International asset allocation in presence of systematic cojumps

Transcription

1 Inenaonal asse allocaon n pesence of sysemac cojumps Mohamed Aou a Oussama M SADDEK b* Duc Khuong Nguyen c Kunaa Pukhuanhong d a CRCGM - Unvesy of Auvegne 49 Boulevad Fanços-Meand B.P Clemon- Feand Fance b CRCGM - Unvesy of Auvegne 49 Boulevad Fanços-Meand B.P Clemon- Feand Fance c IPAG Lab IPAG Busness School 84 Boulevad San-Geman Pas Fance d Tulaske College of Busness Unvesy of Mssou 49 Conell Hall Colomba MO 65 USA Ths veson: Januay 06 Absac The objecve of hs acle s o povde a new paal explanaon of he home bas phenomenon n nenaonal asse holdngs fom an nvesgaon of naday jumps and cojumps. We hypohesze ha global nvesos wll ovewegh domesc asses f nenaonal dvesfcaon benefs ae negavely affeced by a hgh level of synchonzaon and ansmsson of naday jumps acoss makes. Usng naday ndex-based daa fo equy aded funds we povde evdence of sgnfcan sysemac jump sks n nenaonal makes ha dve nvesos o educe he popoon of foegn asses n he dvesfed pofolos. In addon we show a negave lnk beween he demand of foegn asses and he numbe of cojumps beween domesc and foegn asses when consdeng he composon of he opmal pofolo n he sense of mean-vaance and mean-cvar appoaches. Keywods: sysemac jump sk cojumps home bas *Coespondng auho. Tel.: E-mal addesses: mohamed.aou@gmal.com (M. Aou) msaddeko@yahoo.f (O. M saddek) duc.nguyen@pag.f (D.K. Nguyen) pukhuanhongk@mssou.edu (K. Pukhuanhong).

2 . Inoducon Ths pape conbues o he leaue on nenaonal home bas puzzle by examnng whehe jump and cojump sks explan a pa of hs phenomenon. I s now well esablshed n he fnance leaue ha pce dsconnues o jumps should be aken no accoun when sudyng asses pce dynamcs and allocang pofolo acoss asses (Bekae e al. 998; Das and Uppal 004; Gudoln and Ossola 009). The ecen developmen of non-paamec jump denfcaon ess has suppoed he hypohess of jumps n fnancal asse pces. The semnal woks n hs aea nclude Bandoff-Nelsen and Shephad ( ) who es fo he pesence of jumps a he daly level usng measues of bpowe vaaon. The same famly of naday jump denfcaon pocedues ncludes he ess developed by Jang and Oomen (008) Andesen e al. (0) Cos e al. (00) Podolskj and Zggel (00) and Chsensen e al. (04). Andesen e al. (007) and Lee and Mykland (008) have developed echnques o denfy naday jumps usng hgh fequency daa. All of hese jump deecon echnques povde empcal evdence n favo of he pesence of asse pce dsconnues o jumps. Moe ecenly eseaches ae neesed n sudyng cojumps beween asses (Dungey e al. 009; Lahaye e al. 00; Dungey and Hvozdyk 0; Pukhuanhong and Roll 04). Fo nsance Glde e al. (0) examne he fequency of cojumps beween ndvdual socks and he make pofolo and fnd a endency fo a elavely lage numbe of socks o be nvolved n sysemac cojumps. Lahaye e al. (00) look a he elaonshp beween asse cojumps and macoeconomc announcemens and show ha cojumps ae paally assocaed o new macoeconomc announcemens. A-Sahala e al. (05) develop a mulvaae Hawkes jump-dffuson model o capue jumps popagaon ove me and acoss makes and povde song evdence fo jumps o ave n cluses whn he same make and o popagae o ohe wold makes. Bome e al. (05) fnd ha Hawkes one-faco model s moe suable o capue he hgh synchonzaon of jumps acoss asses han he mulvaae Hawkes model. Ou sudy hus exends he pevous sudes n ha nvesgaes empcally cojumps beween nenaonal equy ndces as well as he mpac on nenaonal pofolo allocaon. Moden pofolo heoy suggess ha nenaonal dvesfcaon s an effcen ool o mnmze pofolo sks gven ha nenaonal asses ae ofen less coelaed and dven by dffeen economc facos. Howeve one mgh expec ha sysemac cojumps can lead o an ncease n he coelaon beween hese nenaonal asses and hus affec

3 negavely he benef fom nenaonal dvesfcaon. Invesely f pce jumps ae no sysemac (.e. hey ae specfc o a pacula make due o asse-specfc evens whch ae uncoelaed fom make movemen) hey ae caegozed as dosyncac jumps and wll no affec pofolo allocaon decsons n an nenaonal seng. Accodngly a sk-avese nveso who holds an nenaonal pofolo s exposed o wo ypes of jump sks: sysemac jump sk o cojumps (jumps common o all makes) and dosyncac jump sk (jumps specfc o one make). If hs pofolo s well dvesfed he dosyncac jump sk wll be educed o even elmnaed. Howeve he sysemac jump sk wll pess and could no be elmnaed hough dvesfcaon. Theefoe he denfcaon of sysemac jump sk s cenal o asse pcng pofolo hedgng and asse allocaon. Is pesence and conbuon o he global pofolo sk also epesens a good sk ndcao fo nvesos when selecng pofolo composon among all avalable asses. In hs pape we consde wo hypoheses. The fs poss ha nenaonal equy ndces expeence cojumps and a pa of jump sk s ahe sysemac. The second poss ha he sysemac jump sk could paally explan he lack of nenaonal pofolo dvesfcaon -home bas - obseved n global equy makes. In he pesence of he sysemac cojumps sk beween nenaonal equy ndces we would expec an ncease n he coelaon beween he epesenave makes and hus a decease of he dvesfcaon benef. As a esul nvesos would have pefeence fo domesc asses. Ou empcal ess ely on he use of naday euns fo hee dedcaed nenaonal exchange aded funds SPY EFA and EEM whch especvely am o eplcae he pefomance of hee nenaonal equy ndces: S&P 500 MSCI EAFE (Euope Ausalasa and Fa Eas) and MSCI Emegng Makes. We apply he echnque poposed by Andesen e al. (007) and Lee and Mykland (008) o denfy all naday jumps and cojumps of he hee funds fom Januay 008 o Ocobe 03. A bvaae Hawkes model s employed o capue he me cluseng feaues of naday jumps and he dynamcs of he popagaon acoss makes. Based on he heoecal famewok developed by Todoov and Bolleslev (00) we esmae he sensvy of equy ndex euns owads he sysemac make dffusve and jump sks and show ha jump sk s ahe sysemac and could no be elmnaed by dvesfcaon. S&P 500 ndex s used as a poxy fo he US make. MSCI EAFE ndex s he benchmak fo developed makes excludng he US and Canada wheeas he MSCI Emegng Makes s used o capue he pefomance of emegng equy makes. 3

