Valuing Volatility Spillovers

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1 Valuing Volailiy Spillovers George Milunovich Division of Economic and Financial Sudies Macquarie Universiy Sydney Susan Thorp School of Finance and Economics Universiy of Technology Sydney March 2006 Conac deails: George Milunovich, Division of Economic and Financial Sudies, Macquarie Universiy, NSW, 2109, Ausralia. Tel: Fax: gmilunov@efs.mq.edu.au Susan Thorp, School of Finance and Economics, Universiy of Technology, Sydney, Broadway NSW 2007, Ausralia. Tel: Fax: susan.horp@us.edu.au We hank Lance Fisher, Tony Hall, David Michayluk, Adrian Pagan, Minxian Yang, and paricipans a he 2005 Applied Economerics Conference, Venice, and he 2005 Global Finance Conference, Triniy College, Dublin for helpful commens. Thorp acknowledges he suppor of he Ausralian Research Council

2 ABSTRACT We show ha volailiy spillovers are large enough o maer o invesors. We demonsrae ha sandard deviaions of reurns o mean-variance porfolios of European equiies fall by 1-1.5% a daily, weekly, and monhly rebalancing horizons when volailiy spillovers are included in covariance forecass. We esimae he condiional second momen marix of (synchronized) daily index reurns for he London, Frankfur and Paris sock markes via wo asymmeric dynamic condiional correlaion models (A-DCC): he unresriced model includes volailiy spillovers and he resriced model does no. We combine covariance forecass from he resriced and unresriced models wih a wide range of assumed reurns relaives via a polar coordinaes mehod, and compue ou-of-sample realized porfolio reurns and variances for esing. Diebold-Mariano ess confirm ha mos risk reducions are saisically significan. Sochasic dominance ess indicae ha porfolios accouning for volailiy spillover would be preferred by risk averse agens. Keywords: GARCH, volailiy spillover, porfolio risk, forecasing JEL Classificaion: G11 G15 C53 C32-2 -

3 1. Inroducion There are many empirical sudies of ime-varying second momens bu fewer sudies which acually measure wheher new models will benefi invesors. Since a key ingredien in successful porfolio selecion is an accurae predicion of covariance beween asse reurns, beer forecasing models should generae measurably lower porfolio risk. Volailiy spillovers, for example, have been exensively documened as a feaure of financial daa bu heir imporance for efficien invesmen has no been evaluaed. In his sudy, we incorporae volailiy spillovers ino covariance forecass, form mean-variance porfolios of European equiies, and quanify any resuling benefis o invesors. A volailiy spillover occurs when changes in price volailiy in one marke produce a lagged impac on volailiy in oher markes, over and above local effecs. Volailiy spillover paerns appear o be widespread in financial markes. There is evidence for spillovers beween equiy markes (see for example Hamao, Masulis and Ng 1990, and Lin, Engle and Io 1994), bond markes (Chrisiansen 2003), fuures conracs (Abhyankar 1995, Pan and Hsueh 1998), exchange raes (Engle, Io and Lin 1990, and Baillie and Bollerslev 1990), equiies and exchange raes (Apergis and Reziis 2001), various indusries (Kalenhauser 2002), size-sored porfolios (Conrad, Gulekin and Kaul 1991), commodiies (Apergis and Reziis 2003), and swaps (Eom, Subrahmanyam and Uno 2002). Despie he ineres ha invesors migh have in hese pervasive spillover effecs, we are no aware of any sudy ha invesigaes he quesion of heir impac on efficien asse allocaion. Our firs sep owards answering his quesion is o consruc a covariance model o comprehensively capure he daa while isolaing he impac of volailiy spillovers. Invesors in our sudy hold mean-variance porfolios allocaed among he risk-free asse and equiies in wo of hree major European sock markes, London, Frankfur and Paris

4 Porfolio weighs herefore depend on forecass of he bivariae condiional covariance marix of sock marke reurns. To generae hese forecass while isolaing he impac of volailiy spillovers on porfolio efficiency, we esimae wo nesed models of reurns volailiy using an Asymmeric Dynamic Condiional Correlaion (A-DCC) se up (Cappiello, Engle and Sheppard 2004). The benchmark (resriced) model capures imevarying volailiy and correlaion, including asymmeric effecs, bu omis volailiy spillover erms, which we add o he unresriced model 1. We esimae he models over he firs par of he sample and hen forecas he condiional covariance marix over remaining daa a a range of horizons, compuing opimal porfolio weighs a each forecas. Mean-variance porfolio weighs depend on expeced reurns as well as expeced second-order momens, and i is well known ha ou-of-sample porfolio performance is ofen degraded by a poor choice of expeced reurns (Chopra and Ziemba 1993). A new approach, developed by Engel and Colacio (2004), offers a mehod for minimizing he impac of expeced reurn choice on ou-of-sample porfolio efficiency: in a wo-asse porfolio, relaive raher han absolue reurns maer o opimal porfolio weighing, so by compuing weighs for a wide range of reurns raios, we can beer separae he effecs of covariance forecasing from reurns forecasing. Finally, using opimal weighs, we compue realized porfolio reurns and variances, and hen es for significan difference beween he volailiy spillover formulaion and he benchmark. We find ha accouning for volailiy spillovers in condiional covariance forecass resuls in small bu significan improvemens in porfolio efficiency, relaive o benchmark. The efficiency gains arising from modelling volailiy spillovers range from a 0.02 o a 1.51 per cen reducion in porfolio sandard deviaion. For a porfolio reurning, say, 10 per cen per year, his represens a small risk-adjused improvemen of a mos 0.15 per cen, however ess confirm ha, in he majoriy of cases, hese risk reducions - 4 -

