Country-Specific Idiosyncratic Risk and Global Equity Index Returns

Transcription

1 Counry-Specific Idiosyncraic Risk and Global Equiy Index Reurns C. James Hueng and Ruey Yau Absrac: The idiosyncraic volailiy puzzle arises from he empirical evidence ha socks wih higher pas idiosyncraic volailiies earn lower fuure reurns. Sudies have found ha his puzzle can be explained by cerain ime-series properies of he firm-specific idiosyncraic shocks. In he counry-level marke index daa, however, he puzzle does no exis, which implies ha he ime-series properies of he counry-specific idiosyncraic shocks are differen from hose of he firm-specific idiosyncraic shocks wihin a counry. We find ha he differences are, firs, ha lagged idiosyncraic volailiy is a beer proxy for expeced idiosyncraic risk in he counry-level daa han in he firm-level daa. Second, unlike he firm-specific idiosyncraic skewness, he counryspecific idiosyncraic skewness is no significan enough o play a role in deermining index reurns. Finally, reurn reversals documened in he firm-level daa are no presen in he counry-level index daa. Insead, a momenum effec is found in he counryspecific index reurns. JEL Classificaion: G, G2, G5 Keywords: Idiosyncraic volailiy puzzle; Idiosyncraic skewness; Inernaional asse pricing i

2 . Inroducion The single-facor capial asse pricing model (CAPM) demonsraes ha invesors are able o enjoy he benefi of reducing unsysemaic risk from diversificaion while holding he marke porfolio in equilibrium (Sharpe 964 and Linner 965). However, Meron (987) demonsraes ha a sub-opimally diversified porfolio could be in equilibrium in a capial marke wih incomplee informaion. In realiy, invesors rarely hold well-diversified porfolios. Campbell e al. (200) sugges ha an invesor needs o hold a leas 50 randomly seleced socks o achieve complee porfolio diversificaion. Goezmann and Kumar (2008), however, examine more han 60,000 equiy invesmen accouns from 99 o 996 and find ha less han en percen of he invesors hold more han en socks. The apparen lack of diversificaion among mos invesors indicaes ha idiosyncraic risk should be priced because under-diversified invesors require compensaion in he form of a higher reurn for bearing his risk (Levy 978; Meron, 987). Given he benchmark predicion of a posiive relaionship beween idiosyncraic risk and excess reurns, he relaed empirical evidence is far from being conclusive. In paricular, he "idiosyncraic volailiy puzzle" found by Ang e al. (2006) in he U.S. marke has araced much research. This puzzle arises from he empirical evidence of cross-secional analyses showing ha socks wih high idiosyncraic volailiies in he previous monh have abysmally low average monhly reurns. Ang e al. (2009) poin ou ha he puzzle found by Ang e al. (2006) may be dependen on he paricular sample used. To explore he possibiliy of daasnooping, hey check wheher he puzzle exiss in oher markes. They find he same puzzling evidence ha

3 socks wih high idiosyncraic volailiy end o have low average reurns in each of he G7 equiy markes and in a larger sample of 23 developed markes. Boh sudies (Ang e al., 2006, 2009) invesigae he relaionship beween expeced reurns and he associaed risk wihin a counry using cross-secional firm-level daa. The discussion of he daasnooping problem of he puzzle has never been exended o anoher invesmen avenue - he marke for inernaional equiy indices. Counry-level cross-secional analyses have used hese global index daa o discuss he inernaional CAPM or inernaional marke inegraion/segmenaion (e.g., Li e al., 2003; Driessen and Laeven, 2007; You and Daigler, 200), bu have never addressed he daasnooping issue of he puzzle in hese daa. Sudies suggesing inernaional marke segmenaion such as Bali and Cakici (200) find evidence of a posiive and significan relaionship beween counry-specific idiosyncraic volailiies in he previous monh and fuure index reurns. This absence of he abnormal puzzle in he inernaional equiy index marke simply confirms he normal posiive risk-reurn relaionship and may no deserve much aenion from he lieraure on he idiosyncraic volailiy puzzle. However, he evidence ha he puzzle does no exis in he inernaional equiy index marke provides a new pah of research. Raher han discussing he daasnooping issue of he idiosyncraic volailiy puzzle, we use his evidence o invesigae he differences in he imeseries properies beween he firm-level and counry-level daa. Specifically, some sudies claim ha he puzzle can be explained by cerain ime-series properies of he idiosyncraic shocks. If hese explanaions are legiimae, he ime series properies of he counry-level daa mus differ from hose of he firm-level daa since he puzzle does no exis in he inernaional 2

4 index daa. This would have imporan implicaions for differen invesmen sraegies in firmlevel sock porfolios and in inernaional index porfolios. For example, Huang e al. (200) argue ha he puzzle is caused by he omission of lagged sock reurns as a regressor in he cross-secional regressions. They find ha pas reurns have a negaive effec on curren reurns (reurn reversal). In addiion, here is a posiive conemporaneous correlaion beween realized idiosyncraic volailiy and sock reurns. Therefore, if lagged reurns are omied from he regression, he effec of pas idiosyncraic volailiy on expeced reurns is negaively biased. Once reurn reversals are conrolled for, hey find a significanly posiive relaionship beween he condiional idiosyncraic volailiy and expeced reurns. Fu (2009), on he oher hand, inspecs he second momen of he idiosyncraic shocks. He argues ha since idiosyncraic volailiies are ime-varying wih a small average firs-order auocorrelaion, lagged idiosyncraic volailiy is no a good esimae of expeced idiosyncraic volailiy. Therefore, he negaive relaionship beween lagged idiosyncraic risk and excess reurns does no represen he expeced risk-reurn relaionship. Insead, Fu models he idiosyncraic volailiy as an exponenial GARCH (EGARCH) process, and uses he esimaed condiional idiosyncraic volailiy o proxy for he expeced idiosyncraic volailiy. He finds evidence of a posiive relaionship beween he esimaed condiional idiosyncraic volailiies and sock reurns. Boyer e al. (200) consider he invesors preference for posiive skewness in an aemp o explain he puzzle. They provide evidence supporing he heory ha expeced idiosyncraic skewness and expeced reurns are negaively correlaed (Barberis and Huang, 3

