The Economic Value of Volatility Transmission between the Stock and Bond Markets

Transcription

1 The Economic Value of Volailiy Transmission beween he Sock and Bond Markes Helena Chuliá * Hipòli Torró Sepember 006 Keywords: Volailiy Spillovers, GARCH, Trading Rules JEL Classificaion: C3, C53, G11 * Researcher a he Deparmen of Financial Economics a he Universiy of Valencia. Corresponding auhor Helena Chuliá, Facula d Economia, Universia de València, Avda. dels Tarongers s/n, 460, València (Spain), Tel.: ; Fax: , helena.chulia@uv.es Professor a he Deparmen of Financial Economics a he Universiy of Valencia.

2 Absrac The objecive of his paper is o analyze volailiy ransmission beween socks and bonds in he European marke in an aemp o esablish wheher he observed paern in volailiy can be exploied economically. In order o do so, firsly, we use an asymmeric mulivariae GARCH model ha allows for asymmeries in second momens. Secondly, we design a rading rule o dynamically allocae capial beween equiies and bonds. The resuls indicae ha volailiy spillovers beween boh asses ake place in boh direcions. Moreover, hese volailiy spillovers are economically significan since rading rules based on hem offer profiable reurns afer ransacion coss. This las fac could sugges a failure of he efficien marke hypohesis.

3 1. Inroducion Volailiy linkages beween he sock and bond markes are imporan for invesmen, risk managemen and regulaory policy decisions. According o Fleming e al. (1998), volailiy linkages beween socks and bonds arise from wo main channels. The firs is common informaion: informaion ha affecs boh markes a he same ime [see Harvey and Huang (1991) and Ederingon and Lee (1998)]. The second one is informaion spillover caused by cross-marke hedging. A shock in one asse marke may generae cross-marke asse rebalancing. Therefore, an informaion spillover akes place and his generaes rading and volailiy in boh markes. More recenly, sock marke uncerainy has been provided by Connolly e al. (005) as a key explanaion for he sock-bond correlaion. These auhors use implied volailiies from equiy index opions o reflec sock marke uncerainy, finding a negaive correlaion beween heir uncerainy measure and he fuure correlaion of sock and bond reurns. While here is a vas amoun of works on modeling volailiy spillovers, hese are ofen resriced as hey eiher examine he sock marke or he bond marke separaely 1. Lile aenion has been paid o he ineracion beween he wo markes. Fleming e al. (1998) invesigae he naure of volailiy linkages beween he sock, bond and money markes in he US. They conclude ha volailiy linkages beween he hree markes are srong. Kim e al. (001) carry ou he same analysis for he Ausralian marke obaining similar resuls. Cappiello e al. (003) explore he dynamics and changes in he correlaion of inernaional asse markes. They find ha condiional correlaion beween equiy and bond reurns ypically declines when sock markes suffer from financial urmoil. Wihin he framework of an esimaed mulifacor model, Scruggs and Glabadanidis (003) find, for he US marke, ha condiional sock variance responds asymmerically o boh sock and bond reurn shocks, while condiional bond variance is essenially invarian o sock marke shocks and responds symmerically o bond reurn shocks. Similarly, Goeij and Marquering (004) also find, for he US marke, ha fixed 1 For example, Hamao e al. (1990), Campbell and Henschel (199), Karolyi (1995), Karolyi and Sulz (1996), Kroner and Ng (1998) and Ng (000) iner alia sudy volailiy ransmission beween sock markes and Franses e al. (1997), Scalia (1998) and Chrisiansen (003) iner alia sudy volailiy ransmission beween bond markes. 1

4 income securiies volailiy responds symmerically o is own shocks and show ha asymmeric effecs are presen in he covariance beween socks and bonds. The aim of his paper is o analyze volailiy spillovers beween socks and bonds in he European marke in an aemp o esablish wheher he observed paern in volailiy can be exploied economically. In order o do so, firsly, we use an asymmeric mulivariae GARCH model ha allows for asymmeric volailiy as well as asymmeric covariance. Secondly, we design a rading rule o dynamically allocae capial beween equiies and bonds. Focusing on he inverse relaionship beween expeced reurn and volailiy, Pardo and Torró (006) develop a rading rule ha ry o exploi he asymmeric volailiy spillovers beween large and small firms. Similarly, we design a new rading rule ha ry o ake advanage of he volailiy ransmission beween socks and bonds. Adaping he inuiion behind he sraegies of Pardo and Torró (006), we propose o differeniae beween good volailiy piece of news (an expeced decrease in condiional volailiy) and bad volailiy piece of news (an expeced increase in condiional volailiy). Our sraegy consiss of aking posiions in asse i following he signals given by he volailiy of asse j. In his way, good (bad) volailiy piece of news in he sock marke will imply o ake a long (shor) posiion in he bond marke and he oher way round. The overall resuls of our sudy indicae ha volailiy spillovers beween socks and bonds ake place in boh direcions: a piece of news coming from he sock marke affecs he bond marke volailiy and vice versa, sock marke volailiy is responsive o bond marke reurn shocks. Moreover, hese volailiy spillovers are economically significan since a rading rule based on hem offers profiable reurns afer ransacion coss. This las fac is noeworhy since, as poined by Harris and Pisedasalasai (006), if volailiy spillovers can be used o generae abnormal profis ne of ransacion coss, marke efficiency is no suppored. This sudy innovaes wih respec he exising lieraure in wo ways. Firs, we sudy volailiy ransmission beween socks and bonds in he European markes. As far as we know, previous research on volailiy spillovers beween boh asses focuses on