5 are saisically significan a all forecasing horizons. In addiion, sochasic dominance ess poin o significan improvemens in invesor uiliy arising from volailiy spillover forecasing. Since including volailiy spillover effecs in he porfolio selecion process does no incur any addiional ransacions coss, even small gains can represen improvemen for invesors. This paper proceeds as follows. The nex secion reviews relevan feaures of he volailiy spillover lieraure. We ouline he benchmark and alernaive models in Secion 3, and describe porfolio consrucion in Secion 4. Secion 5 describes he daa and repors parameer esimaes, and presens resuls of ess comparing he performance of porfolios consruced from he benchmark and volailiy spillover models. Secion 6 concludes. 2. Lieraure Review Ineres in volailiy spillovers across inernaional equiy markes inensified afer he Ocober 19, 1987 sock marke crash when a sharp drop in he US equiy markes appeared o creae a widespread volailiy ripple across inernaional markes. In an aemp o explain his, King and Wadhwani (1990) pu forward a marke conagion hypohesis, arguing ha sock price urbulence in one counry is parly driven by urbulence in oher counries, beyond he influence of fundamenals. Idenifying and esing he ransmission of urbulence beween markes has been he focus of he volailiy spillover lieraure. Early sudies of volailiy spillovers ypically focus on developed counry equiy markes, and he ransmission of volailiy from larger o smaller counry markes in paricular. For example, Hamao, Masulis and Ng (1990) find unidirecional volailiy spillovers from US markes o he UK and Japan, and he UK o Japan, while - 5 -

6 Theodossiou and Lee (1993) argue for addiional ransmissions from he US marke o Canada and Germany. Furher, he large-small counry effec appears o be mirrored wihin equiy markes on a firm-size level. Sudies documen volailiy spillover from large o small firms (Conrad, Gulekin and Kaul 1991, and Reyes 2001), alhough bad news may cause spillover in he reverse direcion as well (Pardo and Torro 2003). More recen sudies invesigae spillover effecs beween developed and emerging markes, and among emerging markes hemselves. A ypical finding (see, for example, Wei e al, 1995) is ha volailiy ransmis from developed o emerging markes, and ha he smaller, less developed markes are likely o be more sensiive o ransmied shocks. Geographic localiy, regardless of marke size, is also likely o be a facor in volailiy spillover. Bekaer and Harvey (1997) are able o disinguish beween local and global shocks, sudying volailiy spillovers across emerging sock markes. Regional facors are imporan for Pacific Basin markes, over and above he world-marke effecs of spillovers from he US (Ng 2000). In a relaed sudy, Miyakoshi (2003) goes furher, arguing ha regional effecs are sronger han world marke influence for markes in he Asian region. Europe represens a paricularly ineresing geographic area for volailiy spillover sudies since i encompasses a number of developed markes wih common economic and financial feaures, and overlapping rading hours. Thireen European markes and he US are sudied by Baele (2003), who decomposes volailiy spillovers ino counry specific, regional and world shocks. (The model also allows for regime swiches in he spillover effecs.) Boh regional and world effecs are repored as significan. Furher, spillovers appear o have inensified over he 1980s and 1990s, wih a more pronounced rise among European Union (EU) markes. In a relaed sudy, Billio and Pelizzon (2003) find ha volailiy spillovers o mos European sock markes from boh he world index and he - 6 -

7 German index have increased since he European Moneary Union (EMU) came ino effec. The imporance of regional spillovers for Europe is no resriced o equiy markes. Tesing for volailiy spillover effecs in European bond markes, Chrisiansen (2003) finds evidence of spillover from boh he US and Europe o individual counry s bond markes. The European volailiy spillover effecs are sronger han he US volailiy spillovers in boh bond and equiy markes. An imporan mehodological issue for ransmission sudies is wheher volailiy spillovers can be idenified separaely from lags in informaion ransfer due o nonoverlapping rading hours beween markes. For example, in he foreign exchange marke Engle, Io and Lin (1990) invesigae volailiy spillovers across Tokyo and New York for he Yen/USD exchange rae. Since hese wo markes rade a common securiy, bu operae in differen ime zones, he auhors argue for a Meeor Shower effec, whereby surprises in one marke while he oher is closed show up as soon as he second marke opens. In addiion, by sudying open-o-close agains close-o-open equiy reurns, Lin, Engle and Io (1994) find ha shocks o New York dayime equiy reurns are correlaed wih overnigh Tokyo reurns and vice versa. In he laer case hey conclude ha informaion revealed during he rading hours of one marke has a simulaneous impac on he reurns of he oher marke. Any sudy of volailiy spillovers needs o disinguish beween conemporaneous shocks ha appear lagged because of saggered rading hours, and real-ime lead-lag effecs beween securiy markes (Marens and Poon 2001). Exising empirical research provides evidence of volailiy spillovers boh across and wihin various markes. Our choice of equiy markes (London, Frankfur and Paris) faciliaes invesigaion of larger-smaller marke effecs as well as he inra-regional influences which appear o be srenghening in Europe. In addiion, we resric he sudy - 7 -

8 o synchronous price observaions, avoiding he confusion ha can arise from rading lags. 3. Model Specificaion and Esimaion We build wo bivariae Asymmeric Condiional Correlaion (ADCC) models o capure ime-varying volailiy and asymmeric effecs while also allowing correlaions beween securiy reurns o vary over ime. Recen sudies (Cappiello, Engle and Sheppard 2003, Kearney and Poi 2005) have esablished he imporance of correcly modelling imevarying correlaion, paricularly among European securiy markes. Consider a vecor of reurns for wo equiy markes, r = [ 1 2 ] r r such ha r = c+ u (1) u = Dε, (2) where c is he uncondiional mean vecor of r, D conains condiional sandard deviaions on he main diagonal and zeros elsewhere, ε are he innovaions sandardized by heir condiional sandard deviaions, and se a ime such ha Ψ 1 represens he condiioning informaion ( ε ~ 0R),. (3) Ψ 1 Observe ha E 1 εε = R is also he condiional correlaion marix of he sandardized innovaions. We can herefore specify he condiional covariance marix for he reurns vecor r as ( ) 1 Dε Dε 1 ( )( c) Var( r Ψ 1) = Var 1( r) = E 1 r c r = E = E D εεd, (4) - 8 -