4 We also sudy he mpac of sysemac jumps on nenaonal pofolo allocaon (second hypohess) by consdeng a domesc sk-avese nveso who selecs he pofolo composon based on one domesc asse and wo foegn asses o mnmze he pofolo sk whle equng a mnmum expeced eun. As nvesos ae concened abou negave movemens of asse euns we ake he sk of exeme evens no accoun by usng he Condonal Value-a-Rsk o CVaR (Rockafella and Uyasev 000) as a sk measue n ou pofolo allocaon poblem. Unlke he sandad mean-vaance famewok whch undeesmaes he sk of lage movemens of asse euns he mean-cvar appoach allows us o povde a faly accuae esmae of he downsde sk nduced by sysemac negave jumps of asse euns. Regadng he composon of he opmal pofolo we analyze how jumps and cojumps affec nveso demand fo domesc and foegn asses and show evdence of a song negave lnk beween he demand of foegn asses and he nensy of cojumps beween he domesc and foegn asses. Ou esuls ndcae ha he nenaonal dvesfcaon benef songly depends on he level of he coelaon beween jumps n nenaonal equy makes. The emande of he pape s oganzed as follows. Secon noduces he jump and cojump denfcaon echnques used n ou sudy. Secon 3 pesens he pofolo allocaon poblem n mean-vaance and mean-cvar famewoks. Secon 4 descbes he daa. Secon 5 dscusses ou man empcal fndngs on cojumps and nenaonal asse allocaon. Secon 6 concludes.. Jump and cojump deecon mehodology.. Jump es sascs Ths secon oulnes he heoecal backgound of jump denfcaon echnques. To explan he basc dea behnd hese jump deecon pocedues we begn fo pedagogc easons wh he adonal Bandoff-Nelsen and Shephad ( ) es whch s nally appled o deemne f a day (o a gven me wndow) conans pce jumps. Bandoff-Nelsen and Shephad (BNS) develop a non-paamec jump es based on wo measues of pce vaaon: ealzed vaance (RV) and bpowe vaaon (BV). The fs one measues he vaaon n he pces comng fom boh connuous and jump componens of he oal pce vaaon pocess whle he second one s jump obus and only measues vaaon comng fom he connuous pa of he pocess. 4

5 We assume ha he logahm of he pce of an asse followng jump-dffuson model (Meon 976): p can be geneaed by he dp = µ d + σ W + κ dj 0 T () whee d epesens he me-vayng df componen σ W epesens he me vayng µ volaly componen of he asse pce and W s a sandad Bownan moon. κ dj denoes he me-vayng jump componen of he pocess. κ sands fo he sze of jumps and J s a counng pocess ndependen of of he pce a day : ( p ) =... M W. We consde M equdsan obsevaons of he logahm. The h naday eun of day s hen defned by: = p p =... M The ealzed vaaon of he pce a day s calculaed as followng: M RV = = The ealzed vaance conveges o he negaed vaance (IV) plus a jump componen (called also quadac vaaon) as he samplng fequency of he pce obsevaons nceases ( M ) : M RV σ ds + κ s N j= j whee N epesens he numbe of whn day jump a day and jh jump a day. κ j denoes he magnude of Bandoff-Nelsen and Shephad ( ) popose a jump obus measue of he negaed vaaon called he bpowe vaaon (BV): BV π M M = M = 5

6 The bpowe vaaon conveges o he negaed vaance as he samplng fequency of he pce obsevaons nceases ( M ) : BV M σ ds s Bandoff-Nelsen and Shephad use he dffeence beween ealzed vaance and bpowe vaaon o esmae he sum of jumps whn a day. We consde he elave jump measue defned by: RJ = RV BV RV I measues he conbuon of jumps o he oal whn-day vaaon of he pocess. We also noduce a modfed fom of BNS es sasc poposed by Huang and Tauchen (005) o denfy days wh a leas one jump. Huang and Tauchen pove ha hs pacula fom of he BNS es oupefoms he ohe foms of BNS es appled n he fnancal leaue n ems of sze and powe. The es sasc of Huang and Tauchen (HT) s defned by: Z = RJ π QV + π 5 max M BV whee: QV π = M M M M 3 = 4 3 The null hypohess of absence of jumps a day s sascally ejeced wh a pobably of α f: Z > Φ α 6

7 whee Φ α epesens he nvese of he sandad nomal cumulave dsbuon funcon evaluaed a a cumulave pobably of α. The pevous ess (BNS o HT) ae appled o deemne f a day (o a gven me wndow) conans pce jumps whou povdng fuhe nfomaon abou he numbe he me and he sze of he jumps ha occus dung a gven me wndow. Thus we need o use a es sasc ha wll enable us o denfy all naday jumps fo a gven day and o deemne he occuence me and he sze of each jump. Thee ae essenally wo pocedues n he leaue used o denfy naday jumps such as Andesen e al. (007 hencefoh ABD) and Lee and Mykland (008 hencefoh LM) ess wheeas BNS o HT ess can only check fo he dsconnuy of asse pces a a daly level. The LM and ABD naday pocedues use he same es sasc bu dffe on he choce of he ccal value. ABD assumes ha he es sasc s asympocally nomal wheeas LM povde ccal value fom he lm dsbuon of he maxmum of he es sasc. Moeove Dumu and Uga (0) show ha naday ess of LM and ABD oupefom ohe es pocedues manly f pce volaly s no hgh. The LM es sasc compaes he cuen asse eun wh he bpowe vaaon calculaed ove a movng wndow wh a gven numbe of pecedng obsevaons. The sasc L whch ess a me whehe hee was a jump fom - o s defned as: L = σˆ whee: ( ) σ ˆ = K j = K + j j whee ˆ σ s he ealzed bpowe vaaon calculaed fo a wndow of K obsevaons. I povdes a jump obus esmao of he nsananeous volaly. LM emphasze ha he wndow sze K should be chosen n a way ha he effec of jumps on he volaly esmaon dsappeas. They sugges ha s appopae o choose he wndow sze K beween 5 M and 5 M whee M s he numbe of obsevaons n a day. Unde he null 7

8 hypohess of absence of jumps a anyme n (- ] he LM sasc s asympocally dsbued as: L C S M M M ξ x whee has a cumulave dsbuon funcon P ξ x = exp( e. ξ ( ) ) We noduce C and S : M M log( M ) log( π ) + log(log( M )) C M = c c log( M ) S M = c log( M ) and c = π The null hypohess of absence of jumps a anyme n (-] s ejeced fo a gven sgnfcance level α f: L > log( log( )) S + C α M M ABD povde a es sasc whch s assumed o be nomally dsbued n he absence of jumps. A jump s deeced wh he ABD es on day n naday neval when: BV M f Φ whee Φ epesens he nvese of he sandad nomal cumulave dsbuon funcon evaluaed a a cumulave pobably of epesens he daly sgnfcance level of he es. and ( ) = α M whee α To denfy naday jumps we only apply he naday pocedue of LM-ABD n ou sudy. A jump s deeced wh he LM-ABD es on day n naday neval when: 8

