The Cross Section of Foreign Currency Risk Premia and Consumption Growth Risk


 Godwin Waters
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1 The Cross Secton of Foregn Currency Rsk Prema and Consumpton Growth Rsk By HANNO LUSTIG AND ADRIEN VERDELHAN* Aggregate consumpton growth rsk explans why low nterest rate currences do not apprecate as much as the nterest rate dfferental and why hgh nterest rate currences do not deprecate as much as the nterest rate dfferental. Domestc nvestors earn negatve excess returns on low nterest rate currency portfolos and postve excess returns on hgh nterest rate currency portfolos. Because hgh nterest rate currences deprecate on average when domestc consumpton growth s low and low nterest rate currences apprecate under the same condtons, low nterest rate currences provde domestc nvestors wth a hedge aganst domestc aggregate consumpton growth rsk. (JEL E21, E43, F31, G11) When the foregn nterest rate s hgher than the US nterest rate, rskneutral and ratonal US nvestors should expect the foregn currency to deprecate aganst the dollar by the dfference between the two nterest rates. Ths way, borrowng at home and lendng abroad, or vce versa, produces a zero return n excess of the US shortterm nterest rate. Ths s known as the uncovered nterest rate party (UIP) condton, and t s volated n the data, except n the case of very hgh nflaton currences. In the data, hgher foregn nterest rates almost always predct hgher excess returns for a US nvestor n foregn currency markets. We show that these excess returns compensate the US nvestor for takng on more US consumpton growth rsk. Hgh foregn nterest * Lustg: Department of Economcs, UCLA, Box , Los Angeles, CA 90095, and Natonal Bureau of Economc Research (emal: Verdelhan: Department of Economcs, Boston Unversty, 270 Bay State Road, Boston, MA 02215, and Centre de recherches de la Banque de France, 29 rue Crox des petts champs, Pars France (emal: The authors especally thank two anonymous referees, Andy Atkeson, John Cochrane, Lars Hansen, and Anl Kashyap for detaled comments. We would also lke to thank Rav Bansal, Crag Burnsde, Hal Cole, Vrgne Coudert, Franços Gouro, Martn Echenbaum, John Heaton, Patrck Kehoe, Isaac Kleshchelsk, Chrs Lundbladd, Lee Ohanan, Fabrzo Perr, Sergo Rebelo, Stjn Van Neuwerburgh, and semnar partcpants at varous nsttutons and conferences. The vews n ths paper are solely the responsblty of the authors and should not be nterpreted as reflectng the vews of the Bank of France. 89 rate currences, on average, deprecate aganst the dollar when US consumpton growth s low, whle low foregn nterest rate currences do not. The textbook logc we use for any other asset can be appled to exchange rates, and t works. If an asset offers low returns when the nvestor s consumpton growth s low, t s rsky, and the nvestor wants to be compensated through a postve excess return. To uncover the lnk between exchange rates and consumpton growth, we buld eght portfolos of foregn currency excess returns on the bass of the foregn nterest rates, because nvestors know these predct excess returns. Portfolos are rebalanced every perod, so the frst portfolo always contans the lowest nterest rate currences and the last portfolo always contans the hghest nterest rate currences. Ths s the key nnovaton n our paper. Over the last three decades, n emprcal asset prcng, the focus has shfted from explanng ndvdual stock returns to explanng the returns on portfolos of stocks, sorted on varables that we know predct returns (e.g., sze and booktomarket rato). 1 Ths procedure elmnates the dversfable, stockspecfc component of returns that s not of nterest, thus producng much sharper estmates of the rskreturn tradeoff n equty markets. Smlarly, for currences, by sortng these nto portfolos, we abstract from the currencyspecfc component 1 See Eugene F. Fama (1976), one of the ntal advocates of buldng portfolos, for a clear exposton.
2 90 THE AMERICAN ECONOMIC REVIEW MARCH 2007 Characterstcs of the eght portfolos (excess returns) 2 Mean Std Sharpe rato FIGURE 1. EIGHT CURRENCY PORTFOLIOS Notes: Ths fgure presents means, standard devatons (n percentages), and Sharpe ratos of real excess returns on eght annually rebalanced currency portfolos for a US nvestor. The data are annual and the sample s These portfolos were constructed by sortng currences nto eght groups at tme t based on the nomnal nterest rate dfferental wth the home country at the end of perod t 1. Portfolo 1 contans currences wth the lowest nterest rates. Portfolo 8 contans currences wth the hghest nterest rates. of exchange rate changes that s not related to changes n the nterest rate. Ths solates the source of varaton n excess returns that nterests us, and t creates a large average spread of up to fve hundred bass ponts between low and hgh nterest rate portfolos. Ths spread s an order of magntude larger than the average spread for any two gven countres. As one would expect from the emprcal lterature on UIP, US nvestors earn on average negatve excess returns on low nterest rate currences of mnus 2.3 percent and large, postve excess returns on hgh nterest rate currences of up to 3 percent. The relaton s almost monotonc, as shown n Fgure 1. These returns are large even when measured per unt of rsk. The Sharpe