The Term Structure. of Currency Carry Trade Risk Premia

Transcription

1 The Term Srucure of Currency Carry Trade Risk Premia Hanno Lusig UCLA and NBER Andreas Sahopoulos USC February 05 Adrien Verdelhan MIT and NBER Absrac High ineres rae currencies yield high currency excess reurns on shor-erm Treasury bill invesmens, bu hey end o yield low local excess reurns on long-erm governmen bonds. A longer mauriies, he low erm premium offses he high currency risk premium. Under no arbirage condiions, his exac resul obains when global permanen innovaions o he pricing kernels of differen counries are he same and herefore do no o have permanen effecs on exchange raes. In his case, he uncovered ineres rae pariy holds a long horizons. We derive parameric resricions o mach he downward sloping erm srucure of carry rade risk premia in a large class of affine erm srucure models. Firs Version: May 03. Lusig: UCLA Anderson School of Managemen, 0 Weswood Plaza, Suie C4., Los Angeles, CA (hlusig@anderson.ucla.edu). Sahopoulos: USC Marshall School of Business, 3670 Trousdale Parkway, Hoffman Hall 7, Los Angeles, CA (asahop@marshall.usc.edu). Verdelhan: MIT Sloan School of Managemen, 00 Main Sree, E6-6, Cambridge, MA 039 (adrienv@mi.edu). Many hanks o Bernard Dumas (discussan), Riccardo Colacio (discussan), Mikhail Chernov, Ian Dew-Becker, Ron Giammarino, Lars Hansen, Espen Henriksen, Urban Jermann, Leonid Kogan, Karen Lewis (discussan), Ian Marin, Sefan Nagel, Tarun Ramadorai, Lucio Sarno, Jose Scheinkman, Alan Taylor, Andrea Vedolin (discussan), Mungo Wilson, Fernando Zapaero, Irina Zviadadze, seminar paricipans a he Federal Reserve Board, Georgeown Universiy, LSE, LBS, MIT, UC3 in Madrid, UC Davis, he Said School a Oxford, Cass a Ciy Universiy London, Syracuse Universiy, Universiy of Brisol, UBC, Universiy of Exeer, Universiy of Lausanne, Universiy of Massachuses, Universiy of Michigan, Universiy of Rocheser, USC, and Wharon, as well as he paricipans a he Firs Annual Conference on Foreign Exchange Markes a he Imperial College, London, he 03 Inernaional Macro Finance Conference a Chicago Booh, he 04 Duke/UNC Asse Pricing Conference, and he 04 WFA Meeings.

2 In his paper, we esablish ha he erm srucure of currency carry rade risk premia is downward-sloping: he reurns o he currency carry rade are much smaller for long mauriy bonds han for Treasury bills. Assuming ha markes are complee, we derive a preferencefree condiion ha links foreign and domesic long-erm bond reurns, expressed in a common currency, o he permanen componens of he pricing kernels. The downward-sloping erm srucure of average carry rade reurns is herefore informaive abou he emporal naure of risks ha invesors perceive in currency markes: carry rade reurns are driven by asymmeric exposures o common shocks, bu hese shocks are perceived o be ransiory in naure by bond marke invesors. Building on our preference-free condiion, we derive analyical parameer resricions in a large class of affine erm srucure models wih common facors ha ensure zero carry rade reurns a long mauriies. These parameer resricions rule ou he non-saionariy of he exchange rae in response o common shocks and enforce uncondiional uncovered ineres pariy a very long horizons. Our work is he firs o esablish his connecion beween he saionariy of he exchange rae and he properies of foreign long-erm bond reurns and yields. Carry rades a he shor end of he mauriy curve are akin o selling Treasury bills in funding currencies and buying Treasury bills in invesmen currencies. The exchange rae is here he only source of risk. The se of funding and invesmen currencies can be deermined by he level of shor-erm ineres raes or he slope of he yield curves, as noed by Ang and Chen (00) and Berge, Jordà, and Taylor (0). Likewise, carry rades a he long end of he mauriy curve are akin o selling long-erm bonds in funding currencies and buying long-erm bonds in invesmen currencies. Each leg of he rade is subjec o exchange rae and ineres rae risk. The log reurn on a foreign bond posiion (expressed in U.S. dollars) in excess of he domesic (i.e., U.S.) risk-free rae is equal o he sum of he log excess bond reurn in local currency plus he reurn on a long posiion in foreign currency. Therefore, expeced foreign bond excess reurns convered in domesic currency are he sum of a local bond erm premium and a currency risk premium. Each mauriy componen of he erm srucure of currency carry rade risk premia is defined by he average excess reurn obained by selling and buying bonds

3 of ha mauriy. The differen mauriies describe he whole erm srucure of currency risk. We sudy empirically he erm srucure of currency carry rade risk premia hrough a cross-secion of average porfolio excess reurns. Our daa perain o eiher long ime-series of G0 sovereign coupon bond reurns over he /950 /0 sample, or o a shorer sample (/97 /0) of G0 sovereign zero-coupon yield curves. Using zero-coupon bonds, Figure offers a firs glimpse a he erm srucure of currency risk. The figure, which is sudied in deails laer in he paper, shows he dollar log excess reurns as a funcion of he bond mauriies, using he same se of funding and invesmen currencies. Invesing in shor-erm bills of counries wih fla yield curves (mosly high shor-erm ineres rae) while borrowing a he same horizon in counries wih seep yield curves (mosly low shor-erm ineres rae counries) leads o posiive excess reurns on average. This is he classic carry rade, whose average excess reurn is represened here on he lef hand side of he graph. Invesing and borrowing in long-erm bonds of he same counries, however, deliver negaive excess reurns on average. This is he carry rade a he long end of he erm srucure curve, represened here on he righ hand side of he graph. As he mauriy of he bonds increases, he average excess reurn decreases. Using long ime-series of G0 sovereign coupon bond reurns delivers similar resuls. Beween /950 and /0, he porfolio of fla-slope (mosly high shor-erm ineres rae) currencies yields a one-monh currency risk premium of 3.0% and a local erm premium of.8% per annum (which sum o a bond premium in U.S. dollars of.%). Over he same period, he porfolio of seep-slope (mosly low shor-erm ineres rae) currencies yields a currency risk premium of 0.05% and a local erm premium of 4.0% (which sum o a bond premium of 4.05%). The average spread in dollar risk premia beween he low slope and high slope porfolios is hus.95% (3.0% 0.05%) for Treasury bills, bu i is.85% (.% 4.05%) for he 0- year bond porfolios. Counries wih a high currency risk premium end o have a low bond erm premium. The profiable bond sraegy herefore involves shoring he usual carry rade invesmen currencies and going long in he funding currencies. We obain similar resuls when soring counries by he level of heir shor-erm ineres raes: he risk premia a he long end

