Título artículo / Títol article: Re-examining the risk-return relationship in Europe: Linear or non-linear trade-off?

Transcription

1 Tíulo arículo / Tíol aricle: Re-examining he risk-reurn relaionship in Europe: Linear or non-linear rade-off? Auores / Auors Enrique Salvador Aragó, Chrisos Floros, Vicen Aragó Manzana Revisa: Journal of Empirical Finance, 24, 28: 6-77 Versión / Versió: PDF Preprin Auors Cia bibliográfica / Cia bibliogràfica (ISO 69): url Reposiori UJI: SALVADOR, Enrique; FLOROS, Chrisos; ARAGO, Vicen. Re-examining he risk reurn relaionship in Europe: Linear or non-linear radeoff?. Journal of Empirical Finance, 24, 28: hp://doi:.6/j.jempfin

2 Re-examining he risk reurn relaionship in Europe: linear or non-linear rade-off? Absrac This paper analyzes he risk reurn rade-off in Europe using recen daa from European sock markes. Afer relaxing linear assumpions in he risk-reurn relaion by inroducing a new approach which considers he curren sae of he marke, we are able o obain posiive and significan evidence for a risk-reurn rade-off for low volailiy saes; however, his evidence urns ou o be lower or even non-significan during periods of high volailiy. Mainaining he linear assumpion over he risk-reurn rade-off leads o non-significan esimaions for all cases analyzed. These resuls are robus among counries despie he condiional volailiy model used. This concludes ha he conroversial resuls in previous sudies may be due o srong linear assumpions when modeling he risk reurn rade-off. We argue ha his previous evidence can only be viewed as parial evidence ha fails o cover he global behavior of he relaion beween reurn and risk. Keywords: non-linear risk-reurn radeoff, pro-cyclical risk aversion, Regime-Swiching GARCH, Regime-Swiching MIDAS, risk premium

3 . Inroducion One of he mos discussed opics in financial economics is ha of esablishing a relaionship beween reurn and risk. Several aemps have ried o explain he dynamics and ineracions beween hese wo fundamenal variables. From a heoreical framework, one of he mos cied works analyzing his risk reurn rade-off is he Meron s (973) ineremporal capial asse pricing model (ICAPM). Meron (973) demonsraes ha here is a linear relaionship beween condiional excess marke reurn and is condiional variance, and is covariance wih invesmen opporuniies: r A BX () 2 M f M M, S 2 where M rf is he excess reurn of he marke porfolio over he risk-free asse, is he condiional variance of excess marke reurns (known as idiosyncraic porfolio risk), X M, S is he condiional covariance beween excess marke reurns and he sae variable ha represens he invesmen opporuniies (known as he hedge componen), and A and B are he prices of hese sources of risk. Assuming risk-averse invesors, his model esablishes a posiive relaion beween expeced reurn and marke variance (risk). However, despie he imporan role of his rade-off in he financial lieraure, here is no clear consensus abou is empirical evidence. Campbell (987), Glosen e. al (993), Whielaw (994) and Brand and Wang (24) find a negaive relaion beween hese variables, while oher auhors such as Ghysels, e. al. (25), Leon e. al. (27), Guo and Whielaw (26), Ludvigson and Ng(27) and Lundblad (27) find a posiive rade-off. This paper analyzes he risk reurn rade-off in European counries (, France, Spain, he Unied Kingdom, Swizerland, he Neherlands, Belgium, Denmark, Finland, Sweden and Greece) and ries o shed ligh on he conroversial resuls abou is sign and magniude. We use differen assumpions when modeling condiional volailiies (GARCH and MIDAS approaches) and relax he srong linear assumpion (usually made in previous sudies) by inroducing a Markov Regime-Swiching process. This non-linear mehodology helps us condiion our esimaion upon he curren sae of he marke obaining differen relaionships beween reurn and risk during periods of high and low volailiy. To he bes of our knowledge, his is one of he firs aemps using non-linear models such as Regime-Swiching MIDAS (RS-MIDAS) for analyzing he risk-reurn relaion. In he heoreical framework, all he parameers (he risk prices A and B in ()) and he variables (he sources of risk and X, ) are allowed o be ime varying. However, o make he model 2 M M S empirically racable one should make several assumpions; he mos common is consan risk prices (Goyal and Sana-Clara, 23; Bali e al., 25). I is also necessary o assume specific dynamics for he condiional second momens represening he marke risk. The mos used are he GARCH models (Bollerslev, 986). Finally, he empirical model is esablished in a discree ime economy insead of he coninuous ime economy used in he equilibrium model of he heoreical approach. Anoher common assumpion is ha we consider one se of invesmen opporuniies consan over ime, for example by reaining marke risk as he only source of risk 2 M The firs aemp o use hese models is very recen (Guérin and Marcellino (2)) and i is a specificaion wih poenial applicaions o a large class of empirical sudies in applied economics and finance ye o exploi. 2 Few papers such as Scruggs (998) and Scruggs and Glabadanidis (23) show ha he lack of empirical evidence of a risk-reurn rade-off is due o he omission of he hedge componen. However,