5 2008; Mion and Vorkink, 2007). In addiion, hey find ha pas idiosyncraic volailiy is a srong predicor of fuure idiosyncraic skewness and ha he relaionship beween pas idiosyncraic volailiy and expeced idiosyncraic skewness is posiive. Therefore, invesors may accep lower expeced reurns on socks ha have experienced high idiosyncraic volailiy because hese socks have higher expeced idiosyncraic skewness. This provides a novel explanaion of he idiosyncraic volailiy puzzle. The empirical resuls from hese sudies sugges ha he above ime series properies of he firm-level idiosyncraic shocks mus differ from hose of he counry-specific idiosyncraic shocks in Bali and Cakici (200). Specifically, based on he argumens of Fu (2009), he idiosyncraic volailiies of a specific counry index reurn relaive o he world marke mus be highly auocorrelaed, and herefore pas idiosyncraic volailiy is a good predicor of expeced idiosyncraic volailiy. If no, hen Bali and Cakici's (200) es would be invalid and a beer measure of he expeced idiosyncraic volailiy needs o be used o es he risk-reurn relaionship in he counry-level cross-secional regressions. Furhermore, he relaionships among reurn, idiosyncraic volailiy, and idiosyncraic skewness in he firm-level daa found by Boyer e al. (200) may no hold in he inernaional index marke. On he one hand, Bali and Cakici's (200) resuls may simply imply ha, unlike he preference for loery-like socks in domesic asse porfolios, inernaional invesors do no prefer posiively skewed index securiies in heir under-diversified inernaional porfolios. However, on he oher hand, if hey do prefer posiive skewness, hen pas idiosyncraic volailiy should no posiively predic fuure idiosyncraic skewness in he inernaional index marke; oherwise, he posiive relaionship beween lagged idiosyncraic volailiy and fuure 4

6 reurns found by Bali and Cakici (200) would imply ha porfolios wih higher idiosyncraic skewness earn higher reurns. Using inernaional equiy index daa, we find ha, firs, lagged idiosyncraic volailiy is no a bad predicor of fuure idiosyncraic volailiy in he counry-level index daa, and performs beer han ha in he firm-level daa. I provides useful informaion on expeced idiosyncraic volailiy jus as he condiional idiosyncraic volailiy does. Second, idiosyncraic skewness in he counry-level index daa is essenially zero. Therefore, i does no play a role in deermining he index reurns and can be ignored in he pricing of inernaional porfolios. Finally, reurn reversals are no presen in he counry-level index daa and, insead, a momenum effec is found. This momenum effec, however, does no aler our conclusions on he higher momens of he idiosyncraic shocks. The nex secion discusses he ime-series properies of he counry-specific idiosyncraic volailiy and is relaionship wih he expeced reurns. Secion 3 invesigaes he role of idiosyncraic skewness in inernaional asse pricing. Secion 4 elaboraes on he economic meaning of our saisical findings and concludes he paper. 2. Reurns and idiosyncraic volailiies We sar wih a brief review of he idiosyncraic volailiy puzzle. Le r denoe he d, daily idiosyncraic reurn of sock i on dae d in monh. The idiosyncraic volailiy in monh is defined as he realized monhly sandard deviaion of he daily idiosyncraic shocks: IVOL ( i d ) = Var r. Le R denoe he monhly reurn of sock i in monh. In deriving he,, idiosyncraic shocks from he Fama-French (993) hree-facor model, Ang e al. (2006), using 5

7 U.S. firm-level daa, and Ang e al. (2009), using firm-level daa for several inernaional markes, run a cross-secional regression for he following economeric specificaion in each monh : R = γ + γ IVOL + Γ X + ε, () 0,, where X is a vecor of oher risk measures and firm characerisics. They find ha he ime- series average of he esimaed γ is negaive and saisically significan, indicaing ha, invesors accep a lower reurn on socks wih higher lagged idiosyncraic volailiies. This is he empirical evidence known as "he idiosyncraic volailiy puzzle" because, according o Meron's (987) heory, raional invesors should require higher average reurns o compensae for heir holding imperfecly diversified porfolios. They demand a premium for holding socks wih high idiosyncraic volailiies, and herefore idiosyncraic risk should be priced o compensae raional invesors holding under-diversified porfolios (Malkiel and Xu 2006). Fu (2009) argues ha a valid es for he risk-reurn relaionship should insead be examined by he following regression: R = γ + γ Eˆ [ IVOL ] + Γ Eˆ [ X ] + ε. (') 0,, Tha is, if idiosyncraic volailiy is priced, we expec here o be a posiive empirical relaionship beween expeced reurns and expeced idiosyncraic volailiy. Running () insead of (') would implicily assume ha IVOLi, is a good predicor of E [ IVOLi, ]. To es wheher his assumpion is valid, Fu analyzes he persisence of he realized idiosyncraic volailiies of U.S. socks for he period Using he Fama-French (993) hree- 6