4 6 4 Dollar Excess Reurns Mauriy (in quarers) Figure : Term Srucure of Dollar Bond Risk Premia The figure shows he dollar log excess reurns as a funcion of he bond mauriies. Dollar excess reurns correspond o he holding period reurns expressed in U.S. dollars of invesmen sraegies ha go long and shor foreign bonds of differen counries. The unbalanced panel of counries consiss of Ausralia, Canada, Japan, Germany, Norway, New Zealand, Sweden, Swizerland, and he U.K. A each dae, he counries are sored by he slope of heir yield curves ino hree porfolios. The firs porfolio conains counries wih fla yield curves (mosly high ineres rae) while he las porfolio conains counries wih seep yield curves (mosly low ineres rae counries). The firs porfolio correspond o he invesmen currencies while he hird one corresponds o he funding currencies. The slope of he yield curve is measured by he difference beween he 0-year yield and he 3-monh ineres rae a dae. The holding period is one quarer. The reurns are annualized. The shaded areas correspond o wo sandard deviaions above and below each poin esimaes. Sandard deviaions are obained by boosrapping. Zero-coupon daa are monhly, and he sample window is 4/985 /0. of he mauriy curve are significanly smaller ha hose a he shor end, as he difference in local currency bond erm premia largely offses he currency risk premium. As a resul, he average reurns on foreign long-erm bonds, once convered ino U.S. dollars, are small and rarely saisically differen from he average reurn on U.S. long-erm bonds. We show ha he difference beween domesic and foreign long-erm bond risk premia, expressed in domesic currency, is pinned down by he difference in he enropies of he permanen componens of he domesic and foreign sochasic discoun facors (SDF). The long-erm bond 3

5 risk premia, expressed in domesic currency, are higher on foreign bonds han on domesic bonds when here is less permanen risk in foreign counries pricing kernels han a home. The emporary componens of SDFs play no role because he currency exposure compleely hedges he exposure of he long-shor sraegy in long-erm bonds o he emporary pricing kernel shocks. This heoreical resul speaks direcly o our main empirical finding: while he usual carry rade delivers large and posiive currency excess reurns, he carry rade a he long end of he yield curve does no. Therefore, he permanen componens of he SDF of invesmen currencies mus be a leas as risky as hose of funding currencies, while he opposie is rue for he overall SDFs. This resul produces a novel resricion ha all models in inernaional finance need o saisfy in order o be consisen wih he daa. Our heoreical resuls perain o infinie-mauriy bonds bu our empirical resuls are obained wih bond mauriies up o 5 years. Using he sae-of-he-ar erm srucure model, we show ha he differences in long erm bond reurns are a good proxy for heir counerpar on infinie-mauriy bonds. We simulae he model of Joslin, Singleon, and Zhu (0), using he parameers esimaed o mach he U.S. yield curve, and derive he implied yield curve a all mauriies. The implied infinie-mauriy bond reurns exhibis a correlaion of 0.98 wih he 5-year bond reurns. The heoreical insighs obained from differences in infinie-mauriy bonds are herefore useful o undersand he acual differences in long-erm bond reurns. We hen use our preference-free heoreical resuls o derive parameric resricions in several key affine erm srucure models, from he original Vasicek (997) and Cox, Ingersoll, and Ross (985) o he mos recen muli-facor models. The resricions need o be saisfied o mach our new facs abou he cross-secion of bond reurns and exchange raes. Depending on wheher he shocks are counry-specific or global, we obain counry-specific or cross-counry resricions on he parameers of he models. Through he lenses of hose erm srucure models, we propose a novel inerpreaion of carry rades a he shor end of he yield curve. The insignifican carry rade risk premia a longer mauriies rules ou asymmery in he loadings on he permanen global shocks; he asymmery can only apply o ransiory global shocks. The classic carry rade, a he shor end of he yield curve, herefore compensaes invesors for exposure o ransiory 4

6 global shocks. The res of he paper is organized as follows. Secion rapidly reviews he lieraure. In Secion, we examine he cross-secion of bond excess reurns in local currency and in U.S. dollars and we conras i wih he cross-secion of currency excess reurns. In Secion 3, we derive he no-arbirage, preference-free heoreical resricions imposed on currency and erm risk premia. To do so, we decompose pricing kernels ino a permanen and a emporary componen and link heir properies o he downward erm srucure of currency risk. In Secion 4, we apply his decomposiion o affine erm srucure models and illusrae i quaniaively in a mulicounry affine model. In Secion 5, we presen concluding remarks. The Appendix conains all proofs and an Online Appendix conains supplemenary maerial no presened in he main body of he paper. Relaed Lieraure Our paper is relaed o four large srands of he lieraure: he carry rade reurns, he empirical erm premia across counries, he decomposiion of SDFs, and erm srucure models. Our paper builds on he vas lieraure on UIP condiion and he currency carry rade [Engel (996) and Lewis (0) provide recen surveys]. We are he firs o derive general condiions under which long-run uncondiional UIP follows simply from marke compleeness: if all permanen shocks o he pricing kernel are common, hen foreign and domesic yield spreads in dollars on long mauriy bonds will be equalized, regardless of he properies of he pricing kernel. Our focus is on he cross-secional relaion beween he slope of he yield curve, ineres raes and exchange raes. We sudy wheher invesors earn higher reurns on foreign bonds from counries in which he slope of he yield curve is higher han he cross-counry average. Prior work, from Campbell and Shiller (99) o Bekaer and Hodrick (00) and Bekaer, Wei, and Xing (007), focus mosly on he ime series, esing wheher invesors earn higher reurns on foreign bonds from a counry in which he slope of he yield curve is currenly higher han average for ha counry. Chinn and Meredih (004) documen some ime-series evidence ha 5

