Return Predictability and the Implied Intertemporal Hedging Demands for Stocks and Bonds: International Evidence

Transcription

1 Reurn Predicabiliy and he Implied Ineremporal Hedging Demands for Socks and Bonds: Inernaional Evidence David E. Rapach Deparmen of Economics Sain Louis Universiy 3674 Lindell Boulevard Sain Louis, MO Phone: Fax: Mark E. Wohar* Deparmen of Economics Universiy of Nebraska a Omaha RH-52K Omaha, NE Phone: Fax: mwohar@mail.unomaha.edu November 9, 2004 Absrac In his paper, we invesigae reurn predicabiliy and he implied ineremporal hedging s for socks and bonds in he U.S., Ausralia, Canada, France, Germany, Ialy, and U.K. We firs esimae predicive regression models for domesic bill, sock, and bond reurns in each counry, where reurns depend on he nominal bill yield, dividend yield, erm spread, and lagged reurns. Employing he recenly developed mehodology of Campbell, Chan, and Viceira (2003), we calculae he implied opimal asse s, including heir myopic and ineremporal hedging componens, for domesic bills, socks, and bonds for an invesor wih an infinie horizon, Epsein-Zin-Weil uiliy, and a coefficien of relaive risk aversion equal o 4, 7, or 0 in each counry. We find ha reurn predicabiliy generaes sizable posiive ineremporal hedging s for domesic socks in he U.S. and U.K., while he ineremporal hedging s for domesic socks are decidedly smaller in Ausralia, Canada, and Germany and essenially zero in France and Ialy. The ineremporal hedging s for domesic bonds are negaive and reasonably large in magniude in he U.S., France, Germany, and Ialy, while hey are considerably smaller in magniude in Ausralia, Canada, and he U.K. We also use he Campbell, Chan, and Viceira (2003) approach o calculae opimal asse s for an invesor in he U.S. who, in addiion o domesic bills, socks, and bonds, has access o foreign socks and bonds. We coninue o find a sizable posiive ineremporal hedging for U.S. socks, and an imporan posiive ineremporal hedging for U.K. socks emerges. In anoher exercise, we find ha invesors in Ausralia, Canada, France, Germany, Ialy, and he U.K. who have access o U.S. socks and bonds all display sizable posiive ineremporal hedging s for U.S. socks. Overall, we discover ineresing similariies and differences in he implied ineremporal hedging s for socks and bonds across counries, and our resuls indicae ha reurn predicabiliy implies especially srong ineremporal hedging s for U.S. and U.K. socks. JEL classificaions: C32, G Key words: Asse allocaion; Ineremporal hedging ; Reurn predicabiliy *Corresponding auhor. Rapach acknowledges financial suppor from a John Cook School of Business Summer Research Gran. We hank Tom Miller, Jack Srauss, Bonnie Wilson, and seminar paricipans a Sain Louis Universiy for helpful commens. The usual disclaimer applies. The resuls repored in his paper were generaed using GAUSS 6.0. The GAUSS programs are available a hp://pages.slu.edu/faculy/rapachde/research.hm. Some of he GAUSS programs are based on MATLAB programs available on John Campbell s web page a hp://kuznes.fas.harvard.edu/~campbell/daa.hml.

2 . Inroducion There has been a recen resurgence of ineres in porfolio choice problems. In paricular, ineres in muli-period porfolio choice (dynamic asse allocaion) problems has been reinvigoraed by he body of empirical evidence accumulaed over he las wo decades indicaing ha sock and bond reurns have imporan predicable componens. As iniially recognized by Samuelson (969) and Meron (969, 97), reurn predicabiliy has poenially imporan implicaions for muli-period porfolio choice problems. More specifically, reurn predicabiliy can give rise o ineremporal hedging s for asses, so ha in conras o he canonical saic porfolio choice problem due o Markowiz (952) invesors look beyond one-period-ahead when opimally allocaing across asses. Inuiively, invesors may wan o hedge agains adverse fuure reurn shocks, and reurn predicabiliy provides a emporal mechanism o accomplish his. While reurn predicabiliy can have imporan implicaions for muli-period porfolio choice problems, a difficuly in sudying hese problems is ha exac analyical soluions are generally no available. 2 This has led researchers o use differen approaches in order o solve muli-period porfolio choice problems in empirical applicaions. A number of researchers ake advanage of gains in compuing power and employ compuaionally inensive numerical procedures o approximae he soluions o muliperiod porfolio choice problems in he presence of reurn predicabiliy. For example, Brennan, Schwarz, and Lagnado (997), Barberis (2000), and Lynch (200) use discree-sae approximaions o numerically solve porfolio choice problems for invesors wih long horizons when reurns are predicable. Balduzzi and Lynch (999), Lynch and Balduzzi (2000), and Lynch and Tan (2003) also employ discree-sae approximaions o numerically solve similar ypes of problems when ransacion coss are nonzero. Anoher approach in he empirical lieraure uses approximae analyical mehods o Noe ha for our purposes, i is he exisence of reurn predicabiliy iself and no he reason for is exisence ha has poenially imporan implicaions for muli-period porfolio choice problems, so we can sidesep he horny issue of wheher reurn predicabiliy is due o ime-varying equilibrium reurns or marke inefficiencies (Fama, 99). Campbell (2000) provides a survey of he predicabiliy lieraure. 2 Kim and Omberg (996), Lui (998, 200), and Wacher (2002) obain exac analyical soluions in cerain special classes of muli-period porfolio choice problems.

