.4630 −0.6630

Kurtosis 40.045 8.7132 13.009 12.792 7.6616 11.435 4.7595 13.256 7.2672

J-Btest 12606^† 287.22^† 1021.9^† 885.68^† 206.21^† 665.97^† 39.226^† 1065.6^† 199.85^†

Q²(10) 55.889** 28.251*** 169.10*** 44.717*** 148.68*** 52.724*** 70.006*** 63.381*** 79.549***

ARCH(10) 5.0270* 2.3238* 91.517*** 31.791*** 86.170*** 62.614*** 28.269*** 31.825*** 48.156***

Unitrootteststatistics

ADF −5.9548*** −9.0835*** −9.0401*** −8.1735*** −8.4121*** −8.1858*** −9.2608*** −9.8608*** −7.8933***

KPSS 0.3888 0.0541 0.0342 0.0342 0.035 0.0533 0.1117 0.1296 0.0531

Note:Thistablereportsthesummarystatisticsofdailyreturnsfordifferentassets.J-BistheJarque–Berratestfornormality,Q²(10)istheLjung–Boxstatisticforserial correlationinthesquaredreturns.ARCH(10)isthetestforconditionalheteroskedasticity.ADFandKPSSstandfortheempiricalstatisticsofAugmentedDickey–Fullerand Kwiakowski–Phillips–Shmidt–Shintestsforunitrootandstationarity,respectively.PanelAsummarizestheresultsduringthepre-COVID-19period,i.e.,fromApril29,2013 toMarch10,2020.PanelBsummarizestheresultsduringtheCOVID-19pandemic,i.e.,fromMarch11,2020toJanuary5,2021.

†Normalityisrejectedat5%signicancelevel.

***Nullhypothesisisrejectedat1%condencelevel.

**Nullhypothesisisrejectedat5%condencelevel.

*Nullhypothesisisrejectedat10%condencelevel.

Table2

Pearsoncorrelationmatrix.

PanelA:BeforetheCOVID-19pandemic

BTC 1 0.0069 0.0280 0.0075 −0.0112 0.0091 0.0047 −0.0007 0.0114

GLD 1 −0.0406 −0.0923 −0.1767 −0.0923 0.1915 0.0001 0.0661

SP500 1 0.1983 0.5560 0.5359 −0.0887 0.0050 0.0625

ES50 1 0.3004 0.8246 −0.2226 0.0038 0.1067

N225 1 0.2974 −0.1163 −0.0494 0.0305

F100 1 0.0019 0.0019 −0.0584

EUR 1 0.4522 0.5161

JPY 1 0.1611

GBP 1

PanelB:DuringtheCOVID-19pandemic

BTC 1 0.1099 0.4604 0.4283 0.0787 0.4111 0.0845 −0.0724 0.1832

GLD 1 0.2209 0.1626 0.3026 0.1881 0.1735 0.0942 0.1854

SP500 1 0.6833 0.2435 0.6741 0.1389 −0.2857 −0.2208

ES50 1 0.4513 0.9124 0.1389 −0.2208 0.3187

N225 1 0.4167 0.2355 0.0489 0.3036

F100 1 0.1369 −0.1815 0.2250

EUR 1 0.5159 0.6089

JPY 1 0.4547

GBP 1

Note:ThistablereportsthePearsoncorrelationmatrix.

5. Empiricalresults

5.1. TheA-DCCmodelestimationresults

Tables A.1–A.3 in Appendix A document thefull estimation results of the A-DCC model of Bitcoin and gold with all the returns of the selected assets before and during the COVID- 19 pandemic. The average conditional correlations are weakly positive indicating the ability of Bitcoin and gold to act as

a hedge in times of market stability and also in times of crisis.

Diagnostic testsontheresidualsareperformedtoverifythe qualityoftheempiricalresults.According toTable3,Ljung–Box’s test shows noserial correlationin thesquared residuals. Like- wise, the ARCH test cannot reject the null hypothesis of the absenceofARCHeffects.Therefore,thereisnoevidenceoferro- neousstatisticalspecication.TheA-DCCapproachwithaTARCH specicationiscorrectlyspeciedtodescribethedynamicrelation-

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Table3 Diagnostictests.