8 facor model o obain idiosyncraic reurns, he shows ha he firs-order auocorrelaion of IVOL is small (he average across socks is 0.330). In addiion, he claims ha if IVOL is a good predicor of E [ IVOLi, ], hen IVOL should be modeled as a random walk process. He uses he Augmened Dickey-Fuller -es and shows ha for almos 90% of he socks in his sample, he random walk hypohesis for he realized idiosyncraic volailiy is rejeced a he % significance level. Therefore, he claims ha equaion () run by Ang e al. (2006) is no a valid es of he expeced idiosyncraic risk-reurn relaionship. Raher, a proper measure of condiional volailiy should be used insead of he lagged volailiy. To obain an esimae of E [ IVOLi, ], Fu uses EGARCH models o obain he monhly condiional idiosyncraic variance (denoed as h ). By replacing he lagged realized idiosyncraic volailiy ( IVOLi, ) in () wih h, he finds ha he ime-series average of he esimaed γ is posiive and significan. Tha is, here is a posiive relaionship beween he, expeced reurn and expeced idiosyncraic volailiy, and herefore, he idiosyncraic volailiy puzzle is explained. In a sudy of an inernaional CAPM, Bali and Cakici (200) es wheher he counryspecific idiosyncraic risk is priced by using counry-level aggregae marke index daa from 37 counries and a world marke porfolio index. The daily idiosyncraic shocks in monh are defined as he residuals from a regression of counry i's daily marke porfolio index reurns ( d, R ) on he daily world marke porfolio reurns ( R ): w, d, Spiegel and Wang (2005), who focus on he ou-of-sample predicive power of idiosyncraic volailiy and liquidiy using monhly daa and EGARCH models, also find ha sock reurns are increasing wih he level of idiosyncraic volailiy. 7

9 R = µ + Bea R + r, for d =, 2,..., D, (2) d, w, d, d, where D is he number of rading days in monh and Bea is he condiional world marke bea of counry i in monh. They define he counry-specific idiosyncraic volailiy in monh as he realized monhly sandard deviaion of he daily idiosyncraic shocks: D ( ) 2 IVOL = r r. 2 They run a monhly cross-secional regression of he reurns o d, d, d = counry i s marke porfolio ( R ) on lagged idiosyncraic volailiy ( IVOLi, ) and a vecor of oher risk measures: R = γ + γ IVOL + γ Bea + γ EP + γ DY + ε, (3) 0,, 2, 3, 4, where Bea is he condiional world marke bea from (2), EP is he naural logarihm of he earnings-o-price raio, and DY is he naural logarihm of he dividends-o-price raio in monh. They find ha he ime-series average of he esimaed effec of IVOLi, on R (i.e., ˆ γ, ) is posiive and saisically significan. This indicaes ha inernaional invesors hold under-diversified inernaional equiy-index porfolios because he counry-specific risk is priced, and ha he counry-specific idiosyncraic volailiy predics a posiive index reurn in he nex monh. Therefore, he idiosyncraic volailiy puzzle does no appear in he crosscounry marke index daa. 2 Since we use OLS regressions in (2) including a consan erm, r d, = 0. Noe ha Bali and Cakici (200) ignore he scaling and do no divide he sum of squared residuals by he number of days. We use his specificaion hroughou he paper so ha we can compare our resuls wih heirs. 8

10 According o Fu's argumen, Bali and Cakici's (200) es would be invalid unless he lagged idiosyncraic volailiy were a good predicor of he expeced idiosyncraic volailiy. Therefore, our firs sep is o analyze he ime-series propery of he idiosyncraic volailiy of he counry-level marke index reurns. We use he same daa as hose in Bali and Cakici (200), whose daa end in Sepember 2006, bu updae heir daa o November 200. The daa are obained from Daasream Global indices, and include U.S. dollar-denominaed reurns on sock marke indices for 37 counries plus he world marke porfolio. There are 23 developed markes and 4 developing or emerging markes. 3 Table shows he summary saisics of he monhly marke index reurns for each counry and he world marke, including he means, sandard deviaions, and correlaions wih he world marke index reurns. The nex wo columns show he ime-series averages of EP i, and DY for each counry. We use all available daa o calculae he summary saisics. 4 The saring monh for each counry is shown in he second column. The sample ends in November 200 for all counries. Even wih he updaed daa added, he summary saisics are in general very similar o hose in Bali and Cakici (200): he emerging markes exhibi higher average reurns and higher sandard deviaions of reurns, compared o he developed markes. 3 Based on Bali and Cakici's (200) caegorizaion, he 23 developed markes are Ausralia, Ausria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Ialy, Japan, he Neherlands, New Zealand, Norway, Porugal, Singapore, Spain, Sweden, Swizerland, he Unied Kingdom, and Unied Saes. The 4 developing or emerging markes are Argenina, Brazil, Chile, China, India, Korea, Malaysia, Mexico, he Philippines, Poland, Souh Africa, Taiwan, Thailand, and Turkey. 4 Daa for he earnings-o-price raio ( EP ) and he dividends-o-price raio ( DY ) are in general shorer han he reurns daa. 9

11 To obain he monhly daa of he condiional world marke bea ( Bea ) and idiosyncraic volailiy ( IVOL ), we use daily daa wihin each monh o run ime-series regression (2). This generaes he monhly Bea and he daily idiosyncraic reurns ( r d, ). The daily idiosyncraic reurns are hen used o calculae he realized monhly idiosyncraic volailiy ( IVOL ). The las wo columns of Table show he ime series averages of Bea and IVOL for each counry. Again, he resuls are in general consisen wih hose repored in Bali and Cakici (200): he counry-specific idiosyncraic volailiy is much higher in he emerging markes han in he developed markes, while he cross-secional differences in he sysemaic risk for developed and emerging markes are no as significan as hose in he idiosyncraic risk. To confirm Bali and Cakici's (200) resuls on he idiosyncraic risk-reurn relaionship using he updaed daa, we run several versions of equaion (3) like hey do in heir paper. Table 2 repors he ime-series averages of he esimaed coefficiens, heir P-values based on he Newey and Wes (987) heeroscedasiciy- and auocorrelaion-adjused -saisics, and he ime-series averages of he R-squared. 5 Consisen wih he findings in Bali and Cakici (200), we find ha, firs of all, in all specificaions, almos all he risk measures have a posiive effec on he reurns. Secondly, he world sysemaic risk facor (i.e., he world marke bea) does no affec reurns, indicaing ha counry-specific facors provide more of an 5 The daa availabiliies are differen eiher across variables wihin a counry or across counries. We require a minimum of en observaions in each cross-secional regression. 0