3 2 solve porfolio choice problems for invesors wih infinie horizons when reurns are predicable in neighborhoods of known exac soluions (Campbell and Viceira, 999, 200, 2002). 3 In a recen exension of Campbell and Viceira (999), Campbell, Chan, and Viceira (2003; henceforh, CCV) develop an approach ha combines an approximae analyical mehod wih a relaively simple numerical procedure. This approach has he advanage of being able o accommodae dynamic asse allocaion problems wih a relaively large number of asses and poenial reurn predicors, whereas such problems can quickly become inracable using approaches based on more compuaionally inensive numerical procedures. CCV use heir approach o analyze opimal dynamic asse allocaion across U.S. bills, socks, and bonds when reurn predicabiliy is described by a firs-order vecor auoregressive [VAR()] process fi o real bill reurns, excess sock reurns, excess bond reurns, he nominal bill yield, dividend yield, and erm spread using quarerly U.S. daa for 952:2-999:4. They consider an invesor who maximizes he expeced uiliy of lifeime consumpion over an infinie horizon, where he uiliy funcion is of he Epsein-Zin-Weil (Epsein and Zin, 989; Weil, 989) form. Ineresingly, CCV find ha reurn predicabiliy should lead an invesor in he U.S. o have a sizable posiive mean ineremporal hedging for domesic socks for a range of values of he coefficien of relaive risk aversion (CRRA). They also find ha reurn predicabiliy implies a sizable negaive ineremporal hedging for domesic bonds for an invesor in he U.S. Overall, he empirical resuls in CCV, as well as he oher sudies cied above, indicae ha reurn predicabiliy can generae significan ineremporal hedging s for U.S. asses, especially U.S. socks. While he exising empirical lieraure on muli-period choice problems conains imporan findings relaing o he implicaions of reurn predicabiliy, he lieraure focuses almos exclusively on domesic invesmens in U.S. asses. In he presen paper, we exend he exan empirical lieraure and use he CCV approach o analyze reurn predicabiliy and is dynamic asse allocaion implicaions for 3 Using anoher compuaionally inensive approach, Brand (999) and Aï-Sahalia and Brand (200) use non- and semiparameric procedures o analyze Euler equaions and approximae he soluions o porfolio choice problems for invesors wih long horizons in he presence of reurn predicabiliy. See Brand (2004) for an exensive survey of he lieraure on boh saic and muli-period porfolio choice problems.

4 3 invesors wih Epsein-Zin-Weil uiliy and infinie horizons in he U.S., Ausralia, Canada, France, Germany, Ialy, and U.K. More specifically, we examine he naure of bill, sock, and bond reurn predicabiliy in each of hese counries and he implied ineremporal hedging s for domesic bills, socks, and bonds for invesors in each counry. Following CCV, we assume ha he reurn dynamics in each counry are well-characerized by a VAR() process ha includes hree insrumens: he nominal bill yield, dividend yield, and erm spread. A number of sudies find ha hese variables have predicive abiliy wih respec o sock and bond reurns. 4 Using monhly daa for 952: :05, 5 we esimae VAR processes for each counry and analyze domesic bill, sock, and bond reurn predicabiliy in each counry. Armed wih esimaes of he dynamic processes governing reurns in each counry and plausible assumed values for he parameers relaing o ineremporal preferences including CRRA values of 4, 7, and 0 we use he approximae analyical mehod and numerical procedure developed by CCV o solve he invesor s muli-period porfolio choice problem and esimae he implied mean oal, myopic, and ineremporal hedging s for domesic bills, socks, and bonds in each counry. In order o accoun for sampling uncerainy, we augmen he CCV approach wih a parameric boosrap procedure ha enables us o compue confidence inervals for he mean oal, myopic, and ineremporal hedging s in each counry. We also presen esimaes of he ineremporal hedging s for domesic socks and bonds for each monh over he sample in each counry. In addiion o examining he implied ineremporal hedging s for domesic socks and bonds for invesors in a number of differen counries, we also consider a muli-period porfolio choice problem for an invesor in he U.S. who can inves in socks and bonds from a foreign counry. I is quie feasible o use he CCV approach o solve muli-period porfolio choice problems wih five risky asses and six insrumens. This allows us o exend he empirical applicaion in CCV and analyze a muli-period porfolio choice problem for an invesor in he U.S. who, in addiion o domesic bills, socks, and bonds, has access o socks and bonds from a foreign counry (Ausralia, Canada, France, Germany, Ialy, or he 4 See, for example, Rozeff (984), Campbell (987), Campbell and Shiller (988), Fama and French (988, 989), Hodrick (992), Solnick (993), and Lewellen (2004). 5 Based on daa availabiliy, he sample begins in 96:0 (967:02) for France (Germany).

5 4 U.K.), and where he invesor considers six insrumens (he domesic and foreign nominal bill yields, dividend yields, and erm spreads) ha poenially conribue o reurn predicabiliy. 6 We use he CCV approach o esimae he oal, myopic, and ineremporal hedging s for domesic bills, socks, and bonds and foreign socks and bonds for an invesor in he U.S. when he reurn dynamics are characerized by a VAR() process ha includes he five reurns and six insrumens. 7 In anoher exension, we use he CCV approach o analyze a muli-period porfolio choice problem for an invesor in Ausralia, Canada, France, Germany, Ialy, or he U.K. who, in addiion o domesic bills, socks, and bonds, can inves in U.S. socks and bonds. Previewing our empirical resuls, we find ha reurn predicabiliy generaes sizable posiive ineremporal hedging s for domesic socks for invesors in he U.S. and U.K., while he ineremporal hedging s for domesic socks are decidedly smaller for invesors in Ausralia, Canada, and Germany and essenially zero for invesors in France and Ialy. The ineremporal hedging s for domesic bonds are negaive and reasonably large in magniude for invesors in he U.S., France, Germany, and Ialy, while hey are considerably smaller in magniude for invesors in Ausralia, Canada, and he U.K. We relae similariies and differences in reurn predicabiliy across counries o similariies and differences in he ineremporal hedging s for socks and bonds across counries, and he predicive relaionships beween dividend yields and excess sock reurns in he U.S. and U.K. help o accoun for he sizable ineremporal hedging s for domesic socks for invesors in hese counries. When an invesor in he U.S. has access o foreign socks and bonds, in addiion o domesic 6 Ang and Bekaer (2002) consider a muli-period porfolio choice problem where an invesor in he U.S. can inves in domesic socks and socks from one or wo foreign counries. Unlike mos of he lieraure, Ang and Bekaer (2002) do no characerize reurn predicabiliy using a VAR process ha includes insrumens such as he dividend yield, bu insead use a Markov-swiching process for he momens of he reurns. Also see Cambpell, Viceira, and Whie (2003), who use he CCV approach o sudy a muli-period porfolio choice problem where an invesor in he U.S. has access o domesic bills and bills from a foreign counry (he U.K., Germany, or Japan). The presen paper exends hese sudies by considering a broader range of domesic and foreign asses and a larger number of counries. 7 The size of he parameer space for he VAR() model becomes an issue as addiional asses and insrumens are included in he muli-period porfolio choice problem. Given our relaively long span of daa, i is feasible o reliably esimae a VAR() model composed of five asse reurns and six insrumens. However, i may become necessary o impose resricions on he VAR() model o limi he parameer space if addiional reurns or insrumens are included in he problem.