9 and since D is a funcion only of informaion a 1, we can wrie he condiional covariance marix as H Var ( r) 1 = D E εε D 1 =DRD. (5) (6) The elemens of he D marix are he condiional sandard deviaions, where D = h 11, 0 0 h 22,. (7) We use wo specificaions of condiional variances o separaely capure he effecs of asymmeric dynamics and volailiy spillover: Asymmeric 2 : ( ) 2 h = ω+ α + δi u + βh 1 ii, 1 ii, 1 ii, (8) where I 1 uii, < 0 =. 0 uii, 0 Asymmeric wih volailiy spillover: ( ) 2 h = ω+ α + δi u + βh + γu 2 1 (9) ii, 1 ii, 1 ii, 1 jj, where I 1 uii, < 0 = 0 uii, 0 and ii jj

10 Nex we model he condiional correlaion marix R following Cappiello, Engle and Sheppard (2004). From (1) and (2) above, he sandardized residuals can be calculaed as D u = ε, (10) 1 where he elemens of 1 D have been derived from esimaed equaions for each of he formulaions for h ii, above. By using hese sandardized residuals we are able o esimae a condiional correlaion marix of he form: 1 1 = diag diag R Q Q Q (11) where, Q = Q(1 φ η) ϕm+ φε ε + ϕm m + ηq φ ϕ and η are scalar parameers. The vecor = I [ < 0] m ε o ε (where o is he Hadamard produc) isolaes observaions where sandardized residuals are negaive. Noice ha Q resembles a process in he sandardized volailiies. Finally, we implemen variance argeing, where 1 Q = εε and m = m m. Combining 1 T T esimaes for (6) and (10) resuls in a condiional covariance marix for he reurns vecor r which can be used, along wih a vecor of expeced reurns, o predic opimal porfolio weighs -periods ahead: H = D R D. (12) 4. Porfolio consrucion In his sudy, invesors use shor-horizon mean-variance sraegies o creae porfolios from wo equiy marke indices and he (zero-reurn) risk-free asse, relying on

11 forecass of condiional covariance from dynamic models. On one hand, mean-variance porfolios are no ideal for equiy invesors, since hey maximize uiliy only when asse reurns are ellipically disribued, bu on he oher hand, mean-variance modelling is a well-undersood analyic ool ha maps ino he porfolio performance lieraure, is commonly applied in funds managemen pracice, and can be simply adaped o changing levels of risk aversion. 4.1 Weigh selecion A single-horizon invesor chooses porfolio weighs o minimize porfolio variance subjec o a required reurn µ 0. min whw w (13) s.. wµ = µ (14) o deriving an opimal weighing vecor: 1 1 w = H µ, (15) µh µ µ o where µ is an assumed vecor of expeced reurns o be described below, and µ 0 is he required rae of reurn o he porfolio, here se o uniy. H is he expeced (forecased) covariance marix of reurns. We do no impose full invesmen or shor-sales consrains on he porfolio allocaions, so any wealh no accouned for by w is implicily invesed in he risk-free (assumed zero reurn) asse, and he weigh vecor may include negaive values. The individual variance formulaions described by equaions (8) and (9), in combinaion wih he A-DCC correlaion esimaes, generae wo ses of condiional

12 covariance marices for each pair of marke reurns,{ H } 2, where model i=2 includes i= 1 volailiy spillover effecs and model i=1 does no. We forecas H and rebalance he porfolio a daily, weekly (5 days) wo-weekly (10 days) and monhly (20 days) frequencies, using he A-DCC models described above, esing o see if he impac of volailiy spillover apers off over longer rebalancing horizons. 4.2 Expeced Reurns Engle and Colacio (2004) propose a soluion o he problem of forecasing expeced reurns. Expeced reurn esimaion errors are no only usually large, bu also amplified in he mean-variance opimizaion process, causing poor ou-of-sample porfolio performance. Engle and Colacio poin ou ha, for wo-asse porfolios, opimal weighs are funcions of relaive reurns, no of he absolue size of expeced reurn o each asse. Since i is he reurn raio ha maers, a wide specrum of relaive reurns beween wo asses can be mapped ou over he zero-one inerval. By applying heir mehod, we can es for he impac of volailiy spillover on porfolio efficiency wihou joinly esing a peripheral hypohesis abou expeced reurns. We span a wide range of reurns relaives by choosing pairs of expeced reurns as polar co-ordinaes, µ = π j π j sin,cos and allowing j o vary from 0 o 10, { 0 10} j,...,. The resuling values (lised in Table 1 ) range from zero o one for each asse, including a mid-poin where he expeced reurns of boh asses are equal. Combined wih forecas covariance marices i 2 H i= 1, hese eleven expeced reurn pairs k 11 µ k = 1 allow us o compue opimal porfolio weighs from (15). If one condiional covariance model performs beer for all eleven expeced reurns relaives, we can be confiden ha i is a beer model for any choice of reurns vecor

13 [INSERT TABLE 1 HERE] Since comparison beween eleven porfolios is cumbersome we also derive a Bayesian probabiliy for each value of j and compue a probabiliy-weighed summary measure of porfolio risk and reurn. Again following Engle and Colacio (2004), we compue non-overlapping sample means (using 40 observaions) { l l 1, 2} L µ µ, l=1 from he sample daa for each marke pairing. Any mean pair where eiher value is negaive is dropped, leaving a subse of size d = 1,..., D. From his sample we back ou D values of θ d 2 a cos µ 2, d = π 2 2 µ 2, d+ µ 1, d and use hese values of θ o calculae maximum likelihood parameers of he Bea disribuion â and b ˆ. Finally, we infer he empirical probabiliy j j ( sin 20 cos 20 ) k of each pair of he eleven polar co-ordinae reurns µ = π, π by compuing he value aˆ 1 bˆ 1 1 θ (1 ) j θ j j 1 aˆ 1 bˆ 1 Pr θ = θ =. (16) ϒ () d 0 where 1 ( ) ( ) ϒ is a normalizing consan (and 1 a ˆ 1 bˆ Γ a Γ b () 1 d = 0 Γ ( a + b) ) for each pair of markes. Figure 1 graphs he probabiliy densiy funcions for θ compued from his procedure, wih all showing some skewness across he range of relaive reurns. All bu he mos exreme values of θ have some weigh in he densiy, so focusing on he mos likely value may be misleading. [INSERT FIGURE 1 HERE]