5 oher markes, especially on he US marke. Moreover, hose papers dealing wih he European markes analyse he dynamics and changes in he correlaion beween socks and bonds and no volailiy ransmission beween hem. The second, and mos imporan conribuion of his aricle, lies in designing a rading rule ha can be applied o differen markes and asses in order o analyse he economic significance of he volailiy spillovers observed beween hem. The remainder of his paper is organized as follows. Secion presens he daa and offers some preliminary analysis. Secion 3 deals wih he economeric approach. Secion 4 discusses he empirical resuls. Secion 5 presens he resuls of he rading rule. Secion 6 provides a brief summary and some concluding remarks. Finally, in Secion 7 bibliographical references are compiled.. Daa As proxies for he bond and sock markes in Europe, we use he Euro Bund fuures conrac and he EuroSoxx 50 index. The Euro Bund fuures conrac daa has been provided by CRB Traders, while he daa on he EuroSoxx 50 index has been exraced from The daa cover he period January, 1991, o January 18, 006 and consiss of daily closing prices of he Euro Bund fuures conrac and he EuroSoxx 50 index. The Euro Bund fuures conrac is he mos acively raded derivaive in he world and has become he main insrumen o hedge long-erm ineres rae risk in he Eurozone, herefore i seems o be a good proxy of he European bond marke. Since he fuures conracs expire 4 imes a year 3, in order o creae a coninuous reurn series we need o swich o a new conrac as he nearby conrac approaches o mauriy. To avoid According o he Fuures Indusry Magazine (Annual Volume Survey, March-April 006), he Euro Bund fuures and he EuroSoxx50 fuures are among he op 10 mos acively raded fuures conracs in he world. 3 The Euro Bund fuures conrac is raded on he Eurex fuures and opions exchange. There are four mauriies per year (March, June, Sepember and December) bu he neares o delivery conrac is he mos liquid one. The Euro Bund fuures conrac requires delivery of a German Federal Governmen longerm noional deb securiy wih years o mauriy and a coupon rae of 6%. For he acual delivery, several real-world bonds are possible and cerain conversions facors are applied. Almos 100% of all posiions held in his fuures conrac are closed before selemen, so ha physical delivery is irrelevan. 3

6 delivery effecs, he rollover in he firs o delivery bund fuures conrac is done he day before o he las rading day. We use fuures o avoid shor sale consrains and microsrucure effecs, bu our analysis generalizes o he underlying spo asse via sandard no-arbirage argumens [Fleming e al. (001)]. Trading is much heavier in he fuures han in he spo marke, and Upper and Werner (004) show ha prices in he fuures and spo marke move ogeher very closely, so he main findings should apply o he bond marke as well. The EuroSoxx 50 index represens he blue-chip socks from he Eurozone and capures approximaely 60% of he free floa marke capializaion of he Dow Jones EuroSoxx Toal Marke Index. In conras o he bond marke, we use he EuroSoxx 50 index insead of he fuures conrac on he EuroSoxx 50 since his conrac sared o be raded in I would be beer o use he fuures conrac on he EuroSoxx 50 because fuures reurns are reurns in excess of he risk-free rae and ransacion coss in he fuures marke are lower. We use weekly frequency in order o overcome problems associaed o auocorrelaion. The reurns are compued as log differences using Wednesday o Wednesday closing prices. If a paricular Wednesday happened o be a non-rading day, we use he closing values recorded on he previous rading day. Figure 1 displays he daily evoluion of he EuroSoxx50 index and he Euro Bund Fuure conrac in he sudied period. Table 1 presens some summary saisics on he weekly reurns. The reurns and volailiies are, as expeced, higher for he sock index han for bond fuures. Boh series presen significan kurosis and he Jarque-Bera saisic rejec normaliy in boh markes. The Ljung-Box es indicaes significan auocorrelaion in boh markes in reurns and in squared reurns. Moreover, he ARCH es reveals ha reurns exhibi condiional heeroskedasiciy. Finally, boh he augmened Dickey Fuller (ADF) and Philips and Perron (PP) ess indicae ha boh series have a single uni roo. These observaions are consisen wih well-documened sylized facs on sock and bond reurns [see Connolly e al. (005) and Scruggs and Glabadanidis (003)]. I mus be highlighed ha he reurns of he bond fuures are small relaive o hisoric reurns on he underlying bonds. Since fuures do no require 4