PanelA:BeforetheCOVID-19pandemic

Q²(10) 0.013[1.000] 10.72[0.380] 11.23[0.340] 9.081[0.524] 6.329[0.787] 10.18[0.425] 6.613[0.761] 10.36[0.409] 6.606[0.762]

Q²(20) 0.028[1.000] 16.25[0.700] 15.99[0.719] 14.08[0.826] 16.88[0.663] 24.93[0.204] 14.31[0.814] 17.88[0.595] 10.87[0.949]

ARCH(10) 0.013[1.000] 10.46[0.401] 11.08[0.351] 9.302[0.504] 6.246[0.794] 9.963[0.444] 6.922[0.733] 10.80[0.373] 6.711[0.752]

PanelB:DuringtheCOVID-19pandemic

Q²(10) 0.013[1.000] 3.790[0.956] 2.458[0.992] 2.839[0.985] 8.833[0.548] 5.164[0.880] 9.228[0.511] 3.658[0.962] 3.192[0.977]

Q²(20) 0.028[1.000] 6.318[0.998] 4.958[1.000] 3.839[1.000] 13.61[0.850] 6.001[0.999] 16.10[0.710] 4.342[1.000] 7.540[0.995]

ARCH(10) 0.013[1.000] 5.117[0.883] 4.233[0.936] 2.640[0.989] 11.88[0.293] 6.692[0.754] 11.14[0.346] 6.522[0.770] 6.172[0.801]

Note:Thistablereportsthediagnostictestsformodelmisspecication:Ljung–BoxQ²(10)andQ²(20),andtheARCHtestontheresidualsoftheestimatedA-DCCmodel.

p-valuesassociatedwiththestatisticaltestsarereportedinbrackets.Lowerp-valuescastdoubtonthecorrectspecicationofthemodel.

shipbetweenBitcoinandgoldmarketswiththestockandforex markets.

Figs.2and3illustrateindetailthedynamicsoftheconditional correlationsbetweenBitcoinandgoldreturns,respectively,and thevariousnancialmarkets.Correlationlevelsareveryvolatile throughouttheperiodsandmarkets.Forstockindices,theresults showthattheA-DCCmodelalternatesbetweenlowpositiveand negativevalues inalignmentwithJoy(2011),Klein(2017),Bouri, Shahzad,Roubaud,Kristoufek,andLucey(2020)andCharfeddine, Benlagha,andMaouchi(2020),whichindicatetheroleofBitcoin andgoldinhedgingagainststockmarkets.However,sincethestart ofthepandemic outbreak,thesecorrelationshaveincreasedon average.Thiscanbeattributedtotwoeffects.(i)First,thecollapse ofstockmarkets,whichweresimultaneouslyaffectedbyCOVID- 19.Indeed,thehealthcrisishas turnedintoa seriouseconomic crisisthathashitthemarketsmoredeeplythananyother shock (Bakeretal.,2020).Someshocksweredue tothedirecteffects linkedtothespreadofthevirus,i.e.,infectionrate,mortalityrate,as wellastheeconomicandpsychologicalconsequencesduetosocial distancingandlockdownmeasuresatvariouspartsoftheworld, whichhasledtoamassivesell-offinthenancialmarkets.Some other shocks (whether positive or negative) were due to the direct implicationofcentralbankmonetarypolicyinterventions(FOMC meetings,ECB,BOE,CBJdecisions)atvariouspartsofthenan- cialworldandwithdifferenttiming.(ii)Second,thesharprisein thepricesofBitcoinandgoldduetothecrisisepisodesprompted internationalinvestorstochooseanoptimalallocationstrategyby investingintheseassetstoprotecttheirwealth(Bongeretal., 2020).