14 4.3 Performance measuremen Porfolio performance is a guide o forecasing accuracy, since he bes model of covariance will generae he leas risk. Engle and Colacio (2004) show ha, for a given required rae of reurn µ 0, he porfolio wih he smalles realized sandard deviaion will be a porfolio consruced from he rue covariance marix. We infer ha a covariance forecasing model ha is closer o he underlying daa generaing process (DGP) will predic beer han oher models, and generae lower porfolio risk. So if σ is he porfolio sandard deviaion achieved using he rue covariance marix, and ˆ σ is he sandard deviaion from an inefficienly esimaed covariance marix, hen σ will be less han for ˆ, σ such ha σ ˆ σ <. µ µ o o (17) Consequenly, if including volailiy spillover effecs improves condiional covariance forecass hen porfolios consruced from he beer forecass will have lower realized sandard deviaions. Anoher way of expressing his efficiency gain is by compuing he required rae of reurn we would need in order o mainain a consan risk-o-reward raio while swiching covariance forecass. Le µ * 0 be he required rae of reurn associaed wih he rue covariance marix and ˆµ 0 be he required rae of reurn associaed wih an inefficien covariance marix, and rewrie (17) as an equaliy: σ ˆ σ = (18) µ ˆ µ * 0 0 * where µ 0 < ˆµ 0. Equivalenly we can wrie (18) as:

15 ˆ µ ˆ σ = (19) µ σ 0 * 0 The raio on he lef hand side of equaion (19) measures he addiion o reurns which would compensae he invesor for a less efficien covariance marix. (We repor esimaes of porfolio sandard deviaion raios in Tables 4 6 below.) 5. Empirical Resuls 5.1 Daa and esimaion We esimae 4 he A-DCC models using daily reurns from hree major European sock marke price indices, valued in US dollars: FTSE 100 (London); DAX 30 (Frankfur); and CAC 40 (Paris). Reurns are calculaed as log differences and do no include dividends. No currency hedging is implemened. Trading hours for he London, Frankfur and Paris sock exchanges overlap imperfecly, so o ensure synchronous prices we ake index values a London 16:00 ime (Frankfur and Paris 17:00 ime). 5 The models were esimaed using he firs 2700 observaions of he 3523 size sample, leaving he remaining 823 observaions for esing. The esimaion period runs from 1 January 1992 o 6 May 2002, and predicive power for porfolio formaion is esed over he hree years from 7 May 2002 o 4 July Marens and Poon (2001) poin ou he imporance of synchronous daa for sudies of daily condiional correlaion and volailiy spillover. Subsanial mis-esimaion of reurns correlaion and spillovers can resul from a failure o accoun for iming differences a he daily level. Correlaions will be under-esimaed, and esimaed spillover paerns changed, if non-synchronous daily daa are used in correlaion models. By synchronizing prices we ensure ha esimaed spillovers and correlaions more

16 accuraely expose real-ime ineracions, raher han represening lags in informaion flows, misalignmens in rading, or mismached daa collecion. Table 2 repors key feaures of he daa sample. Average reurns are highes for he DAX 30 index, which also displays he larges sandard deviaion and degree of skewness. The FTSE 100 has annualized reurns around wo per cen lower han he DAX 30 and he leas variance of he hree markes. All hree daily reurns series show considerable non-normaliy manifesed in negaive skewness and excess kurosis. Average skewness is -0.11, and kurosis, [INSERT TABLE 2 HERE] A graph of he daily reurns in Figure 2 shows clusers of volailiy, where groups of large or small changes persis for a number of periods. More frequen periods of urbulence are eviden from 1998 o 2003 (when volailiy begins o drop off) and volailiy paerns appear relaed, as migh be expeced among such closely-aligned equiy markes. [INSERT FIGURE 2 HERE] Table 3 repors esimaes for a oal of six bivariae A-DCC models: wo for each of he hree pairs of reurns series (London-Frankfur, London-Paris and Frankfur-Paris). We compue a benchmark wihou volailiy spillover and an alernaive wih volailiy spillover for each marke pair. (Appendix 1 gives deails of he esimaion mehod.) [INSERT TABLE 3 HERE] The op porion of Table 3 repors parameer esimaes and sandard errors for he variance equaions, and he lower porion repors esimaes of he parameers of he correlaion marices. Wih he excepion of saisically insignifican volailiy spillover parameer from Paris o Frankfur, all parameers have he expeced (posiive) sign. All models show evidence of high levels of volailiy persisence, wih parameers on lagged variables summing o jus below one. Esimaes from he benchmark model (GJR (1,1,1))

17 show significan asymmery effecs ( δ ) in all hree markes. We find ha he asymmeric effec is sronges for he UK marke, dominaing he symmeric volailiy shock componen. In erms of volailiy spillover( γ ), we find significan ransmission from Frankfur and Paris o London, and from Frankfur o Paris, so we observe ha Frankfur is unaffeced by lagged volailiy shocks from he oher markes in his sample. Alhough all volailiy spillover coefficiens are small, Frankfur o Paris shocks are greaes in magniude. Esimaes of volailiy spillover effecs from London o he coninenal markes are posiive, bu smaller and poorly esimaed, a surprising resul given he relaive sizes of he markes. 6 Figure 3 presens graphs of esimaed condiional variance series for he volailiy spillover model. Condiional variances confirm earlier observaions (Figure 2) ha he hree markes have become increasingly volaile since early 1997, possibly in connecion wih he beginning of he Asian crisis. The German marke shows he mos, and he UK marke, he leas, volailiy over he whole sample. 7 [INSERT FIGURE 3 HERE] Condiional correlaion parameer esimaes ( φ, ηϕ, ) for he benchmark and alernaive models differ only slighly. This resul should help us isolae he effecs of volailiy spillovers on he porfolio selecion process. The Frankfur-Paris combinaion displays he mos persisence (η ) in condiional correlaions 8. Asymmeric effecs in condiional correlaions are smaller han heir symmeric counerpars in all hree combinaions, wih he London-Frankfur pair exhibiing he larges asymmeric effec and London-Paris, he smalles. Kearney and Poi (2005) repor weak asymmery effecs for condiional correlaions among Euro-zone equiy markes