7 suppors a condiional version of UIP a longer holding periods, while Boudoukh, Richardson, and Whielaw (03) show ha pas forward rae differences predic fuure changes in exchange raes. Some papers sudy he cross-secion of bond reurns: Koijen, Moskowiz, Pedersen, and Vrug (0) and Wu (0) examine he currency-hedged reurns on carry porfolios of inernaional bonds, sored by a proxy for he carry on long-erm bonds, bu hey do no examine he ineracion beween currency and erm risk premia, he opic of our paper. Ang and Chen (00) and Berge, Jordà, and Taylor (0) have shown ha yield curve variables can also be used o forecas currency excess reurns. These auhors do no examine he reurns on foreign bond porfolios. Dahlquis and Hasselof (03) sudy inernaional bond risk premia in an affine asse pricing model and find evidence for local and global risk facors. Joikashira, Le, and Lundblad (05) sudy he co-movemen of foreign bond yields hrough he lenses of an affine erm srucure model. Our paper revisis he empirical evidence on bond reurns wihou commiing o a specific erm srucure model. We inerpre our empirical findings using a preference-free decomposiion of he pricing kernel, building on recen work in he exchange rae and erm srucure lieraures. On he one hand, a he shor end of he mauriy curve, currency risk premia are high when here is less overall risk in foreign counries pricing kernels han a home (Bekaer, 996; Bansal, 997; and Backus, Foresi, and Telmer, 00). High foreign ineres raes and/or a fla slope of he yield curve mean less overall risk in he foreign pricing kernel. On he oher hand, a he long end of he mauriy curve, local bond erm premia compensae invesors for he risk associaed wih innovaions ha he pricing kernel (Bansal and Lehmann, 997; Hansen and Scheinkman, 009; Alvarez and Jermann, 005; Hansen, 0; Hansen, Heaon, and Li, 008; and Bakshi and Chabi-Yo, 0). In his paper, we combine hose wo insighs o derive preference-free heoreical resuls under he assumpion of complee financial markes. We apply he Alvarez and Jermann (005) and Hansen and Scheinkman (009) decomposiion o a large se of erm srucure models, considering single- and muliple-facor models in he radiion of Vasicek (977) and Cox, Ingersoll, and Ross (985, denoed CIR). Models wih heeroskedasic SDFs, following CIR, are naurally he mos appealing, since currency risk 6

8 premia, when shocks are Gaussian, are simply driven by he differences in condiional volailiies of he log SDFs. This exends earlier work by Backus, Foresi, and Telmer (00), Hodrick and Vassalou (00), Brennan and Xia (006), Leippold and Wu (007), Lusig, Roussanov, and Verdelhan (0) and Sarno, Schneider, and Wagner (0). Lusig, Roussanov, and Verdelhan (0) focused on accouning for shor-run uncovered ineres rae pariy condiion (UIP) deviaions and shor-erm carry rades respecively wihin his class of models. They show ha asymmeric exposure o global innovaions o he pricing kernel are key o undersanding he global currency carry rade premium a shor mauriies. This paper focuses on long-erm carry rades. The Cross-Secion of Long-Term Bond Reurns This secion documens he downward-sloping erm srucure of currency risk premia. We firs define he bond and currency excess reurns and hen urn o he daa and our resuls.. Definiions Our empirical work focuses on hree risk premia: he erm premium in U.S. dollars, he currency risk premium, and he erm premium in foreign currency. Le us firs define hem precisely. Term Premium on Domesic Bonds P (k) denoes he price a dae of a zero-coupon bond of mauriy k. The one-period reurn on he zero-coupon bond is R (k) + = P (k ) + /P (k). The log excess reurns, denoed rx (k) +, is equal o log R(k) + /Rf, where he risk-free rae is Rf = R (0) + () = /P. The expeced log excess reurn on he zero-coupon bond wih mauriy k, or erm premium, is: [ ] [ ] E rx (k) + = E log R (k) + /Rf. The yield spread is he log difference beween he yield of he k-period bond and he risk-free ( rae: y (k) = log R f (k) /(P ) ). /k Taking his reasoning o he daa, hey idenify innovaions in he volailiy of global equiy markes as candidae shocks ha explain he cross-secion of shor-erm currency risk premia, while Menkhoff, Sarno, Schmeling, and Schrimpf (0) propose he volailiy in global currency markes insead. 7

9 Currency Risk Premium The nominal spo exchange rae in foreign currency per U.S. dollar is denoed S. When S increases, he U.S. dollar appreciaes. Similarly, F denoes he oneperiod forward exchange rae, and f is log value. The log currency excess reurn corresponds o: rx F X + = log [ S R f, S + R f ] = (f s ) s +, when he invesor borrows a he domesic risk-free rae, R f, and invess a he foreign riskfree rae, R f,, and where he forward rae is defined hrough he covered ineres rae pariy condiion: F /S = R f, /R f. The currency risk premium is he expeced value of he log currency excess reurn. Term Premium on Foreign Bonds The log reurn on a foreign bond posiion (expressed in U.S. dollars) in excess of he domesic (i.e., U.S.) risk-free rae is denoed rx (k),$ +. I can be expressed as he sum of he log excess reurn in local currency plus he reurn on a long posiion in foreign currency: [ (k), R + = log + rx (k),$ R f S S + ] = log [ R (k), + R f, R f, R f S S + ] = rx (k), + + rxf X +. The firs componen of he foreign bond excess reurn is he excess reurn on a bond in foreign currency, while he second componen represens he log excess reurn on a long posiion in foreign currency, given by he forward discoun minus he rae of depreciaion. Taking expecaions, he oal erm premium in dollars hus consiss of a foreign bond risk premium, E [rx (k), + ], plus a currency risk premium, (f s ) E s +.. Daa We sudy he bond and currency risk premia in he daa. Our benchmark sample consiss of a small homogeneous panel of developed counries wih liquid bond markes. This G-0 panel includes Ausralia, Canada, Japan, Germany, Norway, New Zealand, Sweden, Swizerland, and he U.K. The domesic counry is he Unied Saes. I only includes one counry from he 8