Goodell and Goutte (2021) suggest that Bitcoin may act as a safe haven during the pandemic due to its co-movement with COVID-19 cases. This explains that when the feeling of fearincreases intheU.S. market,investorsturntootherassets such as cryptocurrencies. However, the decrease in correlation appearstobeonlytemporaryandofshortduration.Indeed,fol- lowinggovernmentinterventionswithmonetaryandbudgetary reliefplanstomitigatetheimpactofthecrisis, thecorrelations have shown episodes of positive values in Fig.2,in alignment withBatten,Kinateder,Szilagyi,and Wagner(2021).For example,onMarch17,2020,theU.S. Housepassedthe Coronavirus Aid,Relief, andEconomicSecurity(CARES) actforaneconomic stimulus package exceeding $2 trillion to protect the Ameri- cancitizensfromtheunwantedhealthandeconomicimpactsof COVID-19.

Theseresultsareofparticularinteresttoinvestorswhowish tohedgetheirportfolios.Thefollowingsectionsprovideprecise informationonthehedgeandsafehavenproperties.

5.2. Optimalportfoliodesign

Table4summarizestheoptimaldesignofportfoliosconsisting of Bitcoin (or gold) and other nancial assets before and dur-

ingthe COVID-19 crisis. The results indicate that, to minimize riskgiventheexpectedreturn,theoptimalweightsaregenerally above92%forBitcoinand60%forgoldduringbothsub-periods,on average.Mokni,Ajmi,Bouri,andVo(2020)foundsimilarresults whereinvestors holdmore than80% of theirwealth in Bitcoin tominimizetheriskofaU.S.equityportfolio.Likewise,thiscon- clusionissimilartotheworksofHoodandMalik(2013),Arouri, Lahiani,andNguyen(2015)andChkili(2016)thatrevealtheuse- fulnessofgoldinprovidingdownsideriskhedgingininternational portfolios.

It isworthnotingthatduringthecrisis,theoptimalweights of Bitcoin and gold in conventional currency wallets increased signicantly,suggestingthatforexinvestorsareconsideringcryp- tocurrency and gold like a hedge (Longstaff, 2004). Moreover, duringtheCOVID-19pandemic, the optimal weightsof Bitcoin and gold decreased for all stock market indices. The decrease ismore pronounced withgold.For instance,theweight ofthe gold–S&P500portfoliodecreasedfrom74.52%duringtherst phase to 19.31% during the second phase. Overall, the results concludethatinvestorsshouldallocatelargerproportionsofBit- coin than gold in order to minimize the risk of international wallets.

5.3. Hedgingeffectiveness

Optimalhedgeweightsandratiosprovideageneralunderstand- ingofhowhedgeisconstructedtominimizerisk.However,theydo nothelpidentifytheeffectivenessofthehedgeovertime.There- fore,wecalculatethehedgeeffectiveness(HE)indexusingEq.(15) inTable5.

ThecomparisonbetweenportfoliosincludingBitcoinandthose includinggoldvariesconsiderablydependingontheperiodand markettype. Moving from Panel A (beforeCOVID-19) toPanel B(duringCOVID-19)inTable5,withtheexceptionofthegold- currencypairsandtheBitcoin-Nikkei225 pairthatshowagood hedgingstrategy, theoptimal hedgeratiosfor Bitcoin andgold increasedsignicantly during thecrisis. In fact, for BTC-SP500 market,theratiogoesfromanegativevalueof−0.0117during thepre-COVID-19period toapositivevalueof0.2006duringthe COVID-19period.ThenegativevalueofHRimpliesthatinvestors musttakethesamepositionfortwoassetsinthesameportfolio (shortorlong),whilethepositivevalueindicatesreversepositions arerequiredtohedgeagainsttheriskofeachasset.Forexample, inordertominimizetherisk,alongpositionof$1inS&P500can behedgedwith20.06centsshortpositioninBitcoin.Asnotedby LópezCabreraandSchulz(2016),thelowerthehedgeratio,theless expensivethehedge.Thus,assetcoveragewascheaperbeforethe pandemicthanduringthepandemic(Akhtaruzzaman,Boubaker, Lucey,&Sensoy,2021).Thisndingisconsistentwiththeliterature which reveals higher hedge ratios in times of crisis (Batten et al., 2021),duetothesharpincreaseinuncertaintyovertheeconomic andnancialoutlook.Thevarietyofrisksassociatedwiththisglobal

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Fig.2. DynamiccorrelationsbetweenBitcoin,stockindicesandforeignexchanges.