18 Figure 4 graphs he condiional correlaion series from he volailiy spillover model, showing ha ime-variaion in condiional correlaion is an imporan feaure of he second-momen dynamics. [INSERT FIGURE 4 HERE] 5.2 Porfolio Sandard Deviaions We forecas from esimaed benchmark and volailiy spillover models, generae prediced covariances i 2 H i= 1 a 1, 5, 10 and 20-sep horizons, and compue opimal porfolio weighs ik, 11 w, k = 1 from equaion (15), for wo equiy markes and he riskfree asse. This procedure simulaes realized porfolio reurns from he remaining (823) observaions of he daa se: ik =w ' r ik π,,. (20) where i = 12, corresponds o he benchmark and alernaive porfolios and k = 1,...,11 indicaes he vecor of expeced reurns. As oulined in Secion 4, we expec he more efficien covariance model o produce a lower porfolio risk for any required reurn. (Here, µ = 1. ) Tables 6-8 se ou realized sandard deviaions for he benchmark and volailiy spillover models for London- Frankfur, London-Paris and Frankfur-Paris, respecively. We repor volailiy raios for daily, weekly, en-day and monhly forecasing and rebalancing horizons. In each row, we se he leas sandard deviaion equal o 100, and hen repor he larger sandard deviaion as a proporional increase over he smaller. The las row in each column repors he probabiliy weighed average of he whole column of sandard deviaions, where he weighing applied o each row is given by he Bayesian probabiliies associaed wih each reurn relaive for ha daa. (These are graphed in Figure 1.) For example, in Table 4, o

19 which gives he sandard deviaions for he London-Frankfur marke pairing, he las row under 10-seps-ahead forecass shows ha he porfolio sandard deviaion for he benchmark model was 1.52 per cen bigger han he sandard deviaion for porfolios compued using he volailiy spillover model. 9 On a weighed-average basis, he volailiy spillover model performs beer han he benchmark a every forecas horizon, and for all marke pairs. [INSERT TABLES 4, 5 AND 6 HERE] In erms of economic value, he relaive efficiency gains are no large. The greaes efficiency gain for he volailiy spillover model on a weighed-average basis is for he 5- sep-ahead forecas model for London-Frankfur, where he benchmark model sandard deviaion is , meaning ha neglecing volailiy spillover effecs increases porfolio risk by abou 1.52 per cen of sandard deviaion. Or, in erms of risk-adjused reurns, if invesors who allow for volailiy spillover σ are receiving 10 per cen reurns ( µ = 10), hen invesors who forecas using he benchmark ( ˆ ) σ would need ˆ µ = per cen reurns o equalize he reurn o risk raio such ha * 0 µ ˆ0 µ ˆ σ σ =. The efficiency gains o predicing covariance using he volailiy spillover model hus represen risk-free reurn improvemens around 15 basis poins on a en per cen reurn porfolio. Neverheless hese small efficiency improvemens do no disappear a longer forecas horizons, as can be seen from weekly, fornighly and monhly porfolio sandard deviaions. In fac as Figures (in Appendix 2) sugges, gains seem o peak beween weekly and monhly forecasing horizons before hey sar o diminish a longer horizons where he forecass converge o uncondiional values

20 5.3 Diebold-Mariano Tess We es he saisical significance of any risk reducions by he Diebold and Mariano (1995) mehod for disinguishing beween forecased volailiies. The Diebold-Mariano es saisic is derived from he esimaed difference beween realised variance for he benchmark symmeric and alernaive asymmeric models, calculaed as k 1, k 2 2, k 2 =, v π π (21) forming 11 series for each marke pairing, k 11 v k = 1. Under he null hypohesis he expeced value of k 11 v k = 1 is zero, such ha including volailiy spillover effecs in covariance models does no reduce porfolio variance 10. We conduc a join es of his null hypohesis using a GMM esimae of he parameer β from he regression V = βι+ ε. We firs sack all values of k 11 v k = 1 and esimae a single momen condiion for he coefficien β. We also consruc a sysem of eleven momen condiions, one for each v, again resricing he sysem o a single esimae of β. We k repor -ess of he null hypohesis ha β = 0, using robus Newey-Wes sandard errors from he GMM esimaion. Table 7 repors resuls for each marke pairing and forecas horizon. The majoriy of ess of β (including shor-horizon forecass for London and Paris) rejec he null hypohesis and confirm ha porfolio variances are significanly lower when volailiy spillover is modelled in he condiional covariance marix. Bu he volailiy spillover model does no ge unqualified suppor, wih significan negaive values for β a he longer horizon ess of he London-Paris pair. [INSERT TABLE 7 HERE]

21 5.5 Sochasic dominance ess Tess for second-degree sochasic dominance can ell us wheher risk-reducions are likely o maer o any risk averse invesor. Consider wo samples of porfolio reurns M M { Y } =1 and { X } = 1wih cumulaive disribuions (CDFs) G and F. Second degree I I I I sochasic dominance (SD2) esablishes he condiions under which any risk averse agen prefers one porfolio o anoher: Porfolio Y will be preferred o porfolio X by any agen whose uiliy over porfolio reurns U ( π ) obeys U ( π ) 0, U ( π ) 0 when π π G() d F() d for all o π. o Barre and Donald (2003) derive a Kolmogorov-Smirnov syle es for sochasic dominance of any degree, evaluaing he CDFs a all poins in he suppor. This echnique avoids he problem of choosing an arbirary se of comparison poins which can resul in inconsisency. 11 The null hypohesis o be esed is ha G (weakly) dominaes F o he second degree, agains he alernaive ha i does no. From random samples of equal size, he es saisic is given by: M ( ) sup( ( ) ( ; )), (22) 2 12 / ˆ = I ˆ 2 2 π; G I2 π ˆ M F M π S M M 1 1 where I 2 ( π ; Gˆ ) = 1( Y )( ) 2 ( ˆ i Yi I M) 1( Xi )( ) M F i M π π, π ; = M π π X, and i= 1 i= 1 1( ) is he indicaor funcion, reurning he value 1 when ( X i π ) and zero oherwise. Under he null hypohesis, he es saisic is no greaer han zero. Bald comparisons beween CDFs or heir inegrals are subjec o non-rivial sampling error when he populaion densiy is unknown, so we need some approximaion o he sampling disribuion, here derived by block boosrapping