4 (Glosen e al., 993; Shin, 25; Lundblad, 27). In his paper, we follow hese sudies and analyze he effec of marke reurns given one risk facor defined by he condiional marke volailiy. Given he assumpions menioned above, many papers have inroduced alernaive empirical models o obain favorable evidence following he heoreical inuiion. The mehodology mos commonly used in he empirical analysis of he risk reurn rade-off is he GARCH-M approach (Engle e al., 987). This framework is simple o implemen bu he resuls obained are conroversial. Many sudies fail o idenify a saisically significan ineremporal relaion beween risk and reurn of he marke porfolio (see Baillie and Di Gennaro, 99; Campbell and Hendchen, 992). A few sudies do provide evidence supporing a posiive risk reurn relaion (Bollerslev, 986; Guo and Neely, 28). Several sudies even find ha he ineremporal relaion beween risk and reurn is negaive (examples include Nelson, 99; Li e al., 25). Therefore, alernaive approaches o he simple GARCH-M mehodology have been proposed when analyzing he risk reurn rade-off. The mos imporan frameworks developed as alernaive o GARCH models essenially obain differen esimaions for condiional volailiy. Whielaw (994) uses an insrumenal variable specificaion for he condiional second momens. Harrinson and Zhang (999) use non-parameric echniques in heir sudy in opposie o he parameric approaches used more ofen. Ghysels e al. (25) propose he use of differen daa frequencies o esimae he mean (wih lower daa frequency) and he variance (wih higher daa frequency) equaions. Despie he differences among all he models presened, hey share a srong linear (monoonic) assumpion in he definiion of he relaionship beween reurn and risk. Recenly, Muller e al. (2) use he basic and asymmeric Coinegraed- GARCH (COGARCH) approach o es he Meron s hypohesis. They argue ha he asymmeric COGARCH model is no supporive of Meron s hypohesis, while he symmeric version of COGARCH shows a significan posiive covariance beween he marke risk-premia of boh he CRSP value weighed and equal-weighed excess marke reurns and volailiies over he period However, Meron s model is no he only heoreical approach explaining he risk-reurn relaionship. Whielaw (2) proposes a non-linear relaionship beween reurn and risk based on an equilibrium framework. This heoreical framework is quie differen from Meron s (973) approach because a complex, non-linear, and ime-varying relaionship beween expeced reurn and volailiy is obained. Similarly, Mayfield (24) employs a mehodology in which saes of he marke are essenially defined by volailiy regimes and condiion he riskreurn rade-off upon hese differen saes. Oher auhors also draw alernaive frameworks where is no expeced a monoonic risk-reurn-relaionship (Veronesi, 2) and even some of hem (see Abel,988; Backus and Gregory, 992) develop heoreical models ha suppor a negaive risk-reurn relaion. The main resul obained in our paper is ha a non-linear specificaion is necessary o reflec he posiive and significan rade-off beween reurn and risk. When several volailiy saes are considered, he risk reurn relaionship becomes significan, even ignoring possible changes in he se of invesmen opporuniies. When linear paerns in he risk specificaion (GARCH and MIDAS) are considered, no significan relaionship in any marke is obained. More hey do no find clear evidence a all. Some alernaive approaches use informaion no only abou he marke porfolio bu also abou addiional risk facors such as oher asse porfolios or macroeconomic indicaors, hereby exending heir empirical models o a mulidimensional framework (see Ludvigson and Ng, 27; Bali, 28).

5 specifically, a posiive and significan rade-off beween reurn and risk is obained for low volailiy saes when non-linear paerns are considered (RS-GARCH and RS-MIDAS models). However, for high volailiy saes he magniude of his relaionship becomes lower or nonsignifican. These resuls are robus for all he sock indexes analyzed and show ha he lack of empirical evidence in previous sudies may be due o he srong assumpion of a linear risk reurn relaionship raher han a non-linear one revealing he perils of using linear frameworks o analyze empirically his rade-off. These resuls shed ligh on he conroversial resuls obained in previous sudies using linear models abou he sign and magniude of his relaionship. They also could explain why resuls from linear models appear no o be robus o he sample period used in he analysis. We argue ha sudies using linear models analyzing a sample period corresponding o a low volailiy sae are more likely o find a posiive risk-reurn radeoff, while sudies ha include episodes of crisis or high volailiy are more likely o find a negaive or insignifican rade-off. In boh cases, he conclusions can only be viewed as parial evidence since he omission of non-lineariies may misrepresen he evidence obained. Anoher ineresing resul in he paper is relaed o he magniude of he marke risk price in each regime. During low volailiy periods he risk price is higher han during high volailiy periods. Alhough his resul may seem agains he heoreical inuiion claiming for higher reurns under more volaile markes, here are oher auhors using differen mehodologies who reach a similar conclusion (see Bliss and Panigirzoglou,24; Kim and Lee, 28; and Rossi and Timmerman, 2). The principal conribuions of our paper are as follows. Firs, we sudy he risk-reurn relaion for European sock markes insead of US daa which is more widely used in previous sudies. Second, we develop an empirical framework wih a non-linear risk reurn rade-off by using Markov-Swiching processes for differen specificaions of he condiional variance (RS- GARCH and RS-MIDAS). Third, we show ha a posiive and significan risk reurn rade-off is obained for all European markes analyzed afer considering non-lineariies independenly of he variance specificaion. Besides, we obain a posiive rade-off higher in magniude during low volailiy periods han during high volailiy periods where he relaionship is even nonsignifican or negaive in some cases. Finally, we show he evoluion of he risk premium in Europe during he recen years, including he recen period of he global financial crisis. The remainder of his paper is srucured as follows. Secion 2 describes he daa used in he sudy and develops he mehodology. Secion 3 repors and analyzes he main resuls obained. Secion 4 repors a baery of robusness sudies confirming he main conclusions reached. Finally, secion 5 summarizes. 2. Daa and mehodology For he empirical analysis of he paper, we employ daily sock exchange indexes from European counries 3 for he period Augus 99 - May 22. The sample includes daa from DAX (), CAC (France), IBEX35 (Spain), FTSE (Unied Kingdom), SMI (Swizerland), AEX-Index (The Neherlands), BEL2 (Belgium), OMXC2 (Denmark), OMXH25 (Finland), OMXS3 (Sweden) and Ahex2 (Greece). As a robusness es, we also employ daa from he US marke (SP5 index). These daa allow us o calculae daily and weekly reurns for he same period. Alhough he main conclusions of he paper are reached using weekly daa, we also need daily observaions when esimaing he condiional volailiy of (RS / asymmeric) MIDAS models. All he index daa is obained from Thomson DaaSream. 3 Some counries such as Ialy are no included in he analysis because he main sock index has been changed during he sample period leading o an irregular evoluion of heir quoaions.