10 eurozone, Germany. For hose counries, we gaher discoun bonds and zero-coupon bonds. In order o build he longes ime-series possible, we obain discoun bond indices from Global Financial Daa. The daase includes a 0-year governmen bond oal reurn index, in U.S. dollars and in local currency, for each of our arge counries, as well as Treasury bill oal reurn indices. The 0-year bond reurns are a proxy for he bonds wih he longes mauriy. While Global Financial Daa offers, o he bes of our knowledge, he longes ime-series of governmen bond reurns available, he series have hree key limis. Firs, hey perain o discoun bonds, while he heory developed laer in his paper perains o zero-coupon bonds. Second, hey include defaul risk, while he heory focuses on defaul-free bonds. Third, hey only offer 0-year bond reurns, no he enire erm srucure of bond reurns. To address hese issues, we use zero-coupon bonds obained from he esimaion of erm srucure curves using governmen noes and bonds and ineres rae swaps of differen mauriies; he ime-series are shorer and dependen on he erm srucure esimaions. In conras, bond reurn indices, while spanning much longer ime-periods, offer model-free esimaes of bond reurns. Our resuls urn ou o be similar in boh samples. Our zero-coupon bond daase covers he same benchmark sample of G0 counries from /97 o /0. To consruc our sample, we use he enirey of he daase in Wrigh (0) and complemen he sample, as needed, wih sovereign zero-coupon curve daa sourced from Bloomberg. The panel is unbalanced: for each currency, he sample sars wih he beginning of he Wrigh (0) daase. Yields are available a mauriies from hree monhs o 5 years, in hree-monh incremens. To focus on expeced excess reurns, we sor counries monhly ino porfolios based on variables ha can be used o predic bond and currency reurns. Counries are sored on he level of he shor-erm ineres raes or he slope of heir yield curves (measured by he spread beween he 0-year bond yield and he one-monh ineres rae) and allocaed o hree porfolios. In all cases, porfolios formed a dae only use informaion available a ha dae. The saring daes for each counry are as follows: /987 for Ausralia, /986 for Canada, /973 for Germany, /985 for Japan, /990 for New Zealand, /998 for Norway, /99 for Sweden, /988 for Swizerland, /979 for he U.K., and /97 for he U.S. For New Zealand, he daa for mauriies above 0 years sar in /994. 9

11 The log excess reurns on currency (rx F X ), he log excess reurns on he bond in local currency (e.g., rx (0), ) and in U.S. dollars (e.g., rx (0),$ ) are firs obained a he counry level. Reurns are compued over hree-monh horizons. Then, he porfolio-level excess reurns are obained by averaging hese log excess reurns across all counries in a porfolio. We firs describe resuls obained wih he 0-year bond indices and hen urn o he zero-coupon bandis sudy he whole erm srucure..3 Soring Currencies by Ineres Raes Table : Ineres Rae-Sored Porfolios Panel A: /950 /0 Panel B: /97 /0 Porfolio s Mean f s Mean rx F X Mean s.e. [.0] [0.9] [.0] [.03] [.56] [.37] [.75] [.5] Sd SR s.e. [0.3] [0.3] [0.4] [0.5] [0.6] [0.6] [0.7] [0.7] rx (0), Mean s.e. [0.50] [0.53] [0.64] [0.6] [0.7] [0.77] [0.9] [0.87] Sd SR s.e. [0.4] [0.3] [0.3] [0.] [0.7] [0.6] [0.6] [0.5] rx (0),$ Mean s.e. [.] [.06] [.3] [.] [.85] [.60] [.89] [.77] Sd SR s.e. [0.3] [0.3] [0.3] [0.3] [0.6] [0.6] [0.7] [0.6] rx (0),$ rx (0),US Mean s.e. [.6] [.8] [.5] [.] [.70] [.7] [.6] [.77] Noes: The able repors he average change in exchange raes ( s), he average ineres rae difference (f s), he average currency excess reurn (rx F X ), he average foreign bond excess reurn on 0-year governmen bond indices in foreign currency (rx (0), ) and in U.S. dollars (rx (0),$ ), as well as he difference beween he average foreign bond excess reurn in U.S. dollars and he average U.S. bond excess reurn (rx (0),$ rx US ). For he excess reurns, he able also repors heir annualized sandard deviaion (denoed Sd) and heir Sharpe raios (denoed SR). The holding period is hree monhs. The log reurns are annualized. The balanced panel consiss of Ausralia, Canada, Japan, Germany, Norway, New Zealand, Sweden, Swizerland, and he U.K. The counries are sored by he level of heir shor erm ineres raes ino hree porfolios. The sandard errors (denoed s.e. and repored beween brackes) were generaed by boosrapping 0,000 samples of non-overlapping reurns. Le us sar wih he classic porfolios of counries sored by heir shor-erm ineres raes. 0

12 Table repors summary saisics on currency and bond excess reurns. Clearly, he uncovered ineres rae pariy condiion fails in he cross-secion. As in he lieraure, average currency excess reurns increase from low- o high-ineres-rae porfolios, ranging from 0% o 3.% per year over he las 60 years. The long-shor currency carry rade implemened wih shor-erm Treasury bills herefore delivers a 0.4 Sharpe raio. Should invesors rade long-erm bonds insead of Treasury bills in he same counries? No. Local currency bond risk premia decrease from low- o high-ineres-rae porfolios, from.% o 0.4%. The decline in he local currency bond risk premia parly offses he increase in currency risk premia. As a resul, he average excess reurn on foreign bonds expressed in U.S. dollars measured in he high-ineres-rae porfolio is only slighly higher han he average excess reurns measured in he low-ineresrae porfolio. The long-shor currency carry rade implemened wih long-erm governmen bonds does no deliver a significan average reurn. We obain similar findings over a shorer, pos-breon Woods sample. There is no evidence of saisically significan differences in dollar bond risk premia across he porfolios..4 Soring Currencies by he Slope of he Yield Curve A similar resul appears wih porfolios of counries sored by he slope of heir yield curve. There is subsanial urnover in hese porfolios, more so han in he usual ineres rae-sored porfolios, bu he ypical currencies in Porfolio (fla yield curve currencies) are he Ausralian and New Zealand dollar and he Briish pound, whereas he ypical currencies in Porfolio 3 (seep yield curve currencies) are he Japanese yen and he German mark. In oher words, he fla slope currencies end o be high ineres rae currencies, while he seep slope currencies end o be low ineres rae currencies. Consisen wih his disribuion of ineres raes, average currency excess reurns decrease across porfolios. Table repors he annualized momens of log reurns on he hree slope-sored porfolios. Average currency excess reurns decline from 3.0% per annum on Porfolio o 0.5% per annum on he Porfolio 3 over he las 60 years. Therefore, a long-shor posiion of invesing in seep-yield-curve currencies and shoring fla-yield-curve currencies delivers a currency excess