12 explanaion for he variaion in index reurns. 6 Thirdly, he earnings-o-price raio is he mos significan risk facor, and he effec of he dividends-o-price raio disappears when he earnings-o-price raio is included in he regression. More imporanly, he ime-series average of he esimaed effec of IVOLi, on R, i.e., ˆ γ,, is posiive and saisically significan a he 5% level in all cases. This confirms Bali and Cakici's (200) finding ha he idiosyncraic volailiy puzzle does no exis in he cross-counry marke index daa. However, is he above es valid? Tha is, according o Fu's argumen, is IVOLi, a good esimae of expeced idiosyncraic volailiy E [ IVOLi, ]? The firs column of Table 3 repors he firs-order auocorrelaion coefficiens of IVOL for each counry. The average of he firs-order auocorrelaion coefficiens of IVOL across hese 37 counries is 0.552, which is higher han ha in Fu's U.S. firm-level daa (0.330). Tha is, he idiosyncraic volailiy is more persisen in he counry-level marke index reurns han in he U.S. firm-level daa. The second column of Table 3 repors he es saisics of he Augmened Dickey-Fuller uni-roo - es on IVOL. The resuls show ha he es fails o rejec he null hypohesis of a uni roo in 3 of he 37 counries a he % significance level, compared o 90% rejecion rae in Fu's firm-level daa. These wo pieces of evidence in Table 3, of course, are no srong enough for us o claim ha he lagged idiosyncraic volailiy ( IVOLi, ) is a good proxy for he expeced idiosyncraic volailiy ( E IVOL ) in he counry-level daa. However, i is a beer proxy in he counry-level daa han in he firm-level daa. These observaions may explain why he 6 Bekaer, Hodrick, and Zhang (2009) also show ha local facors explain index reurns hrough a muliple-facor model.

13 idiosyncraic volailiy puzzle in equaion (3) exiss in he firm-level daa bu no in he counry-level daa. Using he realized idiosyncraic volailiy in monh - ( IVOLi, ) as he forecas of he idiosyncraic volailiy in monh ( E IVOLi, ) implicily assumes ha he idiosyncraic volailiy follows a maringale process. The uni-roo es resuls above show ha his may no be a proper assumpion for all he counries in our sample. To relax his resricive assumpion, we follow Huang e al. (200) and use he bes-fi auoregressive inegraed moving average (ARIMA) process o model he monhly condiional idiosyncraic volailiy over a rolling window. Specifically, we use he realized idiosyncraic volailiy over he previous wenyfour monhs o find he bes-fi ARIMA model using he Schwarz crierion (BIC). Then he model is employed o predic he idiosyncraic volailiy in he nex monh. We replace IVOL in (3) wih his esimaed condiional idiosyncraic volailiy ( Eˆ IVOL i, ) from he bes-fi ARIMA model and re-evaluae he relaionship beween he counry-index reurns and he expeced idiosyncraic volailiies: R = γ + γ Eˆ IVOL + γ Bea + γ EP + γ DY + ε. (3') 0,, 2, 3, 4, The resuls are repored in Table 4. The posiive relaionship beween condiional expeced idiosyncraic volailiies and reurns is again confirmed. Ineresingly, he resuls are very similar o hose in Table 2: he effecs of idiosyncraic volailiies on index reurns are all significanly posiive and very similar in magniudes; he world marke bea is he leas significan risk facor; he earnings-o-price raio is he mos significan risk facor; and he effec of he dividends-o-price raio disappears when he earnings-o-price raio is included in 2

14 he regression. Judging from he average R-squared, using he condiional idiosyncraic volailiy esimaed from he bes-fi ARIMA model o replace he lagged idiosyncraic volailiy does no improve he in-sample forecas of he cross-counry marke index reurns. Bali and Cakici (2008), using firm-level daa, show ha, compared o he realized monhly idiosyncraic volailiy calculaed from daily daa, he condiional idiosyncraic volailiy esimaed from a GARCH (,) model or an EGARCH(,) model using monhly daa is a more accurae proxy for expeced fuure idiosyncraic volailiy. 7 As menioned earlier, his is he sraegy used by Fu (2009) o obain he monhly condiional idiosyncraic variance. Therefore, o confirm he posiive idiosyncraic risk-reurn relaionship in he counry-level daa, our nex sep is o follow heir sraegy and esimae he condiional idiosyncraic volailiy from an EGARCH (,) model by using monhly daa. Specifically, we firs regress he monhly counry index reurns on he world reurns, Ri, = µ i + ρirw, + r, o obain he monhly counry-specific idiosyncraic reurns, and hen model he idiosyncraic reurns as an AR-EGARCH(,) process: r = α + α r + ε, (4) 0 i ij j j ε ε ln h = κ + α ln h + β + γ I, (5) + i i i i hi, hi, 7 Foser and Nelson (996) show ha he condiional volailiy esimaed from he GARCH models amouns o he condiional volailiy esimaed from a weighed rolling regression using daa from he preceding monhs. 3

15 where he auoregressive (AR) lag lengh j is chosen by he Ljung-Box Q ess as he minimum lag ha renders he serially uncorrelaed ε (a he 5% significance level) up o 24 lags from an OLS auoregression; h is he condiional variance of ε, ε = hi, vi,, v ~ N (0,), and I + = if ε >0 and I + = 0 oherwise. Table 5 repors he esimaes of he EGARCH process, along wih he means of he esimaed condiional idiosyncraic volailiy h for each counry. In general, he esimaed coefficiens in he EGARCH process are saisically significan a he radiional significance level. For mos counries he condiional idiosyncraic volailiy is highly persisen. The relaive condiional idiosyncraic volailiies ( h ) across counries are consisen wih he relaive realized idiosyncraic volailiies ( IVOLi, ) across counries as repored in Table : he counry-specific idiosyncraic volailiy is much higher in he emerging markes han in he developed markes. We replace IVOLi, in (3) wih h and re-evaluae he relaionship beween he counry-index reurns and expeced idiosyncraic volailiies: R = γ + γ h + γ Bea + γ EP + γ DY + ε. (3'') 0,, 2, 3, 4, The resuls are repored in Table 6. The posiive relaionship beween condiional expeced idiosyncraic volailiies and index reurns is again confirmed. 8 By comparing he resuls in 8 Similar conclusions can also be found in Brockman e al. (2009), who apply Fu s (2009) EGARCH model o anoher se of inernaional index daa and find supporing evidence for a posiive and significan relaionship beween expeced reurns and idiosyncraic volailiy. 4