6 Only few auhors (e.g., Theodossiou and Lee, 995; Li e al., 25; Guo and Neely, 28) have invesigaed he risk-reurn relaion in inernaional sock markes, alhough a sudy considering a wider selecion of counries could help resolve he puzzling resuls obained from U.S. daa. In his paper, we comprehensively analyze he paerns followed by his relaion in he main European sock markes. Because risk-free ineres rae daa are no available o all financial markes under consideraion over he examined period, sock marke volailiy is measured based on sock reurns insead of excess sock reurns (which is equal o sock reurns minus he risk-free ineres rae). Many researchers (Baillie and DeGennarro, 99; Nelson, 99; Choudhry, 996; Li e al., 25) argue ha such a pracice produces lile difference in esimaion and inference in his line of research. All hese auhors sae ha here is virually no difference in eiher he esimaed parameers or he fied variance. 4 In he nex subsecions we develop he mehodology proposed for all empirical models used o analyze he risk reurn rade-off. 2.. Sandard GARCH The firs approach is he radiional GARCH-M model of Engle e al. (987). This framework is he mos used in he financial lieraure o sudy he risk reurn rade-off despie he puzzled resuls from previous sudies. In his approach he mean equaion is defined as follows: r c h ~ N(, h ) (2) where r is he marke reurn, h is he condiional variance, and represens he innovaions, which are assumed o follow a normal disribuion. The condiional volailiy is obained using a sandard GARCH specificaion as in Bollerslev (986): h z z ~ N(,) (3) h h (4) 2 where, and are parameers o esimae and guaranees he saionariy of he process. We esimae his firs model using he quasi maximum likelihood (QML) funcion of Bollerslev and Wooldridge (992), which allows us o obain robus esimaes of sandard errors: T 2 2 2h L ln f r, ; where f r, ; 2 h e (5) However, his approach has no presened favorable evidence on he significance of a risk-reurn radeoff in many previous sudies, such as Baillie and De Gennaro (99), Glosen e al. (993), Shin (25), and Leon e al. (27). Some auhors argue ha he condiional volailiy using his GARCH-M mehodology has almos no explanaory power for realized reurns and ha could be he reason of he non-significan resuls (see Lundblad, 27). Oher auhors claim ha he conroversial resuls are due o wrong modeling of condiional volailiy (see Ghysels e. al, 25; Leon e al., 27) MIDAS regression Recenly, a new mehodology has been developed o capure a significan relaionship beween reurn and risk using daa from differen frequencies o obain expeced reurns and variances, namely he MIDAS (mixed daa sampling) regression (Ghysels e al., 25). They find evidence of a significan posiive rade-off beween reurn and risk and sae he advanages of his mehodology regarding GARCH models. MIDAS models allow he esimaion of smooh 4 Following he insighful suggesions from he referees, we also include a robusness es in se cion 4 considering he risk-free rae for a few represenaive counries.

7 expeced reurn series using low frequency daa and he esimaions of more variable condiional variances using higher frequency daa. We use his specificaion wih weekly reurns ( r ) combined wih D=25 daily lag squared 2 reurns ( R ) o obain he weekly variance. The mean equaion of his model is similar o Equaion 2 wih condiional variance as a explanaory variable for he expeced reurns: r c VAR ( r ) ~ N(, VAR( r )) (6) However, he MIDAS esimaor of weekly condiional variance is no obained hrough a 2 GARCH parameerizaion bu from a funcion of D lag squared daily reurns ( R ): D d 2 VAR( r ) k, k, d R (7) 2 where k, k, d 2 exp D i d exp k d k d 2 2 k i k i 2 2 is he funcion which measure he impac of each lag daily reurn in he variance formaion 5. Assuming normaliy in reurns r ~ N c VAR( r ), VAR( r ), we esimae his model by maximizing he Bollerslev Wooldridge QML funcion, as in Equaion Asymmeric case The symmeric models presened above can easily be exended o he asymmeric case in which he variance responds more heavily afer negaive reurns han i does afer posiive reurns (leverage effec). For he GARCH specificaion, we add a new variable min, in he variance process using he asymmeric GJR model (Glosen e al., 993). These models are esimaed in a similar way o ha presened above, subsiuing Equaions 4 for 9. h h (9) 2 2 We esimae he MIDAS model for he asymmeric case subsiuing Equaion 7 for Equaion : D D 2 2, 2, d d 2, 2, d d () Var r k k d r k k d r d d where, k, k 2, k, k 2 are he parameers o be esimaed and d, d are he indicaor funcions for d r d, respecively. We use Equaion 5 again o esimae hese models. r and 2.4. Non-linear models Some auhors claim ha srong linear assumpion for he risk-reurn relaionship could lead o misleading resuls since imposing his condiion may bias he evidence on his relaionship. To overcome his limiaion, some works develop alernaive heoreical frameworks which assume a more flexible relaionship beween reurn and risk even proposing a non-monoonic relaionship over ime (Rossi and Timmerman, 2). (8) 5 Ghysels e al. (27) develop several weigh funcions for he MIDAS esimaor, bu owing o is racabiliy, he Almon Lag specificaion is he mos frequenly used in he lieraure. 6 Alhough some auhors esimae his specificaion by using non-linear leas squares, Ghysels e al. (25) use he QML esimae in heir original paper.