rato (defned as the rato of the average excess return to ts standard devaton) on the hgh nterest rate portfolo s close to 40 percent, only slghtly lower than the Sharpe rato on US equty, whle the same rato s mnus 40 percent for the lowest nterest rate portfolo. In addton, these portfolos keep the number of covarances that must be estmated low, whle allowng us to contnuously expand the number of countres studed as fnancal markets open up to nternatonal nvestors. Ths enables us to nclude data from the largest possble set of countres. To show that the excess returns on these portfolos are due to currency rsk, we start from the US nvestor s Euler equaton and use consumptonbased prcng factors. We test the model on annual data for the perods and Consumptonbased models explan up to 80 percent of the varaton n currency excess returns across these eght currency portfolos. Are the parameter estmates reasonable? Our results are not consstent wth what most economsts vew as plausble values of rsk averson, but they are consstent wth the evdence from other assets. The estmated coeffcent of rsk averson s around 100, and the estmated prce of US consumpton growth rsk s about 2 percent
3 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 91 per annum for nondurables and 4.5 percent for durables. Consumptonbased models can explan the rsk prema n currency markets only f we are wllng to entertan hgh levels of rsk averson, as s the case n other asset markets. In fact, currency rsk seems to be prced much lke equty rsk. If we estmate the model on US domestc bond portfolos (sorted by maturty) and stock portfolos (sorted by booktomarket and sze) n addton to the currency portfolos, the rsk averson estmate does not change. Our currency portfolos really allow for an outofsample test of consumptonbased models, because the low nterest rate currency portfolos have negatve average excess returns, unlke most of the test assets n the emprcal asset prcng lterature, and the returns on the currency portfolos are not strongly correlated wth bond and stock returns. Consumptonbased models can explan the crosssecton of currency excess returns f and only f hgh nterest rate currences typcally deprecate when real US consumpton growth s low, whle low nterest rate currences apprecate. Ths s exactly the pattern we fnd n the data. We can restate ths result n standard fnance language usng the consumpton growth beta of a currency. The consumpton growth beta of a currency measures the senstvty of the exchange rate to changes n US consumpton growth. These betas are small for low nterest rate currences and large for hgh nterest rate currences. In addton, for the low nterest rate portfolos, the betas turn negatve when the nterest rate gap wth the Unted States s large. All our results buld on ths fndng. Secton I outlnes our emprcal framework and defnes the foregn currency excess returns and the potental prcng factors. Secton II tests consumptonbased models on the uncondtonal moments of our foregn currency portfolo returns. Secton III lnks our results to propertes of exchange rate betas. Secton IV checks the robustness of our estmates n varous ways. Fnally, Secton V concludes wth a revew of the relevant lterature. Data on currency returns and the composton of the currency portfolos are avalable on the authors Web stes. 2 2 See or I. Foregn Currency Excess Returns Ths secton frst defnes the excess returns on foregn TBll nvestments and detals the constructon and characterstcs of the currency portfolos. We then turn to the US nvestor s Euler equaton and show how consumpton rsk can explan the average excess returns on these currency portfolos. A. Why Buld Portfolos of Currences? We focus on a US nvestor who nvests n foregn TBlls or equvalent nstruments. These blls are clams to a unt of foregn currency one perod from today n all states of the world. R t 1 denotes the rsky dollar return from buyng a foregn TBll n country, sellng t after one perod, and convertng the proceeds back nto dollars: R t 1 R, t (E t 1 /E t ), where E t s the exchange rate n dollar per unt of foregn currency, and R, t s the rskfree oneperod return n unts of foregn currency. 3 We use P t to denote the dollar prce of the US consumpton,e basket. Fnally, R t 1 (R t 1 R $ t )(P t /P t 1 )s the real excess return from nvestng n foregn TBlls, and R $ t s the nomnal rskfree rate n US currency. Below, we use lowercase symbols to denote the log of a varable. UIP Regressons and Currency Rsk Prema. Accordng to the UIP condton, the slope n a regresson of the change n the exchange rate for currency on the nterest rate dfferental s equal to one: e t R, t R $ t t 1, and the constant s equal to zero. The data consstently produce slope coeffcents less than one, mostly even negatve. 4 Of course, ths mmedately mples that the (nomnal) expected excess returns, whch are roughly equal to (R, t R $ t ) E t e t 1, are not zero and that they are predcted by nterest rates: hgher nterest rates predct hgher excess returns. 3 Note that returns are dated by the tme they are known. Thus, R t, s the nomnal rskfree rate between perod t and t 1, whch s known at date t. 4 See Lars P. Hansen and Robert J. Hodrck (1980) and Fama (1984). Hodrck (1987) and Karen K. Lews (1995) provde extensve surveys and updated regresson results.