22 We follow Linon, Maasoumi and Whang (2002), and Lim, Maasoumi and Marin (2004), and adjus he boosrapping mehod o keep underlying serial dependence inac. Block size is se a B = 28 where B = α T, α is a posiive consan and T is sample size, here Each se of porfolio reurns is divided ino overlapping blocks of size B, hen a random selecion is made, choosing sufficien (conemporaneous) blocks o creae a disribuion of size es saisic. T. Boosrap samples are used o build an empirical disribuion of he We repor resuls for one-sep-ahead forecass and wo-seps-ahead forecass, since 5 and oher muli-sep forecasing generaes samples oo small for reliable esing. We conduc he es on a weighed average of reurns o he k porfolios, where weighs are he Bayesian probabiliies shown in Figure 1. Resuls in Table 8 show ha he null hypohesis ha he benchmark model dominaes he volailiy spillover model can be rejeced in all bu one of six ess. So we can infer ha in five of six cases, he volailiy spillover forecasing model is preferred by risk averse invesors. [INSERT TABLE 8 HERE] 6. Conclusions Recen advances in modelling ime-varying second momens have highlighed an array of feaures in securiy reurns volailiies ha were previously overlooked. Among hese, volailiy spillovers are boh significan and widespread, well-idenified in a large number of sudies across a range of securiy markes and geographic locaions. However he economic imporance of any aspec of ime series dynamics, including volailiy spillovers, depends no on wheher i can be saisically idenified, bu on wheher i can aler invesmen oucomes

23 In his sudy we value volailiy spillovers for invesors who selec meanvariance equiy porfolios from sock markes in London, Frankfur and Paris. We isolae he porfolio risk reducions ha can be aribued o adding volailiy spillovers o asymmeric dynamic condiional correlaion forecasing models. ADCC models capure boh ime-variaion and asymmery effecs in variance and correlaions, allowing us o idenify volailiy spillovers in a nesed model. We also minimize he impac of expeced reurn choice on ou-of-sample porfolio efficiency by combining covariance forecass wih a full range of assumed expeced reurns relaives using polar co-ordinaes. Porfolio efficiency gains due o volailiy spillover effecs are small, bu significan, measurably reducing sandard deviaions over 1, 5, 10 and 20-sep horizons in he majoriy of cases. In addiion, sochasic dominance ess confirm ha in five of six cases, risk averse invesors will prefer porfolios ha allow for volailiy spillover effecs in covariance forecass. On a porfolio reurning, say 10 per cen p.a., efficiency gains arising from modelling spillovers ranslae o risk-free reurn improvemens close o 0.15 per cen, wihou addiional ransacions coss

24 Appendix 1 A-DCC Esimaion We follow Engle (2002) and esimae he models in wo seps. Assuming ha he sandardized residuals are condiionally normally disribued so ha ε ~ N 0R),, he log likelihood funcion for he vecor of reurns, can be Ψ 1 expressed as ( ε T 1 L = n π log ( 2 ) log H uh u (1.1) = 1 r Now le he mean parameers, c, and he univariae GARCH parameers in D be represened by ψ, and he condiional correlaion parameers in likelihood can be wrien as he sum of a volailiy par and a correlaion par: R, by ζ. The log (, ) = ( ) + ( ) L ψζ L ψ L ζ ψ, (1.2) where he volailiy erm is V C T 1 2 LV( ψ) = nlog( 2π ) + 2log D + ud u, (1.3) 2 = 1 and he correlaion componen is L T 1 = εε + R + εr ε. (1.4) 1 ( ζ ψ ) log C 2 = 1 The procedure is furher simplified by recognizing ha he volailiy par of he log likelihood is jus he sum of he individual univariae GARCH likelihoods:

25 2 1 ( ) = T n ui LV ψ log( 2π ) + log h, 2 i +. = 1 i= 1 h i, (1.5) The wo-sep esimaion mehod involves maximizing each univariae GARCH erm separaely, sandardizing he reurns by esimaed sandard deviaions and hen joinly esimaing elemens of R by maximizing he correlaion componen of he log likelihood L ( ψζ, ). We maximize log likelihoods numerically using he Max SQP C procedure in OX 3.4. This procedure implemens a sequenial quadraic programming echnique o maximize a non-linear funcion subjec o non-linear consrains. Alhough he assumpion of normaliy in ε is convenien for esimaion, i is no necessary for consisency, since quasi-maximum likelihood argumens apply as long as he condiional mean and variance equaions are correcly specified (Hamilon, 1994, p.126). However he sandard errors need o be adjused according o he mehod described for he univariae GARCH volailiy equaions. Sandard errors for he correlaion parameers require a more complicaed process explained in Engle (2002)

26 Appendix 2 Figures graph volailiy raios for differen forecasing horizons (no all repored in he paper) and show he relaive risk reducion as horizon increases. Tables of volailiy raios for addiional forecasing horizons are available from he auhors on reques. [INSERT FIGURES HERE]

27 References Abhyankar, A.H. (1995), Trading round-he-clock: Reurn, volailiy and volume spillovers in he Eurodollar fuures markes, Pacific-Basin Finance Journal, 3, Apergis, N., and Reziis, A. (2001), Asymmeric cross-marke volailiy spillovers: evidence from daily daa on equiy and foreign exchange markes, Mancheser School, 69, Apergis, N., and Reziis, A. (2003), Agriculural price volailiy spillover effecs: he case of Greece, European Review of Agriculural Economics, 30(3), Baillie, R., and Bollerslev, T. (1990), A mulivariae generalized ARCH approach o modelling risk premia in forward foreign exchange rae markes, Journal of Inernaional Money and Finance, 9(3), Baele, L, (2003), Volailiy spillover effecs in European equiy markes, Ghen Universiy Working Paper, Ghen. Bekaer, G., and Wu, G. (2000), Asymmeric volailiy and risk in equiy markes, Review of Financial Sudies, 13(1), Barre, G.F., and Donald, S.G. (2003), Consisen ess for sochasic dominance, Economerica, 71(1), Bekaer, G., and Harvey, C. (1997), Emerging equiy marke volailiy, Journal of Financial Economics, 43(1), Billio M., and Pelizzon, L. (2003), Volailiy and shocks spillover before and afer EMU in European sock markes, Journal of Mulinaional Financial Managemen, 13 (4/5),

