gains from international risk sharing?

Transcription

1 Journl of Inernionl Economics 5 (000) 35 loce/ econbse Why do socks nd consumpion imply such differen q gins from inernionl risk shring? Kren K. Lewis,b, * 300 SH-DH, Whron School, Universiy of Pennsylvni, Phildelphi, PA , USA b Nionl Bureu of Economic Reserch, Cmbridge, MA 038, USA Received 5 April 997; received in revised form 7 Sepember 998; cceped 8 My 999 Absrc Esimes of he gins o inernionl risk-shring bsed upon sock reurns end o find drmiclly higher gins hn do esimes from consumpion-bsed models. In his pper, I exmine he resons for hese differences. Using common heoreicl frmework for boh pproches, I find h he differences re lrgely due o he much higher vribiliy of sock reurns nd is implied ineremporl subsiuion in mrginl uiliy. Also, conrry o convenionl wisdom, he differences in gins from he wo pproches do no rise from reing sock reurns s exogenous rher hn endogenous. 000 Elsevier Science B.V. All righs reserved. Keywords: Inernionl risk shring; Sock reurns; Consumpion JEL clssificion: F4; F36. Inroducion Domesic invesors do no pper o hold sufficien proporion of heir welh in foreign sses o diversify wy domesic idiosyncric risk. This is he q Erlier versions of his pper were circuled under he iles, Consumpion, sock reurns, nd he gins from inernionl risk-shring (NBER Working Pper No. 540) nd Why do socks nd consumpion imply such differen coss of imperfec inernionl risk-shring? *Tel.: ; fx: E-mil ddress: lewisk@whron.upenn.edu (K.K. Lewis) / 00/ $ see fron mer 000 Elsevier Science B.V. All righs reserved. PII: S00-996(99)0007-6

2 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 conclusion of reserch using boh consumpion d nd sock reurn d. Since imperfec risk-shring mens h poenil welfre gins re being foregone, he observion leds direcly o he quesion: how lrge re hese gins? On his issue, he lierure hs been quie divided. Some clculions of risk shring gins bsed upon inernionl consumpion d sugges h hese gins re quie smll. For exmple, Cole nd Obsfeld (99) find h for represenive consumers clibred o US d, he gins re less hn 0.5% of permnen consumpion for plusible prmeer vlues. Tesr (995) nd vn Wincoop (994) repor similrly smll gins from inernionl risk shring. On he oher hnd, clculions of he gins from risk-shring bsed upon sock reurns give much lrger esimes. The pproch ypiclly consrucs combinions of domesic nd foreign porfolios h minimize vrince nd mximize reurns nd sks wheher domesic porfolios re domined by hese porfolios. In ppers les s erly s Levy nd Srn (970), porfolios wih foreign socks were shown o sricly domine domesic US porfolios. Using uiliy funcions similr o hose used in he generl equilibrium lierure, I show below h his simple pril equilibrium frmework implies welfre gins of les 0% of permnen consumpion nd ofen-imes ner 00%. In his pper, I ddress he quesion: why re he mgniudes of he gins bsed upon hese wo pproches so differen? I develop common unifying frmework nd hen show h he differences cn come from hree poenil venues: () he remen of sock reurns s exogenous or endogenous; () he sisicl properies of sock reurns relive o consumpion growh; nd (3) he se of 3 preference prmeers. Since hese hree fcors re he core of his invesigion, I nex discuss he significnce of ech in urn. () The equiy-bsed pproch kes he sock price s exogenous nd sks how n invesor would choose n opiml porfolio given he men nd vrince of his process. Thus, n invesor does no ke ino ccoun he effec h his decision my hve on he sock price. On he oher hnd, he consumpion-bsed pproch kes producion process s exogenous nd sks how opiml risk-shring would ffec he invesor s consumpion ph. This pproch implies h sock prices will chnge s resul of risk-shring. This disincion suggess n inuiive reson why he equiy-bsed pproch leds o significnly higher gins hn he consumpion-bsed pproch: he equiy-bsed pproch does no incorpore he effec of risk-shring on he sock price. For recen discussion of hese wo lierures nd heir relionship, see Lewis (999). A reled quesion is: wh is he rue risk-shring gin? In his pper, I exmine only he nrrower quesion riculed in he ile. 3 Noe h hese venues need no be independen. For exmple, i is well known h by reing sock reurns s endogenous [venue () wih se of plusible preference prmeers [venue (3) implies h he sisicl properies of sock reurns re difficul o reconcile wih consumpion growh [venue (). Below, I discuss nd focus upon his inerdependence.

3 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 3 In his pper, I show h his inuiion is no rue in generl. The reson is simple. When sock prices re endogenous, hese prices mus dus o mke inernionl invesors willing o buy he counry s equiy. This dusmen in prices leds o one-ime ineremporl subsiuion of consumpion from low growh economies o high growh economies h leves ll counries beer off. As resul, he endogenous sock price recion llows for n exr venue of welfre gins h re no presen when sock prices re reed s exogenous. Therefore, he pril equilibrium nure of he equiy pproch does no independenly explin he difference in welfre gins. () The gins from risk-shring depend crucilly upon he benefis of reducing he vribiliy of he mrginl uiliy over ime. In he equiy-bsed pproch, his mrginl uiliy depends upon sock reurns, while in he consumpion-bsed pproch his mrginl uiliy depends upon consumpion. Thus, n obvious reson for he difference in mesuring gins in he wo pproches rises from he greer vribiliy in sock reurns relive o consumpion. In n inernionl growing economy, he poenil gins from moving o n 4 inegred world cpil mrke depends upon he mens s well s he vrinces. To see why, consider he common consumpion pproch of ssuming h men consumpion growh cross counries is equl. This ssumpion hs he effec of mking he deerminisic growh re he sme so h inernionl cpil mrke inegrion only reduces he vribiliy of consumpion round his common world growh re. On he oher hnd, he equiy pproch focuses upon incresing men reurns while minimizing vrince. When he men sock price reurns differ cross counries, hen he differences beween growh res imply h inernionl cpil mrkes llow domesic invesors o move o differen deerminisic growh re in consumpion nd hereby ineremporlly smooh. Therefore, he difference beween he risk-shring gins my pper o rise from he combinion of ssumed: () common consumpion growh cross counries; nd (b) higher vribiliy of sock reurns compred o consumpion growh. In his pper, I show h he difference in consumpion growh hs surprisingly lile effec upon risk-shring gins, while he higher vribiliy of equiy reurns does. (3) Boh consumpion-bsed nd equiy-bsed pproches mus specify preference prmeers h govern risk-version nd ineremporl subsiuion. However, he low vribiliy of consumpion growh relive o equiy reurns hs n 4 Wheher he underlying process hs permnen or rnsiory disurbnces is noher imporn effec. As Obsfeld (994) shows, permnen disurbnces o idiosyncric consumpion imply h riskshring gins re higher. In his pper, I ssume h he disurbnces o idiosyncric consumpion re permnen (i.e. shocks re permnen nd no coinegred cross counries.) Therefore, if he disurbnces re no permnen, I m bising upwrd my gins from he consumpion pproch. Since I find h even he esimes bsed upon permnen shocks o consumpion re drmiclly smller hn he equiy pproch gins, rnsiory shocks o consumpion will only deepen he gp beween gins from he wo pproches.