7 an iniial invesmen heir reurns do no include he risk free rae as compensaion for he cos of an invesmen as i would in he spo marke. On he oher hand, he variance behaviour of bund and is fuures conrac mus be similar. 3. The model The economeric model is esimaed in a hree-sep procedure. Firs, a VAR 4 model is esimaed o clean up any auocorrelaion behavior. Then, he residuals of he model are orhogonalized. These orhogonalized innovaions have he convenien propery ha hey are uncorrelaed boh across ime and across markes. Finally, he orhogonalized innovaions will be used as an inpu o esimae a mulivariae asymmeric GARCH model. Therefore he condiional mean equaions are defined as a VAR(3) process: R 1, R, = µ + 1 = µ + 3 c 1, j j= 1 3 j= 1 c R 1, j R +, j 1, j + 3 j= 1 3 d j= 1 1, j, j d R R, j, j + u 1, + u, (1) where R 1, and R, are he EuroSoxx 50 and Euro Bund fuures reurns respecively, c ij and d ij for i=1, and j=1,, 3 are he parameers o be esimaed and u, µ i 1 and u, are he non-orhogonal innovaions. The VAR lag has been chosen following he AIC crierion. The innovaions u 1, and u, are non-orhogonal because, in general, he = E u u is no diagonal. In order o overcome his problem, in a ' covariance marix ( ) second sep, he non-orhogonal innovaions ( u, 1 and u, ) are orhogonalized ( ε 1, 4 Johansen (1988) es, no repored, shows ha boh series are no coinegraed. 5

8 1 ' 1 andε ). If we choose any marix M so ha M ΣM = I, where I is he ideniy, marix, hen he orhogonalized residuals are obained as: ε = u M 1 () ' and saisfy E( ε ε ) I =. These orhogonalized innovaions have he convenien propery ha hey are uncorrelaed boh across ime and across equaions. Such a marix M can be any soluion of MM ' = Σ. In his sudy we use a srucural decomposiion of he form suggesed by Bernanke (1986) and Sims (1986). In conras o he Cholesky facorizaion, his mehodology does no embody srong assumpions abou he underlying economic srucure. To model he condiional variance-covariance marix we employ a mulivariae GARCH model. The hree mos widely used models are: (1) he VECH model proposed by Bollerslev e al. (1988), () he consan correlaion model, CCORR, proposed by Bollerslev (1990), and (3) he BEKK model of Engle and Kroner (1995). Each model imposes differen resricions on he condiional covariance and gives very differen variance and covariance esimaes. More recenly, Kroner and Ng (1998) have derived anoher mulivariae GARCH model, he Asymmeric Dynamic Covariance Marix model, ADC hereafer. These auhors inroduce asymmeries following he Glosen e al. (1993) approach. An imporan feaure of his model is ha i ness he above models in he sense ha, under cerain resricions, any paricular model can be obained. I mus be noed ha Scruggs and Glabadanidis (003) find ha he ADC model provides he bes fi of condiional second momens for heir sudy of he dynamic covariance beween sock and bond reurns in he US marke. In addiion o his, he ADC model does no impose ad hoc srucures o he correlaion dynamics. The bivariae ADC can be wrien as: 6

9 σ 1, 0 + φ 1 σ 1, θ = σ, 0 φ1 θ 11, 0 11, 0 1 ρ 1 θ11, θ, ρ θ1, θ 11, φ1θ1, + ρ1 = θ, θ, θ θ 0 11,, θ +, (3) where: θ 11, θ θ 1,, ω11 = a + a g + g ω1 b + ω b a a g g 1 1 ' ε 1, ' η 1, 1 b b ε 1 ' σ 1, 1, 1, 1 ε, 1 η ε η 1 1, 1, 1 η, 1 a a σ σ 11 1 g g 1, 1, a a b b 1 g g b b 1 where σ and σ are he EuroSoxx 50 and Euro Bund fuures volailiy respecively, 1,, is he Hadamard produc operaor (elemen-by-elemen marix muliplicaion), ω ij, b ij, aij and ij g for i,j=1, are parameers, ε 1, and, ε are he unexpeced shocks series, η 1, max[0, ε 1, = ] and η, = max[0, ε, ] are he Glosen e al. (1993) dummy series collecing a negaive asymmery from he shocks and, finally, σ ij, for all i,j=1, are he condiional second momen series. The specificaion es proposed by Kroner and Ng (1998) is as follow: (1) if ρ = b = b = a = a = g = g 0, a resriced = asymmeric VECH is obained, () if φ = b = b = a = a = g = g 0, he = asymmeric CCORR model is derived; (3) if φ 1 = 1 and ρ 1 = 0 he asymmeric BEKK model is obained. Equaion (3) allows for boh own-marke and cross-marke influences in he condiional variance, herefore allowing he analysis of volailiy spillovers beween boh markes as well as permiing asymmery in he condiional variance-covariance marix. The parameers of he bivariae ADC sysem are esimaed by maximizing he condiional log-likelihood funcion: 7

10 TN 1 ' 1 L( θ ) = ln( π ) ln H ( θ ) + ε H ( θ ) (4) T ( ε ) = 1 where T is he number of observaions, N is he number of equaions in he sysem and θ denoes he vecor of all he parameers o be esimaed. The log likelihood funcion is esimaed by using he BFGS algorihm via Quasi-Maximum Likelihood Esimaion (QMLE). Bollerslev and Wooldridge (199) show ha he sandard errors calculaed by using his mehod are robus even when he normaliy assumpion is violaed. 4. Resuls Table displays he quasi-maximum likelihood esimaes of he model. The low p-values obained for mos of he parameers show ha he model fis well he daa. As in mos GARCH models, he esimaed parameers, and g i, j for all i,j=1,, c i, j bi, j, ai, j can no be inerpreed individually. Insead, we have o inerpre he non-linear funcions of he parameers which form he inercep erms and he coefficiens of he lagged variances, covariances and error erms. We follow Kearney and Paon (000) and calculae he expeced value and he sandard error of hose non-linear funcions. The expeced value of a non-linear funcion of random variables is calculaed as he funcion of he expeced value of he variables, if he esimaed variables are unbiased. In order o calculae he sandard errors of he funcion, a firs-order Taylor approximaion is used. This linearizes he funcion by using he variance-covariance marix of he parameers as well as he mean and sandard error vecors. Table 3 displays he expeced value and he sandard errors of hese non-linear funcions. I can be seen ha he EuroSoxx 50 volailiy is direcly affeced by is own volailiy ( σ ) bu no by he Euro Bund fuures volailiy ( σ ) 1, EuroSoxx 50 volailiy is affeced by is own shocks ( ) fuures shocks ( ε ). Finally, he coefficiens ( η ) and ( ), 1, 1,,. Ineresingly, he ε as well as by he Euro Bund η are significan, indicaing ha he sock volailiy responds asymmerically o boh sock and bond reurns shock., 8