8 So, in he nex subsecions we develop he mehodologies o analyze if here is a non-linear riskreurn rade-off in he European markes. An explanaion for he conroversial resuls in previous sudies may lie in he wrong specificaion for he relaionship beween risk and reurn which follows non-linear raher han linear paerns. Therefore, an insighful exension is o consider non-lineariies in his rade-off agains he linear framework usually implemened. In order o provide robusness o our resuls, we inroduce non-linearies assuming wo forms for he condiional volailiies. As a resul, RS-GARCH and RS-MIDAS models are developed; heir specificaions are given below (for more deails, see also Appendix A). a) Regime Swiching GARCH model RS-GARCH specificaion is based on he model originally proposed by Hamilon (989); i allows us o disinguish beween differen volailiy saes governed by a hidden sae variable ha follows a Markov process. In his model, he mean equaion is no exacly as shown in Equaion 2 because i is sae-dependen: r c h ~ N(, h ) (), s s s, s, s, s, s where r s,, h s,, and s, are he sae-dependen reurns, variances, and innovaions respecively and s (sae ) or 2 (sae 2). The sae-dependen innovaions follow a normal disribuion, wih wo possible variances depending on he sae of he process. The sae-dependen variances are modeled as in Equaion 4 following a GARCH parameerizaion, bu allowing differen parameers depending on he sae 7. in his case., h, z z ~ N(,) (2) s s h h (3 2, s s s The shifs from one sae o anoher are governed by a hidden sae variable following a Markov process wih a ransiion marix: Pr s s p Pr s s 2 ( q) P Pr s 2 s ( p) Pr s 2 s 2 q Because of his sae-dependence, he model is economerically inracable 8. We mus, herefore, obain sae-independen esimaes of variances and innovaions. We use he recombinaive mehod presened in Gray (996) which assuming condiional normaliy in each regime, uses he definiion of uncondiional variance in reurns mainaining he naure of he GARCH process: (4) ,,,,,2,2,,,,2 h E r E r h h (5) In order o obain sae-independen errors we use he definiion of uncondiional error:,,,,2 r (6) 7 Following Capiello and Fearnley (2) o faciliae convergence, he consan variance erm is no allowed o swich beween regimes. 8 See e.g. Gray (996) and Dueker (997). The main problem is derived from he fac ha saedependence increases exponenially he size of he likelihood funcion.

9 where h and are he sae-independen variances and innovaions, c h s and: mean equaion, s s s, s for a given sae qps p P s ; 2 ; (7) is he ex-ane probabiliy, Where 2,, (8) And he ex-pos (filered) probabiliies are defined as: Ps k ; s k, 2 k f r s k, ; f r s k, ; s k, (9) for k =, 2 s, is he condiional We esimae his model by maximizing he QML funcion of Bollerslev Wooldridge (992) weighed by he filered probabiliy of being in each sae: 2, T 2 2 2hs, ln,, ;, ; 2, sk (2) s L f r where f r s h e k b) Regime Swiching MIDAS model Furher, he Markov Swiching MIDAS incorporaes regime-swiching in he parameers of he mixed daa sampling models (MIDAS) and allows for he use of mixed-frequency daa in Markov-Swiching models. The reason o inroduce his model is o see he role of nonlineariies in an alernaive variance specificaion o GARCH modeling. The modeling of a Regime-Swiching MIDAS model for our purposes analyzing he risk-reurn rade-off is drawn as follows. We define he sae-dependen mean equaion using weekly s reurns ( r, ) which are explained by a sae-dependen consan and a ime-varying saedependen condiional variance using D daily lag squared reurns ( R ); herefore, he mean equaion of his model is: r c VAR( r ) ~ N(, VAR( r )) s, s s s,, s, s s, In his case, alhough he MIDAS esimaor of weekly condiional variance is again a funcion of D=25 lag squared daily reurns ( R ), we le he weigh parameers o swich among saes: D 2 VAR( r ) k, k, d R (22), s, s 2, s d d where k,, k2,, d s s D exp i exp funcion for a given sae s, 2. 2 k d k d, s 2, s, s 2, s 2 k i k i 2 (2) 2 s (23) is he sae-dependen weigh To esimae his model we need he help of Bayesian inference in a similar way explained in he RS-GARCH model. Seing ha he ransiion probabiliy marix, he ex-ane probabiliies and he ex-pos (filered) probabiliies are defined as in equaion (7), (8) and (9) respecively, we can esimae his model by maximizing he following QML funcion.

10 T 2 ln s,, ;, ; 2 (, ) k s L f r where f r s VAR R e k 3. Empirical resuls 2 2, s 2 VAR( Rs, ) This secion displays he esimaions of he condiional mean and volailiy of sock reurns using he models from he previous secion and i discusses he relaionship beween reurn and risk in Europe. Firs, we show he main resuls using our proposed linear models: GARCH and MIDAS. Second, we analyze if he inroducion of an asymmeric effec on volailiy has any impac when analyzing he risk-reurn rade-off. Third, we provide a discussion abou he resuls obained afer relaxing he linear assumpion on he risk reurn rade-off hrough he use of Regime-Swiching models. Finally, we sudy he evoluion of he marke risk premium in all he sock markes considered over he examined period (during he las weny years). 3. Esimaions for linear models We iniially discuss he resuls of he models presened in secion 2. (GARCH) and 2.2 (MIDAS). Alhough hese wo models advocae for a linear relaionship beween reurn and risk, hey are quie differen in heir consrucion and esimaion mehodologies for he condiional variance as i is explained in secion 2. The main resuls for hese models in all he sock markes considered are shown in Table. [INSERT TABLE ] The resuls for he GARCH model (lef side of able ) are qualiaively similar o hose presened in par of he lieraure (Glosen e al., 993; Shin, 25; Leon e al., 27). The resuls indicae a non-significan relaionship beween reurn and risk suggesing here is no linear relaion beween marke reurn and marke risk. This confirms he puzzled resuls of previous sudies which are incapable o provide clear evidence of his rade-off. Furhermore, he variance parameers presen he ypical paerns repored in he lieraure wih a high persisence of he GARCH erm (he persisence varies beween 95% and 99.9% depending on he counry). This fac has led some auhors (Lameroux and Lasrapes, 99; Marcucci, 25) o consider his as a sign for he exisence of differen regimes for he variance process. They sugges ha if hese regime shifs are ignored, GARCH models end o overesimae persisence in periods of financial insabiliy and underesimae i in calm periods. Furher, he righ side of Table shows he resuls obained using he MIDAS mehodology. The main difference beween his and previous models is he way we obain condiional volailiies. We use differen daa frequencies o esimae expeced reurns (weekly daa) and variances (daily daa), as explained in he previous secion. The risk price parameer is again non-significan for his kind of models. The resuls fail o provide a posiive and significan relaionship beween reurn and risk as i would be expeced. Our resuls are differen from hose of previous sudies using his mehodology which obain favorable evidence using his mehodology. These differences may be due o he use of mixed daily and weekly daa, whereas mos sudies use mixed daily (variance) and monhly (reurns) daa, see Ghysels e al. (25) and Leon e al. (27). However, he consideraion of he las financial crisis period (pos-28) in he empirical analysis could blur he evidence of a monoonic risk-reurn rade-off in a linear framework. The variance esimaions using his specificaion also indicae a high degree of persisence because a grea number of daily lags are needed o accuraely esimae he variance. In almos all counries, he weighs corresponding o lag reurns superior o 3 days represen (24)