13 Table : Slope-Sored Porfolios Panel A: /950 /0 Panel B: /97 /0 Porfolio s Mean f s Mean rx F X Mean s.e. [.08] [.03] [0.95] [0.87] [.65] [.55] [.44] [.6] Sd SR s.e. [0.4] [0.3] [0.3] [0.5] [0.7] [0.6] [0.6] [0.8] rx (0), Mean s.e. [0.58] [0.5] [0.58] [0.60] [0.85] [0.75] [0.8] [0.85] Sd SR s.e. [0.3] [0.3] [0.3] [0.3] [0.6] [0.6] [0.7] [0.5] rx (0),$ Mean s.e. [.9] [.7] [.8] [.08] [.8] [.74] [.73] [.56] Sd SR s.e. [0.3] [0.3] [0.3] [0.] [0.6] [0.6] [0.6] [0.5] rx (0),$ rx (0),US Mean s.e. [.4] [.9] [.4] [.08] [.09] [.88] [.6] [.56] Noes: The able repors he average change in exchange raes ( s), he average ineres rae difference (f s), he average log currency excess reurn (rx F X ), he average log foreign bond excess reurn on 0-year governmen bond indices in foreign currency (rx (0), ) and in U.S. dollars (rx (0),$ ), as well as he difference beween he average foreign bond log excess reurn in U.S. dollars and he average U.S. bond log excess reurn (rx (0),$ rx US ). For he excess reurns, he able also repors heir annualized sandard deviaion (denoed Sd) and heir Sharpe raios (denoed SR). The holding period of he reurns is hree monhs. Log reurns are annualized. The balanced panel consiss of Ausralia, Canada, Japan, Germany, Norway, New Zealand, Sweden, Swizerland, and he U.K. The counries are sored by he slope of heir yield curves ino hree porfolios. The slope of he yield curve is measured by he difference beween he 0-year yield and he one-monh ineres rae a dae. The sandard errors (denoed s.e. and repored beween brackes) were generaed by boosrapping 0,000 samples of non-overlapping reurns.

14 reurn of.5% per annum and a Sharpe raio of 0.4. Our findings confirm hose of Ang and Chen (00). The slope of he yield curve predics currency excess reurns very well. Turning o he reurns on local bonds, average bond excess reurns increase across porfolios. Porfolio produces negaive bond excess reurns of 0.9% per annum, compared o 3.3% per annum on Porfolio 3. Imporanly, his sraegy involves long posiions in bonds issued by counries like Germany and Japan. These are counries wih fairly liquid bond markes and low sovereign credi risk. As a resul, credi and liquidiy risk differences are unlikely candidae explanaions for he reurn differences. Here again, he bond and currency excess reurns move in opposie direcions across porfolios. Turing o he reurns on foreign bonds in U.S. dollars, we do no obain significan differences across porfolios. Average bond excess reurns in U.S. dollars end o increase from he firs (fla-yield-curve) porfolio o he las (seep-yield-curve), bu a long-shor sraegy does no deliver a significan excess reurn. Local bond and currency risk premier offse each oher. We ge similar findings when we resric our analysis o he pos-breon Woods sample..5 The Term Srucure of Currency Carry Trade Risk Premia The previous resuls focus on he 0-year mauriy and show ha currency risk premia offse local currency erm premia for coupon bond reurns. We now urn o a full se of reurns in he mauriy specrum, using he zero-coupon bond daase. The erm srucure of currency carry rade risk premia is clearly downward sloping: currency carry rade sraegies ha yield posiive risk premia for shor-mauriy bonds yield lower (or even negaive) risk premia for long-mauriy bonds. As we move from he 4-quarer mauriy o he 60-quarer mauriy, he difference in he dollar erm premium beween Porfolio (fla yield curve, mosly high shor-erm ineres rae) currencies and Porfolio 3 (seep yield curve, mosly low shor-erm ineres rae) currencies decreases from.69% o.77%. The former is significanly differen from zero, whereas he laer is no. While invesing in fla yield curve currencies and shoring seep yield curve currencies provides significan gains in he shor end of he erm srucure, i yields negaive reurns in he long end. 3

15 Excess Reurns Excess Reurns Local Currency Porfolio Mauriy (in quarers) FX Porfolio Mauriy (in quarers) Dollars Excess Reurns Excess Reurns Local Currency Porfolio Mauriy (in quarers) FX Porfolio Mauriy (in quarers) Dollars Excess Reurns Excess Reurns Local Currency Porfolio Mauriy (in quarers) FX Porfolio Mauriy (in quarers) Dollars Excess Reurns Porfolio Mauriy (in quarers) Excess Reurns Porfolio Mauriy (in quarers) Excess Reurns Porfolio Mauriy (in quarers) Figure : Term Srucure of Dollar Bond Risk Premia The figure shows he log excess reurns on foreign bonds in local currency in he op panel, he currency excess reurn in he middle panel, and he log excess reurns on foreign bonds in U.S. dollars in he boom panel as a funcion of he bond mauriies. The lef panel focuses on Porfolio (fla yield curve currencies) excess reurns, while he middle panel repors Porfolio 3 (seep yield curve currencies) excess reurns. The middle panels also repor he Porfolio excess reurns in dashed lines for comparison. The righ panel repors he difference. Daa are monhly, from he zero-coupon daase, and he sample window is 4/985 /0. The unbalanced panel consiss of Ausralia, Canada, Japan, Germany, Norway, New Zealand, Sweden, Swizerland, and he U.K. The counries are sored by he slope of heir yield curves ino hree porfolios. The slope of he yield curve is measured by he difference beween he 0-year yield and he 3-monh ineres rae a dae. The holding period is one quarer. The reurns are annualized. The shaded areas correspond o wo sandard deviaions above and below each poin esimaes. Sandard deviaions are obained by boosrapping. To illusrae his finding, Figure repors he local currency excess reurns (in logs) in he op panel, and he dollar excess reurns (in logs) in he boom panel. The op panel in Figure shows ha counries wih he seepes local yield curves (Porfolio 3, cener) exhibi local bond excess reurns ha are higher, and increase faser wih he mauriy han he fla yield curve counries (Porfolio, on he lef-hand side). Thus, ignoring he effec of exchange raes, invesors should inves in he shor-erm and long-erm bonds of seep yield curve currencies. Considering he effec of currency flucuaions by focusing on dollar reurns radically alers 4