4 92 THE AMERICAN ECONOMIC REVIEW MARCH 2007 Currency Portfolos. To better analyze the rskreturn tradeoff for a US nvestor nvestng n foregn currency markets, we construct currency portfolos that zoom n on the predctablty of excess returns by foregn nterest rates. At the end of each perod t, we allocate countres to eght portfolos on the bass of the nomnal nterest rate dfferental, R, t R $ t, observed at the end of perod t. The portfolos are rebalanced every year. They are ranked from low to hgh nterests rates, portfolo 1 beng the portfolo wth the lowest nterest rate currences and portfolo 8 beng the one wth the hghest nterest rate currences. By buldng portfolos, we flter out currency changes that are orthogonal to changes n nterest rates. Let N j denote the number of currences n portfolo j, and let us smply assume that currences wthn a portfolo have the same UIP constant and slope coeffcents. Then, for portfolo j, the change n the average exchange rate wll reflect manly the rsk premum component, 0 j j 1 (1/N j ) (R, t R $ t ), the part we are nterested n. We always use a total number of eght portfolos. Gven the lmted number of countres, especally at the start of the sample, we dd not want too many portfolos. If we choose fewer than eght portfolos, then the currences of countres wth very hgh nflaton end up beng mxed wth others. It s mportant to keep these currences separate because the returns on these very hgh nterest rate currences are very dfferent, as wll become more apparent below. Next, we compute excess returns of foregn j,e TBll nvestments R t 1 for each portfolo j by averagng across the dfferent countres n each portfolo. We use E T to denote the sample mean for a sample of sze T. The varaton n average excess returns E T [R j,e t 1 ] for j 1,..., 8 across portfolos s much larger than the spread n average excess returns across ndvdual currences, because foregn nterest rates fluctuate over tme: the foregn excess return s postve (negatve) when foregn nterest rates are hgh (low), and perods of hgh excess returns are canceled out by perods of low excess returns. Our portfolos shft the focus from ndvdual currences to hgh versus low nterest rate currences, n the same way that the Fama and French portfolos of stocks sorted on sze and booktomarket ratos shft the focus from ndvdual stocks to small/value versus large/growth stocks (see Fama and Kenneth R. French 1992). B. Data Wth these eght portfolos, we consder two dfferent tme horzons. Frst, we study the perod, whch spans a number of dfferent exchange rate arrangements. The Euler equaton restrctons are vald regardless of the exchange rate regme. Second, we consder a shorter tme perod, 1971 to 2002, begnnng wth the demse of BrettonWoods. Interest Rates and Exchange Rates. For each currency, the exchange rate s the endofmonth average daly exchange rate, from Global Fnancal Data. The foregn nterest rate s the nterest rate on a threemonth government securty (e.g., a US TBll) or an equvalent nstrument, also from Global Fnancal Data ( We used the threemonth nterest rate nstead of the oneyear rate, smply because fewer governments ssue blls or equvalent nstruments at the oneyear maturty. As data became avalable, new countres were added to these portfolos. As a result, the composton of the portfolo as well as the number of countres n a portfolo change from one perod to the next. Secton A.1 of the Appendx contans a detaled lst of the currences n our sample. Two addtonal ssues need to be dealt wth: the exstence of expected and actual default events, and the effects of fnancal lberalzaton. Default. Defaults can have an mpact on our currency returns n two ways. Frst, expected defaults should lead ratonal nvestors to ask for a default premum, thus ncreasng the foregn nterest rate and the foregn currency return. To check that our results are due to currency rsk, we run all experments for a subsample of developed countres. None of these countres has ever defaulted, nor were they ever consdered lkely canddates. Yet, we obtan very smlar results. Second, actual defaults modfy the realzed returns. To compute actual returns on an nvestment after default, we used the dataset of defaults compled by Carmen M. Renhart, Kenneth S. Rogoff, and Mguel A. Savastano (2003). We appled an (ex ante) recovery rate of 70 percent. Ths number
5 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 93 TABLE 1 US INVESTOR S EXCESS RETURNS Portfolo mean SR mean SR Notes: Ths table reports the mean of the real excess returns (n percentage ponts) and the Sharpe rato for a US nvestor. The portfolos are constructed by sortng currences nto eght groups at tme t based on the nomnal nterest rate dfferental at the end of perod t 1. Portfolo 1 contans currences wth the lowest nterest rates. Portfolo 8 contans currences wth the hghest nterest rates. The table reports annual returns for annually rebalanced portfolos. reflects two sources, Manmohan Sngh (2003) and Moody s Investors Servce (2003), presented n Secton A.2 of the Appendx. If a country s stll n default n the followng year, we smply exclude t from the sample for that year. 5 Captal Account Lberalzaton. The restrctons mposed by the Euler equaton on the jont dstrbuton of exchange rates and nterest rates make sense only f foregn nvestors can n fact purchase local TBlls. Denns Qunn (1997) has bult ndces of openness based on the codng of the IMF Annual Report on Exchange Arrangements and Exchange Restrctons. Ths report covers 56 natons from 1950 onward and 8 more startng n Qunn s (1997) captal account lberalzaton ndex ranges from zero to 100. We chose a cutoff value of 20, and we elmnated countres below the cutoff. In these countres, approval of both captal payments and recepts s rare, or the payments and recepts are at best only nfrequently granted. 5 In the entre sample from 1953 to 2002, there are 13 nstances of default by a country whose currency s n one of our portfolos: Zmbabwe (1965), Jamaca (1978), Jamaca (1981), Mexco (1982), Brazl (1983), Phlppnes (1983), Zamba (1983), Ghana (1987), Jamaca (1987), Trndad and Tobago (1988), South Afrca (1989, 1993), and Pakstan (1998). Of course, many more countres actually defaulted over ths sample, but those are not n our portfolos because they mposed captal controls, as explaned n the next paragraph. C. Summary Statstcs for the Currency Portfolo Returns Ths secton presents some prelmnary evdence on the currency portfolo returns. The frst panel of Table 1 lsts the average excess returns n unts of US consumpton E T [R j,e t 1 ] and the Sharpe rato for each of the annually rebalanced portfolos. The largest spread (between the frst and the seventh portfolo) exceeds 5 percentage ponts for the entre sample, and close to 7 percentage ponts n the shorter subsample. The average annual returns are almost monotoncally ncreasng n the nterest rate dfferental. The only excepton s the last portfolo, whch conssts of very hgh nflaton currences: the average nterest rate gap wth the Unted States for the eghth portfolo s about 16 percentage ponts over the entre sample and 23 percentage ponts post Bretton Woods. As Rav Bansal and Magnus Dahlqust (2000) have documented, UIP tends to work best at hgh nflaton levels. Countres change portfolos frequently (23 percent of the tme), and the tmevaryng composton of the portfolos s crtcal. If we allocate currences nto portfolos based on the average nterest rate dfferental over the entre sample nstead, then there s essentally no pattern n average excess returns. Exchange Rates and Interest Rates. Table 2 decomposes the average excess returns on each portfolo nto ts two components. For each portfolo, we report the average nterest