16 Table 6 wih hose in Table 2 and Table 4, we can see ha regressions (3), (3'), and (3'') generae very similar resuls. Alhough (3'') yields a slighly higher R-squared han (3) and (3'), he similariies in hese resuls show ha he lagged realized idiosyncraic volailiy is a good proxy for he condiional idiosyncraic volailiy in he inernaional marke index daa. I provides as useful informaion as he condiional volailiy does in forecasing inernaional index reurns. As argued by Huang e al. (200), if reurn reversals exis and he lagged reurn is omied from he regressions, he effec of idiosyncraic volailiy on expeced reurns would be under-esimaed. Even if his is he case, however, our conclusion above would no change because we have found a posiive and significan effec. To see he role of reurn reversals in our analysis, we add he lagged reurns o regressions (3), (3'), and (3'') and repor he resuls in Panels (A)-(C) of Table 7, respecively. Ineresingly, reurn reversals do no exis in he counry-level index daa. Raher, we find a saisically significan momenum effec. Therefore, omiing he lagged reurns acually over-esimaes he effec of condiional idiosyncraic volailiy on he expeced reurns. The magniude of he momenum effec is, however, relaively small. Afer conrolling for he momenum effec, he size and he level of significance of he effec of idiosyncraic volailiy on he expeced reurns are slighly reduced. However, he effec is sill posiive and marginally significan a he 0% level in almos all cases. In sum, our resuls are consisen wih Fu's argumen. As long as lagged realized idiosyncraic volailiies proxy well for expeced idiosyncraic volailiies, invesors will expec higher reurns on porfolios ha have experienced high idiosyncraic volailiies. The lagged realized idiosyncraic volailiy is as good a predicor of he expeced idiosyncraic volailiies 5

17 as he condiional idiosyncraic volailiy is for he inernaional marke index daa. Our analysis serves o validae Bali and Cakici's (200) use of lagged idiosyncraic volailiy o es he risk-reurn relaionship in he inernaional marke index daa. These conclusions remain robus afer conrolling for he momenum effec. 3. Reurns, idiosyncraic volailiies, and idiosyncraic skewness By incorporaing he idea ha invesors prefer posiive skewness, Boyer e al. (200) consider he role of idiosyncraic skewness in explaining he idiosyncraic volailiy puzzle. Using firm-level daa in he U.S. sock marke, hey find ha firs, pas idiosyncraic volailiy is, compared o he pas idiosyncraic skewness, a srong predicor of fuure idiosyncraic skewness; and second, expeced idiosyncraic skewness and expeced reurns are negaively correlaed. Therefore, invesors may accep lower expeced reurns on socks ha have high lagged idiosyncraic volailiies because hese socks have higher expeced idiosyncraic skewness. The absence of he idiosyncraic volailiy puzzle in he global equiy index daa implies ha Boyer e al.'s argumens may no hold in he inernaional index marke. If inernaional invesors prefer posiively skewed index securiies in heir under-diversified inernaional porfolios, hen pas idiosyncraic volailiy should no posiively predic fuure idiosyncraic skewness in he inernaional index marke; oherwise, he posiive relaionship beween lagged idiosyncraic volailiy and index reurns found in Bali and Cakici (200) implies ha porfolios wih higher idiosyncraic skewness earn higher reurns. To verify Boyer e al.'s firs argumen (i.e., pas idiosyncraic volailiy is a srong predicor of fuure idiosyncraic skewness) in he counry-level index marke, we follow heir seps and esimae he cross-secional regression separaely for each monh : 6

18 ISK = β + β ISK + β IVOL + υ, (6) 0,, T 2, T where T is he forecas horizon. We use our counry-level index daa and equaion (2) o obain he idiosyncraic shocks ( r d, ). By denoing S() as he se of rading days from he firs day of monh -T+ hrough he end of monh, Boyer e al. define he hisorical esimae of idiosyncraic volailiy a ime as he sandard deviaion of he daily idiosyncraic shocks in IVOL = r r. 9 Therefore, he definiion of IVOL is differen from ha in S(): ( ) 2 d, d, d S ( ) our earlier analyses unless he forecas horizon T =. The se of rading days S(-T) would cover he firs day of monh -2T+ hrough he end of monh -T. The hisorical esimae of idiosyncraic skewness a ime is hen defined as: ( ) 3 3 ISK = ri, d, ri, d, IVOLi, d S ( ). Using U.S. firm-level daa, Boyer e al. find ha if IVOLi, is included, he ime series average of T ˆ, β is insignifican and he ime series average of β is significan. Tha is, pas idiosyncraic 2, ˆ volailiy is, compared o he pas idiosyncraic skewness, a sronger predicor of fuure idiosyncraic skewness. Using he counry-level index daa, Table 8 repors, for horizons T =, 6, 2, 24, and 60, he ime-series averages of he esimaed coefficiens, heir P-values based on Newey and Wes (987) -saisics, and he ime-series averages of he R-squared. Apparenly, all he esimaes are no only small in magniudes bu also highly insignifican. Neiher lagged 9 Boyer e al. (200) divide he sum of squared residuals by he number of rading days. We do no do his scaling because he curren expression is consisen wih our analyses earlier in he paper when he forecas horizon is one monh (i.e. T=). See Foonoe 3. 7