16 he resuls. Figure shows ha he dollar excess reurns of Porfolio are higher han hose of Porfolio 3 a he shor end of he yield curve, consisen wih he carry rade resuls of Ang and Chen (00). Ye, an invesor who would aemp o replicae he shor-mauriy carry rade sraegy a he long end of he mauriy curve would incur losses on average: he long-mauriy excess reurns of fla yield curve currencies are lower han hose of seep yield curve currencies, as currency risk premia more han offse erm premia. This resul is apparen in he lower panel on he righ, which is he same as Figure in he inroducion. The erm srucure of currency carry rade risk premia slopes downwards..6 Robusness Checks We consider many robusness checks, sudying (i) differen ime windows, (ii) differen lenghs of he bond holding period, and (iii) differen samples of counries. All he resuls are repored in he Online Appendix. Here we simply describe he main findings. The difference beween currency excess reurns and foreign bond excess reurns appears robus across ime windows. Figure 3 presens he cumulaive hree-monh log reurns on invesmens in foreign Treasury bills and foreign 0-year bonds, saring in 950. Counries are sored ino porfolios based on he slope of heir yield curves. The reurns correspond o an invesmen sraegy going long in Porfolio (fla yield curves, mosly high shor-erm ineres raes) and shor in he Porfolio 3 (seep yield curves, mosly low shor-erm ineres raes). Even when dividing he sample ino wo, hree, or four sub periods, he main resul remains: an invesor buying shor-erm Treasury bills of counries wih fla yield curves, mosly high shorerm ineres raes, while selling shor-erm Treasury bills of counries wih seep yield curves, mosly low shor-erm ineres raes, enjoys posiive reurns; invesing in long-erm bonds of he same counries resuls in negaive reurns. Again, currency and local erm premia offse each oher, and hus average carry rade reurns are differen a he shor-end and he log end of he erm srucure. Our resuls appear robus o he choice of he bond holding period. We consider invesmens of one, hree, and welve monhs. The paerns are similar. We someimes obain significan 5

17 The Slope Carry Trade 4 FX Premium Local Bond Premium cumulaive log reurns Figure 3: The Carry Trade and Term Premia: Condiional on he Slope of he Yield Curve The figure presens he cumulaive one-monh log reurns on invesmens in foreign Treasury bills and foreign 0-year bonds. The benchmark panel of counries includes Ausralia, Canada, Japan, Germany, Norway, New Zealand, Sweden, Swizerland, and he U.K. Counries are sored every monh by he slope of heir yield curves ino hree porfolios. The slope of he yield curve is measured by he spread beween he 0-year bond yield and he one-monh ineres rae. The reurns correspond o an invesmen sraegy going long in Porfolio and shor in he Porfolio 3. The sample period is /950 /0. 6

18 dollar erm premia when invesors only inves for one monh, bu such a sraegy would enail large ransacion coss, which would likely wipe ou he reurns. For longer holding periods, he dollar erm premia are no significan. Our findings appear also robus across samples of counries. Wih discoun bonds, we consider wo addiional ses of counries: firs, a larger sample of 0 developed counries (Ausralia, Ausria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Ialy, Japan, he Neherlands, New Zealand, Norway, Porugal, Spain, Sweden, Swizerland, and he U.K.), and second, a large sample of 30 developed and emerging counries (he same as above, plus India, Mexico, Malaysia, he Neherlands, Pakisan, he Philippines, Poland, Souh Africa, Singapore, Taiwan, and Thailand). In he sample of developed counries, he seep-slope (low yielding) currencies are ypically counries like Germany, he Neherlands, Japan, and Swizerland, while he fla-slope (highyielding) currencies are ypically Ausralia, New Zealand, Denmark and he U.K. A he onemonh horizon, he.4% spread in currency excess reurns obained in his sample is more han offse by he 5.9% spread in local erm premia. This produces a saisically significan 3.5% reurn on a posiion ha is long in he low yielding, high slope currencies and shor in he high yielding, low slope currencies. These resuls are essenially unchanged in he pos-breon- Woods sample. A longer horizons, he currency excess reurns and he local risk premia almos fully offse each oher. In he enire sample of counries, including he emerging marke counries, he difference in currency risk premia a he one-monh horizon is 3.04% per annum, which is more han offse by a 8.37% difference in local erm premia. As a resul, invesors earn 5.33% per annum on a long-shor posiion in foreign bond porfolios of slope-sored currencies. As before, his involves shoring he fla-yield-curve currencies, ypically high ineres rae currencies, and going long in he seep-slope currencies, ypically he low ineres rae ones. The annualized Sharpe raio on his long-shor sraegy is We also consruc an exended version of he zero-coupon daase which, in addiion o he counries of he benchmark sample, includes he following counries: Ausria, Belgium, he 7

19 Czech Republic, Denmark, Finland, France, Hungary, Indonesia, Ireland, Ialy, Malaysia, Mexico, he Neherlands, Poland, Porugal, Singapore, Souh Africa, and Spain. The daa for he aforemenioned exra counries are sourced from Bloomberg. 3 Considering a large cross-secion of counries does no change our main resul. An invesor who buys he one-year bonds of fla-yield curve currencies and shors he one-year bonds of seep-yield-curve currencies realizes a dollar excess reurn of 4.% per year on average. However, a he long end of he mauriy srucure his sraegy generaes negaive and insignifican excess reurns: he average annualized dollar excess reurn of an invesor who pursues his sraegy using 5-year bonds is 0.4%. The erm srucure of currency carry rade risk premia remains downward-sloping. We urn now o he inerpreaion of his empirical finding. 3 The Foreign Term Premium and he Properies of SDFs We begin by defining noaion and hen derive our main heoreical resuls. 3. Noaion In order o sae our main resuls, we firs need o inroduce he domesic and foreign pricing kernels and sochasic discoun facors. Pricing Kernel and Sochasic Discoun Facor The nominal pricing kernel is denoed Λ (ϖ); i corresponds o he marginal value of a dollar delivered a ime in he sae of he world ϖ. The nominal SDF is he growh rae of he pricing kernel: M + = Λ + /Λ. The price of a zero-coupon bond ha maures k periods ino he fuure is given by: P (k) ( ) Λ+k = E. Λ 3 The saring daes for he addiional counries are as follows: /994 for Ausria, Belgium, Denmark, Finland, France, Ireland, Ialy, he Neherlands, Porugal, Singapore, and Spain, /000 for he Czech Republic, 3/00 for Hungary, 5/003 for Indonesia, 9/00 for Malaysia, 8/003 for Mexico, /000 for Poland, and /995 for Souh Africa. 8