6 5 bills, socks, and bonds, he posiive ineremporal hedging for domesic socks remains sizable, and an imporan posiive ineremporal hedging for U.K. socks emerges. The ineremporal hedging s for socks from foreign counries beside he U.K. are ypically small, as are he ineremporal hedging s for foreign bonds from all counries. When invesors in Ausralia, Canada, France, Germany, Ialy, and he U.K. have access o U.S. socks and bonds, we find subsanial posiive (negaive) ineremporal hedging s for U.S. socks (bonds). Overall, we discover imporan similariies and differences in he opimal ineremporal hedging s for socks and bonds across counries, and our resuls indicae ha reurn predicabiliy implies especially srong ineremporal hedging s for U.S. and U.K. socks. I is imporan o emphasize ha he asse s derived from muli-period porfolio choice problems in he recen empirical lieraure (including he presen paper) are parial equilibrium in naure, as he reurn processes are reaed as exogenous. Tha is, given an exogenous reurn process (usually calibraed o U.S. daa), researchers calculae he implied opimal asse s for an individual invesor wih a long horizon and an assumed se of preferences; no aemp is made o use he implied model of invesor behavior o explain observed asse reurns. 8 Two ways of inerpreing he esimaed asse s in he exan empirical lieraure have been offered. Firs, Campbell and Viceira (2002) sugges viewing he esimaed asse s as normaive descripions of invesor behavior, so ha he esimaed asse s are hose ha an invesor wih an assumed se of preferences should have for a given reurn process. In line wih his, CCV (p. 42) moivae he developmen of heir approach by observing ha while [a]cademic research in finance has had a remarkable impac on many aspecs of he financial services indusry academic financial economiss have hus far provided surprisingly lile guidance o financial planners who offer porfolio advice o long-erm invesors. Alernaively, we can follow he suggesion of Lynch (200) and view he esimaed asse s as a posiive descripion of he 8 Explaining observed asse reurns requires embedding a represenaive invesor of his ype in a general equilibrium framework where all markes clear. Lynch (2003) makes imporan progress in his area by embedding a represenaive invesor wih a long horizon and access o hree sock porfolios sored on book-o-marke raios in a general equilibrium model and comparing he quaniaive properies of asse reurns implied by he model o acual U.S. asse reurns.

7 6 behavior of a unique individual or small group (raher han a represenaive agen) in he economy who explois he reurn predicabiliy creaed by a large number of oher invesors wih differen preferences. These differen preferences may be creaed by habi persisence, as in Campbell and Cochrane (999), or hey may be of he ype assumed in models of behavioral finance, such as Barberis, Huang, and Sanos (2000). The res of he paper is organized as follows: Secion 2 describes our empirical approach, including he CCV framework and our parameric boosrap procedure; Secion 3 presens our empirical resuls; Secion 4 concludes wih suggesions for fuure research. 2. Empirical Approach 2.. The Muli-Period Porfolio Choice Problem Consider an invesor who has access o n risky asses. 9 Le R be he real reurn on a, + benchmark asse (usually a Treasury bill) from ime o ime +, and le R, i = 2,, n, be he real i, + reurns on he expressed as n addiional asses. 0 The real reurn on he invesor s porfolio ( ) can be R p, + R n p, + = α i, ( Ri, + R, + ) + R, + i= 2, () where α i, is he porfolio weigh on asse i a ime. Leing r i, + = log( Ri, + ), define he vecor of log excess reurns as x r r,, r r ]'. In addiion o he n reurns, a vecor of + = [ 2, +, + n, +, + insrumens s + helps o deermine he dynamics of he complee sysem of sae variables. Gahering he reurns and insrumens ino an m-vecor, he sae vecor is given by 9 We adop he noaion of CCV hroughou his secion. 0 While we follow CCV and use a 3-monh Treasury bill as he benchmark asse, he designaion of he benchmark asse is arbirary. n i = 2 i, The porfolio weigh on he benchmark asse a ime is α. Noe ha he CCV framework does no impose borrowing or shor-sales consrains. The use of equaion (8) below o approximae he log real reurn on he porfolio has he effec of ruling ou he possibiliy of bankrupcy; see Campbell and Viceira (2002, pp ).

8 7 z r, x, s ]'. (2) + = [, As in a number of oher sudies, 2 CCV assume ha he dynamics of he sysem of sae variables are well-characerized by a VAR() process, so ha he daa-generaing process for he sae vecor given by z + is z, (3) + = Φ 0 + Φz + v+ where Φ 0 is an m-vecor of VAR inerceps; Φ is an m m marix of VAR slope coefficiens; is v + an m-vecor of VAR innovaions ha are independenly and idenically disribued as N (0, Σ ). For m v some of he expressions used below, i is useful o pariion Σ v such ha Σ v 2 σ = σ x σ s σ Σ Σ ' x xx xs ' σ s ' Σ xs, (4) Σ ss where 2 σ is he variance of he innovaion o he benchmark asse reurn; σ is an ( n ) -vecor of x covariances beween innovaions o he benchmark asse reurn and innovaions o he excess reurns on he remaining asses; σ is an ( m n) -vecor of covariances beween innovaions o he benchmark s asse reurn and innovaions o he insrumens; Σ is he ( n ) ( n ) variance-covariance marix for xx he innovaions o he excess reurns; Σ is he ( m n) ( n ) marix of covariances beween xs innovaions o he excess reurns and innovaions o he insrumens; Σ is he ( m n) ( m n) ss variance-covariance marix for he innovaions o he insrumens. Noe ha he vecor of VAR innovaions is assumed o be homoskedasic. CCV argue ha his is a reasonable assumpion, as sudies such as Campbell (987), Harvey (989, 99), and Glosen, Jagannahan, and Runkle (993) find ha relaive o heir effecs on expeced reurns, sae variables have only a limied abiliy o predic risk. 3 2 See, for example, Campbell (99), Balduzzi and Lynch (999), Kandel and Sambaugh (996), Campbell and Viceira (999), Lynch and Balduzzi (2000), Barberis (2000), and Lynch (200). 3 Assuming ha he vecor of VAR innovaions is homoskedasic is sandard in much of he lieraure, such as he sudies cied in foonoe 2 above. Chacko and Viceira (2003) solve a muli-period porfolio choice problem wih sochasic volailiy.