9 σˆ > θ The ccal value θ s calculaed fo dffeen sgnfcance levels. Fo a daly sgnfcance level of 5% and a samplng fequency of 5 mnues whch coesponds o 77 naday euns pe day n ou sudy we oban a ccal value of 3.40 usng ABD mehod and 4.40 usng LM mehod. Followng Bome and al. (05) we ake a ccal value θ = 4. We also sudy dffeen ccal values (3 4 and 5) o vefy he obusness of ou esuls... Cojump denfcaon pocedue Fnancal asses can jump smulaneously and we call hs pce jump a cojump. To denfy a cojump n he pces of a pa of asses we use a wo-sep pocedue: we fs denfy he naday jumps of each asse usng he LM-ABD mehod descbed n he pevous secon. We hen apply he followng co-exceedance ule o deec a cojump: m σˆ m : a cojump = n > θ > θ 0 : no cojump σˆ n A cojump n he pces of he pa of asses (mn) s deeced on day n naday neval when boh asses jump a he same naday me neval. In ou sudy we dsngush beween an dosyncac jump of a sngle sock ha occus ndependenly of he make movemen and a sysemac jump ha happens a he make level. As we lm ou empcal sudy o hee nenaonal equy ndces ha cove hee dffeen makes (US developed counes excludng US and emegng counes) a jump s consdeed o be nenaonally sysemac f he hee ndces jump smulaneously. If only one o wo ndces ae nvolved n an naday jump hs jump s no classfed as an nenaonal sysemac jump. I s only consdeed as sysemac whn s coespondng make. 9

10 3. Pofolo allocaon poblem We consde a sk-avese nveso who selecs hs pofolo composon based on n asses: one domesc sky asse and n- foegn sky asses. We suppose ha all asses ae expessed n he nveso s domesc cuency. The nveso allocaes funds acoss n asses n a way o mnmze he sk whle equng a mnmum expeced eun. We fs consde he sandad mean-vaance (MV) appoach nally fomulaed by Makowz (95). The Makowz appoach defnes he sk as he vaance of he pofolo eun. The MV pofolo opmzaon poblem s fomulaed as follows: mn ( wσw) w subjec o : w k 0 k=... n e w = () (P) µ w = µ () whee : w= ( w w... w ) s he veco of pofolo weghs µ = ( µ... ) n µ µ s he mean veco of n euns and cov( s he vaance-covaance max of euns. e = (... ) Σ = k ) k n denoes he veco of ones. The fs consan () s he budge consan. The second one () consans pofolo's expeced eun o be equal o a gven value µ. The opmzaon poblem (P) can be solved usng quadac pogammng echnques. The MV appoach s based on he fs wo momens of he eun dsbuon. I s well esablshed ha f asse euns ae nomal he pofolo opmzaon poblem could be educed o he mean-vaance famewok. Howeve n he case of non-nomal euns he opmzaon poblem depends on he pefeence of he nveso. If he only caes abou he mean and he vaance of he pofolo he MV famewok could be used o oban he opmal pofolo weghs. Howeve n mos cases he whole dsbuon of he eun should be consdeed n he opmzaon poblem. Moeove he vaance as a symmec sk measue fals o dffeenae beween he upsde and downsde sks. The vaance ofen leads o an oveesmaon of he sk fo posvely skewed dsbuon and an undeesmaon of he sk fo negavely skewed dsbuon. The vaance also fals o esmae he sk of exeme 0

11 evens (fa al dsbuon): lage losses as well as lage gans. As nvesos ae moe concened abou negave movemens of asse euns s mpoan o pay aenon o he downsde sk when selecng pofolo asses. The ssue of he pofolo allocaon unde he non-nomaly of asse euns has been wdely suded and seveal alenaves o he sandad MV famewok have been poposed (Jondeau and Rocknge 006; Gudoln and Tmmemann 008). Boh sudes have exended he MV famewok o cove hghe momens of asse euns by appoxmang he expeced uly usng Taylo sees expansons. Moe ecenly a pacula aenon has been gven o he downsde sk n pofolo allocaon and seveal pecenle sk measues have been poposed as an alenave o he vaance such as Value a Rsk VaR (Basak and Shapo 00; Gavoonsk and Pflug 999) and Condonal Value-a-Rsk CVaR whch s also known as mean excess loss mean shofall o al VaR (Rockafella and Uyasev 000; Kokhmal Palmqus and Uyasev 00). The VaR s an esmae of he uppe pecenle of loss dsbuon. I s calculaed fo specfed confdence level ove a cean peod of me. The VaR s wdely used by fnancal pacones o manage and conol sks; howeve s use n pofolo opmzaon emans vey lmed. Indeed he VaR has some undesable popees (Azne e al ) whch affec s effcency as a sk measue such as he lack of sub-addvy mplyng ha VaR of a pofolo wh wo nsumens may be geae han he sum of he ndvdual VaRs of hese wo nsumens. Also he VaR s usually calculaed usng scenaos. In hs case VaR s non-convex and non-smooh and he opmzaon becomes vey unsable and leads o mulple local exema. Conay o he VaR he CVaR has moe aacve fnancal and mahemacal popees. I s sub-addve and convex (Rockafella and Uyasev 000). I s also consdeed as a coheen sk measue (Pflug 000) n he sense of Azne e al. ( ). The CVaR of a pofolo epesens he condonal expecaon of losses ha exceeds he VaR. Ths defnon ensues ha VaR s neve hghe han he CVaR. The CVaR and VaR opmzaon poblems ofen lead o smla opmal pofolo as boh measues ae lnked by defnon. The followng descbes he CVaR opmzaon appoach nally developed by Rockafella and Uyasev (000).We fs defne he loss funcon of a pofolo composed of N asses. I s defned fo a gven veco of weghs w by:

12 N f ( w ) = w n= = w n n whee s he andom veco of asse euns. The pobably of f ( w ) no exceedng a heshold α s gven by: Ψ( w α ) = f ( w ) α p( ) d whee p() s he densy funcon of he veco of euns. Ψ s a funcon of α fo a fxed veco of weghs w and epesens he cumulave dsbuon funcon fo he loss assocaed wh he veco of weghs w. We denoe α ( w) and φ ( w) assocaed of w and a confdence level. as he values of he VaR and he CVaR of he loss funcon α ( w ) = mn( α R : Ψ( w α ) ) ( w) = ( ) φ f ( w ) α f ( w ) p( ) d ( w) Followng Rockafella and Uyasev (000) we povde he expesson ( w) funcon F defned as follows: φ usng he ( w α ) α + ( ) [ f ( w ) α] F = p( ) d + + whee[ x ] = max(x;0) Rockafella and Uyasev (000) demonsae ha F ( w ) α s convex and connuously dffeenable as a funcon of α. I s also elaed o he CVaR of he loss funcon by he followng fomula: φ ( ) ( w) = mn F ( w α ) α R

13 Rockafella and Uyasev (000) also pove ha mnmzng φ ( w) equvalen o mnmzng F ( w ) ove all ( w α ) R N R α. ove all N w R s mn N w R φ ( w ) ( α ) = mn F w N ( w α ) R R To smplfy he expesson of F we need o appoxmae he negal n he defnon of F. One possble soluon s o geneae a andom collecon of he veco of euns () () ( q) (... ) and appoxmae F as he followng: ~ F q ( ) ( w α ) = α + [ f ( w ) α ] q( ) = + Replacng he loss funcon by s expesson gves: ~ F q ( ) ( w α ) = α + [ w α] q( ) = + By noducng he auxlay vaable u he mnmzng of F ~ s equvalen o he lnea equaon: α + q u q( ) = ( ) subjec o: u 0 u + w + α 0 fo = q If we add he budge and he expeced age eun consans he mean-cvar opmzaon poblem s gven by: q α + u α ω q( ) = mn (P) subjec o : e w = µ w = µ w 0 k fo k = n ( ) u 0 u + w + α 0 fo = q 3