20 Enropy Bond reurns and SDFs are volaile, bu no necessarily normally disribued. In order o measure he ime-variaion in heir volailiy, i is convenien o use enropy. 4 The dynamics of any random variable X + are hus measured hrough he condiional enropy L, defined as: L (X + ) = log E (X + ) E (log X + ). The condiional enropy of a random variable is deermined by is condiional variance, as well as is higher momens; if var (X + ) = 0, hen L (X + ) = 0, bu he reverse is no generally rue. If X + is condiionally lognormal, hen he enropy is simply he half variance of he log variable: L (X + ) = (/)var (log X + ). The relaive enropy of he permanen and ransiory componens of he domesic and foreign SDFs urns ou o be key o undersanding he erm srucure of carry rade risk. Under regulariy condiions, here is a higher-order expansion of L (X + ) = κ /! + κ 3 /3! + κ 4 /4! +... where κ i are he cumulans of log X. Exchange Raes in Complee Markes When markes are complee, he change in he exchange rae corresponds o he raio of he domesic o foreign SDFs: S + S = Λ + Λ Λ Λ +, where denoes a foreign variable. The no-arbirage definiion of he exchange rae comes direcly from he Euler equaions of he domesic and foreign invesors, for any asse R expressed in foreign currency: E [M + R + S /S + ] = and E [M + R + ] =. When markes are complee, he SDF is unique, and hus he change in exchange rae is he raio of he wo SDFs. As Bekaer (996) and Bansal (997) show, in a lognormal model, he log currency risk premium equals he half difference beween he condiional volailiies of he log domesic and foreign SDFs. Higher order momens are criical for undersanding currency reurns. 5 When 4 Backus, Chernov, and Zin (04) make a convincing case for he use of enropy in assessing macro-finance models. 5 The lieraure on disaser risk in currency markes concurs. In earlier work, Brunnermeier, Nagel, and Pedersen (009) show ha risk reversals increase wih ineres raes. Jurek (008) provides a comprehensive empirical invesigaion of hedged carry rade sraegies. Gourio, Siemer, and Verdelhan (03) sudy a real business cycle model wih disaser risk. Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (03) esimae a no-arbirage model wih crash risk using a cross-secion of currency opions. Chernov, Graveline, and Zviadadze 9

21 higher momens maer and markes are complee, he currency risk premium is equal o he difference beween he enropy of he domesic and foreign SDFs (Backus, Foresi, and Telmer, 00): ( ) ( [ ] E rx F X Λ+ Λ ) + = (f s ) E ( s + ) = L L + Λ Λ. 3. Main Theoreical Resuls In his secion, we presen our wo key heoreical resuls on (i) he erm srucure of yields, (ii) he erm srucure of carry rade premia; and (iii) he long-erm bond reurn pariy condiion. 3.. Term Srucure of Ineres Raes and Foreign Currency Risk Premia We sar by racing ou he relaion beween yields as we increase he holding period. The per period risk premium in logs on a long posiion in foreign currency over k periods consiss of he yield spread minus he expeced rae of depreciaion over he holding period: E [rx fx +k ] = yk, y k (/k)e [ s +k ]. The implied expeced rae of depreciaion over k periods is given by he expeced shor-erm ineres rae differences minus he fuure shor-erm currency risk premia, summed up over he holding period: k ( ) k E s +k = E r +j r +j E rx F +j X. The implied log currency risk premium over k periods is given by: j= j= E [rx fx +k ] = (yk, y k ) + (/k)e k ( k r+j r+j ) + (/k) E rx F +j X. j= The firs wo erms measure he deviaions from he expecaions hypohesis over he holding period k. The las erm measures he deviaions from shor-run uncovered ineres rae mauriy (0) sudy jump risk a high frequencies. Gavazzoni, Sambalaiba, and Telmer (0) show ha lognormal models canno accoun for he cross-counry differences in carry reurns and ineres rae volailiies. j= 0

22 over he holding period k. Proposiion. The per period risk premium on foreign bonds held over k periods equals he difference in he per period enropy of he pricing kernels over he holding period k: [ ( ) ( E [rx fx +k ] = (/k) Λ+k Λ )] L L +k Λ Λ. Hence, he risk premium on a long posiion in foreign currency is governed by how quickly enropy of he pricing kernel builds up a home and abroad. Proof of Proposiion : Proof. In any no-arbirage model, he yield spread equals he log of he expeced rae of depreciaion of he foreign currency over he holding period k: y k, y k = (/k) log E ( exp[ log Λ +k ] ) + (/k) log E ( exp[ log Λ +k ] ). Using he definiion of enropy, we can resae his expression as: y k, [ ( ) ( y k Λ+k Λ )] = (/k)e s +k + (/k) L L +k Λ Λ. Following Backus, Chernov, and Zin (04), we use I(k) = (/k)e log E ( exp[ log Λ +k ] ) o denoe he average per period enropy. In a saionary environmen, our resul implies ha he average yield spread equals he average per period enropy difference plus he expeced rae of depreciaion per period: E(y k, y k ) = E s + + I(k) I (k). If m + is i.i.d., hen I(k) = I() for all k. In ha simple case, he average risk premium over k holding periods is idenical o he average one-period foreign currency risk premium: E[rx fx +k ] = E(yk, y k ) (/k)e s +k does no depend on k. Backus, Chernov, and Zin (04) refer o his as he case of no horizon dependence. Noe ha s + is non-saionary in he i.i.d. case.