9 8 The invesor is assumed o have Epsein-Zin-Weil uiliy, which she maximizes over an infinie horizon. The recursive preferences ha characerize Epsen-Zin-Weil uiliy are given by UC [, E( U + )] = {( δ) C + δ[ E( U + )] }, (5) ( γ ) / θ γ / θ θ /( γ ) where C is consumpion a ime ; E ( ) is he expecaion operaor condiional on informaion available a ime ; γ > 0 is he CRRA; ψ > 0 is he elasiciy of ineremporal subsiuion (EIS); 0 < δ < is he ime discoun facor; θ = ( γ ) /( ψ ). As emphasized by CCV, Epsein-Zin-Weil uiliy severs he igh link beween he CRRA and EIS ha characerizes he popular ime-separable power uiliy funcion. 4 This is a nice feaure of equaion (5), as he CRRA and EIS are concepually disinc noions relaing o ineremporal preferences. A each ime, he invesor selecs C and α, α in order o maximize equaion (5), using all available informaion a ime, subjec o he 2,, n, ineremporal budge consrain, W W C R, (6) + = ( ) p, + where W is wealh a ime. The Euler equaion for consumpion for his problem is given by (Epsein and Zin, 989, 99) / ψ θ ( θ ) + p, + i = E {[ δ ( C / C ) ] R R, + }, (7) for any asse i. Wih ime-varying invesmen opporuniies, exac analyical soluions for his problem are generally no available. CCV combine an exension of he Campbell and Viceira (999) approximae analyical soluion wih a relaively simple numerical procedure o compue he invesor s opimal asse allocaion and consumpion policies. A key approximaion used by CCV involves he equaion for he log real reurn on he invesor s porfolio. We approximae he log real reurn on he porfolio using r ' ' 2 p, + = r, + + α x α ( σ x Σ xx α ), (8) 4 When γ =ψ, equaion (5) reduces o he familiar case of ime-separable power uiliy; when γ = ψ =, equaion (5) reduces o log uiliy.

10 9 where α = α,, α ]', and σ is he vecor of diagonal elemens in [ 2, n, 2 x Σ xx. This approximaion is exac in coninuous ime, and CCV observe ha i is highly accurae for shor ime inervals. 5 CCV also employ firs- and second-order log-linear approximaions of he budge consrain and Euler equaion, respecively, yielding E ( r i, + γ [cov ( r r i, +, +, r w + = rp, + [ (/ ρ )]( c w ) + k, (9) + ) var ( r p, + ) cov ( r i, +, + r, r, + p, + ) = ( θ / ψ ) cov ( r )] [cov ( r i, +, r i, +, +, c + w ) var ( r +, + ) +, (0) )] where c and w are he log-levels of C and W, respecively; ρ = exp[ E( c w )] ; k = log( ρ ) + ( ρ) log( ρ) / ρ. The approximaions o he budge consrain and Euler equaion are exac when ψ =, so ha he soluion o he approximae model is appropriae when ψ is near uniy. The policy funcions for α and c w and quadraic, respecively, in : z, which consiue he soluion o he approximae model, are linear α, () = A0 + A z ' ' c w = B0 + B z + z B2 z, (2) where A0 [( n ) ], A [ ( n ) m ], B 0 ( ), B ( m ), and B2 ( m m ) are coefficien marices ha are consan hrough ime and funcions of γ, ψ, δ, ρ, Φ 0, Φ, and Σ v. 6 We are primarily ineresed in he parameers in equaion (), which govern he invesor s opimal asse allocaions Numerical Soluion Procedure In order o compue esimaes of,,,, and B, CCV use a numerical procedure. Firs, hey A0 A B0 B 2 se Φ 0 = ˆΦ 0, Φ = ˆΦ, and Σ v = Σˆ v, where ˆΦ 0, ˆΦ, and Σˆ v are esimaes of he VAR parameers in 5 Our use of monhly daa in Secion 3 below should help o ensure he accuracy of he approximaion in our empirical applicaions. 6 The coefficien marices are consan hrough ime due o he infinie-horizon assumpion. This assumpion means ha we do no have o solve he problem backward recursively saring from he erminal dae.

11 0 equaion (3). They also se δ = on an annual basis (so ha he discoun facor equals 0.92 on a monhly basis) and consider differen values for γ and ψ. The following resul is useful in implemening he numerical procedure: xx H x 2 x A 0 = ( / γ ) Σ [ Φ σ + ( γ ) σ ] + [ (/ γ )] Σ [ Λ 0 /( ψ )], (3) x A = ( / γ ) Σ xx H xφ + [ (/ γ )] Σ xx[ Λ /( ψ )], (4) where H 0, I,0 ] is a marix ha selecs x from he sae vecor ; and Λ are x = [ ( n ) n ( n ) ( m n) xx /2 z Λ 0 marices ha depend on he parameers of equaion (2) ( B 0, B, and B2 ), as well as γ, ψ, δ, ρ, Φ 0, Φ, and Σ. Noe ha Λ /( ) and Λ /( ) are independen of ψ for a given ρ and ha we v 0 ψ ψ can define a nonlinear sysem, B =Ξ, =, and Φ ' B Ξ ( B2 ) = Ξ 2 funcions of γ, ψ, δ, ρ,, Φ,,,, B,, and. Σ v A0 A 0 vec, where,, and B B 7 2 Ξ0 Ξ 2 Ξ are As described in Campbell, Chan, and Viceira (2002), we implemen he ieraive process of he numerical procedure as follows. For a given value of γ, we se ρ = δ and selec an arbirary value for ψ, as well as iniial values for,, and. We plug he iniial values for,, and B ino equaions (3) and (4) o obain an iniial se of esimaes for and. Using he iniial, B, and hrough he se of equaions, B =Ξ, =, and B0 B B2 B0 B 2 A0 A B0 B2 values and he iniial A0 and A esimaes, we hen obain a new se of esimaes for B0, B, and B2 0 0 ' B Ξ vec ( B2 ) = Ξ 2. We begin he process again by plugging he new,, and esimaes ino equaions (3) and (4) o obain a new se of A and B0 B B2 0 A esimaes. We ierae in he manner unil he B0, B, and B2 esimaes (and hus he A0 and A esimaes) converge. Following CCV, we assume ψ =. In his case, he opimal consumpion-wealh 7 The complee expressions for Λ 0, Λ, Ξ 0, Ξ, and Ξ 2 are given in Campbell, Chan, and Viceira (2002).