11 more han he 4% of he oal volailiy. This emphasizes he imporance of considering more han a monh of daily reurns for measuring MIDAS condiional variance. In conclusion, neiher sandard GARCH models nor sandard MIDAS are able o show a significan rade-off beween reurn and risk in Europe. 3.2 Esimaions for asymmeric case We now discuss he resuls of he models presened in secion 2.3 (Asymmeric GARCH and MIDAS). There is evidence in he lieraure (Nelson, 99; Glosen e al., 993) of a differen effec of shocks of differen sign in he volailiy formaion. Negaive shocks usually presen a greaer impac on volailiy known as leverage effec. The inclusion of his effec on he condiional volailiy esimaion could have effecs on he sign and significance of he riskreurn rade-off observed in Europe. The esimaions for he asymmeric models including he leverage effec for all he sock markes considered are shown in Table 2. [INSERT TABLE 2] The resuls for he asymmeric GARCH model (lef side of able 2) are similar o hose for he symmeric case. The resuls show again a non-significan relaionship beween reurn and risk. The only counry wih a significan rade-off is Greece which exhibis a negaive relaion beween reurn and risk. The variance parameers presen again a high level of persisence bu i is slighly lower han he sandard case (he persisence varies beween 9% and 99.9% depending on he counry). The esimaions for he asymmeric MIDAS also fail o show a significan risk-reurn rade-off for mos of he counries. Using his model we find he abnormal case of Greece wih a negaive and significan risk-reurn rade-off. Regarding he variance persisence, his specificaion le us disinguish beween he impac of posiive and negaive shock and how long hey persis. I is very difficul o give a general inerpreaion of he paerns followed by posiive or negaive shocks for all counries since each marke volailiy seems o follow idiosyncraic paerns. However, in mos of he counries negaive shocks are less persisen han posiive shocks. These findings could be in line wih hose of Marcucci (25) since during periods of marke jiers, here is an increase of he number of negaive shocks; his increase in he number of innovaion reduces heir impac over ime. So, asymmeric GARCH and MIDAS models are no able o show a significan risk-reurn relaionship for Europe during he sample period analyzed confirming he incapabiliy of linear models and quesioning he heoreical framework supporing hem. Therefore, i seems imporan o go beyond his seup and relax he srong assumpion of a linear risk-reurn radeoff Esimaion for non-linear models The resuls repored in he previous secions do no suppor he linear assumpion of he riskreurn rade-off. Even Meron (98) remarks ha his relaionship does no have o be linear. For his reason, we inroduce in he previous models a Regime-Swiching process which relaxes he linear assumpion aken when analyzing he risk-reurn rade-off by condiioning our resuls o saes of high and low volailiy. Nex we show he resuls for our wo volailiy specificaions in he case we inroduce a non-linear risk-reurn rade-off; i.e. we discuss he resuls obained from RS-GARCH and RS-MIDAS specificaions.

12 a) Regime-Swiching GARCH model Table 3 presens he esimaions for he non-linear model assuming a GARCH process in he volailiy formaion (RS-GARCH 9 ). The resuls for his model le us shed ligh o he dynamics followed by he risk-reurn relaion. In paricular, we can associae sae wih low volailiy periods and sae 2 wih high volailiy periods using he medians of he esimaed volailiy in each sae. For s, corresponding o he low volailiy sae, here is a significan posiive relaionship beween reurn and risk for almos all counries (a % for, France, Spain,, Swizerland, Belgium, Denmark and Sweden; a 5% for Finland and a % for he Neherlands). The only counry wih no significan relaionship beween reurn and risk is Greece. [INSERT TABLE 3] However, when we look a he resuls for he sae s 2 we obain less evidence for a posiive and significan relaionship beween reurn and risk. Only for, Spain and Sweden he relaionship is significan a 5% and in Denmark, Belgium and France a %. In he res of he counries he rade-off beween hese wo variables is no significan during high volailiy saes. Besides, and even more ineresing, he risk price coefficien during high volailiy saes ( s 2 ) is lower han i is for he low volailiy regime. This finding is no consisen wih he spiri of he heoreical linear models ha sugges ha higher volailiy should be compensaed wih higher reurns. However, some papers such as Lundblad (27), Kim and Lee (28), and Rossi and Timmerman (2) repor he same evidence. This fac indicaes ha in high volailiy periods he invesor s risk appeie is lower. One poenial explanaion for his resul may be due o he exisence of a differen risk price depending on he volailiy regime. An invesmen considered oo risky in calm periods (low volailiy) is less risky when here is a period of marke insabiliy wih more uncerainy and any invesmen involving risk. This finding could also be explained by invesors characerisics in high volailiy saes. In hese periods, more risk-averse invesors leave he marke, leing less risk-averse invesors adjus he price of risk according o heir less demanding preferences (Bliss and Panigirzoglou, 24). Auhors such as Kim and Lee (28) find a pro-cyclical behaviour in he invesor risk appeie. During low volailiy periods invesors are more relucan o ake risk and during high volailiy periods invesors have more will o accep he same risk. However, a recen sudy developed by Rossi and Timmerman (2) shows ha he risk-reurn rade-off may follow non-monoonic paerns. These auhors sae ha a low-medium levels of condiional volailiy here is a posiive risk-reurn rade-off bu his relaionship ges invered a high levels of volailiy. Our resuls seems o suppor hese sudies observing a srong evidence of a posiive risk-reurn radeoff during periods of low volailiy bu he evidence is differen depending on he counry analysed during periods of high volailiy. The persisence of he GARCH erm during low volailiy saes is higher han he observed for high volailiy saes. This fac confirms he evidence from he lieraure (Marcucci, 25). He 9 Following he insighful suggesion from one anonymous referee we check he gains from adding regime swiches in he consan parameer c compared o he case where he only parameer swiching would be he one enering before he condiional variance. The corresponding LR ess shows ha he bes model is he second one. Therefore, we show in able 3 he resuls of his specificaion. Resuls of his analysis are available from auhors upon reques. For breviy, he able conaining his informaion is no displayed in his version of he paper bu is available upon reques.