23 Horizon dependence arises when we deviae from he i.i.d. assumpion. I(k) I() measures deviaions from he i.i.d. assumpion. For example, posiive auocorrelaion in he log of he pricing kernel ends o increase I(k) above I(), while negaive auocorrelaion ends o decrease i below I(). More posiive (negaive) auocorrelaion in he domesic han in he foreign pricing kernel ends o creae a upward (downward) sloping erm srucure of foreign currency risk premia. Ineremporal dependence in higher-order momens maers as well. Nex, we consider specific deviaions from he i.i.d. assumpion. Corollary. If he log of he pricing kernels log Λ and log Λ are weakly saionary, hen U.I.P. holds for large k: lim k (yk, y k ) = (/k)e s +k Proof of Corollary : Proof. For a saionary variable x, (/k)l (exp( x +k )) 0 as k. Corollary. If he log of he exchange rae s is weakly saionary, hen U.I.P. holds for large k: lim k (yk, y k ) = (/k)e s +k Proof of Corollary : Proof. If he log of he exchange rae s is weakly saionary, hen log Λ and log Λ are coinegraed wih co-inegraing vecor (, ). These wo variables hen mus share he same sochasic rend: log Λ = µ + z and log Λ = µ + z where (z, z ) I(0). We refer o his sochasic rend as µ. The per period enropy of he foreign and domesic pricing kernels hen converge in he limi o ha of he sochasic rend: lim k (/k)l ( Λ+k Λ ) = lim k (/k)l ( Λ ) +k Λ = lim k (/k)l (exp( µ +k )).

24 3.. Carry Trade Term Premia Nex, we fix he holding period, bu insead we increase he mauriy of he bonds. Thus, we characerize carry rade risk premia a long mauriies. To do so, we appeal o resuls ha decompose he pricing kernel ino a ransiory and a saionary componen. Permanen and Transiory Innovaions Following Alvarez and Jermann (005), Hansen, Heaon, and Li (008), and Hansen and Scheinkman (009), we decompose each pricing kernel ino a ransiory (Λ T ) componen and a permanen (Λ P ) componen such ha: Λ = Λ P Λ T, where Λ T = lim k δ +k P (k) The consan δ is chosen o saisfy he following regulariy condiion: 0 <. P lim (k) k δ k < for all. We also assume ha, for each +, here exiss a random variable x + wih finie expeced value E (x + ) such ha a.s. Λ + P (k) + δ + δ k infinie mauriy bond reurn is hen: x + for all k. Under hose regulariy condiions, he R ( ) + = lim k R(k) + = lim P (k ) + /P (k) = ΛT k Λ T. + The permanen componen, Λ P, is a maringale. 6 I is an imporan componen of he pricing kernel. Alvarez and Jermann (005) derive a lower bound on is volailiy, and, given he size of he equiy premium relaive o he erm premium, conclude ha he permanen componen of he pricing kernel is large and accouns for mos of he risk. 7 In oher words, a lo of persisence 6 Noe ha Λ P is equal o: P (k) E (Λ +k ) δ +k. Λ P = lim Λ = lim k δ+k k The second regulariy condiion ensures ha he expression above is a maringale. 7 Alvarez and Jermann (005) derive he following lower bound: where R + denoes any posiive reurn and R ( ) + ( Λ P ) ( ) L + E Λ P (log R +) E log R ( ) +, is he reurn on a zero-coupon bond of infinie mauriy. 3

25 in he pricing kernel is needed o deliver a low erm premium and a high equiy premium. In he absence of arbirage, Alvarez and Jermann (005) show ha he local erm premium in local currency is given by: E [ rx ( ) + ] [ ] ( ) ( ) = lim E rx (k) Λ+ Λ P + = L L + k Λ Λ P. The SDF decomposiion defined here is subjec o imporan limiaions ha need o be highlighed. Hansen and Scheinkman (009) poin ou ha his decomposiion is no unique under he assumpions used in Alvarez and Jermann (005). Bu building on heir own work on semigroup operaors and eigenfuncions, Hansen and Scheinkman (009) also provide condiions under which a muliplicaive decomposiion of he SDF exiss and is unique. We review heir resuls in he Appendix and he link beween he wo approaches. When working on specific erm srucure models laer in he paper, we show ha he eigenfuncion decomposiion of Hansen and Scheinkman (009) leads o exacly he same resuls as he consrucive decomposiion of Alvarez and Jermann (005). They show ha here exiss a posiive eigenvalue λ and an eigenfuncion e such ha E [M + e + ] = λe. The muliplicaive decomposiion implies ha he maringale componen can be saed as: Λ P + Λ P = M +e + λe, In addiion, he emporary (or ransien) and permanen componens are poenially correlaed, which may complicae heir inerpreaion. Despie his limiaion, we show ha his decomposiion proves o be paricularly useful when analyzing foreign bond reurns a longer mauriies. We will use i o derive parameric resricions in a large class of affine erm srucure models in which permanen and ransiory shocks are orhogonal. Following he decomposiion of he pricing kernel discussed above, exchange rae changes can also be decomposed ino a permanen and a ransiory componen, defined below: S + S = ( Λ P + Λ P ) ( ) Λ P, Λ T + Λ T, Λ P, Λ T + Λ T, = SP + S P + S+ T S T. 4

26 Exchange rae changes capure he differences in boh he ransiory and he permanen componen of he wo counries SDFs. Noe ha S+ P, he raio of wo maringales, is iself no a maringale. However, if wo counries share he same maringale componen of he pricing kernel, hen he resuling exchange rae is saionary. Proposiion. The foreign erm premium in dollars is equal o he domesic erm premium plus he difference beween he domesic and foreign enropies of he permanen componens of he pricing kernels: E [ rx ( ), + ] + (f s ) E [ s + ] = E [ rx ( ) + ] + L ( Λ P + Λ P ) ( ) Λ P, + L Λ P,. In case of an adverse emporary innovaion o he foreign pricing kernel, he foreign currency appreciaes, bu his is exacly offse by he capial loss suffered on he longes mauriy zerocoupon bond, as a resul of he increase in foreign ineres raes. Hence, ineres rae exposure compleely hedges he emporary componen of he currency risk exposure, and he only source of priced currency risks in he foreign bond posiions are he permanen innovaions. bonds. This is he analogue of proposiion for he one period risk premium on longer mauriy In order o produce a currency risk premium a longer mauriies, enropy differences in he permanen componen of he pricing kernel are required. If here are no such differences and domesic and foreign pricing kernels are idenically disribued, hen high local currency erm premia coincide wih low currency risk premia and vice-versa, and dollar erm premia are idenical across currencies. The uncondiional version of his resul for one-period reurns is equivalen o uncovered ineres rae pariy for very long holding periods. Noe ha he k-period holding reurn in excess of he U.S. risk-free rae on a foreign k-period bond denoed in dollars is k(y k, ) s +k + k(f s ). The uncondiional version of long-run UIP implies ha, as k, E[y k, ] E[lim k k k j= s +j] = E[y k ]. 5