12 raio is consan and equal o δ (Giovannini and Weil, 989), so ha ρ = δ, and he numerical procedure can sop. 8 We can also use equaions (3) and (4) o idenify he myopic and ineremporal hedging componens of asse, following Meron (969, 97). The firs erm on he righ-hand-side (RHS) of equaions (3) and (4) represens he myopic par of asse. The myopic componen focuses solely on a single-period-ahead and essenially corresponds o he asse generaed under he saic Markowiz problem. The second erm on he RHS of equaions (3) and (4) represens he ineremporal hedging par of asse. In conras o he saic Markowiz problem, an ineremporal hedging can arise in a muli-period porfolio choice problem, as a risk-averse invesor in a muli-period seing may look beyond a single-period-ahead and be ineresed in hedging her exposure o adverse fuure reurn shocks. Noe ha a muli-period choice problem is a necessary, bu no sufficien, condiion for he exisence of an ineremporal hedging. For example, when γ =, he second erm on he RHS of equaions (3) and (4) vanishes, so ha here is no ineremporal hedging. In his case, he invesor is no sufficienly risk-averse o generae an ineremporal hedging. If he marix of VAR slope coefficiens ( Φ ) is a zero marix so ha here is no reurn predicabiliy he second erm on he RHS of each equaion will also vanish. 9 Thus, in order for an ineremporal hedging o exis in a muli-period porfolio choice problem, he invesor mus be sufficienly risk-averse and reurns mus be predicable. 20 In our applicaions in Secion 3 below, we use he CCV procedure o esimae equaions () and (2) for a infiniely lived invesor in he U.S., Ausralia, Canada, France, Germany, Ialy, and U.K. (in urn) who can inves in domesic 3-monh Treasury bills, a broad domesic sock marke index, and 8 If ψ, an addiional ieraive loop is necessary o find he opimal value of ρ. In heir empirical applicaions, CCV noe ha he soluions o problems wih ψ = 0. 5 are similar o he soluions o problems wih ψ =. 9 CCV observe ha and are zero marices when Φ is a zero marix, so ha he second erm on he RHS of Λ 0 Λ equaions (3) and (4) vanishes. 20 Acually, an addiional condiion needs o be saisfied: he variance-covariance marix for he VAR innovaions, Σ v, canno be diagonal; see, for example, Brand (2004, Secion 2.3). The imporance of his condiion will become eviden in he discussion of he empirical resuls in Secion 3 below.

13 2 domesic 0-year governmen bonds. We follow CCV in he basic se-up of he model. Namely, we rea he log real reurn on a 3-monh Treasury bill ( rbr ) as he reurn on he benchmark asse, so ha he wo log excess real reurns are hose on he sock marke index and a 0-year governmen bond ( xsr and xbr, respecively). In addiion o lagged reurns, hree domesic insrumens serve as poenial reurn predicors: he nominal yield on a 3-monh Treasury bill ( bill ), he log of he dividend yield on he sock marke index ( div ), and he erm spread ( spread ). Given hese reurns and insrumens, he sae vecor is z rbr, xsr, xbr, bill, div, spread ]'. We assume ψ = (so ha ρ = δ ) and δ = = [ + + r on an annual basis, and we esimae he VAR parameers in equaion (3) using maximum likelihood, yielding ˆΦ 0, ˆΦ, and Σˆ v. 2 We consider hree values for γ : 4, 7, and 0. These γ values are similar o hose considered in oher sudies, 22 and hey represen plausible values for he CRRA. We repor esimaes of he mean asse s for domesic 3-monh Treasury bills, socks, and 0-year governmen bonds over he sample for each γ value using α = ˆ + Aˆ z, where and are he esimaes of and A, A0 Â0 Â A0 respecively, in equaion () obained using he numerical procedure described above and z T = = z, where T is he number of available sample observaions for he sae vecor. In addiion o he oal mean asse s, we use equaions (3) and (4) o esimae he mean myopic and hedging s for each asse and each value of γ. Given our ineres in ineremporal hedging s in he presen paper, we also presen figures showing he hedging s for domesic socks and bonds for each monh over he sample when γ = 7 in each counry. To ge a sense of he sampling uncerainy associaed wih our poin esimaes of he mean oal, myopic, and hedging s for each asse in each counry, we consruc 68% confidence inervals for 2 OLS esimaion of ˆΦ 0 and ˆΦ in equaion (3) is equivalen o maximum likelihood esimaion. 22 For example, CCV include abulaed resuls for γ = 5 ; Balduzzi and Lynch (999) consider γ = 6 ; Barberis (2000) considers γ = 5, 0 ; Lynch (200) considers γ = 4.

14 3 he mean s using he following parameric boosrap procedure. 23 We assume ha observaions for he sae vecor z + are generaed by equaion (3) wih he parameers of he VAR se o heir maximumlikelihood esimaes. In order o generae a series of innovaions o use in consrucing a pseudo-sample, we make T + 00 independen draws from a N ˆ m v (0, Σ ) disribuion. Using he randomly drawn innovaions, equaion (3) wih = ˆΦ and Φ = ˆΦ, and seing he iniial z observaions o zero, we Φ 0 0 can build up a pseudo-sample of T + 00 observaions for z. We drop he firs 00 ransien sar-up observaions in order o randomize he iniial z observaions, leaving us wih a pseudo-sample of T observaions for z, maching he original sample. For he pseudo-sample, we use he numerical procedure described above o esimae equaions () and (2) and he mean oal, myopic, and hedging s for each asse. We repea his process,000 imes, giving us an empirical disribuion for each of he mean asse s. We consruc 68% confidence inervals for each mean asse from he empirical disribuions using he percenile mehod described in Davidson and MacKinnon (993, p. 766) Predicive Regression Model Esimaion Before presening esimaes of he oal, myopic, and ineremporal hedging s for each asse in each counry in Secion 3.3 below, we repor OLS esimaion resuls for he VAR() model, equaion (3), for each counry in Secion 3.2 below. The VAR model capures he exen of reurn predicabiliy in each counry ha provides he basis for he ineremporal hedging s. Leing Φ = { φ 0 } and Φ = }, equaion (3) can be expressed in more deail as 0 i rbr xsr xbr { φ i, j 0 + = φ + φ,rbr + φ,2 xsr + φ,3 xbr + φ,4bill + φ,5div + φ,6 spread + v, = φ2 + φ2,rbr + φ2,2 xsr + φ2,3xbr + φ2,4bill + φ2,5div + φ2,6spread + v2, = φ3 + φ3,rbr + φ3,2 xsr + φ3,3xbr + φ3,4bill + φ3,5div + φ3,6spread + v3, +, (5), (6), (7) 23 Primarily due o compuaional coss, mos exan sudies (including CCV) repor only poin esimaes of asse s.