4 4 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 imporn effec upon he mesured gins from risk-shring. This observion coupled wih plusible preference prmeers leds o well-known inconsisencies beween consumpion-bsed models nd observed finncil mrke d. To invesige he impornce of reconciling preference prmeers in he consumpion bsed pproch wih he observed behvior of sock prices, I conduc wo ses of experimens o solve for risk version nd ineremporl subsiubiliy endogenously. Firs, I se he mens of sock reurns nd he risk-free re o equl heir vlues implied by he consumpion model. In his cse, risk version nd ineremporl subsiuion re high. Since higher risk version nd ineremporl subsiubiliy boh increses he vlue of reducing vribiliy in he fuure, he risk-shring gins re quie high, consisen wih he equiy pproch gins. Second, I se he vrinces of sock reurns equl o heir vlues implied by he consumpion model. To explin he high vrince, risk version nd ineremporl subsiubiliy mus be low. In his cse, risk-shring gins re quie low, even lower hn hose implied by he sndrd consumpion pproch. The srucure of he pper is s follows. In Secion, I describe he welfre gin funcion. In Secion, I use sock reurns o provide mesures of risk-shring gins using he equiy-bsed pproch. In Secion 3, I use consumpion d o clcule risk-shring gins using sndrd consumpion-bsed pproch. In Secion 4, I use sock reurn d o clcule gins using he consumpion-bsed pproch. In Secion 5, I use momens of sock reurn d o bck ou implied preference prmeers nd re-exmine he consumpion-bsed gins. Concluding remrks follow.. The gin funcion.. The bsic frmework To clcule welfre gins, I follow sndrd prcice nd clcule he equivlen vriion of curren uiliy h brings he invesor/ consumer up o he 5 sme uiliy level s he would enoy under opiml risk-shring. In he consumpion-bsed lierure his uiliy depends upon he consumpion level. In he equiy-bsed lierure, uiliy depends upon welh direcly. For now, I simply denoe he rgumen in uiliy ime s X for generliy. Below, I ssume h X is log-normlly disribued: 6 x 5 x m s where N(0, s ) () 5 For exmple, see Lucs (987) nd Obsfeld (994,b). 6 In simulion experimens, serilly correled consumpion nd equiy reurns gve similr resuls o hose found below. I focus upon he nlyicl soluions in he ex, however.

5 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 5 nd where boh here nd below he lower-cse leers refer o he nurl logrihm of he vrible [i.e. x 5 ln(x), unless noed oherwise. Furhermore, he opiml ph for X is denoed s X while is counerprs o m nd s re defined s m nd s, respecively. Thus, he welfre gin d is defined by he equion: U(X ( d ), m, s) 5 U(X, m, s) () where U is he uiliy funcion. Where possible below, I denoe he uiliy condiioned on he ime vrible s simply U. Clculing he gins requires specifying uiliy funcion. Consn-relive risk version (CRRA) is sndrd uiliy funcion used in sse pricing s well s clculing welfre gins. However, his uiliy funcion ssumes h he coefficien of relive risk version is he sme s he inverse of he ineremporl elsiciy of subsiuion. As shown in Obsfeld (994), risk version nd he inverse of ineremporl subsiubiliy hve opposie effecs upon welfre gins. Therefore, i is imporn o use uiliy funcion h does no impose his consrin upon preferences. 7 For his reson, I use he Epsein nd Zin (989) uiliy funcion: u g (u )/(g ) /(u ) U 5 hx b[e (U ) for g, u. 0, ± (3) The prmeer u cn be inerpreed s he inverse of he ineremporl elsiciy of subsiuion in consumpion. On he oher hnd, g is he prmeer of relive risk version. The sndrd ime-ddiive uiliy funcion resuls when g 5 u. When X is log-normlly disribued, Appendix B shows h ime uiliy is: (/(u )) S F S DGD U 5 U(X, m, s) 5 X b exp ( u ) m gs (4) Similrly, uiliy under opiml risk-shring is given by: (/(u )) S F S DGD U 5 U(X, m, s) 5X b exp ( u ) m gs (49) where risk-shring suggess h s, s. This relionship will be deermined endogenously below. The wo prmeers, g nd u, re imporn in he welfre gin nlysis. The role of hese wo key prmeers herefore wrrns inspecion. Firs, noe h uiliy is incresing in he ceriny equivlen log consumpion growh ph, m gs. Therefore, reducions in he vrince of his ph will 7 While I use he Epsein-Zin funcion becuse i provides more prsimonious represenion of reduced-form uiliy, Obsfeld (994,b) uses he Weil (990) uiliy funcion. However, he Weil funcion is monoonic rnsformion of he Epsein-Zin funcion so h he gin funcion nd ll sse pricing relionships re idenicl using he wo uiliies.