14 The mean-cvar opmzaon (P) poblem can be solved usng lnea pogammng echnques. We noe ha f asse euns ae nomally dsbued and wh 0. 5 value of he Mean-Vaance and mean-cvar appoaches ae equvalen and gve he same opmal pofolo weghs (Rockafella and Uyasev 000). In hs pape we apply boh appoaches o deemne he opmal pofolo composon and sudy how he depaue fom he nomaly caused by he pesence of jumps affecs he opmal pofolo composon. 4. Daa descpon We use naday daa of hee nenaonal exchange aded funds n ou empcal nvesgaon: SPDR S&P 500 (SPY) Shaes MSCI EAFE (EFA) and Shaes MSCI Emegng Makes (EEM). SPDR S&P 500 ETF ams o eplcae he pefomance of S&P 500 ndex by holdng a pofolo of he common socks ha ae ncluded n ndex wh he wegh of each sock n he pofolo subsanally coespondng o he wegh of such sock n he ndex. The S&P 500 ndex s a US sock make ndex conanng he socks of 500 lage- Cap copoaons and hus a poxy fo he whole US sock make. The Shaes MSCI EAFE ETF ams o eplcae he pefomance of he MSCI EAFE ndex. The MSCI EAFE Index s a sock make ndex ha capues he sock make pefomance of developed makes ousde of he U.S. & Canada and hus a poxy fo he nenaonal equy. I ncludes Euope Ausala and Fa Eas equy makes. The Shaes Emegng Makes ETF ams o eplcae he pefomance of he MSCI Emegng Makes ndex. The Shaes Emegng Makes Index s a sock make ndex ha capues he sock make pefomance of emegng makes. I coves ove 800 secues acoss makes and epesens appoxmaely % of wold make cap. Ou empcal analyss s based on naday pces of he hee funds fom Januay 008 o Ocobe 03. Pces ae sampled evey fve mnues fom 9:30 o 5:55 o smooh he mpac of make mcosucue nose. 5. Empcal fndngs 5.. Inaday jump denfcaon Ths secon summazes he esuls fom applyng LM-ABD naday jump deecon es. A pacula aenon s gven o he naday volaly paen (Rognle 00 and Dumu and Uga 0) whch can lead o spuous jump deecon. To ameloae he 4

15 obusness of ou jump deecon pocedue we coec he naday volaly paen usng a jump obus coeco (Fgue ) poposed by Bolleslev e al. (008). Appendx A povdes a dealed descpon of he volaly paen coeco used n ou sudy. *** Inse Fgue hee *** We esmae he ealzed bpowe vaaon usng a wndow of 55 naday euns whch coesponds o wo days of naday euns sampled a a fequency of fve mnues. Jumps ae deeced wh a ccal value θ = 4 (Bome e al. 05) whch means ha he naday jump eun sze s a leas fou mes geae han he esmae of he local volaly. We also apply ccal values of 3 and 5 o sudy he obusness of ou esuls. In hs secon we only povde empcal esuls of LM-ABD jump deecon pocedue foθ = 4 esuls fo θ = 3 and 5 ae avalable upon eques.. Dealed Table povdes he numbe of oal posve and negave naday jumps deeced ove he peod of sudy. We especvely denfy 9 4 and 04 naday jumps fo SPY EFA and EEM funds whch coesponds especvely o 0.989% 0.986% and 0.900% of he oal numbe of naday euns ove he peod of sudy. Thus he numbe of deeced naday jumps s slghly hghe n developed makes (US and EFA) han n emegng makes. The dffeence beween he numbes of naday jumps denfed whn each egon of he wold s n lne wh he pevous esuls n he leaue ndcang ha he degee of na-egon comovemen and negaon s hghe n developed makes han emegng ones. A posve (negave) jump s a jump wh posve (negave) eun. The esuls show he numbe of negave jumps s moe han 56% of oal numbe of deeced jumps fo each fund whch s slghly geae han posve ones. Sock makes end o be moe lnked ogehe when pces ae deceasng. Table also shows basc sascs abou he dsbuon of naday jump euns. The mean of naday jump euns of SPY (-4.3e-04) whch s n absolue value wo mes hghe han EFA and EEM (-.5e-04 and -.e-04 especvely) ndcaes ha he negave movemens of naday pces ae moe sevee fo he US make dung he peod of ou sudy. The naday jump eun volaly s hghe fo emegng makes (0.0058) han fo developed ones (aound ). A a daly level Table shows he pecenage numbe of days wh a leas one naday jump s aound 40% of he oal numbe of days of he peod of sudy (468 days) fo hee ETFs. *** Inse Tables and hee *** 5

16 Table 3 shows some sascs of deeced cojumps. We fnd ha SPY and EFA funds cojump 585 mes ove he peod of sudy. Ths epesens 5% of he oal numbe of deeced jumps. The SPY and EEM funds cojump 509 mes ove he same peod. The numbe of cojumps of EEM wh SPY s slghly geae o he cojumps wh EFA (458) whch ndcaes ha emegng equy makes ae moe lnked o he U.S make han o ohe developed makes coveed by he EFA fund. The hee funds ae nvolved n 365 cojumps dung he peod of sudy. As fo jumps he numbe of negave cojumps s hghe han posve cojumps fo hee funds. Fo nsance negave cojumps epesen 6% of oal cojumps nvolvng he hee suded funds. Table 4 pesens he pobably o have a leas one cojump beween SPY and EFA s 0.7 a a daly level. Ths pobably s lowe fo SPY and EEM (0.3) o EFA and EEM (0.). The pobably o have he hee funds nvolved n a one smulaneous jump o moe s 0.8. Fgues and 3 show he vaaon of he daly nensy of jumps (JI) and cojumps (CJI) dung he peod of sudy. These me-vayng jump (cojump) nenses ae calculaed each day usng a ollng sx-monh wndow of obsevaons as follows: JI = N Jump n days and CJI = Jump m N days Jump n whee N Jump n days s he numbe of days of he obsevaon peod (0 days n ou case). s an ndcao funcon of jump occuence fo he asse n a he naday neval. s an ndcao funcon of cojump occuence fo he asse m and n a he naday Jump m Jump n neval. *** Inse Tables 3 and 4 and Fgue hee *** We noce ha he daly jump and cojump nenses have nceased sgnfcanly dung he fnancal css of fo he hee funds. Thee s a paen ha he US make was he fs o each he peak of he jump nensy dung he css followed by developed makes and hen emegng makes. The skng evdence s dung a peak n Januay 00 a dop n June 0 and a jump n Decembe 0. The US make seems o be followed moe closely by he developed makes. The emegng makes ae he mos 6