28 Black, F. (1976), Sudies of sock price volailiy changes, Proceedings of he 1976 Meeings of he American Saisical Associaion, Business and Economical Saisics Secion, Bollerslev, T., and Wooldridge, J.M. (1992), Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariances, Economeric Reviews, 11, Campbell, J.Y., and Henschel, L. (1992), No news is goods news: An asymmeric model of changing volailiy in sock reurns, Journal of Financial Economics, 31, Cappiello, L., Engle, R.F., and Sheppard, K. (2004), Asymmeric dynamics in he correlaions of global equiy and bond reurns, European Cenral Bank Working Paper No.204, European Cenral Bank, Frankfur. URL: hp:// Chrisiansen, C. (2003), Volailiy-spillover effecs in European bond markes, Working Paper Series No. 162, November 2003, Cenre for Analyical Finance, Aarhus School of Business, Universiy of Aarhus, Aarhus. Chopra, R., and Ziemba, W. (1993), The effec of errors in means, variances and covariances on opimal porfolio choice, Journal of Porfolio Managemen, 6-11 Conrad, J., Gulekin, M., and Kaul, G. (1991), Asymmeric predicabiliy of condiional variances, Review of Financial Sudies, 4, Diebold, F.X., and Mariano, R.S. (1995), Comparing predicive accuracy, Journal of Business and Economics Saisics, 13(3), Engle, R.F. (2002), Dynamic condiional correlaion - A simple class of mulivariae GARCH models, Journal of Business and Economic Saisics, 20(3), Engle, R.F., and Colacio, R. (2004), Tesing and valuing dynamic correlaions for asse allocaions, unpublished manuscrip, New York Universiy, New York

29 Engle, R.F., Io, T., and Lin, W. (1990), Meeor showers or hea waves? Heeroskedasic inra-daily volailiy in he foreign exchange marke, Economerica, 58(3), Eom, Y., Subrahmanyam, M., and Uno, J. (2002), The ransmission of swap spreads and volailiies in he inernaional swap markes, Journal of Fixed Income, 12(1), Glosen, L., Jagannahan, R., and Runkle, D. (1993), On he relaionship beween he expeced value and he volailiy of he nominal excess reurn on socks, Journal of Finance, 48(5), Hamao, Y., Masulis, R., and Ng, V. (1990), Correlaions in price changes and volailiy across inernaional sock markes, Review of Financial Sudies, 3, Hamilon, J. D. (1994), Time Series Analysis, Princeon Universiy Press, Princeon. Harju, K., and Hussain, S.M. (2005) Inraday linkages across inernaional equiy markes, Unpublished manuscrip, Deparmen of Finance, Hanken-Swedish School of Economics and Business Adminisraion, Vasa, Finland. Kalenhauser, B. (2002), Reurn and volailiy spillovers o indusry reurns: does EMU play a role?, Cener for Financial Sudies Working Paper No , an der Johann Wolfgang Goehe-Universia Frankfur am Main. Kearney, C., and Poi, V. (2005) Correlaion dynamics in European equiy markes, Research in Inernaional Business and Finance, (forhcoming). King, M., and Wadhwani, S. (1990), Transmission of volailiy beween sock markes, Review of Financial Sudies, 3, Koumos, G. (1992), Asymmeric volailiy and risk rade off in foreign sock markes, Journal of Mulinaional Financial Managemen, 2,

30 Lim, G.C., Maasoumi, E., and Marin, V.L. (2004), Discouning he equiy premium puzzle, unpublished manuscrip, Economics Deparmen, Universiy of Melbourne, Melbourne. Lin, W., Engle, R., and Io, T. (1994), Do bulls and bears move across borders? Inernaional ransmission of sock reurns and volailiy, Review of Financial Sudies, 7, Linon, O., Maasoumi, E., and Whang, Y-J. (2002), Consisen ess for sochasic dominance: a subsampling approach, Cowles Foundaion Discussion Paper no. 1356, Cowles Foundaion for Research in Economics, Yale Universiy, New Haven. Marens, M., and Poon, S.-H. (2001), Reurns synchronizaion and daily correlaion dynamics beween inernaional sock markes, Journal of Banking and Finance, 25, Miyakoshi, T. (2003), Spillovers of sock reurn volailiy o Asian equiy markes from Japan and he US, Inernaional Financial Markes, Insiuions & Money, 13, Nelson, D. (1991), Condiional heeroskedasiciy in asse reurns: A new approach, Economerica, 59, Ng, A. (2000), Volailiy spillover effecs from Japan and he U.S. o he Pacific-Basin, Journal of Inernaional Money and Finance, 19, Pan, M., and Hsueh, P. (1998), Transmission of sock reurns and volailiy beween he U.S. and Japan: evidence from sock index fuures markes, Asia-Pacific Financial Markes, 5, Pardo, A., and Torro, H. (2003), Trading wih asymmeric volailiy spillovers, Working paper, Universia de Valencia, Valencia

31 Poon, S., and Taylor, S. (1992), Sock reurns and sock marke volailiy: An empirical sudy of he U.K. sock marke, Journal of Banking and Finance, 16, Reyes, M.G. (2001), Asymmeric volailiy spillover in he Tokyo Sock Exchange, Journal of Economics and Finance, 25(2), Theodossiou, P., and Lee, U. (1993), Mean and volailiy spillovers across major inernaional sock markes: furher empirical evidence, Journal of Financial Research, 16, Wei, K.C.J., Liu, Y.J., Yang, C.C., and Chaung, G.S. (1995), Volailiy and price change spillover effecs across he developed and emerging markes, Pacific-Basin Finance Journal, 3, Wu, G. (2001), The deerminans of asymmeric volailiy, Review of Financial Sudies, 14(3),