13 concludes ha in high volailiy periods here are a higher number of shocks affecing he variance formaion and reducing heir impac over ime. Furher, he persisence is overesimaed in high volailiy periods if regime-swiching is ignored (Marcucci, 25). In addiion, he expeced duraion for he low volailiy sae is approximaely 2 weeks, abou four imes higher han ha obained for he high volailiy sae. Figure shows he smoohed probabiliies 2 of being in sae for he sample period. [INSERT FIGURE ] Ineresingly, he smooh probabiliies of being in low/high volailiy saes are very close wih he economic cycles (boom / crisis) during he sample period and could be associaed wih hem. For example, in figure we can observe clearly he downurn in worldwide economic aciviy around 2 and he effec of he las financial crisis in 28 wih he high volailiy regime becoming he mos imporan during hese periods. Alhough here seems o be a clear link beween marke volailiy saes and business cycles his relaionship does no always hold (Lusig and Verdelhan, 22). Therefore, we should inerpre our resuls as condiioned o he sae of he marke (more risk price during low volailiy periods, less risk price during high volailiy periods) raher han o he sae of he economy (more risk price during boom periods, less risk price during crisis periods). b) Regime-Swiching MIDAS model In he previous secion we repor a posiive risk-reurn relaionship during low volailiy saes under a non-linear specificaion wih GARCH variances. Here, we provide robusness for he claim of a non-linear relaion beween reurn and risk by using an alernaive variance model o GARCH. Table 4 displays he esimaions for he RS-MIDAS model 3 for all counries considered. The inerpreaion of regimes in he MIDAS specificaion is no as sraighforward as in he RS- GARCH specificaion. In his specificaion he esimaor of volailiy does no explicily model a swich in he level of volailiy. So, we inerpre he regime as high and low based on he probabiliy chars of being in a given regime 4. Supporing he resuls of he las secion, during low volailiy saes he relaion beween reurn and risk is posiive and significan for all counries considered. Also, he value for he risk price coefficien is higher han he obained for high volailiy periods. The resuls for high volailiy periods are very differen beween counries alhough for he majoriy of markes he relaionship ges invered in periods of high volailiy. For insance in he Neherlands, Finland and Sweden we obain posiive and We obain he expeced duraion of being in each sae s =,2 as and, respecively. p q 2 The smoohed probabiliy is defined as he probabiliy of being in each sae considering he enire informaion se Ps T ; Ps 2 T ; Ps T ; Ps ; p p Ps ; Ps 2 ; 3 Following he insighful suggesion from one anonymous referee we check he gains from adding regime swiches in he weigh funcion k, k 2 compared o he case where he only parameer swiching would be he one enering before he condiional variance. The corresponding LR ess show ha he bes model is he second one. Therefore, we show in able 4 he resuls of his specificaion. Resuls of his analysis are available from auhors upon reques. 4 We wan o hank an anonymous referee for his commen. The probabiliy chars for he RS-MIDAS models are no displayed in he paper since hey follow very closely he ones for RS-GARCH models. However, hey are available from auhors upon reques.

14 significan esimaion for he risk-reurn rade-off in high volailiy saes (bu of a lower magniude han during low volailiy saes). For markes such as Swizerland he relaionship is no significan. For he res of he markes (, France, Spain, he, Belgium, Denmark and Greece) he rade-off during high volailiy periods is negaive. [INSERT TABLE 4] The resuls for his family of models 5 seem o suppor again he inerpreaions abou he exisence of a pro-cyclical behavior in he invesor risk appeie. The invesors rading during high volailiy periods are more willing o ake risk han he invesors rading during low volailiy periods. They also suppor he exisence of a non-monoonic (and non-linear) relaionship beween reurn and risk depending on he sae of he marke. During low volailiy periods a posiive and significan rade-off is observed, bu his relaionship urns o be differen during high volailiy periods Risk premium evoluion in Europe In his las secion we analyze he evoluion followed by he marke risk premium during he las years in each European marke. The risk premium demanded by he invesors is given by he non-diversifiable risk exising in he marke. Figure 2 shows he risk premium evoluion in Europe during he sample period for wo represenaive counries (he res can be found in he online appendix). The marke risk premium is measured by he ime-varying variable h in our models. For he Regime-Swiching models, we obain he independen esimaion for he condiional variance a each period hrough a weighed average using he filer probabiliies,,,,2 h h h where, is he exane probabiliy of being in he sae and h, for s,2 are he sae-dependen variances. s [INSERT FIGURE 2] These figures represen he weekly marke risk premiums (expressed in basis poins) for all he sample period, he differen mehodologies and he counries considered. The figures show similar paerns for he risk premium evoluion. I seems ha risk exposure is similar for all mehodologies wih sligh differences beween hem. However, he evoluion is quie differen depending on he counry analyzed. Alhough all markes have been hi by a similar crisis, i seems ha in some of hem (Finland a he beginnings 2s, Greece over las years) he effec was worse han in he res. However, all he counries share a huge increase of he demanded risk premium in he recen years coinciding wih he las financial crisis. Table 5 shows he median 6 of he esimaed weekly risk premiums series for all he European sock marke indexes considered. Almos all he obained risk premiums vary beween 2%-4% depending on he counry. Only in he cases of Greece and Finland he demanded premiums exhibis higher values (around 5.5% for Finland and 7% for Greece). The differences in he risk premiums among mehodologies are slighly. Mos of hese premiums are similar han he 3% o 5% obained in oher sudies for US daa (Bali, 28). [INSERT TABLE 5] 5 For all MIDAS specificaions we repeaed he esimaions using a bea funcion insead of an exponenial. The resuls lead o he same conclusion and are available from auhors upon reques. 6 We use he median insead of he mean as a proxy for he average non-diversifiable risk in each period because i is less affeced by ouliers.