6 6 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 increse his ceriny equivlen ccording o he prmeer of risk version, g. Clerly, hen, higher risk version g will led o greer welfre gins. Second, noe h ineremporl elsiciy declines s u increses. Therefore, he uiliy vlue of gins long he CE ph in he fuure declines. For his reson, higher vlues of u will led o lower welfre gins from reducion in vrince. To solve explicily for he gin funcion, subsiue Eq. (4) nd Eq. (49) ino he 8 gin definiion Eq. () nd solve for d n iniil ime period 0. This implies: /(u ) 0S F S DGD /(u ) X b exp ( u ) m gs d 5 (5) X0S b exp F( u ) Sm gs DGD Thus, he welfre gins depend upon, firs, he curren level of he uiliy deerminn relive o he opiml X 0/X 0, nd, second, he relionship beween he wo ceriny equivlen growh phs, m gs nd m gs evlued wih he ineremporl elsiciy of subsiuion prmeer, u... Componens of he gin funcion Below, I exmine he equiy-bsed nd consumpion-bsed lierure on riskshring gins using he gin funcion Eq. (5). To describe he experimens below, I wrie his funcion generlly s: d 5 d(m, m, s, s; g, u, b; X 0/X 0) 5 d(m; V; I) (59) where M is momen mrix of mens nd vrinces of disurbnces long he urky nd opiml growh ph; V is he se of preference prmeers; nd I ; X /X. 0 0 I consider wo ses of vlues for he momen mrix M. For he equiy-bsed pproch, M corresponds o he momens of sock-reurns defined s M. On he s oher hnd, for consumpion-bsed clculions, M is comprised of momens of consumpion growh res, defined s M c. I evlue he welfre gins over wo rnges of he prmeer vecor V: se of plus plusible prmeers, denoed V, nd se of prmeers h mch cerin mch momens of sse prices, denoed V. plus To deermine vlues for V, I consul he lierure. Plusible vlues for risk 9 version re considered o be beween nd 0. On he oher hnd, u is ypiclly 8 This is he sme gin funcion s used in Obsfeld (994) for he cse where X5X. 9 Risk version coefficiens wihin his rnge re exmined in sudies for he welfre gins of inernionl risk-shring, such s Obsfeld (994), Cole nd Obsfeld (99), nd Tesr (995). Mehr nd Presco (985) consider risk version of 0 o be oo high.

7 K.K. Lewis / Journl of Inernionl Economics 5 (000) ssumed o be rher high. Finlly, b is usully ssumed o be less hn. For plus V, I ssume b equls 0.98, following Obsfeld (994), nd llow u nd g o vry beween nd 5. Since hese vlues for prmeers cnno explin sse pricing relionships, I mch lso invesige he se of prmeers V which includes vlues h do mch cerin relionships. To deermine hese vlues, I choose he prmeers o eque he mens nd vrinces of sock reurns nd he risk-free re in he d o heir heoreiclly prediced vlues. Finlly, he gin funcion depends upon he vrible I which depends upon he rio of he deerminn of uiliy in urky o is counerpr under risk-shring..3. Ouline of he remining nlysis Below, I begin by clculing he gin funcion for wo benchmrk cses ssuming plusible preference prmeers. In Secion, I exmine he firs benchmrk cse. I show h he equiy-bsed model implies he gin vlue: plus d 5 d(m ; V ; ) (6) s plus Th is, for plusible prmeers, V, he gins depend upon he mens nd vrinces of sock reurns, M s. Also, since iniil welh, W 0, is unffeced by risk-shring, I 5 W 0/W 05 X 0/X 05. In Secion 3, I sudy he second benchmrk cse. I show h he consumpionbsed pproch implies he gin vlue: plus c 0 0 d 5 d(m ; V ; C /C ) (7) where he endogenous deerminion of sock prices implies h iniil urky consumpion does no equl iniil consumpion under risk-shring. As described in he inroducion, he gins in Eq. (6) re much lrger hn he gins obined from Eq. (7). I hen invesige he resons for hese differences. In Secion 4, I focus upon he endogenous equiy gin funcion Eq. (7) nd sk wh ssumpion cn mke he gins mch hose of he exogenous equiy gin funcion Eq. (6). I firs relx he common ssumpion h men consumpion growh res re common cross counries. I nex use exogenous equiy reurn momens o counerfcully clcule he endogenous equiy welfre gins. Thus, he gin is: plus s 0 0 d 5 d(m ; V ; W /W ) (8) 0 For exmple, Hll (988) rgues h u is probbly no less hn 0. See Kocherlko (990), however, for n rgumen h b cn exceed.

8 8 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 Finlly, in Secion 5, I sudy he effecs of preference prmeers using he consumpion momens. In his cse, he gins re he sme s in Eq. (7) bu wih differen preference prmeers: mch c 0 0 d 5 d(m ; V ; C /C ) (9) These experimens in Eqs. (8) nd (9) demonsre he imporn role plyed by high vribiliy in mrginl uiliy of consumpion, wheher in he form of cul equiy reurns s in Eq. (8) or in risk version s in Eq. (9). 3. Equiy-bsed gins using equiy reurns 3.. The bsic frmework The gins from inernionl diversificion in socks hve been noed since les he 970s. A sndrd pproch for exmining hese gins is o clcule he hisoricl mens nd vrinces of porfolios h include foreign socks nd deermine wheher hey domine porfolios of domesic socks lone. I follow his pproch below lhough clerly his pproch ignores he poenil for esimion risk o ffec porfolio decisions. These more diversified porfolios genere lower vrince nd/ or higher mens hn do domesic equiies lone. To illusre, Fig. depics combinion of men reurns nd sndrd deviions of porfolio combinions h llow for differen weighs on foreign socks in he porfolio of domesic US invesor. In priculr, I ke he reurns on he sock mrke indices from Morgn Snley Inernionl Cpil Mrke Perspecives for he G-7 from 969 o 993. I hen consruc muul fund by king populion-weighed verge of he non-us counry equiies, convering 3 he foreign reurns ino dollrs nd hen defling by he US price level. Deils re provided in Appendix A. In Fig., Poin A represens he men nd sndrd deviion (S.D.) of he US sock mrke over he period, corresponding o zero weigh on foreign socks. Moving long he curve represens higher weighs o he foreign sock. Clerly, he US sock mrke is domined by porfolios include foreign socks. However, he Lewis (999) discusses recen empiricl lierure h considers esimion risk in inernionl porfolio llocion nd hs even quesioned he presence of equiy home bis. For differen evidence on his issue, see Gormn nd Jorgensen (996), Beker nd Uris (996), Smbugh (997), nd Psor (999). The quesion ddressed in his pper perins only o he convenionl nlysis of welfre gins bsed upon hisoricl mens nd vrinces. 3 I choose populion-weighed verge in order o mke he nlysis consisen wih he consumpion-bsed represenive gen frmework of he nex secion nd hereby o fcilie he experimens using momens of sock reurns below. However, similr resuls were obined using, lernively, cpilizion-weighed verge nd simple verge of equiies.