17 lagged fom he ohe wo. The esuls suppo he evdence found by A-Sahala e al. (05). Fgue 3 shows he cojump nensy beween he US and developed make s hghes followed by he US and emegng makes and he developed make and emegng makes. The nensy hee makes jump smulaneously s lowes. Oveall he lead/lag of jumps s smla o paen of lead/lag n fnancal css. Tha s he unusual ncease of he naday jumps (boh posve and negave) seems o be nally ggeed n he US makes and hen popagaed o ohe makes n he wold. *** Inse Fgue 3 hee *** 5.. Tme and space cluseng of naday jumps We pevously show ha naday jumps seem o be nally ggeed n he US make and hen popagaed o ohe makes n he wold. The paen of naday jumps also suggess ha jumps end o appea n cluses whn he same egon (Fgue 4). Thus seems ha nenaonal naday jumps have endency o popagae boh n me (n he same make) and n space (acoss makes). Ths dependency beween he occuences of jumps ha we obseve n he daa canno be epoduced by he sandad Posson Pocess whch s on he conay based on he hypohess of ndependence of he ncemens meanng ha he numbes of jumps on dsjon me nevals should be ndependen. So we need o use an alenave model o epoduce he me and space popagaon of asse jumps. We employ he Hawkes pocess (Hawkes 97) whch s a self-exced pon pocess whose nensy depends on he pah followed by he pon pocess. Ths pocess has been appled n dffeen scences such as sesmology and neuology and moe ecenly n fnance (n modelng he adng acvy fo example). To ou knowledge hee ae only wo papes whch have employed he Hawkes pocesses o model he dynamcs of asse jumps n fnancal makes. A-Sahala e al. (04) ae neesed n capung he paens of conagon n he nenaonal equy makes usng a mulvaae Hawkes jump-dffuson model bu esan he empcal analyss o he daly daa of majo nenaonal equy ndces. The Hawkes pocesses wee also appled n Bome e al. (05) o epoduce he me cluseng of jumps. Howeve hey fnd ha he exenson of he Hawkes pocesses o a mul-asse famewok s nconssen wh daa and nsead popose o use Hawkes faco models o capue he coss-seconal dependences of jumps. They conduc he empcal sudy based on he hgh fequency daa of 0 hgh cap Ialan socks. 7

18 *** Inse Fgue 4 hee *** The followng descbes he Hawkes pocesses ha we use n he cuen pape o model he dynamcs of naday jumps of he hee nenaonal equy funds. We fs begn wh he unvaae Hawkes pocess whch can be used o capue he me cluseng of naday jumps fo one asse. The nensy pocess s defned by: ( λ λ ) d α dn d λ + = whee N s he numbe of jumps occung n he me neval [0 ]. A jump occuence a a gven me wll ncease he nensy whch nceases he pobably of anohe jump (self excaon). The nensy nceases by α wheneve a jump occus and hen decays back owads a level λ a a speed. These paamees can be esmaed usng he mehod of maxmum lkelhood. Gven he jump aval mes n he lkelhood funcon s wen as: n ( n ) ( e ) + λ A( ) α L(... n) = λ log α whee A( ) = n n + + = = ( ) j e jp and A ( ) = 0 fo ( ) esmaon ae povded n Ogaa (978) and Ozak (979).. The deals of he maxmum lkelhood Now we consde he muldmensonal famewok of he Hawkes pocess defned by: n ( λ λ ) d + d λ dn and j = n = α j j = j Unde hs model a jump n make j nceases he jump nensy whn he same make by α j j (self excaon) and coss makes by α j (coss excaon). Then he jump nensy of make eves exponenally o s aveage level λ a a speed. In he empcal analyss we fnd ha he numecal esoluon of he muldmensonal Hawkes model n he case of hee makes s poblemac as he numbe of paamees s oo lage o esmae. So we esan he calbaon pocedue o he bvaae model gven by: 8

19 dλ dλ = = ( λ λ ) d + αdn + αdn ( λ λ ) d + α dn + α dn In hs case we only have 8 paamees o esmae: = ( λ λ α α α α ) Θ. Table 5 Panels A B and C show he esuls of he maxmum lkelhood esmaon of he bvaae Hawkes model fo SPY/EFA SPY/EEM and EFA/EEM especvely. We fs emak ha he model paamees ae all sascally sgnfcan mplyng ha he bvaae Hawkes model fs he daa of naday jump occuences of hee funds. The values of he paamees measung he degee of self-excaonα andα ae lage. Ths esul povdes clea evdence ha he US make developed makes (excludng he US) and emegng makes ae songly self-exced meanng ha he occuence of a jump a a gven me nceases he pobably of ohe jumps whn he same make. The compason beween hee funds suggess ha he self-excaon acvy s hghe n he US make han ohe makes. The values of he paamees α andα measung he degee of ansmsson of jumps beween makes ae elavely small compaed o he self-excaon paamees. The degee of ansmsson of jumps beween makes seems o be asymmec wh a songe ansmsson fom he US make o developed makes ex-us (.40 e-03) and emegng makes (.3 e-03). The degee of he evese ansmsson of jumps fom developed makes (.79 e-03) o emegng makes (.6 e-03) o he US make ae sascally dffeen fom zeo bu emans elavely small. The degee of ansmsson of jumps fom he developed makes ex-us o emegng makes (.58 e-03) s hghe han ha fom emegng makes o developed makes ex-us (.6 e-03) Sysemac dffusve and jump sks *** Inse Table 5 hee *** Ths secon sudes he conbuon of naday jumps of asse pces o he oal sysemac sk n equy makes. We have shown n pevous secons ha nenaonal equy makes ae chaacezed by a hgh level of synchonzaon beween jumps as well as a hgh degee of ansmsson of hese jumps acoss makes. Such neconnecon beween jumps wll ncease he sysemac jump sk n he nenaonal equy makes. 9

20 In he followng we befly evew he heoecal famewok ha we use o dsenangle and esmae he sensvy owads sysemac dffusve and jump sks n he conex of faco models. The naday log-pce of he make pofolo p ae assumed o follow connuous-me jump dffuson pocesses: p 0 and he h asse = 0 T dp 0 µ 0 d + σ 0 W 0 + κ 0 dj 0 c d dp = µ d + σ 0 dw0 + κ 0 dj 0 + ε = n whee W 0 s a sandad Bownan moon κ 0 dj 0 denoes he jump componen of he make pofolo pocess. κ 0 sands fo he sze of make jumps and J 0 s a counng pocess ndependen of W 0. o he asse. The wo beas ε sands fo he connuous and dsconnuous pce movemens specfc and measue especvely asse 's sensves o c d connuous and dsconnuous movemens of he make. Aggegang he naday pce movemens of he make pofolo and he asse ove a peod [0 T] we oban he lnea wo-faco model: c c d d = µ ε whee s he eun of he asse ove he peod [0T] µ s s df em c 0 and d 0 ae he connuous and he dsconnuous pas of he make eun especvely. Ove he me neval [0T] suppose ha asse pces ae obseved a dscee me gds m τ whee m s he numbe of obsevaons pe one me un and τ =... mt and p τ = p τ p τ m m s he naday eun of asse ove he τ m τ m naday me neval. Followng Todoov and c d Bolleslev (00) we esmae and usng he obseved dscee naday euns of he make pofolo and he asse { p p } 0 τ τ ove he peod [0; T]: 0