11 The behavior of he Euro Bund fuures volailiy differs slighly from ha of he EuroSoxx 50. The Euro Bund fuures volailiy is affeced by is own volailiy (, ) bu no by he sock index volailiy ( σ 1, ) affeced by he EuroSoxx 50 shocks ( ε ) and also by is own shocks ( ε ) 1, σ,. Our findings sugges ha he bond volailiy is,. Finally, he bond volailiy responds symmerically o bond shocks ( η, is no significan) and asymmerically o sock reurn shocks ( η 1, is significan). Similarly, Cappiello e al. (003) and Scruggs and Glabadanidis (003) also find ha bond marke condiional volailiy does no exhibi an asymmeric response o is own unexpeced shocks. Overall, he resuls show ha fixed income securiies do no show asymmeric effecs in condiional second momens and ha volailiy spillovers beween sock and bond markes in Europe are asymmeric and bidirecional. In he nex secion we will ry o es if hese spillovers are economically significan in he sense ha we can pu forward a profiable rading sraegy based on hem. Table 4 displays he Wald es for he resricions imposed on he ADC model o obain he encompassed models. Given ha ρ 1 0, φ 1 1 and φ 1 0, all special cases of mulivariae GARCH models nesed wihin he ADC model can be rejeced. As φ 1 0, he asymmery in he condiional covariance is no purely driven by he asymmery in he condiional variance. The residuals analysis displayed in Table 5 shows no evidence of linear and non-linear dependence in he sandardized residuals. Figure displays he condiional annualized variances of he EuroSoxx 50 and Euro Bund fuures reurns. I can be seen ha he bond marke is much less volaile han he equiy marke. The condiional annualized covariance and correlaion beween he EuroSoxx 50 and he Euro Bund fuures reurns, ploed in Figures 3 and 4 respecively, follow a similar paern. Noe ha boh are posiive in he beginning of he sample, become negaive in he subsequen years and increase slighly in he end of he sudied period. According o Connolly e al. (005), here exiss a negaive relaion beween measures of sock marke uncerainy and he fuures correlaion of sock and bond reurns. Since he second par of our sample period coincides wih he European Moneary Union and he Telecom bubble, he negaive correlaion beween boh asses 9

12 classes during his period may be due o he invesor s uncerainy abou hese wo evens. 5. Trading rule In his secion, we ry o exploi volailiy ransmission beween socks and bonds designing a rading rule based on he inverse relaionship beween expeced reurn and expeced volailiy. We disinguish wo kinds of piece of news: bad volailiy piece of news (an expeced increase in condiional volailiy) and good volailiy piece of news (an expeced decrease in volailiy). Our sraegy is a crossed one because i consiss of aking posiions in one asse according o he signals given by he volailiy of a relaed asse. Concreely, if an increasing (decreasing) volailiy spillover is forecased in asse j hen, a rader should sell (buy) asse i. Table 6 displays he rading rule and shows how o ake posiions in he marke in -1 afer any ype of piece of news. I mus be noed ha his rading rule can be applied o differen markes and asses in order o analyse he economic significance of he volailiy spillovers observed beween hem. In order o es he profiabiliy of he sraegy we choose he period Sepember 13h, 000 o January 18h, 006 (80 observaions). Moreover, we also es he profiabiliy of his sraegy in wo sub-periods according o he behaviour of he EuroSoxx 50 index. We disinguish a bearish period from Sepember 13h, 000 o March 3rd, 003 and a bullish period from March 10h, 003 o January 18h, 006. Alhough he model is esimaed by using he EuroSoxx 50 index, he rading rule is implemened wih is fuures conrac since i is more flexible and he daa is available from Anyway, he resuls are similar o hose obained by using he EuroSoxx 50 index. To implemen he dynamic rading sraegy, we generae ou-of-sample one-weekahead forecass for he condiional covariance marix re-esimaing he model each ime 10