6 94 THE AMERICAN ECONOMIC REVIEW MARCH 2007 TABLE 2 EXCHANGE RATES AND INTEREST RATES Portfolo E T ( R j ) E T ( e j ) E T ( p j ) E T ( R j ) E T ( e j ) E T ( p j ) Notes: Ths table reports the tmeseres average of the average nterest rate dfferental R j t (n percentage ponts), the average j rate of deprecaton e t 1 (n percentage ponts), and the average nflaton rate p j (n percentage ponts) for each of the portfolos. Portfolo 1 contans currences wth the lowest nterest rates. Portfolo 8 contans currences wth the hghest nterest rates. Ths table reports annual nterest rates, exchange rate changes, and nflaton rates for annually rebalanced portfolos. rate gap (E T ( R j )) n the frst row of each panel and the average rate of deprecaton (E T ( e j )) n the second row. 6 If there were no average rsk premum, these should be dentcal. Table 2 shows they are not. Investors earn large negatve excess returns on the frst portfolo because the low nterest rate currences n the frst portfolo deprecate on average by 34 bass ponts, whle the average foregn nterest rate s 2.46 percentage ponts lower than the US nterest rate. On the other hand, the hgher nterest rate currences n the seventh portfolo deprecate on average by almost 2.18 percentage ponts, but the average nterest rate dfference s on average 4.7 percentage ponts. The thrd row n each panel reports the nflaton rates. As mentoned, for the very hgh nterest rate currences n the last portfolo, much of the nterest rate gap reflects nflaton dfferences. Ths s not the case for low nterest rate portfolos. Our currency portfolos create a stable set of excess returns. In order to explan the varaton n these currency excess returns, we use consumptonbased prcng kernels. 6 j R t s the average nterest rate dfferental (1/N j ) (R, t R $ t ) for portfolo j at tme t. The average rsk premum s approxmately equal to the dfference between the frst and the second row. Ths approxmaton does not exactly lead to the excess return reported n Table 1, because Table 1 reports the real excess return (based on the real return on currency and the real US rskfree rate), and because of the log approxmaton. D. US Investor s Euler Equaton We turn now to a descrpton of US nvestor preferences. We use M t 1 to denote the US nvestor s real stochastc dscount factor (SDF) or ntertemporal margnal rate of substtuton, n the sense of Hansen and Rav Jagannathan (1991). Ths dscount factor prces payoffs n unts of US consumpton. In the absence of shortsale constrants or other frctons, the US nvestor s Euler equaton for foregn currency nvestments holds for each currency and thus for each portfolo j: j,e (1) E t M t 1 R t 1 0. Preferences. Our consumptonbased asset prcng model s derved n a standard representatve agent settng, followng Robert E. Lucas (1978) and Douglas T. Breeden (1979), and ts extenson to nonexpected utlty by Larry G. Epsten and Stanley E. Zn (1989) and to durable goods by Kenneth B. Dunn and Kenneth J. Sngleton (1986) and Martn Echenbaum and Hansen (1990). We adopt Motohro Yogo s (2006) setup whch convenently nests all these models. The standn household has preferences over nondurable consumpton C t and durable consumpton servces D t. Followng Yogo (2006), the standn household ranks stochastc streams of nondurable and durable consumpton {C t, D t }accordng to the followng utlty ndex: U t 1 u C t, D t 1 1/ 1 E t U t 1 1/ } 1/ 1 1/,
7 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 95 TABLE 3 NESTED MODELS Parameters CCAPM DCAPM EZCCAPM CAPM 1/ 1/ 3 Lnear factor model loadngs b c (1/ ) / 0 b d 0 (1/ 1/ ) 0 0 b m Notes: s the coeffcent of rsk averson, s the ntratemporal elastcty of substtuton between nondurables C and durables D consumpton, s the elastcty of ntertemporal substtuton, (1 )/(1 1/ ), and s the weght on durable comsumpton. where (1 )/(1 1/ ); s the subjectve tme dscount factor; 0 governs the household s rsk averson; and 0 s the elastcty of ntertemporal substtuton (EIS). The oneperod utlty kernel s gven by a CESfuncton over C and D: u C, D 1 C 1 1/ D 1 1/ 1/ 1 1/, where (0, 1) s the weght on durable consumpton, and 0 s the ntratemporal elastcty of substtuton between nondurables and durables. Yogo s (2006) model, whch we refer to as the EZ DCAPM, nests four famlar models. Table 3 lsts all of these. On the one hand, f we mpose 1/, the Durable ConsumptonCAPM (DCAPM) obtans, whle mposng produces the EpstenZn ConsumptonCAPM (EZCCAPM). When 1/ and, the standard BreedenLucas CCAPM obtans. As shown by Yogo (2006), the ntertemporal margnal rate of substtuton (IMRS) of the standn agent s gven by (2) M t 1 C t 1 C t 1/ v(d t 1/C t 1 ) 1/ 1/ v(d t /C t ) 1/ (R w t 1 ) 1, where R w s the return on the market portfolo and v s defned as v D/C 1 D 1 1/ C 1/ 1 1/. E. Calbraton We start off by feedng actual consumpton and return data nto a calbrated verson of our model, and we assess how much of the varaton n currency excess returns ths calbrated model can account for. To do so, we take Yogo s (2006) estmates of the substtuton elastctes and the durable consumpton weght n the utlty functon. 7 Next, we feed the data for C t, D t, and R w t, the market return, nto the SDF n equaton (2), and we smply evaluate the prcng errors E T [M t 1 R j,e t 1 ] for each portfolo j; was chosen to mnmze the mean squared prcng error on the eght currency portfolos. 8 Table 4 reports the mpled maxmum Sharpe rato (frst row), the market prce of rsk (row 2), the standard error (row 3), the mean absolute prcng error (MAE, n row 4), as well as the R 2. The benchmark model n the last column explans 65 percent of the crosssectonal varaton wth equal to 30. To understand ths result, t helps to decompose the model s predcted excess return on currency portfolo j n the prce of rsk and the rsk beta: j,e E T R t 1 j,e cov T M t 1, R t 1 var T M t 1 M j var T M t 1 E T M t 1 prce of rsk 7 We fx at 0.023, at 0.802, and at These parameters were estmated from a US nvestor s Euler equaton on a large number of equty portfolos (Yogo 2006, 552, table II, All Portfolos). 8 As a result of these hgh levels of rsk averson n a growng economy, our model cannot match the rskfree rate..