9 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 9 Fig.. Risk reurn rdeoff for US invesor. US ggrege proporion of welh held in foreign equiies ppers o be only bou 8%, ccording o Bohn nd Tesr (996). The figure lso noes he world porfolio where he shres of socks equl heir shres in he world index. The porfolios represened by Fig. re herefore combinions of he US sock mrke nd fixed porfolio of foreign socks. A fully opiml combinion of foreign socks would be he porfolios providing he lowes vrince for ny given men reurn, he so-clled efficien fronier. Since he porfolios of his efficien fronier would imply even higher uiliy hn hose given by his more resricive risk-reurn rdeoff, he rue gins from sock diversificion will be even higher hn hose mesured by Fig Clculing welfre gins Wih he uiliy funcion in Eq. (3), I clcule he gins from moving from he uiliy of porfolio of 00% US socks poin A o he uiliy he opiml combinion. I follow he sndrd men-vrince ssumpion h welh is he deerminn of uiliy. Thus, if welh is log-normlly disribued, he uiliy funcions he urky poin A nd he opimum re given by Eqs. (4) nd (49), respecively, where X5X 5 W, iniil US welh. The mens nd vrinces of he urky nd opiml porfolios re differen, however, nd re given by: us us us w 5 w m s z where z N(0, s ) (0)

10 0 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 w 5w m s z where z N(0, s ) (09) The men nd reurn of he foreign-lloced welh porfolio depends upon he men nd vrince of he overll porfolio. The evoluion of his porfolio cn be pproximed s: us w 5w ( f)r fr * () us where R nd R* re he reurns in he US nd foreign equiy mrkes, respecively, nd where f is he porfolio shre in he foreign equiy. Tble, Pnel (A) gives he summry sisics for he US nd foreign equiy mrkes. Then, he opiml choice of he foreign equiy shre f cn be deermined by mximizing he uiliy funcion Eq. (49) subec o he consrin h he men nd he vrince of welh in Eq. () deermines m nd s in Eq. (09). Subsiuing he men nd vrince of Eq. () ino Eq. (09), he prmeers of he welh disribuion cn be expressed in erms of he porfolio shre in he foreign equiy, m(f) nd s(f). Thus, he firs-order condiions from mximizing Eq. (49) wih respec o he porfolio shre f subec o he consrin h m 5m(f) nd Tble plus Equiy-bsed model gins from US diversificion using equiy d d(m ; V ; ) Men S.D. Correlion (A) Summry sisics US Foreign Foreign shre Men S.D. (B) Opiml porfolio g g g g u 5 u 53 u 54 u 55 (C) Gins g g g g NA NA NA u NA, no vilble, becuse diversified uiliy is no defined: b #M (see Eq. (3) in he ex). s

11 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 s 5s(f) is: ( m/ f) 5 g( s/ f) () Solving Eq. () for f deermines he opiml porfolio. Noe h he porfolio llocion decision depends only upon he degree of risk version nd no oher preference prmeers. Deils re given in Appendix C. Pnel (B) repors he opiml llocion ino he foreign equiy s he risk version prmeer vries from o 5. For g equl o, he invesor lloces his porfolio 00% in foreign socks. However, s risk version rises up o g equl o 5, he invesor reduces he vribiliy of his porfolio by moving his foreign llocion shre down o 44%. Correspondingly, he men of his porfolio lso declines. Given hese opiml porfolios bsed upon g, he gin funcion cn be clculed s in Eq. (5) where X05X 0 nd m nd s, nd m nd s re deermined by he US sock mrke nd he opiml porfolio of US nd foreign sock mrkes, respecively. Tble repors he welfre gins in Pnel (C). As noed bove, hese mesures represen he lower bounds for he rue gins since he fesible se of porfolios is resriced o liner combinions of he US sock mrke nd fixed muul fund of foreign socks. Ech row of Pnel (C) repors he gins for given level of g, nd herefore given porfolio llocion. From lef o righ, s u increses he welfre gins decrese since he elsiciy of subsiuion decreses nd he invesor plces less uiliy on fuure gins in ceriny equivlen consumpion. For exmple, for g 5, he gins re 8.8% of curren welh when u 5, bu only 0% of welh when 4 u 55. For higher risk version, he gins generlly increse. For exmple, when g 55 nd u 5, he gins re bou 5% of welh. For high levels of g nd u, expeced uiliy is no defined. To see why uiliy is no defined s u nd/or g imes he vrince s increses, noe h uiliy in Eq. (4) depends upon he condiion h: b. exp F( u ) Sm gs DG (3) This is becuse he inverse of b exp[( u )( m gs ) cs s n overll 5 discoun re for fuure reurns. When Eq. (3) does no hold, s cn hppen when he ceriny equivlen growh re ( m gs ) is negive, uiliy does no converge s goes o infiniy nd his discoun re exceeds uniy. This possibiliy is more likely s g or s increse. In he presen cse, he condiion is violed 4 Noe h welfre gins re sricly incresing in g for given disribuion prmeers: m, s, m nd s. However, he disribuion prmeers for he opiml porfolio vry wih g s noed bove. 5 See he discussion in Obsfeld (994).

12 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 Fig.. Log consumpion profile for US invesor fcing exogenous sock reurns. becuse he vribiliy of sock reurns is high. In Secion 5 below, he condiion will be violed in some circumsnces becuse risk version is oo high Grphicl descripion of he gins from diversificion Fig. illusres he gins o n individul US invesor o moving from he growh ph ssocied wih domesic reurns lone o he ph ssocied wih he opiml porfolio. The op wo lines show he difference in growh phs ssocied wih holding he foreign relive o he domesic porfolio in he cse where here is no unceriny. Clerly, he higher reurns, lbeled m, correspond o he invesor s wih he opiml porfolio hving much higher consumpion profile hn he lower men US reurns, lbeled m. s The lower wo lines show he ceriny equivlen phs in he presence of unceriny. Boh phs re lower hn heir counerprs in he bsence of unceriny. However, he porfolio wih foreign reurns lso hs lower vrince hn he domesic porfolio. 4. Consumpion-bsed gins using consumpion growh I now provide simple frmework for ssessing gins from risk-shring using he consumpion-bsed pproch. Following he lierure, I ssume h idenicl represenive gens in counry for ech of N counries receive heir own counry s per cpi oupu srem, Y. For simpliciy, I ssume h he producion