21 c ( p + p ) ( p p ) mt τ 0 τ τ 0 τ τ = p + p + p p = τ 0 τ θ 0 τ τ 0 τ θ 0 τ mt ( p0 τ ) τ = p 0 τ θ 0 τ d = sgn mt τ = sgn { p p }( p p ) τ 0 τ τ 0 τ mt τ = sgn { p p }( p p ) τ mt ( p0 τ ) τ = 0 τ 4 τ 0 τ The ndcao funcons p τ + p0 τ θ+ 0 τ p τ p0 τ θ 0 τ and p0 τ θ 0 τ ae noduced o fle ou jumps when calculang he connuous bea. The naday jumps ae deeced usng he LM-ABD pocedue fo he hee consdeed pocesses. The make ndex s consuced by abung an equal wegh o each fund. We fx he samplng fequency of naday euns used n he esmaon of he connuous and dsconnuous beas a 5 mnues wh euns spannng fom 9:35 am o 4:00pm fo each adng day. We calculae boh beas on a monhly bass based on he daa fom he pevous monh (abou 67 naday euns fo each fund pe monh). Fgue 5 Panels A B and C show he vaaon of he monhly connuous and dsconnuous beas especvely of SPY EFA and EEM. We fnd ha he monhly esmaed beas (connuous and dsconnuous) ae sascally sgnfcan fo hee funds. The dsconnuous beas ae n aveage hghe han connuous beas fo hee funds. Boh dffusve and jump sks ae sgnfcanly dffeen fom zeo and hus hey ae sysemac. Moeove make jumps seem o be moe sensve n aveage han connuous make moves. These gaphs also sugges ha he connuous beas ae less dspesed han dsconnuous beas acoss me and funds. Ths esul s confmed by he values of he sandad devaon of connuous and dsconnuous beas ha we fnd fo each fund (dealed fgues ae epoed n Table 6). Fgue 6 shows he auocoelogams of dffusve and jump beas aveaged acoss hee funds. Ths fgue suggess ha he degee of pessence of connuous bea s hghe han he dsconnuous bea fo a leas he fs hee odes of auocoelaon whch s conssen wh he evdence found by Bandoff-Nelsen and Shephad ( ) Andesen e al. (007) and Bolleslev e al. (05). ***Inse Table 6 and Fgues 5 and 6 hee ***

22 5.4. The mpac of asse cojumps on opmal pofolo composon Ths secon sudes he composon of he opmal nenaonal pofolo n MV and CVaR famewoks and deemnes how he demand of foegn asses vaes n funcon of he numbe of cojumps beween domesc and foegn asses. The se of nsumens o nves n s composed of he hee funds pevously noduced. We use weekly hsocal euns (a oal of 94 obsevaons fo each fund) o un he MV and CVaR opmzaon. The weekly euns ae calculaed as he log dffeence of closng pces. The opmzaon pocedue s pefomed each week usng a ollng wndow of abou 50 weekly euns (one yea) ha mmedaely pecede he opmzaon day. We need o mnmze he pofolo sk (sandad devaon o CVaR) fo a gven level of expeced eun o fnd he opmal pofolo weghs. The composon of he opmal pofolo s hus a funcon of he age expeced eun. The se of opmal pofolos obaned fo dffeen level of expeced euns s called he effcen fone. We choose o sudy one pacula pofolo fom he effcen fone ha maxmzes he expeced eun pe un of sk known as he angency pofolo. Ths pofolo has he advanage of beng moe dvesfed han he global mnmum sk pofolo whch s ofen concenaed on asses ha mnmze he sk of he pofolo. Fgue 7 Panel A shows he vaaon of he opmal popoon of he foegn asses (EFA and EEM) fo MV and CVaR appoaches. The pofolo s composed of one domesc asse (SPY fund) and wo foegn asses (EFA and EEM). We fs emak ha MV and CVaR appoaches almos lead o he same pofolo composons dung he peod of sudy. Fgue 7 Panels B and C sugges sandad devaon and CVaR ae hus equvalen sk measues when hey ae calculaed usng ou hsocal daa. The fgues also show ha he sandad devaon o he CVaR of he domesc asse (SPY) ae ofen lowe han hose of foegn funds (EFA and EEM). Indeed he vaably of foegn asses ncludes boh he changes of he sock pces and he exchange aes. *** Inse Fgue 7 hee *** Once he composon of he opmal pofolo s deemned we wll show how cojumps beween domesc and foegn asses affec he benef fom nenaonal dvesfcaon. Ou hypohess poss ha a hghe nensy of cojumps beween domesc and foegn asses leads o an ncease of he coelaon beween hese asses. The benef fom nenaonally dvesfyng s hus educed and heefoe he domesc nveso wll be encouaged o nves moe on domesc asses. To confm ou analyss we fs calculae he

23 coelaon beween he daly nensy of cojumps and he opmal popoon of foegn asses ha we fnd usng MV and CVaR appoaches. The man esuls fo he MV appoach ae summazed n Table 7. The coelaon beween he demand of foegn asses and he daly nensy of cojumps beween he domesc asse (SPY n ou sudy) and each of foegn asses s negave. We fnd a coelaon of fo EFA and fo EEM. If we ake he cojumps ha nvolve he hee funds we fnd a negave coelaon of aound whch ndcaes ha he popoon of foegn asses s moe sensve o he cojumps ha nvolve hee funds han he cojumps beween wo funds. The coelaon s moe negave fo negave cojumps han posve cojumps. The coelaons beween he demand of foegn asses and he negave cojumps beween SPY and EFA SPY and EEM and all hee ae and compaed o and -0. fo posve cojumps of he same pa. We also pefom a es of sgnfcance of he Peason coelaon wh a null hypohess of no coelaon pesen beween wo vaables agans an alenave hypohess ha hee s lnea coelaon pesen. The coelaon s sascally sgnfcan fo posve negave and all makes cojumps whch suppos he hypohess of he exsence of negave coelaon beween he numbe of cojumps beween domesc and foegn asses and he demand of he foegn asses n he nenaonal pofolo composon. *** Inse Table 7 hee *** The esuls hus fa have ndcaed ha he mpac of asse cojumps on he demand of foegn asses dffes fom one asse o anohe. Indeed he popoon of foegn asses n he pofolo seems o be moe sensve o he numbe of naday cojumps of he domesc asse wh he fund ha poxes he developed makes (EFA). The popoon of foegn asses s educed he mos when jumps n hee makes occu smulaneously. We also sudy he elaonshp beween he daly nensy of cojumps and he opmal demand of he foegn asses by pefomng a lnea egesson. The objecve s o deemne f he vaaon of he popoon of he foegn asses n he pofolo could be explaned by he vaaon of he numbe of asses cojumps. We apply he OLS o povde an esmae of he egesson coeffcen. Table 8 summazes he esuls of he lnea egesson. We fnd a egesson coeffcen of -.8 (-.6 and -.59 especvely) when he daly nensy of cojumps beween SPY/EFA (SPY/EEM and SPY/EFA/EMM) s used as explanaoy vaable. The esuls of he CVaR appoach ae no pesened because MV and CVaR appoaches lead o almos he same opmal pofolo composon. The esuls ae avalable upon eques. 3

24 All of hese coeffcens ae sascally dffeen fom zeo wh he p-value nfeo o e-. Ths means ha f he aveage daly nensy of cojumps beween SPY and EFA nceases by % he opmal popoon of he foegn asses wll decease by -.8%. The cojumps ha nvolve he hee funds have a geae effec as an ncease of % of he daly nensy of cojumps of hee asses leads o a educon of -.59% of he popoon of foegn asses n he pofolo. *** Inse Table 8 hee *** 5.5. The mpac of asse cojumps on he benefs of pofolo dvesfcaon Ths secon sudes he mpac of asses cojumps on he benefs of pofolo dvesfcaon. We expec ha sysemac cojumps wll affec negavely he gan of he dvesfcaon. We use he measue of dvesfcaon benef developed by Chsoffesen e al. (0) o sudy how he dvesfcaon benef of he nenaonal pofolo composed of hee funds (SPY EFA and EEM) evolve ove me. Ths measue s based on he Condonal Value-a-Rsk measue (CVaR) and s defned as: CDB ( w) φ φ = φ α ( w) ( w) whee α ( w) and ( w) φ ae he values of he VaR and he CVaR of he pofolo loss funcon assocaed of he veco of weghs w and a confdence level. φ s he uppe bound on he pofolo CVaR defned as he weghed aveage of he asses ndvdual CVaRs 3 : φ N = w n φ n n= whee ( w) wn and φ n denoe especvely he wegh and he CVaR of he nh asse. The VaR α s by defnon he lowe bound on he pofolo CVaR. As a esul he condonal dvesfcaon benef measue (CDB) s a posve funcon vayng beween 0 and and s nceasng n he level of dvesfcaon benef. The CDB measue does no depend on he esmae of expeced euns. 3 The CVaR s a coheen sk measue n he sense of Azne e al. ( ). 4