13 ha new weekly reurns are known. This process goes on unil he end of he sample is reached 5. To obain some idea of he number of simulaneous volailiy piece of news in he quadrans Q(+,+), Q(+,-), Q(-,+) and Q(-,-), figure 5 displays combined iems of volailiy piece of news on he Euro Bund fuures and he EuroSoxx 50 index. Volailiy piece of news in he quadrans Q(+,+) and Q(-,-) refer o bad and good piece of news since hey reflec an increase and decrease in volailiy, respecively. We see from his figure ha combined volailiy piece of news have many observaions in he quadrans Q(-,+) and Q(+,-). Thus, here is a considerable amoun of volailiy piece of news wih differen sign; when here exiss a good volailiy piece of news (decrease of volailiy) in he Euro Bund fuures, he volailiy piece of news in he EuroSoxx 50 is bad (increase of volailiy) and he oher way round. Moreover, i mus be noed ha combined good volailiy piece of news in boh asses, Q(-,-), are much less common and smaller. Table 7 displays he profiabiliy of he rading rule based on he inverse relaionship beween expeced reurn and expeced volailiy (Panels B, C and D) along wih he buy-and-hold sraegies for he EuroSoxx 50 fuures conrac and he Euro Bund fuures conrac (Panel A) used as benchmarks in boh markes. In order o es he economic value of he rading sraegies we ake ino accoun marke fricions (bid-ask spread and commissions). Profiable sraegies afer ransacion coss are highlighed wih an aserisk 6. 5 We have also esimaed a dynamic condiional correlaion (DCC) form of he mulivariae EGARCH model [see Engle (00)] in order o compare he predicive accuracy of boh models, he ADC and he EGARCH. The mean squared error (MSE) and he mean absolue error (MAE) crieria indicae ha he ADC model ouperforms he EGARCH model. Moreover, he Diebold and Mariano (1995) es rejecs he null hypohesis of equaliy of forecas accuracy a he 5% confidence level. This means ha here is saisical significan difference beween forecass of he wo compeing models. 6 Transacion cos on Euro Bund fuures and Eurosoxx50 fuures are approximaed by a 0.01% and 0.0% per ransacion, respecively. Furhermore, he number of ransacions is lower han he number of observaions since abou 60% of imes he rading rule mainains is long or shor posiion in he fuures markes. 11

14 The resuls show ha, in all cases (6 ou of 6), he oal reurn of he sraegy is posiive, no maer in which period we are. Moreover, he sraegy is profiable even when we accoun for ransacion coss 7. A close look o he resuls shows ha swiching decisions based on volailiy piece of news coming from he Euro Bund fuures seem o be more profiable han hose based on volailiy piece of news coming from he EuroSoxx 50. They bring posiive reurns in he bearish period, when he buy and hold sraegy delivers high negaive reurns. Moreover, in he bullish period hey do no bea he marke bu provide posiive reurns. If we focus on swiching decisions based on volailiy piece of news coming from he EuroSoxx 50 fuures conrac we see ha he sraegy brings reurns similar o hose of he buy and hold sraegy bu i mus be noed we are analysing a period in which he German moneary conex was quie sable. Figures 6A and 6B displays he accumulaive reurn of he rading sraegy and confirm he las resuls. Overall, our resul suppors ha, in a bearish period, relying on signals from he Euro Bund fuures volailiy we can consisenly ouperform he marke. This resul is noeworhy since i means ha in periods of high volailiy and low reurns, he reurn obained by rading on he EuroSoxx 50 index, he European sock marke benchmark, can be improved wih an asse ha reflecs macroeconomic informaion, he Euro Bund fuures. Sock markes around he world are more closely relaed whereas bond markes seem o be more locally influenced. Neverheless, he resul of he rading rule suggess ha he Euro Bund fuures conrac has an informaion role on he EuroSoxx 50 index. This resul has imporan implicaions for porfolio managemen. 6. Conclusions The focus of his sudy deals wih he issue of volailiy ransmission beween socks and bonds in he European marke and wheher we can pu forward a profiable dynamic rading sraegy beween boh asses. 7 The dynamic rading rules have also been implemened by using he forecas volailiy obained wih he mulivariae EGARCH model. The resuls show ha he profiabiliy of he rading rules is smaller when we use hese forecass. Therefore, he more accurae forecas, he more profiable rading rules. 1

15 Firsly, we analyze volailiy ransmission beween boh markes. In order o do so, we use an asymmeric mulivariae GARCH model ha allows for asymmeric volailiy as well as asymmeric covariance. The overall resuls indicae ha volailiy spillovers ake place in boh direcions: a piece of news coming from he sock marke affec he bond marke volailiy and he oher way round, sock marke variance is responsive o bond marke reurn shocks. However, while condiional sock variance responds asymmerically o boh bond and sock reurn shocks, condiional bond variance responds asymmerically o sock reurn shocks bu symmerically o bond reurn shocks. Secondly, we ry o exploi volailiy ransmission beween socks and bonds by using a rading rule wih an innovae approach ha disinguishes bad volailiy piece of news and good volailiy piece of news. This sraegy is based on he inverse relaionship exising beween expeced reurn and volailiy. I consiss of aking posiions in one asse according o he signals given by he volailiy of a relaed asse. Concreely, if an increasing (decreasing) volailiy spillover is forecased in bonds hen, a rader should sell (buy) socks and he oher way round. Furhermore, his rading rule can be applied o differen markes and asses in order o analyse he economic significance of he volailiy spillovers observed beween hem. Our rading rule confirms ha volailiy spillovers are economically significan since i offers profiable reurns afer ransacion coss. However, swiching decisions based on volailiy piece of news coming from he Euro Bund fuures seem o be more profiable han hose based on volailiy piece of news coming from he EuroSoxx 50. Concreely, hey bring posiive reurns in he bearish period, when he buy and hold sraegy delivers high negaive reurns. The finding ha here exis volailiy spillovers beween he EuroSoxx 50 and he Euro Bund fuures and ha a rading rule based upon hem generae economic profis afer ransacion coss is imporan for porfolio managemen decisions and, moreover, i is inconsisen wih marke efficiency. 13