13 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 3 of his oupu is given s n exogenous endowmen process. As before, I clcule welfre gins by solving for nd compring he ceriny equivlen consumpion phs in he bsence of risk-shring nd under perfec risk-shring. To do so, I firs briefly review he closed economy cse in he bsence of inernionl risk shring before consrucing he diversified equilibrium. 4.. Aurky The equilibrium consumpion process wihou ccess o inernionl mrkes is rivilly given by he endowmen process. For counries wih idenicl preferences, he pricing of risk will differ in he N closed economies if he oupu processes differ cross counries. This closed economy equilibrium is well-known nd 6 herefore is only briefly summrized here. Defining s s he se of he economy ime, including relizions of he endowmen, he represenive gen will mximize uiliy in Eq. (3) such h his budge consrin holds. Specificlly, he will consume ech period nd buy shres in he domesic equiy. This opimizion is given by he Bellmn equion: s.. u g (u )/(g ) (/(u )) C,k V(W, s ) 5Mx[(C ) be (V(W, s ) ) (4) C k Q 5 (Q Y )k (5) where V is he vlue funcion bsed upon uiliy Eq. (3) which is in urn funcion of he welh of counry invesor ime, W, nd he se vecor ime, s. In priculr, W ; k Q where k re he shres held of socks pying ou he per cpi endowmen of counry nd Q is he sock price. This mximizion implies he firs-order condiion: (g )/(u ) u(g )/(u ) (g )/(u ) b Eh(C /C ) ( R ) 5 (6) where R is he reurn on he domesic sock pying domesic per cpi endowmens, Y. I ssume h he endowmen process is log-normlly disribued s bove. Defining y ; ln(y ), y 5 y m s z where z N(0, s ) (7) In his cse, Appendix B shows h he sock price hs n nlyicl soluion h depends upon he curren level of oupu s well s he disribuion nd preference prmeers: Q 5 Q (Y ; m, s ; g, u, b ). In equilibrium, k 5 so h ech invesor holds one shre of per cpi oupu. 6 See Epsein (988) nd, for he ime ddiive cse, Lucs (98).

14 4 K.K. Lewis / Journl of Inernionl Economics 5 (000) Risk-shring wih open finncil mrkes Suppose now h inernionl cpil mrkes re opened so h ech counry cn rde in he equiies of ll oher counries. In he new equilibrium, invesors in i counry hold k shres in socks of counry i oupu. Insed of Eq. (5), he budge consrin becomes: N N i i i i i i5 i5 C O k Q #O k (Y Q ) (8) where he mximizion in Eq. (4) is now over C nd k, ;i 5,..., N. Since ll counries hve he sme homoheic uiliy funcion, hen ech counry 7 holds he sme porfolio llocion in equilibrium. Therefore, he problem cn be rewrien in erms of world muul fund pying ou he world per cpi endowmen, defined s Y. A ime, shres of he muul fund held by counry nd is price re defined s k nd Q, respecively. Rewriing he budge consrin Eq. (8) wih hese definiions implies: C k #k (Y Q ) (9) This mximizion implies he firs-order condiion, (g )/(u ) u(g )/(u ) (g u )/(u) b E h(c /C ) (R ) (R ) 5 (0) where R nd R re, respecively, he reurns on he world muul fund nd he domesic equiy world mrke prices. The world endowmen process is log- 8 normlly disribued s bove, y 5y m s z where z N(0, s ) () Appendix B shows h he world muul fund price, Q, nd he domesic sock price in world mrkes, Q, hve nlyicl soluions of he form: Q 5Q (Y ; m, s, m, s; g, u, b ) nd Q 5Q (Y ; m, s; g, u, b ) where s is he covrince of he world producion process nd he domesic producion process. The equilibrium llocion of shres of he world muul fund depends upon hese sock prices. Invesors in ech counry sell off clims o curren nd fuure oupu from heir own counry in exchnge for clims o curren nd fuure world oupu. Residens in counry sell heir equiy price Q nd for ech uni of i 7 See, for exmple, Ingersoll (987). 8 Noe h, s wih equiy, his specificion implies n pproximion h boh he sum of he oupu processes nd he oupu processes hemselves re log-normlly disribued. See Lewis (996) for Mone Crlo simulions h sugges his pproximion is relively innocuous.

15 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 5 good, hey will purchse he quniy /Q of shres in he muul fund. Therefore, in equilibrium, k 5Q /Q Clculing welfre gins Using he urky equilibrium in Secion (3.) nd he open economy equilibrium in Secion (3.), welfre gins cn be clculed using Eq. (5). Noe h, under urky, he iniil level of consumpion, C05 Y 0, while he opiml iniil level of consumpion depends upon he vlue of he economy s shre in world oupu, C 05k 0 Y05 (Q 0/Q 0)Y 0. Therefore, he gin funcion cn be rewrien s in Eq. (5) where X /X 5 (Y Q )/(YQ ). Since hese prices depend only upon he prmeers of he disribuion nd he uiliy funcion, he gins cn be clculed for ech counry given he consumpion mens nd vrinces s deiled in Eq. (7) plus d 5 d(m c; V ; C 0/C 0). Tble repors he resuls of clculing he welfre gins for ech of he G-7 counries gins he world. Following Obsfeld (994b), I use n upded version hrough 99 of he d in he Penn World Tbles (Summers nd Heson, 99). For he purpose of comprison wih he sock mrke d, he smple begins in 969. Pnel (A) repors summry sisics for he G-7 counries s well s world index h corresponds wih he populion-weighed world sock mrke index discussed in Secion. Mny consumpion-bsed mesures of welfre gins from risk-shring re consumpion growh res s equl cross counries. For his reson, Pnel (B) repors he welfre gins from risk-shring when ll counries re ssumed o shre he sme growh re s he world. To conserve spce, he gins re repored for he wo exreme cses exmined in Tble, i.e. for g nd u equl o nd 5. The lowes welfre gins re represened by he cse where risk version is lowes g 5 nd where he inverse of ineremporl subsiubiliy is highes u 55. The gins re slighly lrger for he ime-ddiive cses, g 5u 5 nd 5, nd re lrges when risk version nd ineremporl subsiuion is high wih g 55 nd u 5. However, he gins re significnly lower hn hose in Tble bsed upon similr uiliy prmeers. In mos cses, he gins re quie low. For he US, he mximum gin is 0.5% of curren consumpion. Cerin counries mesure higher gins from risk-shring relive o oher counries. The higher gins ypiclly rise from wo sources. Firs, for counries such s Cnd or he UK, heir urky consumpion phs re more vrible hn he res of he world. Second, counries such s Ily hve lower correlion wih he res of he world, hereby incresing he vlue of heir equiy on world mrkes. Pnel (C) exmines he effecs of llowing he men growh res o differ. The overll mgniude of he gins end o increse wih differing mens. For he lowes gin prmeers, he US gins increse o 0.06% from 0.04%. Srikingly,