32 Table 1: Pairs of expeced reurns Range of expeced reurns used o calculae porfolio weighs where j µ(1) µ(2) θ πj πj µ = sin,cos

33 Table 2: Summary saisics- daily sock index reurns, % p.a. Daily reurns calculaed as r = 100ln( p / p 1) from price indices synchronized a London 16:00 ime, 2 January 1992 o 4 July All indices are in USD, unhedged. Daa supplied by Daasream. FTSE 100 DAX 30 CAC 40 Mean Sd. Dev Skewness Kurosis Jarque-Bera Observaions

34 Table 3: Parameer esimaes, A-DCC models. Columns show esimaed parameers for GJR ADCC and GJR-ADCC wih Volailiy Spillover condiional covariance models. P-values are in brackes. GJR and GJR(volailiy spillover) equaions were compued for every marke using de-meaned reurns, and hen sandardised residuals were used o compue esimaes for he ADCC and ADCC(volailiy spillover) models. Esimaed over 2700 daily reurns, sampling 2/1/1992 6/5/2002. Parameer London-Frankfur London-Paris Frankfur-Paris GJR (1,1,1) GJR (1,1,1) Volailiy spillover GJR (1,1,1) GJR (1,1,1) Volailiy spillover GJR (1,1,1) GJR (1,1,1) Volailiy spillover UK DE UK DE UK FR UK FR DE FR DE FR ω (0.079) (0.028) (0.018) (0.048) (0.079) (0.099) (0.044) (0.220) (0.028) (0.099) (0.042) (0.079) α (0.304) (0.001) (0.894) (0.002) (0.304) (0.006) (0.862) (0.063) (0.001) (0.006) (0.004) (0.887) β (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) δ (0.000) (0.042) (0.000) (0.053) (0.000) (0.011) (0.000) (0.020) (0.042) (0.011) (0.036) (0.0018) γ (0.033) (0.819) (0.077) (0.374) (0.803) (0.026) φ η ϕ

35 Table 4: Porfolio sandard deviaions, London Frankfur Noes: Smalles porfolio sandard deviaion for each pair of expeced reurns is scaled o 100. Values over 100 represen proporional increases in sandard deviaions. The final row is a weighed average of he preceding rows where weighs are he Bayesian probabiliies repored in Figure 1. One-sep-ahead forecass Five-seps-ahead forecass Ten-seps-ahead forecass Tweny-seps-ahead forecass J VOLATILITY SPILLOVER VOLATILITY SPILLOVER VOLATILITY SPILLOVER VOLATILITY SPILLOVER

36 Table 5: Porfolio sandard deviaions, London Paris Noes: Smalles porfolio sandard deviaion for each pair of expeced reurns is scaled o 100. Values over 100 represen proporional increases in sandard deviaions. The final row is a weighed average of he preceding rows where weighs are he Bayesian probabiliies repored in Figure 1. One-sep-ahead forecass Five-seps-ahead forecass Ten-seps-ahead forecass Tweny-seps-ahead forecass J VOLATILITY SPILLOVER VOLATILITY SPILLOVER VOLATILITY SPILLOVER VOLATILITY SPILLOVER

37 Table 6: Porfolio sandard deviaions, Frankfur Paris Noes: Smalles porfolio sandard deviaion for each pair of expeced reurns is scaled o 100. Values over 100 represen proporional increases in sandard deviaions. The final row is a weighed average of he preceding rows where weighs are he Bayesian probabiliies repored in Figure 1. One-sep-ahead forecass Five-seps-ahead forecass Ten-seps-ahead forecass Tweny-seps-ahead forecass J VOLATILITY SPILLOVER VOLATILITY SPILLOVER VOLATILITY SPILLOVER VOLATILITY SPILLOVER

38 Table 7: Diebold-Mariano ess for porfolio variance equaliy. GMM esimaes of coefficiens and robus p-values for he es ha difference in porfolio variances (u) is joinly zero for all expeced reurns. An aserisk indicaes rejecion a he 1% (***), 5 % (**) or 10 % (*) level. Significan posiive values for β indicae ha porfolio variances are less under he volailiy spillover model, negaive values indicae ha hey are more. Single momen condiion Muliple momen condiions Marke pairing 1 sep ahead 5-seps ahead 10 seps ahead 20-seps ahead 1 sep ahead 5-seps ahead 10 seps ahead 20-seps ahead London Frankfur 0.009** (0.02) 0.152* (0.07) 0.315** (0.01) 0.174* (0.10) 0.002** (0.02) 0.060*** (0.00) 0.122* (0.06) (0.00) London - Paris ** (0.05) (0.34) (0.44) (0.72) 0.001* (0.10) 0.031*** (0.00) -0.02*** (0.00) *** (0.00) Frankfur - Paris 0.010** (0.021) (0.34) 0.291*** (0.01) 0.552* (0.10) 0.006*** (0.00) 0.032*** (0.00) 0.159*** (0.01) 0.386*** (0.00)

39 Table 8: Sochasic Dominance relaions, one-sep-ahead and wo-seps-ahead forecass. Boosrapped P-values for ess of second degree sochasic dominance relaions beween pairs of porfolio reurns where porfolios are formed on he basis of one- or wo-sep-ahead forecass from he benchmark and volailiy spillover models. Porfolio reurns are a weighed average over all values of θ where weighs are he Bayesian probabiliies repored in Figure 1. An aserisk indicaes rejecion a he 1%(***), 5 % (**) or 10 % (*) level when he reverse null is no rejeced. Failure o rejec boh nulls is inconclusive. Marke pairing Volailiy Spillover dominaes Benchmark 1-sep-ahead Null Hypohesis Benchmark dominaes Volailiy Spillover Volailiy Spillover dominaes Benchmark 2-seps ahead Null Hypohesis Benchmark dominaes Volailiy Spillover London Frankfur * * London Paris *** ** Frankfur Paris 0.06* ***

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