16 7. References Bernanke, B. (1986), Alernaive Explanaions of he Money-Income Correlaion, Carnegie-Rocheser Conference Series on Public Policy 5, pp Bollerslev, T., R. F. Engle, and J. M. Wooldridge (1988), A Capial Asse Pricing Model wih Time Varying Covariances, Journal of Poliical Economy 96, 1, pp Bollerslev, T. (1990), Modelling he Coherence in Shor-Run Nominal Raes: A Mulivariae Generalized ARCH Approach, Review of Economics and Saisics 7, 3, pp Bollerslev, T. and J. M. Wooldridge (199), Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time-Varying Covariances, Economeric Review 11,, pp Campbell, J. Y. and L. Hensschel, (199), No News is Good News: An Asymmeric Model of Changing Volailiy in Sock Reurns, Journal of Financial Economics 31, 3, pp Cappiello, L., R. F. Engle, and K. Sheppard (003), Asymmeric Dynamics in he Correlaions of Global Equiy and Bond Reurns, European Cenral Bank Working Paper Series, 04. Chrisiansen, C. (003), Volailiy Spillover Effecs in European Bond Markes, Aarhus School of Business Working Paper. Connolly, R., C. Sivers, and L. Sun (005), Sock Marke Uncerainy and he Sock- Bond Reurn Relaion, Journal of Financial and Quaniaive Analysis 40, 1, pp Dickey, D. A. and W. A. Fuller (1981), Likelihood Raios Saisics for Auorregresive Time Series wih a Uni Roo, Economerica 49, 4, pp Ederingon, L.H. and J. H. Lee (1993), How markes process informaion: news releases and volailiy, Journal of Finance 48, 4, pp Engle, R. F. and K. F. Kroner (1995), Mulivariae Simulaneous Generalized Arch, Economeric Theory 11, 1, pp Engle, R. F. (00), Dynamic Condiional Correlaion-A Simple Class of Mulivariae GARCH Models, Journal of Business and Economics Saisics, 0, 3, pp

17 Diebold, F. and R. S. Mariano (1995), Comparing predicive accuracy, Journal of Business & Economic Saisics, 13, 3, pp Fleming, J., C. Firby and B. Osdiek (1998), Informaion and Volailiy Linkages in he Sock, Bond and Money Markes, Journal of Financial Economics 49, 1, pp Fleming, J., C. Firby and B. Osdiek (001), The Economic Value of Volailiy Timing, Journal of Finance 56, 1, pp Franses, P.H., R. van Ieperen, M. Marens and B. Menkveld (1997), Volailiy ransmission and paerns in bund fuures, Journal of Financial Research 0, 4, Glosen, L. R., R. Jagannahan and D. E. Runkel (1993), On he Relaion beween he Expeced Value and Volailiy of Nominal Excess Reurn on Socks, Journal of Finance 48, 5, pp De Goeij, P. and W. Marquering (004), Modeling he Condiional Covariance Beween Sock and Bond Reurns: A Mulivariae GARCH Approach, Journal of Financial Economerics, 4, pp Hamao, Y., R., W. Masulis, and V. Ng (1990), Correlaions in Price Changes and Volailiy across Inernaional Sock Markes, Review of Financial Sudies 3,, pp Harris, R. D. F. and A. Pisedasalasai (006), Reurn and Volailiy Spillovers Beween Large and Small Socks in he UK, Forhcoming in Journal of Business Finance and Accouning. Harvey, C. R. and R. D. Huang (1991), Volailiy in he foreign currency fuures marke, Review of Financial Sudies 4, 3, pp Johansen, S. (1988,) Saisical Analysis of Coinegraion Vecors, Journal of Economic Dynamics and Conrol 1, -3, pp Karolyi, G.. A. (1995), A Mulivariae GARCH Model of Inernaional Transmissions of Sock Reurns and Volailiy: he Case of he Unied Saes and Canada, Journal of Business and Economic Saisics 13, 1, pp Karolyi, G.A. and R. M Sulz (1996), Why Do Markes Move Togeher? An Invesigaion of US-Japan Sock Reurn Comovemens, Journal of Finance 51, 3, pp