16 6 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 Tble plus Consumpion-bsed model gins from diversificion using consumpion d d(m c; V ; C 0/C 0) Men S.D. Correlion mrix Cnd US Jpn Frnce Germny Ily UK World (A) Summry sisics Cnd US Jpn Frnce Germny Ily UK World u 55 u 55 u 5 u 5 g 5 g 55 g 5 g 55 (B) Sme mens US Cnd Jpn Frnce Germny Ily UK (C) Differing mens US Cnd Jpn Frnce Germny Ily UK he mximum welfre gins for Jpn increse o 5.5% relive o only 0.95% in Pnel (B). Thus, much of he gins o Jpn derive from he srong equiy vlue of is high growh re. I illusre his effec grphiclly below. Alhough hese vlues re lrger hn when he mens re ssumed o be he sme, hey remin significnly smller hn he sock reurn gins in Tble Grphicl represenion of he gins Fig. 3 shows he source of hese gins wih differing mens for he low gin cse where g 5 nd u 55. The oher prmeer vlues imply similr rde-off. For ech counry, he figure shows he ceriny equivlen consumpion growh phs for urky s he dshed line (lbeled s m gs ) nd for risk-shring s he

17 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 7 Fig. 3. Risk-dused consumpion profile for endogenous sock reurns using consumpion mens nd vrinces (g 5, u 55).

18 8 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 solid line (lbeled s m gs ). The inerceps show he level of consumpion ssocied wih he ph ime 0. This figure mkes cler why Jpn hs relively lrge gins while he US hs low gins. The Jpnese sell off heir clims o heir high growh economy by receiving relively high price in he beginning of ime. The Jpnese pricipe becuse hey subsiue curren consumpion for fuure consumpion. The res of he world wns he higher growh nd provides his ineremporl subsiuion. Therefore, Jpn enoys lrge gins. On he oher hnd, he US hs low gins becuse Americns rde off curren consumpion for fuure consumpion. Furhermore, since he US oupu srem is highly correled wih he world oupu srem, he price of US equiy is relively low. Similr rde-offs re shown for he oher counries s well Conclusions from consumpion gins Overll, he evidence in Tbles nd shows h he equiy-bsed pproch generes significnly higher gins hn he consumpion-bsed pproch. While boh mesures re bsed upon similr gin funcion, he nlysis so fr poins o wo min poenil resons. Firs, he consumpion pproch res sock prices s endogenous nd hereby llows for n ineremporl subsiuion cross counries s well s he opporuniy o reduce risk. This ineremporl subsiuion is missing in he exogenous sock-bsed pproch. Clerly, his ineremporl subsiuion cn poenilly excerbe he gins for some counries such s Jpn wih high growh res. Therefore, his explnion lone does no pper o explin he puzzle. Second, he mesures of risk-dused growh phs re differen. The consumpion pproch clcules his ph from he mens nd vrinces of consumpion growh. The equiy pproch generes his ph using he mens nd vrinces of equiy reurns. This pproch res he underlying uiliy s genered from he sock reurn process. This disincion suggess simple, counerfcul experimen for sudying he implicions of his ssumpion, s I describe nex. 5. Consumpion-bsed gins using equiy reurns To invesige he impornce of reing he underlying deerminn s uiliy of sock reurns s opposed o consumpion, I reclcule he consumpion pproch gins using sock reurn d insed of consumpion d. Alhough clculing he gins in his wy is clerly counerfcul, i ddresses he quesion: re he higher gins mesured by equiy genered by he higher vribiliy nd/ or differences in men reurns in socks s opposed o consumpion growh?

19 5.. Mesuring he gins K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 9 I clcule he generl equilibrium gin using Eq. (5) where s for he generl equilibrium resuls in Tble, X 0/X 05 (Y 0Q 0)/(YQ 0 0) ±. In his cse, however, he sock prices depend upon he mens nd vrinces of sock reurns s opposed o consumpion nd he iniil consumpion level is reed s he welh level, X 0/X 05 (Y 0Q 0)/(Y 0Q 0) 5 W 0/W 0. Furhermore, he uiliy level iself depends upon he mens nd vrinces, so h he gin funcion cn be wrien in he plus generl form given in Eq. (8): d 5 d(m s; V ; W 0/W 0). Pnel (A) of Tble 3 repors he summry sisics used in he clculion. As wih consumpion, he correlion of US equiy reurns wih he world porfolio is higher hn h of he oher counries. Pnels (B) nd (C) give he gins for he exreme cse of low nd high gin prmeer combinions: u 5 5, g 5 nd u 5, g 5 5. Pnel (B) repors he gins for he cse where he world men reurn is ssumed he sme for ll counries, he Tble 3 plus Consumpion-bsed model gins from diversificion using equiy d d(m s; V ; W 0/W 0) US Cnd Jpn Frnce Germny Ily UK World Ineres re (A) Summry sisics Men S.D Corr(X, World) u 55 u 55 u 5 u 5 g 5 g 55 g 5 g 55 (B) Sme mens US 4.5 NA.79 NA Cnd 9.5 NA 6.5 NA Jpn 5.03 b NA b NA Frnce.98 NA 5.96 b NA Germny 3.80 NA 6.3 NA Ily 43.6 NA 79.6 b NA UK 7.44 NA 6.46 b NA (C) Differing mens US 3.85 NA Cnd 4.80 NA b NA Jpn 6.49 NA NA Frnce.0 NA NA Germny 4.04 NA NA Ily NA b NA NA b NA UK 5.8 NA 63. NA u No vilble, becuse domesic sock price world prices is no defined: b #M H (see Eq. (B.5) in Appendix B). b u No vilble, becuse urky uiliy is no defined: b #M (see Eq. (3) in he ex).