8 96 THE AMERICAN ECONOMIC REVIEW MARCH 2007 TABLE 4 CALIBRATED NONLINEAR MODEL TESTED ON EIGHT CURRENCY PORTFOLIOS SORTED ON INTEREST RATES CCAPM DCAPM EZCCAPM EZDCAPM std T [M]/E T [M] var T [M]/E T [M] MAE R Notes: Ths table reports the rsk prces and the measures of ft for a calbrated model on eght annually rebalanced currency portfolos. The sample s (annual data). The frst two rows report the maxmum Sharpe rato (row 1) and the prce of rsk (row 2). The last two rows report the mean absolute prcng error (n percentage ponts) and the R 2. Followng Yogo (2006), we fxed at (EZCCAPM and EZDCAPM), at (DCAPM and EZDCAPM), and at (DCAPM, EZDCAPM). s fxed at to mnmze the mean squared prcng error n the EZDCAPM. s set to There s a large dfference n rsk exposure between the frst and the seventh portfolos: M 1 s 2.54, whle M 7 s When multpled by the prce of rsk of 28 bass ponts, ths translates nto a 3percentagepont spread n the predcted excess return between the frst and the seventh portfolo, about 65 percent of the actual spread. The low nterest rate portfolo provdes the US nvestor wth protecton aganst hgh margnal utlty growth, or hgh M, states of the world, whle the hgh nterest rate portfolos do not. Ths varaton n betas s the focus of the next secton. II. Does Consumpton Rsk Explan Foregn Currency Excess Returns? So far, we have engneered a large crosssectonal spread n currency excess returns by sortng currences nto portfolos, and we have shown that a calbrated verson of the model explans a large fracton of ths spread. In ths secton, startng from the Euler equaton and followng Yogo (2006), we derve a lnear factor model whose factors are nondurable US consumpton growth c t, durable US consumpton growth d t, and the log of the US market return r t m. Usng standard lnear regresson methods, we show that US consumpton rsk explans most of the varaton n average excess returns across the eght currency portfolos, because, on average, low nterest rate currences expose US nvestors to less nondurable and durable consumpton rsk than hgh nterest rate currences. We start by dervng the factor model, then we descrbe the estmaton method, and we present our results n terms of ft, factor prces, and preference parameters. A. Lnear Factor Model The US nvestor s uncondtonal Euler equaton (approxmately) mples a lnear threefactor model for the expected excess return on portfolo j: 9 (3) E R j,e b 1 cov c t, R t j,e b 2 cov d t, R j,e t b 3 cov r w j,e t, R t 1. The vector of factor loadngs b depends on the preference parameters,, and : [1/ (1/ 1/ )] (4) b (1/ 1/ ). 1 The expected excess return on portfolo j s governed by the covarance of ts returns wth nondurable consumpton growth, durable consumpton growth, and the market return. When b 1 0 (the case that obtans when 1 and 9 Ths lnear factor model s derved by usng a lnear approxmaton of the SDF M t 1 around ts uncondtonal mean: M t 1 E M t 1 1 m t 1 E m t 1, where lower letters denote logs. Snce we use excess returns, we normalze the constant n the SDF to one, because we cannot dentfy t from the estmaton.