15 So, alhough he evoluion of he marke risk premium in Europe has followed similar paerns during he las years for mos of he counries analyzed, here are cerain differences among counries (due o idiosyncraic characerisics of each marke) which lead o differen levels in he demanded risk premium. Especially, counries such as Finland or Greece seem o deparure from he sandard European risk premium. 4. Robusness In his secion we show several addiional analyses for a seleced group of represenaive counries 7 in order o check he robusness of he evidence obained. 4.. US daa Almos all he previous evidence and empirical sudies abou he risk-reurn radeoff are obained using US daa. Alhough he aim of he paper is essenially he European marke, i is always ineresing comparing our resuls o his benchmark marke 8. Table 6 shows he empirical resuls for all he models presened in he paper using US daa. In his case, he proxy used for he marke reurns is he S&P5 index colleced from Daasream for he period Augus 99 May 22. [INSERT TABLE 6] The resuls presened in Table 6 suppor he evidence obained in he European markes. In oher words, we canno find favorable evidence for he symmeric or asymmeric GARCH or MIDAS models. However, when we relax he assumpion of linear radeoff beween reurn and risk, we obain posiive and significan esimaions in low volailiy saes. Again, he esimaed coefficiens during hese periods are much higher han he ones in high volailiy saes. Also ineresingly, he coefficien during high volailiy saes is negaive when considering regimeswiching models suggesing an invered risk-reurn relaion in he US marke a high levels of volailiy; his evidence is in line wih Rossi and Timmerman (2) Excess marke reurns The main heoreical model expressed in Equaion () links excess marke reurns wih risk. For he reasons saed horough he paper, we have been forced o use simple marke reurns in our empirical sudy. In his subsecion we see ha if we accoun for he risk free rae in our reurns series we do obain almos he same resuls. Table 7 shows he esimaion resuls for all he models presened in he paper considering hree represenaive markes (, and ). Excess marke reurns are simply consruced by subracing he marke reurns o he risk free rae. The choice of he proxy for he risk free rae is he local 3-monh T-bill suiable compounded a he corresponding frequency. [INSERT TABLE 7] If we compare he resuls from Table 7 o he ones using simple marke reurns we can only observe sligh differences in he esimaed coefficiens. The conclusions abou he sign and significance of he relaionships are clearly mainained. Given his evidence,, we are confiden o claim ha he no consideraion of he risk-free rae when consrucing he excess marke 7 The choice of he counries in hese analyses is due o daa limiaions in some of he variables used and o he aim of mainaining breviy. 8 We wan o hank he commens from one anonymous referee which le us became aware of an independen and simulaneously wrien paper by Ghysels, Guérin and Marcellino (23) which also analyse regime-swiches in he risk-reurn radeoffs for he US marke.

16 reurns series does no lead o differen conclusions abou he risk-reurn rade-off. This is in line wih Baillie and DeGennarro (99), Nelson (99), Choudhry(996) and Li e al. (25) Condiioning o macro variables Anoher assumpion aken in mos of he previous empirical sudies analyzing reurn and risk, considers one se of invesmen opporuniies consan over ime leaving he marke risk as he only source of risk. A sandard criique in he esimaion of he risk-reurn relaion is his lack of condiioning variables (Scruggs, 998; Whielaw, 26) o conrol for some predicors. As a furher robusness analysis o our resuls we run a series of models of he form: r c VAR( r ) X ~ N(, VAR( r )) (25) for linear models and 2 r c VAR( r ) X ~ N(, VAR( r )) (26) for non-linear models, s s,, s, s 2, s, s, s, s Where he variance specificaions follow he symmeric, asymmeric and regime-swiching GARCH and MIDAS dynamics presened in he paper for a seleced group of counries (, and ). The conrol variables represening X in Equaions 25 and 26 are he sock marke dividend yield series, he local 3-monh T-bill, he local -year Governmen Bond and he yield spread beween he year and he 3-monh raes 9. For he sake of breviy, we place he resuls of his analysis in an online appendix. As a brief overview, we argue ha he resuls do no change significanly he conclusions reached so far. We obain significan evidence for low volailiy saes when non-linear models are considered; however, his is no rue when imposing a linear srucure. We encourage he readers o visi he online appendix for furher deails Daa frequency Selecing he daa frequency for boh excess reurns and marke variance is an issue ha previous sudies do no give a clear answer. While some papers prefer low frequency observaions free of shor-erm noise o deec his radeoff (Guo and Whielaw, 26; Lundblad, 27), oher papers use weekly or even daily observaions (Guo and Neely, 28). One of he advanages of he MIDAS specificaion presened in his paper is ha i allows for using a mixure of differen frequencies for he esimaion of he mean and he variance equaion. In he previous secions of his paper, we displayed a mixure beween weekly and daily observaions. In his final subsecion we repor he resuls obained when a monhly frequency is considered for he marke reurns in he mean equaions (Equaion 6 and 2), while observaions a a daily frequency are used for modeling he variance equaion (Equaions 7 and 22). We also esimae he remaining GARCH models using monhly observaions. All hese addiional analyses can be found in he aforemenioned online appendix. Again, he main resuls of he paper hold for all counries Conclusion This sudy proposes new evidence o he well-known conroversy abou he empirical relaionship beween reurn and risk. From he basis of he heoreical works explaining his rade-off, he ineracion beween hese wo variables is esed empirically using European 9 We ener hese predicors in he risk-reurn regressions boh sequenially and joinly (see online appendix for deails). 2 The resuls for he RS-GARCH specificaion are weaker since using monhly observaions in a GARCH conex leads o a sample of 267 observaions which may presen some problems relaed o small sample properies.