20 0 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 counerpr of Pnel (B) in Tble. Pnel (C) repors he gins when he mens re llowed o differ by counry, he sndrd ssumpion in he equiy pproch lierure. Clerly hese gins re subsnilly lrger hn hose in Tble. As wih he consumpion-bsed gins, he gins re lower for he US hn he oher counries since is correlion wih he world is higher. For he oher counries, he gins re quie lrge. Even for he low gin prmeer cse, he benefis o hese counries from risk-shring rnge from bou 5% o 6% of permnen consumpion. 5.. Grphicl represenion of he gins Fig. 4 shows why hese gins re so much lrger when sock reurn mens nd vrinces re used. As before, he figure shows he urky ceriny equivlen growh ph of consumpion s he dshed line, while he risk-shring ph is depiced by he solid line. As in Fig. 3, he prmeer vlues re he low gins cse: g 5 nd u 55. For counries such s Cnd, he differences in vrinces bsed upon riskshring relive o urky imply significn increses in he il o risk-dused consumpion growh. Since Ily s urky risk-dused growh re is negive, he gins from moving o risk-shring re unbounded nd undefined. For Frnce, Germny nd he UK, risk-shring no only pus he economies on higher risk-dused consumpion growh ph, i lso rises iniil consumpion s well. The reson is h hese counries obin high price for heir equiy on world mrkes becuse of heir relively high men nd low correlion wih he world reurn. Even hough heir urky deerminisic men reurns re higher hn he world, hese counries chieve higher risk-dused growh re of consumpion becuse of he lower vrince resuling from diversificion. This improvemen in vrince mesured wih consumpion d is significnly lower since consumpion vrinces re so much smller. As in Fig. 3, Jpn gins by ineremporlly subsiuing higher curren consumpion levels for fuure consumpion growh. Using sock reurn mens nd vrinces excerbes his effec, however. The significnly higher men of Jpnese socks ogeher wih is diversificion benefis imply h Jpn cn commnd significnly high price of socks on world mrkes. Wih growh re of nerly % in he sock index, he Jpnese sock price is so high h he risk-shring consumpion levels re bove he urky levels for over 5 yers. As resul, he gins o Jpn re subsnil. As his figure shows, n imporn conribuing fcor o why sock reurns sugges such differen gins of inernionl risk-shring is he drmiclly higher vrince of socks. This high vrince implies boh h invesors re willing o py significnly for clims on equiies h reduce overll consumpion risk nd h risk-dused consumpion growh is drmiclly incresed by reducions in vrince.

21 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 Fig. 4. Risk-dused consumpion profile for endogenous sock reurns using sock reurns mens nd vrinces (g 5, u 55). These clculions re counerfcul, however. They re sock reurns s hough hey chrcerize he consumpion process. I is well-known h he generl equilibrium frmework cnno explin he mens or vrinces of sock reurns wih plusible uiliy prmeers. Therefore, n lernive explnion of he puzzle my be h he preference prmeer vlues for he consumpion-bsed

22 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 model re inconsisen wih he behvior of sock reurns. I consider his possibiliy nex. 6. Consumpion-bsed gins using preference prmeers h mch equiy reurns The inconsisency beween sock reurn behvior nd is predicions from consumpion-bsed models using convenionl uiliy prmeer vlues hs been esblished repeedly in he lierure. For exmple, Mehr nd Presco (985) show h, wih ime-ddiive uiliy, he men excess reurn on equiy over he risk-free re in he US requires very high levels of risk-version. This relionship is verified in Weil (989) who shows h high levels of risk-version lso genere implusibly high levels of he risk-free re. Kndel nd Smbugh (99) rgue h, while he equiy premium nd he risk-free re cn be explined by high levels of risk-version, oher feures of sock reurns cnno. In his secion, I sk wheher prmeer vlues h mch feures of reurn behvior cn help reconcile he difference in implied gins beween equiy nd consumpion risk-shring. 6.. Mching he men equiy reurns nd risk-free re I begin by solving for he risk-version prmeer g nd he ineremporl elsiciy prmeer u h mch he men of he equiy reurns nd he risk-free 9 re implied by he model o heir counerprs in he d. Th is, he consumpion-bsed model in Secion 3 implies sock reurns given he momens of consumpion, M c, nd prmeers, V. The form of hese reurns is derived in Appendix D. Tble, Pnel (A) repors he mens nd vrinces of hese reurns. Therefore, solving he wo equions for he men of equiy nd he men of he risk-free re for he wo unknown prmeers, g nd u, provides he vlues for he mch mching prmeers, V. Tble 4 repors hese combinions of g nd u mching hree differen equiy reurns: he urky US sock price in Pnel (A), he US sock price in inegred world mrkes in Pnel (B), nd he world muul fund price in Pnel (C). For ech combinion of preference prmeers, he ble provides summry sisics for ech counry s implied reurns from he model. In ll hree pnels of Tble 4, u is quie low while g is quie high. In fc, o explin boh he high world muul fund price of 5.86% nd he 3% risk-free re 9 For he purposes of clculing he risk-free re, I use he dollr London-Inerbnk Offer Re (LIBOR) defled by US inflion. See Appendix A.

23 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 3 Tble 4 Preference prmeers h mch men reurns nd heir implied inernionl reurns US Cnd Jpn Frnce Germny Ily UK World (A) Mching US equiy for urky nd risk-free re Implied prmeers: u 5.69, g 55.7 Implied reurns: Men S.D (B) Mching US equiy for inegred mrkes nd risk-free re Implied prmeers: u 50.9, g Implied reurns: Men S.D (C) Mching world equiy nd risk-free re Implied prmeers: u 5.6, g 53.5 Implied reurns: Men S.D Corr(X, World) in Pnel (C), he risk version prmeer is 3.5 while he inverse of he 0 ineremporl elsiciy of subsiuion is only.6. Nobly, even when he mens of reurns re mched s in Tble 4, he vrinces re no. In ll cses, he implied sndrd deviions re significnly lower hn he cul sndrd deviions in he d. I will reurn o his problem below. 6.. Implied welfre gins from inernionl diversificion I now sk wheher he consumpion-bsed gins implied by preference prmeers h mch he equiy reurns re consisen wih he equiy-bsed gins. Tble mch 5, Pnel (A) herefore repors he gins s given in Eq. (9) d 5 d(m c; V ; C 0/C 0). These gins correspond o he sme gins clculed s in Tble, bu using vlues of u nd g h mch reurns. The firs column of Tble 5 Pnel (A) gives welfre gin esimes when he uiliy prmeers re se o mch he US reurns under urky nd in Pnel (A) of 0 The differences beween he closed US economy cse in Pnel (A) nd oher esimes in he lierure lrgely derive from differences in he risk-free re nd he equiy premium over his period. Using cenury-long d, Mehr nd Presco (985) repor n esime of 0.9 for he risk-free re nd bou 6% for he excess of equiy over he risk-free re.