15 4 bill div spread 0 + = φ4 + φ4,rbr + φ4,2 xsr + φ4,3xbr + φ4,4bill + φ4,5div + φ4,6spread + v4, = φ5 + φ5,rbr + φ5,2 xsr + φ5,3xbr + φ5,4bill + φ5,5div + φ5,6spread + v5, = φ6 + φ6,rbr + φ6,2 xsr + φ6,3xbr + φ6,4bill + φ6,5div + φ6,6spread + v6, +, (8), (9), (20) for =,, T. The firs hree equaions of he VAR, equaions (5)-(7), can be viewed as predicive regression models for real bill, excess sock, and excess bond reurns, respecively. I is well-known ha here are a number of economeric difficulies associaed wih esimaing predicive regressions for sock and bond reurns (Mankiw and Shapiro, 986; Sambaugh, 986, 999; Nelson and Kim, 993; Kirby, 997; Bekaer, Hodrick, and Marshall, 997). Essenially, hese difficulies lead o size disorions in ess of he significance of he slope coefficiens in predicive regressions. 24 In order o help correc for possible size disorions when assessing he predicive power of he lagged reurns and insrumens wih respec o bill, sock, and bond reurns in each counry, we repor p-values corresponding o he -saisics for he slope coefficiens in equaions (5)-(7) using a parameric boosrap procedure similar o he one described in Secion 2.2 above, wih he excepion ha we assume real bill, excess sock, and excess bond reurns are generaed by rbr xsr xbr ~ ~ 0 + = φ + v, + ~ ~ 0 + = φ2 + v2, + ~ ~ 0 + = φ3 + v3, +, (5 ), (6 ), (7 ) respecively, under he null hypohesis of no reurn predicabiliy. Using his resriced VAR() model as he daa-generaing process, we can simulae a pseudo-sample of T observaions for z and calculae he -saisic for each of he slope coefficiens in equaions (5)-(7) for he pseudo-sample. 25 We repea his process,000 imes, giving us empirical disribuions for he -saisics for each of he slope coefficiens 24 Relaedly, OLS esimaes can be subjec o small-sample biases (Sambaugh, 986, 999). Given ha smallsample bias correcions can be very complicaed in he sysem defined by equaions (5)-(20), we follow CCV and assume ha invesors rea he OLS esimaes of he VAR coefficiens as given and known. 25 Like he boosrap procedure described in Secion 2.2 above, we randomize he iniial values by including 00 ransien sar-up observaions in each pseudo-sample ha we subsequenly discard. z

16 5 in equaions (5)-(7) under he null hypohesis of no reurn predicabiliy. In order o generae p-values corresponding o one-sided significance ess, if he -saisic for a given slope coefficien for he original sample is posiive, hen he p-value is he proporion of he boosrapped -saisics ha are greaer han he -saisic for he original sample; if he -saisic for he original sample is negaive, hen he p-value is he proporion of he boosrapped -saisics ha are less han he -saisic for he original sample. Inoue and Kilian (2003) argue ha more powerful one-sided ess should be preferred in predicive regressions, as heory frequenly suggess he sign of a coefficien. 26 For each reurn equaion, we also repor a boosrapped p-value corresponding o a Wald es of he null hypohesis ha he slope coefficiens are joinly zero. 3. Empirical Resuls 3.. Daa The daa for he U.S., Ausralia, Canada, France, Germany, Ialy, and U.K. are from Global Financial Daa. Following CCV, our sample begins in 952:04 for each counry, wih he excepions of France and Germany, where, due o daa availabiliy, he sample begins in 96:0 and 967:02, respecively. 27 The sample ends in 2004:05 for each counry. We measure he log real reurn on a 3-monh Treasury bill for a given monh as he difference in he logs of he oal reurn index for bills for he given and previous monhs minus he difference in he logs of he consumer price index for he given and previous monhs. 28 The log excess sock (bond) reurn for a given monh is he difference in he logs of he oal reurn index for socks (0-year governmen bonds) for he given and previous monhs minus he difference in he logs of he oal reurn index for bills for he given and previous monhs. The nominal 26 While we repor p-values for one-sided ess, we can simply double he p-values o conver hem o p-values for wo-sided ess under he assumpions ha he disribuions are approximaely symmeric. 27 We had originally planned o include Japan in order o include all of he G-7 counries, bu daa for all of he necessary series for Japan are no available for a sufficienly long period from Global Financial Daa. 28 Due o daa availabiliy, we use he wholesale price index for Ausralia.

17 6 bill yield is he yield on a 3-monh Treasury bill, 29 and he erm spread is he difference beween he yields on a 0-year governmen bond and 3-monh Treasury bill. Names and descripions of he Global Financial Daa files used o consruc all of he variables are provided in he Daa Appendix. Table repors summary saisics (mean, sandard deviaion, and firs-order auocorrelaion coefficien) for he hree risky asse reurns and hree insrumens for each of he seven counries we consider. The mean and sandard deviaion for he reurns are expressed in annualized percenage unis, and we include he Sharpe raio (he raio of he annualized mean o he annualized sandard deviaion) for he excess sock and bond reurns. The mean excess sock reurns for he U.S., Ausralia, and U.K. are beween 5% and 6%. Canada, Germany, and France exhibi lower mean excess reurns, while Ialy has he lowes mean excess reurn of.89%. Mean excess sock reurns for he U.S., Ausralia, and U.K. are approximaely 3 o 4 percenage poins higher han mean excess bond reurns for hese counries. Mean excess sock reurns are jus over 2 percenage poins higher han mean excess bond reurns for Canada, and mean excess sock reurns are acually less han percenage poin higher han mean excess bond reurns for France, Germany, and Ialy. For all of he counries, he sandard deviaion of excess sock reurns is approximaely 2 o 4 imes larger han he sandard deviaion of excess bond reurns, and he sandard deviaion of he real bill reurn is always considerably below ha of excess bond reurns for each counry. The Sharpe raios for excess sock reurns are he highes for he U.S., Ausralia, and U.K. (0.39, 0.32, and 0.29, respecively), while Ialy has he smalles raio (0.09). The Sharpe raios for excess bond reurns are very similar across all of he counries (wih he excepion of Germany), ranging from 0.5 o 0.9. For Germany, he Sharpe raio for excess bond reurns is considerably higher a Observe ha he Sharpe raio for excess sock reurns is approximaely.5 o 2.5 imes larger han he Sharpe raio for excess bond reurns for he U.S., Ausralia, Canada, and U.K., while for France, Germany, and Ialy, he Sharpe raio for excess sock reurns is acually less han he Sharpe raio for excess bond reurns. All else 29 Following a number of oher sudies, we use deviaions in he nominal 3-monh Treasury bill yield from a -year backward-looking moving average.