19 idiosyncraic skewness nor lagged idiosyncraic volailiy predics fuure idiosyncraic skewness. Therefore, Boyer e al.'s observaion ha pas idiosyncraic volailiy predics fuure idiosyncraic skewness is no presen in he counry-level daa. Nex, using he counry-level daa, we verify he second argumen by Boyer e al. ha he expeced idiosyncraic skewness and expeced reurns are negaively correlaed. Their cross-secional regression is specified as: R = γ + Λ Z + γ Eˆ [ ISK ] + ε, (7) 0, + T where Z is a vecor of risk measures and E[ ISK + T ] is he expeced idiosyncraic skewness. 0 Boyer e al. consruc he measure of E[ ISK + T ] from he condiional regression (6). However, as shown in Table 8, he model fis badly, and one herefore canno expec he condiional idiosyncraic skewness from (6) o be a good measure of he expeced idiosyncraic skewness. To find a beer measure of he expeced idiosyncraic skewness, we adop Fu's (2009) imeseries sraegy and esimae he condiional idiosyncraic skewness from an EGARCH model. For comparisons wih our earlier resuls and for he sake of simpliciy, we only focus on esimaing condiional idiosyncraic skewness wih he horizon T = in he following analyses. ε Recall ha in he previous secion, we follow Fu and assume ha v = follows he h sandard normal disribuion. Alhough he EGARCH specificaion accommodaes he asymmeric propery of volailiy (whereby negaive shocks increase volailiy more han 0 Boyer e al. (200) acually run his regression wih porfolio reurns and skewness formed by soring sock based on expeced skewness. Here we insead use individual index reurns and skewness because we only have 37 indices. 8

20 9 posiive shocks), ha model is unable o explicily esimae he skewness of he idiosyncraic shocks. To model he skewness of he disribuion, we relax he assumpion of normaliy and adop a more flexible disribuion, namely, he skewed suden- (ST) disribuion proposed by Hansen (994). The ST disribuion is a parsimonious wo-parameer disribuion, bu also a flexible one. I is able o model no only lepokurosis bu also asymmery. The densiy funcion of he ST disribuion is: < = + +, for 2, for 2 ), (, 2 2,, 2 2,, σ µ λ µ σ η σ σ µ λ µ σ η σ λ η η η i i i i i ST v v c v v c v g where < η < 2, < < λ, 2 4 = η η λ µ c, µ λ σ + =, and 2. ( 2) 2 c η Γ η π η Γ + = The skewness M sk µσ µ σ =, where he hird raw momen ( )( 2). ( )( 3) c M λ λ η η η + = The parameer η conrols he ails and he peak of he densiy and λ conrols he rae of descen of he densiy around v = 0. For deails on he ST densiy, see he appendix in Hansen (994). To incorporae he condiional idiosyncraic skewness in our ime-series model, we specify he idiosyncraic shock in an auoregressive condiional densiy (ARCD) model suggesed by Hansen (994). Hansen s ARCD modeling sraegy is o model he parameers in he condiional densiy funcion as funcions of he elemens of he informaion se so ha he higher momens also depend on he condiioning informaion. He conjecures ha since

21 GARCH models make he condiional second momen a funcion of he lagged errors, i is reasonable o believe ha his sraegy could also work well for he oher momens. Therefore, we follow his suggesion and model he skewness parameer λ in he densiy funcion of he ST disribuion as:.99 (.99) λ =.99 +, ω = a + b ε + c ε, 2 + exp( ω ) (8) where he logisic ransformaion is used o se consrains ha λ lies beween.99 and -.99, even hough ω is allowed o vary over he enire real line. We call his exension of he model in Secion 2 he AR-EGARCH-ARCD model. The resuling condiional skewness is denoed by sk. The esimaed condiional idiosyncraic skewness is hen used as an addiional regressor in (3'') o es wheher expeced idiosyncraic skewness and expeced reurns are negaively correlaed in he counry-level index marke: R = γ + γ h + γ Bea + γ EP + γ DY + γ sk + ε (3''') 0,, 2, 3, 4, 5,, where h is he condiional variance esimaed from he new (AR-EGARCH-ARCD) model. Panel (A) of Table 9 presens he resuls from (3'''). In Panel (B), we add he lagged reurns as an addiional regressor o conrol for he momenum effec. The resuls in boh panels show ha he relaion beween he reurns and he expeced idiosyncraic skewness is highly insignifican. The oher esimaed coefficiens are very similar o hose repored in Table 6 and The esimaion resuls are no repored o save space bu are available from he auhors upon reques. 20

22 Panel (C) of Table 7. Therefore, idiosyncraic skewness does no play a significan role in he counry-level index marke. One possible explanaion of he above resuls is ha inernaional invesors do no exhibi a pro-loery preference in heir under-diversified inernaional porfolios. However, by furher invesigaing he idiosyncraic skewness, we find ha he above resuls are more likely o be due o he symmery of he counry-specific idiosyncraic shocks. In Table 0, we show he means of he realized idiosyncraic skewness for T=, 6, 2, 24, and 60. I can be seen ha he averages of ISK across hese 37 counries are very small in magniude. I is only 0.0 for T= in our counry-level daa, which is relaively small compared o ha in he firm-level daa repored in Boyer e al., which is abou For he oher horizons, he magniudes are even smaller. In our esimaions of he AR-EGARCH-ARCD model, alhough no repored here o save space, many of he esimaed coefficiens ( â, ˆb, and ĉ ) in he condiional densiy funcion (8) are saisically insignifican. This observaion also cass doub on he role of idiosyncraic skewness in he inernaional index pricing. To see how appropriae his modeling sraegy is, we furher experimen wih wo alernaive specificaions. The firs one removes he ime-varying propery of he skewness and imposes a consan skewness parameer by seing b=c=0 in (8), which we denoe as he AR-EGARCH-ST model. 3 The 2 Boyer e al. (200) repors a mean of 0.85 for he idiosyncraic skewness in he U.S. firmlevel daa. However, heir measure is ours muliplied by a scale of are 22 rading days (D =22) yields a mean of D. Assuming ha here 3 We find ha only 9 ou of he 37 indices have a saisically significan (a he 5% level) skewed disribuion. The resuls are available from he auhors upon reques.