17 sock markes under wo main differen frameworks. The firs framework considers a linear relaion beween reurn and risk, while he second one relaxes his assumpion and allows for non-linear dynamics. Linear empirical models show a non-significan evidence of his basic rade-off. However, when his srong linear assumpion is relaxed, we are able o idenify a significan relaionship beween expeced reurn and risk. One of our claims is ha he risk-reurn rade-off presens differen paerns depending on he sae of he marke. The dynamics of his relaion observed during low volailiy saes (which suppors he heoreical inuiion) are differen from hose observed during high volailiy saes. This fac leads o a non-monoonic relaion over ime which is oally agains he linear assumpion made in many previous sudies. One of he oher main resuls of our paper is ha i provides a relaionship beween volailiy regimes and aiudes owards risk. The risk price level in sock markes ends o be higher in low volailiy saes and lower in high volailiy saes. The invesor profile in each conex may have an influence on his lower risk appeie during high volailiy periods. During low volailiy periods invesors are more relucan o assume risk bu during high volailiy periods hey have more will o accep he same levels of risk. The evidence obained in he paper is also robus for US daa, when condiioning he esimaions o addiional predicors and o differen daa frequencies. Above all, hese resuls highligh he perils of srong linear assumpions when analyzing he ineracions beween reurn and risk and sugges ha previous sudies using linear models were likely o fail on he aemp o capure he global behavior beween hese wo variables.

18 APPENDIX A All he models are esimaed by maximum likelihood. The compuaions of he Regime-Swiching models are carried ou using he Opimizaion Library FMINCON of Malab R2b selecing he BFGS algorihm. Alhough Regime-Swiching GARCH models and Regime-Swiching MIDAS models are differen (as i is discussed in secion 2.4 and 2.5), once he parameerizaion of he variance equaion is defined in each case, he algorihm o implemen is similar for boh specificaions. he ex-ane Le be he vecor of parameers of he differen models; s, Pr k s k is he informaion se up o -); Ps k ; probabiliy of being in he sae k (where filered probabiliy of being in he sae k; and, ; 2, s likelihood vecor (where he main difference beween GARCH 2 parameerizaion of he ime-varying variance h s, ). The algorihm we used is described by he following seps: 2 2, s 2hs, he f r s h e he sae- dependen and MIDAS specificaion is in he ) Give iniial values for he parameers of he model and he ex-ane probabiliies:, s, Pr k s k 2) Implemen Hamilon (989) filering procedure using his firs observaion. P s 2 k ; k f r s k, ; s k, f r s k, ; s k, 3) Compue he value of he log-likelihood funcion for = 2 ln s,, ; k f r k for k =, 2 4) Repea seps 2 and sep 3 for all observaions and compue he log-likelihood funcion unil =T L T 2 ln s,, ; k f r k 5) Maximize he log-likelihood funcion o obain an updae version of he vecor of parameers j : T 2 j j L ln s,, ; k f r k 6) Ierae seps 2-5 wih he updaed parameers unil achieving convergence. Hamilon (989) claims ha his algorihm is a special case of he EM algorihm: he expecaion (E) sep corresponds o sep 2 and he Maximizaion (M) sep o sep 3. During he Expecaion sep he algorihm is able o guess he values for he laen variable given he daa and he up daed parameers while he values of he parameers ha maximize he log-likelihood funcion are driven in he Maximizaion sep. 2 Due o he recursive naure of he GARCH parameerizaion, we use a recombinaive mehod o obain independen variances and errors for each. See secion 2.4 for furher deails.

19 TABLE. Esimaed parameers for GARCH and MIDAS models for all he European sock markes considered Parameer (sd. error) France Spain Swizerland Neherlands Belgium Denmark Finland Sweden Greece c.27 ** (.2).8 (.5).9 (.3). (.).9 ** (.9).23 ** (.9).5 (.).26 ** (.2).25 (.6).27 * (.4).29 * (.6) GARCH-M Persisence c.98 (.42) (.85) (.65) (.22) (.73) (.42) (.75) (.25) (.287) (.57) (.66) ** (.2).2 (.2).28 ** (.2).6 * (.9).9 (.9).9 ** (.9).7 * (.9).28 ** (.).5 (.4).25 ** (.2).3 ** (.5) MIDAS 5 3 > (.68) 5,6% 36,34% 48,6% (.76) 4.74% 38.4% 47.2% (.79) 4.64% 36.62% 48.74% (.84) 4.58% 38.8% 47.34% -.59 (.875) 9.73% 4.38% 39.89% -.66 (.635) 6.34% 36.47% 47.9% (.829) 5.79% 37.9% 47.% -.88 (.929) 4.92% 35.46% 49.62%.228 (.547) 2.75% 35.93% 5.32% (.745) 8.62% 4.97% 39.4% ** (.6) 4.74% 22.6% 73.2% This able shows he esimaed parameers for he sandard GARCH and sandard MIDAS models presened in he paper (robus sandard errors in parenheses). ***, ** and * represen significance a %, 5% and % respecively. The las hree columns represen he percenage of he oal weighs assigned o he -5 firs observaions, 6-3 firs observaions and 3-25 observaions respecively when esimaing he condiional variance in a MIDAS framework.