9 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 97 1), then an asset wth hgh nondurable consumpton growth beta must have a hgh expected excess return. Ths turns out to be the emprcally relevant case. When the ntratemporal elastcty of substtuton s larger than the EIS, b 2 0 obtans. In ths case, an asset wth a hgh durable consumpton growth beta also has a hgh expected excess return. In ths range of the parameter space, nondurables and durable goods are substtutes and, as a result, hgh durable consumpton can offset the effect of low nondurable consumpton on margnal utlty. Our benchmark asset prcng model, denoted EZDCAPM, s descrbed by equaton (3). Ths specfcaton, however, nests the CCAPM wth c t as the only factor, the DCAPM wth c t and d t as factors, the EZCCAPM wth c t and r t w, and, fnally the CAPM as specal cases, as shown n the bottom panel of Table 3. Beta Representaton. Ths lnear factor model can be restated as a beta prcng model, where the expected excess return s equal to the factor prce tmes the amount of rsk of each portfolo j : (5) E R j,e j, where ff b, and ff E(f t f )(f t f ) s the varancecovarance matrx of the factors. A Smple Example. A smple example wll help n understandng what s needed for consumpton growth rsk to explan the cross secton of currency returns. Let us start wth the planvanlla CCAPM. The only asset prcng factor s aggregate, nondurable consumpton growth, c t 1, and the factor loadng b 1 equals the coeffcent of rsk averson. We can restate the expected excess return on portfolo j as the j product of the portfolo beta c [cov( c t, R j,e t )/var( c t )] and the factor prce c b 1 var( c t ): (6) E R j,e t cov c t, R j,e t b var c t 1 var c t c j c, j The factor prce measures the expected excess return on an asset that has a consumpton growth beta of one. Of course, the CCAPM can explan the varaton n returns only f the consumpton betas are small/negatve for low nterest rate portfolos and large/postve for hgh nterest rate portfolos. Essentally, n testng the CCAPM, we gauge how much of the varaton n average returns across currency portfolos can be explaned by varaton n the consumpton betas. If the predcted excess returns the rghthandsde varable n equaton (5) lne up wth the realzed sample means, we can clam success n explanng exchange rate changes, condtonal on whether the currency s a low or hgh nterest rate currency. A key queston, then, s whether there s enough varaton n the consumpton betas of these currency portfolos to explan the varaton n excess returns wth a plausble prce of consumpton rsk. The next secton provdes a postve answer to ths queston. B. An Asset Prcng Experment To estmate the factor prces and the portfolo betas, we use a twostage procedure followng Fama and James D. MacBeth (1973). 10 In the frst stage, for each portfolo j, we run a tmeseres regresson of the currency returns j,e R t 1 on a constant and the factors f t, n order to estmate j. In the second stage, we run a crosssectonal regresson of the average excess returns E T [R e t ] on the betas that were estmated n the frst stage, to estmate the factor prces. Fnally, we can back out the factor loadngs b and hence the structural parameters from the factor prces. We start by testng the consumptonbased US nvestor s Euler equaton on the eght annually rebalanced currency portfolos. Table 5 reports the estmated factor prces of consumpton growth rsk for nondurables (row 1), durables (row 2), and the prce of market rsk (row 3). Each column looks at a dfferent model. We also report the mpled estmates for the preference parameters,, and (rows 4 to 6). The 10 Chapter 12 of John H. Cochrane (2001) descrbes ths estmaton procedure and compares t to the generalzed method of moments (GMM) appled to lnear factor models, followng Hansen (1982). We present results obtaned wth GMM as a robustness check n Secton IV.
10 98 THE AMERICAN ECONOMIC REVIEW MARCH 2007 TABLE 5 ESTIMATION OF LINEAR FACTOR MODELS WITH EIGHT CURRENCY PORTFOLIOS SORTED ON INTEREST RATES CCAPM DCAPM EZCCAPM EZDCAPM Factor prces Nondurables [0.917] [0.915] [0.845] [0.830] Durables [0.987] [0.968] Market [7.916] [7.586] Parameters [6.158] [6.236] [5.440] [5.558] [0.003] [0.056] [0.048] [0.001] Stats MAE R p value [0.025] [0.735] [0.024] [0.628] Notes: Ths table reports the FamaMacBeth estmates of the rsk prces (n percentage ponts) usng eght annually rebalanced currency portfolos as test assets. The sample s (annual data). The factors are demeaned. The standard errors are reported between brackets. The last three rows report the mean absolute prcng error (n percentage ponts), the R 2 and the pvalue for a 2 test. standard errors are n parentheses. 11 Fnally, the last three rows report the mean absolute prcng error (MAE), the R 2, and the pvalue for a 2 test. The null for the 2 test s that the true prcng errors are zero and the pvalue reports the probablty that these prcng errors would have been observed f the consumptonbased model were the true model. C. Results We present results n terms of the factor prces, the ft, the preference parameters, and the consumpton betas. 11 These standard errors do not correct for the fact that the betas are estmated. Jagannathan and Zhenyu Wang (1998) show that the FamaMacBeth procedure does not necessarly overstate the precson of the standard errors f condtonal heteroskedastcty s present. We show n Secton IVE that these standard errors are actually close to the heteroskedastctyconsstent ones derved from GMM estmates. Factor Prces. In our benchmark model (EZDCAPM), reported n the last column of Table 5, the estmated prce of nondurable consumpton growth rsk c s postve and statstcally sgnfcant. An asset wth a consumpton growth beta of one yelds an average rsk premum of around 2 percent per annum. Ths s a large number, but t s qute close to the market prce of consumpton growth rsk estmated on US equty and bond portfolos (see Secton IVC). The estmated prce of durable consumpton growth rsk d s postve and statstcally sgnfcant as well, around 4.6 percent. These factor prce estmates do not vary much across the dfferent models. Fnally, market rsk s prced at about 3.3 percent per annum, but t s not sgnfcantly dfferent from zero. Model Ft. We fnd that consumpton growth rsk explans a large share of the crosssectonal varaton n currency returns. The EZ DCAPM explans 87 percent of the crosssectonal varaton n annual returns on the 8