23 second alernaive specificaion assumes a symmeric disribuion by seing a=b=c=0 in (8), denoed as he AR-EGARCH- model, which is essenially an AR-EGARCH model wih a symmeric -disribuion. 4 Table shows he esimaion resuls from he cross-secional regression (3'') wih he condiional idiosyncraic volailiies esimaed from hese hree alernaive models (AR-EGARCH-ARCD, AR-EGARCH-ST, and AR-EGARCH-). The resuls are very similar across hese models. Therefore, wheher we ignore he idiosyncraic skewness or no, he cross-secional evidence from Bali and Cakici's (200) is no affeced. We conclude ha he disribuion of he idiosyncraic shocks in he inernaional index marke is mosly symmeric and he idiosyncraic skewness is no significan enough o affec he index reurns. 4. Discussions and Conclusions Theories show ha under-diversified invesors should be compensaed for assuming idiosyncraic risks. Empirical sudies esing his hypohesis ofen use a cross-secional regression of reurns on realized idiosyncraic volailiies in he previous monh and check for a posiive coefficien. The empirical evidence ha socks wih higher pas idiosyncraic volailiies earn lower fuure reurns has evoked a series of sudies rying o explain his empirical puzzle. Sudies researching he ime series properies of he idiosyncraic reurns for answers claim ha he puzzle can be explained by he fac ha, firsly, pas idiosyncraic volailiy is no a good predicor of expeced idiosyncraic volailiy; secondly, he esimae is 4 No repored here bu available upon reques, he mean log-likelihoods from hese hree specificaions are so similar ha he null of a symmeric disribuion canno be rejeced by he likelihood raio es. 22

24 biased downward because of an omied variable (pas reurns); and finally, invesors' aversion o idiosyncraic risk is ouweighed by he preference for idiosyncraic skewness. This paper finds ha hese ime series properies of he idiosyncraic reurns in he firm-level daa are differen from hose in he counry-level index daa, which explains why he idiosyncraic volailiy puzzle found in he former does no exis in he laer. Firs of all, in he marke for global equiy indices, pas idiosyncraic volailiy is a good predicor of expeced idiosyncraic volailiy. I provides jus as useful informaion as he condiional idiosyncraic volailiy in predicing fuure reurns. Tha is, counry-specific volailiies are persisen. The lieraure on inernaional equiy pricing suggess ha he undiversifiable counry-specific risk may be due o facors such as purchasing power pariy deviaions, i.e., exchange rae and inflaion risk (Adler and Dumas, 983), governmen resricions on capial movemens in emerging markes (Henry, 2000), and asymmeric informaion across markes (Brennan and Cao, 997). Since hese counry-specific risk facors are mos likely relaed o policies adoped by individual counries, i is reasonable for uncerainy o be more persisen in hese daa, especially in he emerging markes (Lewis, 20). For example, before an emerging economy announces ha i is liberalizing is capial marke, uncerainy regarding capial mobiliy and exchange rae policies is persisenly high. Afer he announcemen, his ype of uncerainy should remain a a lower level. In addiion, as informaion flows more slowly across borders han across firms wihin a counry, i is no surprising ha he counry-specific risk possesses a higher degree of clusering, compared o he firm-specific risk relaive o he individual counry's aggregae marke risk. Therefore, pas idiosyncraic volailiy serves as a beer predicor of he expeced idiosyncraic volailiy in he global index marke han in he sock marke wihin a counry. 23

25 Secondly, in conras o he finding of reurn reversals in he firm-level daa, we repor ha he inernaional index reurns exhibi reurn momenum. Posiive auocorrelaions of index reurns and shor-erm profis of momenum sraegies in he inernaional index marke have been well documened in he lieraure (e.g., Ahn e al., 2002; Bhojraj and Swaminahan, 2006). Anoniou e al. (2005) aribue his index reurn momenum o he inroducion of index fuures, which has increased posiive feedback rading in he spo markes. As posiive feedback raders respond o pas index changes, posiive auocorrelaions of index reurns are shown over shor horizons. This momenum effec indicaes ha he esimae of he expeced idiosyncraic risk-reurn relaionship is acually biased upward in a sandard inernaional CAPM model like equaion (2) (Bali and Cakic 200) if he pas reurn is omied from he regression. Afer adjusing he bias, however, our conclusions on he higher momens of he counry-specific idiosyncraic shocks remain robus. Finally, we find ha counry-specific idiosyncraic skewness is no significan enough o affec he counry-level index reurns. Unlike individual sock reurns, he counry-specific idiosyncraic reurns are mosly symmerically disribued. Considering he globalizaion of he financial markes, i is no reasonable o expec ha an inernaional equiy index would earn an abnormal exreme reurn. Therefore, under-diversified inernaional porfolio invesors have no incenive o search for loery-like equiy indices. As a resul, insignifican differences in counry-specific idiosyncraic skewness do no creae significan differences in equiy index pricing. Index invesors' preferences for idiosyncraic skewness do no ouweigh heir aversion o idiosyncraic risk. Along wih he findings of he momenum effecs and more persisen idiosyncraic volailiies, our resuls sugges ha differen invesmen sraegies should be adoped in forming global and domesic equiy porfolios. 24

26 References Adler, M. & Dumas, D. (983). Inernaional porfolio choice and corporae finance: A synhesis. Journal of Finance, 38, Ahn, D., Boudoukh, J., Richardson, M., & Whielaw, R. F. (2002). Parial adjusmen or sale prices? Implicaions from sock index and fuures reurn auocorrelaions. Review of Financial Sudies 5, Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2006). The cross-secion of volailiy and expeced reurns. Journal of Finance 6, Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2009). High idiosyncraic volailiy and low reurns: Inernaional and furher U. S. evidence. Journal of Financial Economics 9, -23. Anoniou, A., Koumos, G., & Pericl A. (2005). Index fuures and posiive feedback rading: Evidence from major sock exchanges. Journal of Empirical Finance 2, Bal T. G. & Cakic N. (2008). Idiosyncraic volailiy and he cross-secion of expeced reurns. Journal of Financial and Quaniaive Analysis 43, Bal T.G. & Cakic N. (200). World marke risk, counry-specific risk and expeced reurns in inernaional sock markes. Journal of Banking and Finance 34, Barberis, N. & Huang, M. (2008). Socks as loeries: The implicaions of probabiliy weighing for securiy prices. American Economic Review 98(5),