18 Kearney, C. and A. J. Paon (000), Mulivariae GARCH Modeling of Exchange Rae Volailiy Transmission in he European Moneary Sysem, Financial Review 35, 1, pp Kroner, K. F. and V. K. Ng (1998), Modelling Asymmeric Comovemens of Asse Reurns, The Review of Financial Sudies 11, 4, pp Kim, S., F. In, and C. Viney (001), Modelling Linkages beween Ausralian Financial fuures Markes, Ausralian Journal of Managemen 6, 1, pp MacKinnon, J. G. (1991), Criical Values for Coinegraion Tess, In R.F. Engle and C.W.J. Granger, eds., Long-run Economic Relaionships: Readings in Coinegraion. Oxford: Oxford Universiy Press, Chaper 13. Ng, A. (000), Volailiy Spillover Effecs from Japan and he US o he Pacific- Basin, Journal of Inernaional Money and Finance 19,, pp Pardo, A. and H. Torró (006), Trading wih Asymmeric Volailiy Spillovers, Universiy of Valencia Working Paper. Available a SSRN: hp://ssrn.com/absrac=47081 Phillips, P. C. B. and P. Perron (1988), Tesing for a Uni Roo in Time Series Regression, Biomerica 75,, pp Scalia, A. (1998), Informaion Transmission and Causaliy in he Ialian Treasury Bond Marke, Journal of Empirical Finance 5, 4, pp Scruggs, J. T. and P. Glabadanidis (003), Risk Premia and he Dynamic Covariance beween Sock and Bond Reurns, Journal of Financial and Quaniaive Analysis 38,, pp Sims, C. A. (1986), Are Forecasing Models Usable for Policy Analysis?, Federal Reserve Bank of Minneapolis Quarerly Review 10, 1, pp Upper, C. and T. Werner (004), Time Variaion in he Tail Behavior of Bund fuures Reurns, The Journal of fuures Markes 4, 4, pp

19 8. Tables Table 1. Summary saisics R 1, P-value R, P-value Mean Volailiy Skewness [0.0001] [0.0000] Kurosis [0.0000] [0.0000] Bera-Jarque [0.0000] [0.0000] Q(10) [0.0000] [0.0175] Q²(10) [0.0000] [0.0000] ARCH(10) [0.0000] [0.0000] ADF(4) PP(6) Noes: P-values displayed as [.]. R 1, and R, represen he log-reurns of he Dow Jones EuroSoxx 50 index and he Euro Bund fuures. The Bera-Jarque saisic ess for he normal disribuion hypohesis and has an asympoic disribuion χ (). Q(10) and Q²(10) are Ljung-Box ess for sixh order serial correlaion in he reurns and squared reurns. ARCH(10) is Engle s es for sixh order ARCH, disribued as χ (10). The ADF (number of lags) and PP (runcaion lag) refer o he Augmened Dickey and Fuller (1981) and Phillips and Perron (1988) uni roo ess. Criical value a 5% significance level of Mackinnon (1991) for he ADF and PP ess (process wih inercep bu wihou rend) is

20 Table. Esimaes of he VAR-ADC model Panel (A). Esimaed coefficiens for he VAR model µ (0.03) R , 1 (0.00) R , (0.06) R , 3 (0.00) R , 1 (0.3) R , (0.45) R , 3 (0.45) R 1, R, Panel (B). Esimaed coefficiens for he ADC model (0.06) (0.17) (0.55) (0.15) (0.75) (0.6) (0.01) w 11 w w (0.00) = (0.0) (0.00) b b 11 1 b b (0.00) = (0.40) (0.34) (0.00) a a 11 1 a a (0.00) = (0.00) (0.00) 0.74 (0.00) g g 11 1 g g (0.00) = (0.00) (0.00) (0.56) φ 1 = ρ1 = (0.00) (0.00) This able displays he esimaion of he model defined in equaions (1) o (3) carried ou in a hree-sep procedure. In Panel (A) ordinary leas squares esimaes of he proposed VAR are displayed. In Panel (B), he quasi-maximum likelihood esimaes of he ADC model fied o he orhogonalised residuals are showed. P-values appear in brackes. The necessary condiions for he saionariy of he process are saisfied. The log-likelihood value is

21 Table 3. Resuls of he linearised mulivariae ADC model EuroSoxx 50 condiional variance equaion -5 σ 1, = 1.81x σ 1, σ 1, σ, ε 1, ε 1, -1ε, ε, η 1, η 1, -1η, η, x ( ) (80.885) ( ) ( 0.430) (3.3959) (.3899) (.1858) ( ) (-7.384) ( 5.311) Euro Bund fuures condiional variance equaion σ, = 1.39x10 + x σ 1, σ 1, σ, ε 1, ε 1, 1 ε, ε, η 1, η 1, 1 η, η, x x (1.7754) (0.479) ( ) (9.981) (5.1744) (4.9847) (3.854) (.4939) ( ) Noe: σ 1, and σ, denoe he condiional variance for he EuroSoxx 50 and he Euro Bund fuures reurn series, respecively. Below he esimaed coefficiens are he sandard errors, wih he corresponding -values given in parenheses. (0.87) The expeced value is obained aking expecaions o he non-linear funcions, herefore involving he esimaed variance-covariance marix of he parameers. In order o calculae he sandard errors, he funcion mus be linearized using firs order Taylor series expansion. This is someimes called he dela mehod. When a variable Y is a funcion of a variable X, i.e., Y =F(X), he dela mehod allows us o obain approximae formulaion of he variance of Y if: (1) Y is differeniable wih respec o X and () he variance of X is known. Therefore: V Y X Y X ( Y ) ( Y ) ( X ) V ( X ) When a variable Y is a funcion of variables X and Z in he form of Y = F(X, Z), we can obain approximae formulaion of he variance of Y if: (1) Y is differeniable wih respec o X and Z and () he variance of X and Z and he covariance beween X and Z are known. This is: Y Y Y Y V, X Z X Z Once he variances are calculaed i is sraighforward o calculae he sandard errors. ( Y ) V ( X ) + V ( Z ) + Cov( X Z ) 19