18 7 equal, he Sharpe raios lead us o expec a higher myopic on average for socks in he U.S., Ausralia, Canada, and U.K. This is borne ou in he empirical resuls repored in Secion 3.3 below. Excess sock reurns exhibi fairly limied persisence in all counries (firs-order auocorrelaion coefficiens beween 0.03 and 0.2). Excess bond reurns ypically appear somewha more persisen han excess sock reurns, wih Ialy and he U.K. displaying he mos persisen excess bond reurns. Real bill reurns are moderaely persisen for all counries, ranging from 0.2 o 0.5. In conras o he reurns, he insrumens appear very persisen for all counries, wih he firs-order correlaion coefficiens ranging from 0.88 o 0.93 for he nominal bill yield, 0.98 o 0.99 for he dividend yield, and 0.94 o 0.97 for he erm spread VAR Esimaion Resuls Tables 2 hrough 8 repor esimaion resuls for he VAR() model for each counry. The op par of each able repors esimaes of he slope coefficiens and heir corresponding -saisics, as well as he 2 R goodness-of-fi measure, for each equaion of he VAR. Boosrapped p-values are also repored for he coefficiens in he reurn equaions, where he p-values are compued using he boosrap procedure described in Secion 2.3 above. In addiion, boosrapped p-values for Wald ess of he null hypohesis ha he explanaory variables are joinly zero in each of he reurn equaions (ha is, no reurn predicabiliy) are also repored below he 2 R measures. The boom par of each able repors he crosscorrelaions of he VAR residuals. We briefly discuss he VAR esimaion resuls. The 2 R measures for he esimaed real bill reurn equaions in Tables 2 hrough 8 range from (Canada) o 0.3 (France), and we are easily able o rejec he null hypohesis of no real bill reurn predicabiliy for each counry a convenional significance levels according o he boosrapped p-values. The coefficiens on he lagged bill reurn are posiive and significan a convenional levels for all counries. For Ausralia, Canada, Ialy, and he U.K., he hree lagged insrumens all have negaive

19 8 coefficiens ha are significan (or nearly significan) a convenional levels. Two (one) of he lagged insrumens are significan a convenional levels for France (Germany). Wih respec o he esimaed excess sock reurn equaions, we see ha he 2 R measures are beween 0.06 (Ialy) and (U.K.). These measures are in line wih he exan empirical lieraure, which finds ha he degree of predicabiliy in excess sock reurns is limied. Neverheless, he Wald es easily rejecs he null hypohesis of no predicabiliy for he U.S., Ausralia, Canada, France, and U.K. according o he boosrapped p-values, and as indicaed in he exan empirical lieraure and as we will see below even a limied degree of predicabiliy can have quaniaively imporan asse allocaion implicaions. We canno rejec he null hypohesis of no predicabiliy using he Wald es for Germany and Ialy a convenional significance levels. Looking a he individual coefficiens, a leas one of he lagged reurn coefficiens is significan a convenional levels for each counry. The lagged nominal bill yield is significan a convenional levels for he U.S. and Germany, where i eners wih a negaive coefficien. For he U.S., Ausralia, and U.K., he lagged dividend yield has a posiive and significan coefficien in he excess sock reurn equaion, while he lagged erm spread is only significan for Canada. The 2 R measures are somewha higher for he fied excess bond reurn equaions han he excess sock reurn equaions for each counry, wih he excess bond reurn equaion measures ranging from (Ausralia) o 0.55 (Ialy). According o he boosrapped p-values, we can rejec he null hypohesis of no predicabiliy for he excess bond reurn equaion for each counry a convenional significance levels. Eiher wo or hree of he lagged reurns are significan a convenional levels for he U.S., Canada, France, Germany, Ialy, and U.K. The lagged nominal bill yield has a negaive and significan coefficien for Ausralia, France, Germany, and Ialy, while he erm spread has a posiive and significan (or nearly significan) coefficien for he U.S., Canada, France, Germany, Ialy, and U.K.

20 9 The auoregressive coefficiens end o dominae he esimaed equaions for each of he insrumens for each counry. This is in line wih he large auocorrelaion coefficiens repored for he insrumens in Table. The 2 R measures are quie high for hese equaions, ranging from 0.79 o Wih respec o he cross-correlaions of he VAR innovaions for each counry, one noable feaure is he srong negaive correlaion beween innovaions o excess sock reurns and he dividend yield for each counry. The sronges correlaion is (U.S.), while he weakes is sill (Ialy). There are also sizable negaive correlaions beween innovaions o he nominal bill yield and erm spread for each counry ( o 0.748), as well as a fairly large negaive correlaion beween innovaions o excess bond reurns and nominal bill yields for each counry, ranging from (U.S.) o (Germany). Innovaions o excess sock and bond reurns are posiively correlaed, and while ypically smaller han he oher correlaions we have menioned in absolue value, hey sill appear reasonably large (0.32 o 0.29). Summarizing he VAR esimaion resuls repored in Tables 2 hrough 8, real bill reurns and excess bond reurns appear significanly predicable a convenional levels for each of he counries. Excess sock reurns appear significanly predicable a convenional levels for he U.S., Ausralia, Canada, France, and U.K., bu no for Germany and Ialy. In addiion, here are consisen paerns in he correlaions of he VAR innovaions, wih he srong negaive correlaion beween innovaions o excess sock reurns and he dividend yield a noable feaure for each counry Domesic Asse Demands for Invesors in Differen Counries Table 9 repors he mean oal, myopic, and ineremporal hedging s (in percenages) for domesic bills, socks, and bonds and γ values of 4, 7, and 0 in each counry. To ge a sense of sampling uncerainy, he able also repors 68% confidence inervals for he mean asse s generaed using he parameric boosrap procedure described in Secion 2.2 above. Of course, he oal mean s