11 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 99 3 CCAPM 7 3 DCAPM EZ CCAPM 7 3 EZ DCAPM FIGURE 2. CONSUMPTIONCAPM Notes: Ths fgure plots the actual versus the predcted excess returns for eght currency portfolos. The predcted excess returns are on the horzontal axs. The FamaMacBeth estmates are obtaned usng eght currency portfolos sorted on nterest rates as test assets. The flled dots (1 8) represent the currency portfolos. The data are annual and the sample s currency portfolos, aganst 74 percent for the DCAPM and 18 percent for the smple CCAPM. For the EZDCAPM, the mean absolute prcng error on these 8 currency portfolos s about 32 bass ponts over the entre sample, compared to 65 bass ponts for the DCAPM, and 200 bass ponts for the smple CCAPM. Ths last number s rather hgh, manly because of the last portfolo, wth very hgh nterest rate currences. When we drop the last portfolo, the mean absolute prcng error on the remanng 7 portfolos drops to 109 bass ponts for the smple CCAPM, and the R 2 ncreases to 50 percent. The smple CCAPM and the EZCCAPM are rejected at the 5percentsgnfcance level, but the DCAPM and the EZDCAPM are not. Durable consumpton rsk plays a key role here as the models wth durable consumpton growth produce very small prcng errors (less than 15 bass ponts) on the frst and the seventh portfolo. Ths s clear from Fgure 2, whch plots the actual excess return aganst the predcted excess return (on the horzontal axs) for each of these models. Preference Parameters and Equty Premum Puzzle. From the factor prces, we can back out the preference parameters. The ntratemporal elastcty of substtuton between nondurables and durables cannot be separately dentfed from the weght on durable consumpton. We use Yogo s (2006) estmate of to calbrate the elastcty of ntratemporal substtuton when we back out the other preference parameter estmates. The EIS s estmated to be 0.2, substantally larger than 1/, and the weght on durable consumpton s estmated to be around 1.1, close to the 0.9 estmate reported by Yogo (2006), obtaned on quarterly equty portfolos. Snce the EIS estmate s sgnfcantly smaller than the calbrated, margnal utlty growth decreases n durable
12 100 THE AMERICAN ECONOMIC REVIEW MARCH 2007 TABLE 6 ESTIMATION OF FACTOR BETAS FOR EIGHT CURRENCY PORTFOLIOS SORTED ON INTEREST RATES Portfolos Panel A: Nondurables Durables * Market 0.066* * Panel B: Nondurables Durables * 2.032* 1.225* * Market 0.106* 0.099* * Notes: Each column of ths table reports OLS estmates of j n the followng tmeseres regresson of excess returns on the j,e j factor for each portfolo j: R t 1 0 j j 1 f t t 1. The estmates are based on annual data. Panel A reports results for and Panel B reports results for We use eght annually rebalanced currency portfolos sorted on nterest rates as test assets. * ndcates sgnfcance at 5percent level. We use NeweyWest heteroskedastctyconsstent standard errors wth an optmal number of lags to estmate the spectral densty matrx followng Donald W. K. Andrews (1991). consumpton growth, and assets whose returns covary more wth durable consumpton growth trade at a dscount (b 2 0). In the benchmark model, the mpled coeffcent of rsk averson s around 114 and ths estmate s qute precse. In addton, these estmates do not vary much across the four dfferent specfcatons of the consumptonbased prcng kernel. Ths coeffcent of rsk averson s of course very hgh, but t s n lne wth stockbased estmates of the coeffcent of rsk averson found n the lterature, and wth our own estmates based on bond and stock returns. For example, f we reestmate the model only on the 25 FamaFrench equty portfolos, sorted on sze and booktomarket, the rsk averson estmate s 115. In addton, the lnear approxmaton we adopted causes an underestmaton of the market prce of consumpton rsk for a gven rsk averson parameter. These hgh estmates are not surprsng. The standard devaton of US consumpton growth (per annum) s only 1.5 percent n our sample. Ths s Rajnsh Mehra and Edward C. Prescott s (1985) equty premum puzzle n dsguse: there s not enough aggregate consumpton growth rsk n the data to explan the level of rsk compensaton n currency markets at low levels of rsk averson, as s the case n equty markets, but there s enough varaton across portfolos n consumpton betas to explan the spread, f the rsk averson s large enough to match the levels. We now focus on ths cross secton of consumpton betas. Consumpton Betas. Consumptonbased models can account for the cross secton of currency excess returns because they mply a large cross secton of betas. On average, hgher nterest rate portfolos expose US nvestors to much more US consumpton growth rsk. Table 6 reports the OLS betas for each of the factors. Panel A reports the results for the entre sample. We fnd that hgh nterest rate currency returns are strongly procyclcal, whle low nterest rate currency returns are acyclcal. For nondurables, the frst portfolo s