27 Bekaer, G., R. J. Hodrick, and X. Zhang Inernaional Sock Reurn Comovemens. Journal of Finance 64, Bhojraj, S. & Swaminahan, B. (2006). Macromomenum: Reurns predicabiliy in inernaional equiy indices. Journal of Business 79, Boyer, B., Mion, T., & Vorkink, K. (200). Expeced idiosyncraic skewness. Review of Financial Sudies 23, Brennan, M. & Cao, H. (997). Inernaional porfolio invesmen flows. Journal of Finance 52, Brockman, P., M.G. Schue, and W. Yu Is idiosyncraic risk priced? The inernaional evidence. Working paper. Campbell, J.Y., Leau, M., Malkiel, B.G., & Xu, Y. (200). Have individual socks become more volaile? An empirical exploraion of idiosyncraic risk. Journal of Finance 56, -43. Driessen, J. & Laeven, L. (2007). Inernaional porfolio diversificaion benefis: Crosscounry evidence from a local perspecive. Journal of Banking and Finance 3, Fama, E. & French, K. (993). Common risk facors in he reurns on socks and bonds. Journal of Financial Economics 33, Foser, D. P., and D. B. Nelson Coninuous Record Asympoics for Rolling Sample Variance Esimaors. Economerica 64:

28 Fu, F. (2009). Idiosyncraic risk and he cross-secion of expeced sock reurns. Journal of Financial Economics 9, Goezmann, W.N. & Kumar, A. (2008). Equiy porfolio diversificaion. Review of Finance 2, Hansen, B. (994). Auoregressive condiional densiy esimaion. Inernaional Economic Review 35, Henry, P. B. (2000). Sock marke liberalizaion, economic reform, and emerging marke equiy prices. Journal of Finance 55, Huang, W., Liu, Q., Rhee, S.G., & Zhang, L. (200). Reurn reversals, idiosyncraic risk, and expeced reurns. Review of Financial Sudies 23, Hueng, C. J. & McDonald, J. B. (2005). Forecasing Asymmeries in Aggregae Sock Reurns: Evidence from Condiional Skewness. Journal of Empirical Finance 2, Hueng, C. J. & R. Yau. (2006). Invesor s Preference and Porfolio Selecion: Is Diversificaion an Appropriae Sraegy? Quaniaive Finance 6 (3), Levy, H. (978). Equilibrium in an imperfec marke: A consrain on he number of securiies in he porfolio. American Economic Review 68, Lewis, K. (20). Global Asse Pricing. NBER Working Paper, No L K., Sarkar, A., & Wang, Z. (2003). Diversificaion benefis of emerging markes subjec o porfolio consrains. Journal of Empirical Finance 0,

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30 Table : Summary Saisics of Inernaional Marke Indices (Monhly Daa) The sample ends in November 200 for all counries. The variable EP is he naural logarihm of he earnings-o-price raio, and DY is he naural logarihm of he dividends-o-price raio. The idiosyncraic volailiy IVOL ( ) 2 = ri, d, ri, d, D d = equaion (2): Ri, d, = µ + Bea Rw, d, + r. d, Marke Index Reurns Daa Counry sar Mean Sd Correlaion wih WORLD 29, where he idiosyncraic shocks are esimaed by DY EP Bea IVOL Argenina Aug Ausralia Jan Ausria Jan Belgium Jan Brazil Jul Canada Jan Chile Jul China Jul Denmark Jan Finland Mar France Jan Germany Jan Greece Jan Hong Kong Jan India Jan Ireland Jan Ialy Jan Japan Jan Korea Sep Malaysia Jan Mexico May Neherlands Jan New Zealand Jan Norway Jan Philippines Nov Poland Mar Porugal Jan Singapore Jan Souh Africa Jan Spain Mar Sweden Jan Swizerland Jan Taiwan May Thailand Jan Turkey Jun UK Jan US Jan WORLD Jan

31 Table 2: Cross-Secional Regressions This able repors he cross-secional regression resuls for (3): R = γ + γ IVOL + γ Bea + γ EP + γ DY + ε. 0,, 2, 3, 4, The average inerceps, average slope coefficiens, and average R 2 are presened. The numbers in parenheses are P-values calculaed based on Newey and Wes (987) -saisics. A P-value of indicaes ha he P-value is nonzero, bu smaller han Consan IVOLi, Bea EP DY Avg. R (0.004) (0.034) (0.025) (0.024) (0.596) (0.000) (0.022) (0.000) (0.309) (0.042) (0.00) (0.000) (0.005) (0.553) (0.000) (0.42) (0.025) (0.575) (0.027) (0.00) (0.006) (0.527) (0.00) (0.967) 30

32 Table 3: Firs-Order Auocorrelaion Coefficiens and he Augmened Dickey-Fuller -es Saisics for Monhly Realized Idiosyncraic Volailiy ADF- is he Augmened Dickey-Fuller -saisic. The aserisk * indicaes ha he es fails o rejec he null hypohesis of a uni roo a he % significance level. Counry s -order auocorrelaion ADF- Argenina Ausralia Ausria Belgium Brazil Canada Chile China * Denmark Finland * France Germany * Greece * Hong Kong India Ireland Ialy Japan * Korea Malaysia * Mexico * Neherlands New Zealand Norway Philippines * Poland Porugal * Singapore Souh Africa Spain Sweden * Swizerland Taiwan * Thailand * Turkey * UK US