21 20 across he hree asses sum o 00; he mean myopic s across asses also sum o 00, while he mean hedging s sum o 0. For he U.S., here are large posiive mean oal and ineremporal hedging s for socks for each repored γ value. As we would expec, he mean oal for socks he mos risky asse decreases as γ increases. While he mean hedging for socks also decreases as γ increases, he mean hedging for socks as a share of he oal acually increases as γ increases. The mean oal s for bonds are noiceably smaller han he mean oal s for socks. The mean hedging s for bonds are negaive and fairly large in magniude, conribuing o he smaller oal s for bonds vis-à-vis socks. The mean oal for bills is negaive for each repored γ value, so ha he invesor ypically shors bills. There is also a fairly sizable negaive mean hedging for bills for each repored γ value. The resuls in Table 9 for he mean hedging s for sock in he U.S. are similar o he mean hedging for socks (00.84) repored in CCV for he U.S. using quarerly daa for 952:2-999:4 and γ = 5 ; he mean hedging s for bonds in he U.S. in Table 9 are smaller in magniude han he mean hedging for bonds ( 22.57) repored in CCV. The mos sriking resul for he U.S. in Table 9 (and in CCV) is he subsanial posiive oal and ineremporal hedging s for domesic socks for an invesor in he U.S. Wha explains he sizable ineremporal hedging for domesic socks in he U.S.? While a number of facors are a work in his mulivariae analysis, as emphasized by CCV and ohers, 30 wo facors appear o play especially imporan roles: (i) he posiive coefficien on he lagged dividend yield in he excess sock reurn equaion of he VAR; (ii) he srong negaive correlaion beween innovaions o excess sock reurns and he dividend yield. To see how hese facors generae a srong ineremporal hedging for socks, consider a negaive innovaion o excess sock reurns nex period. Due o he large Sharpe raio for socks in he U.S., invesors are usually long in socks, so ha he negaive innovaion o excess sock reurns represens a worsening of he invesor s invesmen opporuniies nex 30 See, for example, he discussion in Brand (2004, Secion 2.2.).

22 2 period. However, a negaive innovaion o excess sock reurns nex period ends o be accompanied by a posiive innovaion o he dividend yield nex period, and according o he posiive coefficien on he lagged dividend yield in he excess sock reurn equaion of he VAR, he higher dividend yield nex period leads o higher expeced sock reurns wo periods from now. 3 Thus, by looking beyond oneperiod-ahead as an invesor wih γ > will do and aking ino accoun reurn predicabiliy, as well as he negaive correlaion beween innovaions o sock reurns and he dividend yield, socks become a good hedge agains hemselves, in ha hey hedge exposure o fuure adverse reurn shocks. As a cauionary noe, observe ha he 68% confidence inervals for he mean asse s end o be quie wide for he U.S. in Table 9, especially wih regard o he mean oal s for each asse and he mean myopic s for bonds and bills. This suggess ha he reporing of poin esimaes alone can mask considerable sampling uncerainy in empirical muli-period porfolio choice problems. 32 Neverheless, i is imporan o observe ha while many of he confidence inervals for he U.S. are quie wide in Table 9, he confidence inervals for he mean hedging s for socks appear igh enough o conclude ha he mean hedging s for socks are posiive and sizable in he U.S. for he repored γ values. The confidence inervals for he mean hedging s for bonds in he U.S. also appear igh enough o conclude ha he mean hedging s for bonds are close o zero or negaive in he U.S. for he repored γ values. While here is ofen subsanial sampling uncerainy regarding mean asse s, i is reasonable o view he empirical evidence as supporive of a sizable posiive implied ineremporal hedging for domesic socks and a small or negaive implied inerermporal hedging for domesic bonds in he U.S. In order o glean addiional insigh ino he ineremporal hedging s for domesic socks and bonds in he U.S., Panel A of Figure porrays he esimaed hedging s for socks and bonds for each monh over he sample in he U.S. when γ = 7. Overall, he hedging for socks appears 3 Furhermore, he large auoregressive coefficien in he div + equaion in Table 2 means ha here will be a persisen increase in he expeced dividend yield, leading o a persisen increase in expeced excess sock reurns. 32 Subsanial sampling uncerainy can be a problem in general wih regard o asse allocaion problems; see Brand (2004, Secion 3..2).

23 22 considerably less volaile han he hedging for bonds. The hedging for socks is ypically well above he hedging for bonds over he sample, wih he excepion ha he hedging for bonds does move above he hedging for socks during he lae 990s and Turning o he resuls for Ausralia in Table 9, while he mean oal s for domesic socks are moderaely large, hey are considerably smaller han he mean oal s for domesic socks in he U.S. The mean hedging s for socks in Ausralia are much smaller han he corresponding s in he U.S., and he 68% confidence inervals for he mean hedging s for socks in Ausralia do no lead o rejecion of he null hypohesis of a zero mean hedging. The mean oal s for bonds in Ausralia are very similar o hose in he U.S., while he mean hedging s for bonds in Ausralia are much closer o zero han in he U.S. Again, he 68% confidence inervals for he mean hedging s for bonds in Ausralia do no indicae rejecion of he null hypohesis of zero mean hedging for bonds. From Panel B of Figure, we see ha he hedging for socks is always above he hedging for bonds when γ = 7, alhough a drop in he average hedging s for boh socks and bonds in Ausralia is eviden in he early 980s. The resuls for Canada in Table 9 are similar o hose for Ausralia, in ha he mean oal and hedging s for socks are posiive bu considerably smaller han he corresponding s for he U.S., while he mean oal s for bonds are similar o, and he mean hedging s for bonds are considerably smaller in magniude han, hose for he U.S. We canno rejec he null hypohesis of zero mean hedging s for bonds in Canada according o he 68% confidence inervals. The hedging for bonds is much more volaile han he hedging for socks in Canada when γ = 7 (see Panel C of Figure ). Turning o he resuls for Germany in Table 9, he mean oal and hedging s for socks are similar o he corresponding s in Ausralia and Canada, while he mean hedging s for bonds in Germany are similar o hose in he U.S. From Panel E of Figure, we also see ha he hedging for bonds is more volaile han he hedging for socks in Germany when γ = 7.