25 Ths measue akes no accoun he nonnomaly feaues of he pce euns especally n pesence of jumps and cojumps. I wll also enable us o assess he benef of he dvesfcaon n he case of hgh coelaon beween lage down moves n nenaonal makes 4. We calculae he value of he CDB on a monhly bass based on he naday daa fom he pevous monh (abou 67 naday euns fo each fund pe monh). The confdence level s se a 5%. The weghs of asses ae chosen monhly n a way o maxmze he level of he CDB. The fgue 8 shows he vaaon of he opmal level of he dvesfcaon benef fo he pofolo composed of hee funds SPY EFA and EEM. The condonal dvesfcaon benef vaes beween 0.06 (Augus 0) and 0.3 (Augus 008 and Januay 03). The dvesfcaon benef s hgh a he begnnng and he end of he peod of ou sudy compaed he mddle of he peod (009-0). I seems ha he dvesfcaon benef was paculaly educed dung he fnancal css peod. *** Inse Fgue 8 hee *** To assess he mpac of sysemac and dosyncac jumps on he benef of he dvesfcaon we measue he coelaon beween he opmal level of he CDB and he daly nensy of especvely sysemac and dosyncac jumps. We also noduce a measue of he coelaon beween he jumps ha akes no accoun boh sysemac and dosyncac jumps. Ths measue s defned as follows 5 : ρ j = mt τ = mt p τ p mt 4 p τ τ = τ = j τ p 4 j τ whee denoes s he naday eun of asse ove he τ τ naday me neval p τ pevously defned n secon 5.3. Obvously he coelaon beween he jumps nceases wh he numbe of sysemac cojumps and deceases wh dosyncac jumps. We calculae he max of coelaon of hee funds on a monhly bass based on he naday daa fom he pevous monhs. We hen calculae he aveage value of he coelaon max o oban a sngle jump coelaon value pe monh. m m 4 The hgh coelaon of lage down moves n nenaonal makes s documened by Longn and Solnk (00) and Ang and Bekae (00). 5 See Jacod and Todoov (009) fo moe deals. 5

26 The esuls of he measues of he coelaon beween he opmal level of he dvesfcaon benef and especvely he coelaon of jumps he daly nensy of cojumps and dosyncac jumps ae epoed n Table 9. We fnd a negave coelaon of fo he daly nensy of cojumps and fo he coelaon of cojumps. The song dependence beween he dvesfcaon benef and he coelaon of jumps s also shown n Fgue 9. Conay o he sysemac cojumps he dvesfcaon benef s posvely coelaed o he nensy of dosyncac jumps. The coelaon level s aound 0.4. Oveall hese esuls ndcae ha he nenaonal dvesfcaon benef nceases wh he nensy of dosyncac jumps and deceases wh he level of sysemac cojumps obseved n he nenaonal makes. *** Inse Fgue 9 and Table 9 hee *** 6. Conclusons In hs pape we sudy how asse jumps and cojumps affec he benef fom he nenaonal dvesfcaon. Usng a nonpaamec naday jump deecon echnque developed by Lee and Mykland and Andeson e al. we fnd ha nenaonal equy funds have endency o be nvolved n sysemac cojumps. We fuhe show ha hese jumps ae ansmed boh n me (n he same make) and n space (acoss makes). The hgh degee of neconnecon beween jumps n equy makes suggess ha he jump sk s ahe sysemac and hus couldn' be elmnaed by he dvesfcaon. By dsenanglng and esmang he sensvy owads sysemac dffusve and jump sks we show evdence ha sysemac jump sk s sgnfcan and heeby should be aken no consdeaon by nvesos when selecng he composon of he pofolos among all avalable asses. Sudyng he lnk beween he composon of he opmal nenaonal pofolo (n he sense of Mean-Vaance and Mean-CVaR appoaches) and he cojumps beween domesc and foegn asses we fnd ha a domesc nveso s encouaged o educe he popoon of hs wealh nvesed n foegn asses when he numbe of cojumps nceases. In fac he coelaon of asse euns nceases wh he nensy of cojumps and by consequence he benef fom dvesfyng he pofolo aboad s exemely educed when he cojumps of nenaonal equy ndces ae fequen. Ths wok opens neesng pespecves fo fuue eseach. I would be of gea nees o boaden he scope of hs sudy by ncludng a lage numbe of nenaonal equy 6

27 ndces and sudyng he mpac of asse cojumps on he demand of foegn asses fo a lage panel of counes. I s also neesng o see f he mechansms of jumps ansmsson suded n hs pape ae also vald fo ndvdual couny makes. Refeences A-Sahala Y. Cacho-Daz J. and Laeven R. 05. Modelng fnancal conagon usng muually excng jump pocesses Fohcomng Jounal of Fnancal Economcs do:0.06/j.jfneco Andesen T.G Bolleslev T. and Dobev D No-abage sem-mangale escons fo connuous-me volaly models subjec o leveage effecs jumps and..d. nose: Theoy and esable dsbuonal mplcaons. Jounal of Economecs 38():5-80. Andesen T.G. Dobev D. and Schaumbug E 0. Jump obus volaly esmaon usng neaes neghbo uncaon. Jounal of Economecs 69() Ang A. and Bekae G. 00 Inenaonal Asse Allocaon wh Regme Shfs Revew of Fnancal Sudes Azne P. Delbaen F. Ebe J.M. Heah D. 997: Thnkng coheenly. Rsk Vol. 0 No.. p Azne P. Delbaen F. Ebe J. and Heah D Coheen measues of sk Mahemacal Fnance Bandoff-Nelsen O. E. and Shephad N. 004a. Powe and bpowe vaaon wh sochasc volaly and jumps. Jounal of Fnancal Economecs -37. Bandoff-Nelsen O. E. and Shephad N. 004b. Measung he mpac of jumps n mulvaae pce pocesses usng bpowe covaaon.wokng Pape Unvesy of Oxfod. Bandoff-Nelsen O. E. and Shephad N Economecs of esng fo jumps n fnancal economcs usng bpowe vaaon. Jounal of Fnancal Economecs