24 4 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 Tble 5 mch Consumpion-bsed model gins from diversificion using mching prmeers d(m c; V ; C 0/C 0) US equiy (urky) US equiy (inegred) World equiy g 55.7 g g 53.5 u 5.69 u 50.9 u 5.6 (A) Mching mens US Cnd b NA Jpn Frnce Germny Ily 4.4 NA 89.5 UK b NA US equiy (urkey) US equiy (inegred) World equiy g g g 5.98 u u u (B) Mching vrinces US Cnd Jpn Frnce Germny Ily UK No vilble, becuse domesic sock price world prices is no defined: b #M H. No vilble, becuse urky uiliy is no defined: b # M (see Eq. (3) in he ex). u b u Tble 4. While he gins for some counries such s Ily, he UK nd Cnd re quie lrge in excess of 33%, he gin for he US is lower bou 7%. The second column uses he prmeers mched o he US equiy reurn under open mrkes. Agin, he coss for he US re lrger, bu remin less hn hose implied by sock reurns bou %. Finlly, he hird column mches he equiy premium o he reurn on he world equiy muul fund. In his cse, wih risk version prmeer of 3.5, he welfre gins re in he rnge suggesed by he sock reurns. Furhermore, he risk-dused growh res for he high vrinces counries of he UK nd Cnd now become negive nd he effecive discoun re exceeds uniy, implying h uiliy is no defined. Overll, he evidence suggess h when uiliy prmeers mch he world equiy premium, even consumpion d imply lrge gins o risk-shring Grphicl represenion of he gins Fig. 5 shows why he gins from inernionl diversificion re so lrge wih high risk version. For he purposes of illusrion, I use he prmeers g 53.5

25 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 5 Fig. 5. Risk-dused consumpion profile for endogenous sock reurns using consumpion mens nd vrinces (g 53.5, u 5.6).

26 6 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 nd u 5.6 which mch he world equiy premium nd he risk-free re [Tble 4, Pnel (C). Wih high risk version, he urky consumpion profiles re significnly fler hn heir low risk version counerprs depiced in Fig. 3. For he US, in fc, he high risk version implies h he urky risk-dused consumpion profile is negive. Furhermore, he risk-dused consumpion phs for he UK nd Cnd re sufficienly negive o viole he condiion in Eq. (3) h he discoun re is less hn. As resul, uiliy is undefined for hese wo counries hese prmeer vlues. The risk-shring equilibrium leds o significn welfre gins. Wih high risk-version, invesors in ech counry vlue grely even smll reducions in consumpion vrinces. As resul, he socks from counries such s Ily hving low covrinces wih he res of he world become much more vluble. This phenomenon lso shows why he US hs lower gins hn he oher counries. Since he US hs he highes covrinces wih he res of he world, is sock is less vluble on he world mrke. This is depiced in Fig. 5 by drop in he inercep, represening decline in iniil consumpion. However, he gins from buying n incresing risk-dused consumpion ph significnly ouweigh he losses from he low equiy vlue. Compring Figs. 4 nd 5 shows h he profiles mesured using sock reurn d nd plusible preference prmeers re quie differen from hose mesured using consumpion d nd preference prmeers h mke consumpion d mch sock reurns in he model. The sock reurn clculions in Fig. 4 using low risk version imply seeply rising consumpion phs. The gins derive from higher growh re, n ineremporl subsiuion owrd higher curren consumpion, or boh. On he oher hnd, Fig. 5 show h high risk version coupled wih consumpion d lower he risk-dused consumpion growh re boh in urky nd under risk-shring. Thus, for counries oher hn he US, he gins derive lrgely from higher vlue of domesic equiy on world mrkes, depiced by he shif in inercep Mching vrinces Above, I described he gins from diversificion using preference prmeers h mch he mens of sock reurns. However, Tble 4 demonsres h he models imply vrinces h re oo low o be consisen wih cul sock reurn vrinces. I herefore consider differen se of prmeers h mch he vrince, insed of he men, of sock reurns. These clculions re deiled in Appendix D. Pnel (B) of Tble 5 repors he resuls for he hree sock reurns described previously: he urky domesic sock reurn, he domesic sock reurn world prices, nd he world equiy reurn. As he ble shows, he ineremporl subsiubiliy mus be very low o mch he sock reurn vrinces. u, he inverse of his subsiuion prmeer, rnges from 94.6 in he cse of domesic equiy

27 K.K. Lewis / Journl of Inernionl Economics 5 (000) 35 7 world prices o 3.96 for domesic equiy urky. Inuiively, for invesors o be willing o ccep such high vribiliy, hey mus be relively indifferen o chnges in mrginl uiliy over ime. For his indifference o be rue, he elsiciy of subsiuion mus be exremely low (u high). A he sme ime, risk version is modere wih g rnging from bou 6 o. As described bove, low ineremporl subsiuion miiges he gins from risk-shring. In his cse, invesors do no vlue srongly he fuure gins of lower vribiliy in he consumpion profile. Thus, some counries such s he US h mus subsiue curren for fuure consumpion in he risk-shring equilibrium gin less hn 0.0% of permnen consumpion. These gins re clerly lower hn he more sndrd consumpion-bsed gins mesured in Tble. 7. Conclusion In his pper, I hve exmined he sources of differences beween equiy-bsed nd consumpion-bsed clculions of he welfre gins from inernionl riskshring. The differences lrgely come from differences in he uiliy gin of relized vribiliy. This uiliy gin cn rise in wo wys. Firs, he vribiliy of he deerminn of uiliy cn iself be high. Thus, when he deerminn of uiliy is ssumed o be equiy, he high reurn vrince implies significnly higher vribiliy of uiliy over ime compred o he cse when his deerminn is ssumed o be consumpion. Second, he vlue of reducing he vribiliy my be high. For exmple, when risk version is sufficienly high o explin he equiy premium on n inernionl diversified world muul fund, he implied gins from risk-shring mesured from consumpion is comprble o he gins bsed upon equiy direcly. The pper shows h eiher of hese fcors cn explin he differences beween he high gins from equiy-bsed risk-shring compred o consumpion-bsed risk-shring. However, he nlysis lso demonsres h hese wo effecs opere hrough somewh differen chnnels. Using sock reurn d in consumpion-bsed model, he gins lrgely rise from increses in he growh re of he ceriny equivlen ph of consumpion. However, using high risk version implies fler ceriny equivlen consumpion phs bu greer gins from ineremporl subsiuion. The pper lso rules ou he impornce of some oher plusible explnions for differences in mesured gins. Convenionl wisdom suggess h clculions bsed upon exogenous sock reurns do no llow he invesor o inernlize he effecs of his cions on he sock reurn nd hereby excerbe he gins from inernionl diversificion. However, I show h inernlizing hese effecs cn cully increse he gins from diversificion for high growh counries. This pper pinpoins he resons for significn differences in ppren gins o risk shring beween equiy-bsed nd consumpion-bsed mesures. Therefore,