22 Table 4. Tesing resricions for nesed models BEKK (0.00) VECH 6.67x10 (0.00) CCORR 1.11x10 7 (0.00) This able displays he Wald es for he resricions imposed on he ADC model o obain he encompassed models. P-values appear in brackes. The specificaion es proposed by Kroner and Ng (1998) is as follows: (1) If ρ 1 = b1 = b1 = a1 = a1 = g1 = g 1 = 0, a resriced asymmeric VECH is obained; () if φ 1 = b1 = b1 = a1 = a1 = g1 = g 1 = 0, he asymmeric CCORR is derived; (3) if φ and ρ 0 he asymmeric BEKK model is obained. 1 = 1 1 = Table 5. Summary saisics for he sandardised residuals of he model ε 1, / h 11, ε, / h, Q(10) [0.136] [ ] Q²(10) [ ] [0.7347] ARCH(10) [ ] [ ] Noe: Q(10) and Q²(10) are Ljung-Box ess for sixh order serial correlaion in he sandardized residuals and squared residuals. ARCH() is Engle s es for sixh order ARCH, disribued as χ (10). The p- value of hese ess are displayed as [.]. Table 6. Trading rules according o he inverse relaionship beween expeced reurn and volailiy Crossed sraegies on asse i E[ σ Ω 1] > 0 Bad volailiy news j, (Shor asse i ) Crossed sraegies on asse j [ Ω 1 ] > 0 E σ i, (Shor asse j ) Good volailiy news [ Ω 1] < 0 E σ j, (Long asse i ) [ Ω 1 ] < 0 E σ i, (Long asse j ) Noe: σ and σ are he condiional variance of he asse i and asse j respecively, Ω 1 refers o he i, j, informaion se available in -1 and, finally, [ Ω ] E 1 is he expecaion operaor condiioned o informaion available in -1. Below each sraegy appears he posiion aken in -1. 0

23 Table 7. Ex-ane profiabiliy of rading rules Panel (A). Buy and hold sraegies Toal Period 13/09/00-18/01/06 (80 observaions) Firs Sub-period 13/09/00-05/03/03 (130 observaions) Second Sub-period 1/03/03-18/01/06 (150 observaions) EuroSoxx 50 reurn Euro Bund fuures reurn Risk free ineres rae Panel (B). Trading Profiabiliy in he oal period (13/09/00-18/01/06) Type of volailiy news Euro Bund fuures ** EuroSoxx50 *** Bad Good Toal * 33.6 * Panel (C). Trading Profiabiliy in he firs sub-period (13/09/00-05/03/03) Type of volailiy news Euro Bund fuures ** EuroSoxx50 *** Bad Good Toal * 7.99 * Panel (D). Trading Profiabiliy in he second sub-period (1/03/03-18/01/06) Type of volailiy news Euro Bund fuures ** EuroSoxx50 *** Bad Good Toal 5.80 * 5.47 * (*) Indicaes profiable sraegies ne of ransacion coss. These are, approximaely, 1.70, 0.90 and 0.80 in he Euro Bund fuures conrac and 3.10, 1.50 and 1.50 in he EuroSoxx 50 fuures conrac for he oal period, firs sub-period and second sub-period, respecively. (**) This column shows he resul of rading in he Euro Bund fuures conrac afer a volailiy piece of news on he EuroSoxx 50 fuures conrac. (***) This column shows he resul of rading in he EuroSoxx 50 fuures conrac afer a volailiy piece of news on he Euro Bund fuures conrac. 1

24 9. Figures Figure 1. Evoluion of he Euro Bund fuures and he EuroSoxx 50 index in he sudied period Euro Bund Fuure Dow Jones EuroSoxx 50 index Figure. Annualized condiional and uncondiional volailiy of he Euro Bund fuures and he EuroSoxx 50 index in he sudied period Euro Bund Fuures anualised condiional and uncondiional volailiy Dow Jones EuroSoxx 50 anualised condiional and uncondicional volailiy The dashed line corresponds o he uncondiional volailiy while he solid line corresponds o he condiional volailiy.

25 Figure 3. Condiional and uncondiional covariance beween he Euro Bund fuures and he EuroSoxx 50 index in he sudied period Condiional and uncondiional covariance 1,5 1 0,5 0-0,5-1 -1,5 - -, The dashed line corresponds o he uncondiional covariance while he solid line corresponds o he condiional covariance. Figure 4. Condiional and uncondiional correlaion beween he Euro Bund fuures and he EuroSoxx 50 index in he sudied period Condiional and uncondiional correlaion 0,8 0,6 0,4 0, 0-0, -0,4-0,6-0, The dashed line corresponds o he uncondiional correlaion while he solid line corresponds o he condiional correlaion. 3

26 Figure 5. Simulaneous iems of volailiy news 8 6 Euro Bund fuures volailiy piece of news EuroSoxx 50 volailiy piece of news The Y-axis represens Euro Bund fuures annualised volailiy piece of news and he X-axis represens EuroSoxx 50 annualised volailiy piece of news. Figure 6A. Accumulaive reurn of he rading sraegy on Euro Bund fuures Accumulaive reurn rading on Euro Bund fuures Bearish period Bullish period Figure 6B. Accumulaive reurn of he rading sraegy on EuroSoxx 50 Accumulaive reurn rading on EuroSoxx Bearish period Bullish period 4