28 Basak S. and Shapo A. 00.Value-a-sk based sk managemen: Opmal polces and asse pces Revew of Fnancal Sudes Bekae G. Eb C. Havey C. Vskana T The dsbuonal chaacescs of emegng make euns and asse allocaon. Jounal of Pofolo Managemen Wne 0-6. Bolleslev T. Law T. H. Tauchen G Rsk jumps and dvesfcaon. Jounal of Economecs44() Bolleslev T. and Todoov V. 0. Tal feas and sk pema. Jounal of Fnance Bolleslev T. L S.Z. and Todoov V. 05. Roughng up Bea: Connuous vs. Dsconnuous Beas and he Coss-Secon of Expeced Sock Reuns. Wokng Pape. Bome G. Calcagnle L.M. Teccan M. Cos F. Mam S. Lllo F. 05. Modellng sysemc pce cojumps wh hawkes faco models. Quanave Fnance Volume 5 pp (0). Chsensen K. Oomen R. and Podolskj M. 04. Fac o fcon: Jumps a ula hgh fequency. Jounal of Fnancal Economcs 4 (3) Chsoffesen P. Eunza V. Langlos H. and Jacobs K. 0. Is he Poenal fo Inenaonal Dvesfcaon Dsappeang? Revew of Fnancal Sudes Cos F. Pno D. and Reno R. 00. Theshold bpowe vaaon and he mpac of jumps on volaly foecasng. Jounal of Economecs Das S. Uppal R Sysemc Rsk and Inenaonal pofolo choce. Jounal of Fnance Dumu A.M. and Uga G. 0. Idenfyng jumps n fnancal asses: A compason beween nonpaamec jump ess. Jounal of Busness & Economc Sascs 30 () Dungey M. McKenze M. and Smh V Empcal evdence on jumps n he em sucue of he US easuy make. Jounal of Empcal Fnance 6 (3) Dungey M. Hvozdyk L. 0. Cojumpng: Evdence fom he US Teasuy bond and fuues makes. Jounal of Bankng and Fnance Gavoonsk A.A. and Pflug G Fndng opmal pofolos wh consans on Value a Rsk n B. Geen ed. Poceedngs of he Thd Inenaonal Sockholm Semna on Rsk Behavou and Rsk Managemen Sockholm Unvesy. 8

29 Gudoln M. and Ossola E. Do Jumps Mae n Emegng Make Pofolo Saeges? chape n Fnancal Innovaons n Emegng Makes (Gegoou G. edo) Chapman Hall London 009 pp Gudoln M. and Tmmemann A Inenaonal asse allocaon unde egme swchng skew and kuoss pefeences. Revew of Fnancal Sudes () Glde D. Shackleon M. and Taylo S. 0.Cojumps n Sock Pces: Empcal Evdence. Jounal of Bankng and Fnance Has L A ansacon daa sudy of weekly and nadaly paens n sock euns. Jounal of Fnancal Economcs Hawkes A.G. 97. Speca of some self-excng and muually excng pon Pocesses Bomeka Huang X. and Tauchen G The elave conbuon of jumps o oal pce vaance. Jounal of Fnancal Economecs 3(4) Jacod J. and Todoov V. (009) Tesng fo Common Avals of Jumps fo Dsceely Obseved Muldmensonal Pocesses The Annals of Sascs Jang G. J. and Oomen R. C. A. 008.Tesng fo jumps when asse pces ae obseved wh nose - A swap vaance appoach. Jounal of Economecs Jondeau E. and Rocknge M. 006.Opmal pofolo allocaon unde hghe momens Euopean Fnancal Managemen () Kokhmal. P. Palmqus J. and Uyasev. S. 00. Pofolo opmzaon wh condonal Value-A-Rsk objecve and consans. The Jounal of Rsk 4-7. Lahaye J. Lauen S. and Neely C.J. 00. Jumps cojumps and maco announcemens. Jounal of Appled Economecs Lee S. S. and Mykland P.A Jumps n fnancal makes: A new nonpaamec es and jump dynamcs. Revew of Fnancal Sudes Longn F. and Solnk B. 00 Exeme Coelaon of Inenaonal Equy Makes Jounal of Fnance Makowz H.M. 95. Pofolo Selecon. Jounal of fnance 7() Meon R Opon pcng when undelyng sock euns ae dsconnuous. Jounal of Fnancal Economcs Ogaa Y The asympoc behavou of maxmum lkelhood esmaes fo saonay pon pocesses Annals of he Insue of Sascal Mahemacs Ozak T Maxmum lkelhood esmaon of Hawkes self-excng pon pocesses. Annals of he Insue of Sascal Mahemacs 3()

30 Pukhuanhong K. and Roll R. 04. Inenaonally coelaed jumps. Fohcomng Revew of Asse Pcng Sudes do:0.093/apsu/au009. Rockafella R. T. and Uyasev S Opmzaon of Condonal Value-a-Rsk. Jounal of Rsk -4. Rockafella R. T. and Uyasev S. 00. Condonal Value-a-Rsk fo Geneal Loss Dsbuons. Jounal of Bankng and Fnance 6 (7) Rognle M. 00. Spuous Jump Deecon and Inaday Changes n Volaly wokng pape MIT. Todoov V. and Bolleslev T. 00. Jumps and Beas: A New Famewok fo Dsenanglng and Esmang Sysemac Rsks. Jounal of Economecs Wood R A. McInsh A. Thomas H. and Od K An nvesgaon of ansacon daa fo NYSE socks. Jounal of Fnance

31 3 Appendx A: Inaday volaly paen I s wdely documened (Wood e al. (985) and Has (986)) ha naday euns show a sysemac seasonaly ove he adng day also called he U-shaped paen. The naday volaly s paculaly hghe a he open and he close of he adng han he es of he day. To mnmze he effecs of naday volaly on ou jump deecon es we modfy ou pocedue by escalng naday euns wh a volaly jump obus coeco noduced by Bolleslev and al. (008). The escaled ˆ s defned by: ς ˆ = whee: + + = = = = + = = + T M T M M T T M ς fo = M- + + = = = = + = = + T M T M M T T M ς fo = + + = = = = + = = T M T M M T T M ς fo = M

32 Table : Summay sascs of jump occuences and jump szes The numbe of oal posve (pecenage) and negave (pecenage) deeced jumps ae epoed. The mean sandad devaon skewness and kuoss of jump szes ae shown. Resuls epoed below ae obaned usng LM-ABD pocedue wh a ccal value θ = 4. SPY EFA EEM Inaday jumps Posve jumps 475 (4%) 495 (44%) 455 (44%) Negave jumps 644 (58%) 69 (56%) 569 (56%) Mean (jump eun) -4.3e e-04 -.e-04 Sandad devaon Skewness Kuoss Table : Summay sascs of jump occuences a day level The numbe of days wh no jumps one jump and wo jumps up o moe han 5 jumps ae epoed. The las ow shows he pecenage of days wh a leas one jump. SPY EFA EEM Moe han A leas one jump 4% 4% 40% 3

33 Table 3: Summay sascs of cojump occuences The numbe of oal (pecenage of cojumps compaed o he oal numbe of deeced jumps) posve and negave deeced co jumps beween SPY and EFA (column ) SPY and EEM (column ) EFA and EEM (column 3) and SPY EFA and EEM (column 4) ae epoed. SPY / EFA SPY/EEM EFA/EEM SPY/EFA/EEM Inaday cojumps 585 (53%) 509 (50%) 458 (45%) 365 (36%) Posve cojumps Negave cojumps Table 4: Summay sascs of cojump occuences a day level The numbe of days wh no cojumps one cojump wo cojumps up o moe han 4 cojumps. The las ow shows he pecenage of days wh a leas one cojump. SPY/EFA SPY/EEM EFA/EEM SPY/EFA/EEM Moe han A leas one cojump 7% 3% % 8% 33