consumpton beta s 10 bass ponts, and the seventh portfolo s consumpton beta s 110 bass ponts. For durables, the spread s also about 100 bass ponts, from 24 bass ponts to 129 bass ponts. In the second post Bretton Woods subsample, reported n Panel B, the spread n consumpton betas ncreases to 150 bass ponts between the frst and the seventh portfolo (wth betas rangng from zero bass ponts to 154 bass ponts for nondurables, and from 50 to 210 bass ponts for durables). Fnally, the market betas of currency returns are much smaller overall. Next, we estmate the condtonal factor betas, condtonng on the nterest rate gap wth the Unted States, and we fnd that low nterest rate currences provde a consumpton hedge for US nvestors exactly when US nterest rates are hgh and foregn nterest rates are low. D. Condtonal Factor Betas We can go one step further n our understandng of exchange rates by takng nto account the
13 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 101 TABLE 7 ESTIMATION OF CONDITIONAL CONSUMPTION BETAS FOR CHANGES IN EXCHANGE RATES ON CURRENCY PORTFOLIOS SORTED ON INTEREST RATES j,c 1 j,c 2 j,d 1 j,d 2 j,m 1 j,m Panel A. Nondurables [0.73] [1.20] [1.28] [1.99] [0.91] [1.00] [0.75] [0.90] [0.10] [0.19] [0.17] [0.30] [0.17] [0.14] [0.07] [0.03] Panel B. Durables [1.01] [1.47] [1.39] [1.44] [0.92] [0.67] [0.51] [0.53] [0.10] [0.17] [0.17] [0.19] [0.14] [0.08] [0.06] [0.01] Panel C. Market [0.13] [0.19] [0.14] [0.24] [0.10] [0.09] [0.06] [0.08] [0.02] [0.02] [0.02] [0.03] [0.02] [0.01] [0.01] [0.00] Notes: Each column of ths table reports OLS estmates of j,k n the followng tmeseres regresson of nnovatons to returns j for each portfolo j ( t 1 ) on the factor f k j and the nterest rate dfference nteracted wth the factor: t 1 j,k 0 j,k k 1 f t 1 j,k 2 R tj k f t 1 j,k t 1. We normalzed the nterest rate dfference R tj to be zero when the nterest rate dfference R j t s at a j mnmum and hence postve n the entre sample. t 1 are the resduals from the tme seres regresson of changes n the j exchange rate on the nterest rate dfference (UIP regresson): E t 1 /E j j t 0 j 1 R j j t t 1. The estmates are based on annual data and the sample s We use eght annually rebalanced currency portfolos sorted on nterest rates as test assets. The prcng factors are consumpton growth rates n nondurables (c) and durables (d) and the market return (w). The NeweyWest heteroskedastctyconsstent standard errors computed wth an optmal number of lags to estmate the spectral densty matrx followng Andrews (1991) are reported n brackets. tme varaton n the condtonal consumpton growth betas. 12 It turns out that low nterest rate currences offer a consumpton hedge to US nvestors exactly when the US nterest rates are hgh and foregn nterest rates are low. To see ths, we consder a smple twostep procedure. j We frst obtan the UIP resduals t 1 for each portfolo j. We then regress each resdual on each factor f k, controllng for the nterest rates varatons n each portfolo: j t 1 j,k 0 j,k k 1 f t 1 j,k 2 R tj k f t 1 j,k t 1, where for expostonal purpose we ntroduce the normalzed nterest rate dfference R tj, whch 12 There s a condtonal analogue of the threefactor model n equaton (3): E t R,e,e b 1 cov t c t 1, R t 1,e b 2 cov t d t 1, R t 1 w b 3 cov t r t 1,e, R t 1. Snce the nterest rate s known at t, these covarances terms nvolve only the changes n the exchange rate e t 1. s zero when the nterest rate dfference R t s at a mnmum, and hence postve n the entre sample. We use the nterest rate dfferental as the sole condtonng varable, because we know from the work by Rchard A. Meese and Kenneth Rogoff (1983) that our ablty to predct exchange rates s rather lmted. The results are reported n Table 7. Each bar n Fgure 3 reports the condtonal factor betas for a dfferent portfolo. The frst panel reports the nondurable consumpton betas, the second panel the durable consumpton betas, and the thrd panel the market betas. When the nterest rate dfference wth the Unted States hts the lowest pont, the currences n the frst portfolo apprecate, on average, by 287 bass ponts when US nondurable consumpton growth drops 100 bass ponts below ts mean, whle the currences n the seventh portfolo deprecate, on average, by 96 bass ponts. Smlarly, when US durable consumpton growth drops 100 bass ponts below ts mean, the currences n the frst portfolo apprecate by 174 bass ponts, whle the currences n the seventh portfolo deprecate by 105 bass ponts. Low nterest rate
14 102 THE AMERICAN ECONOMIC REVIEW MARCH Nondurables 0.4 Nondurables*nterest gap Durables 0.4 Durables*nterest gap Market 0.02 Market*nterest gap FIGURE 3. CONDITIONAL FACTOR BETAS OF CURRENCY Notes: Each panel shows OLS estmates of j,k 1 (panels on the left) and j,k 2 (panels on the rght) n the followng tmeseres regresson of nnovatons to changes n exchange rate for each j portfolo j on the factor and the nterest rate dfference nteracted wth the factor: t 1 j,k 0 j,k k 1 f t 1 j,k 2 R tj k f t 1 j,k t 1. R j s the normalzed nterest rate dfference on portfolo j. The data are annual and the sample s currences provde consumpton nsurance to US nvestors, whle hgh nterest rate currences expose US nvestors to more consumpton rsk. As the nterest rate gap closes on the currences n the frst portfolo, the low nterest rate currences provde less consumpton nsurance. For every 4percentagepont reducton n the nterest rate gap, the nondurable consumpton betas decrease by about 100 bass ponts. 13 Interest Rates as Instruments. To test whether the representatve agent s ntertemporal margnal rate of substtuton (IMRS) can ndeed explan the tme varaton n expected returns on these portfolos, n addton to the crosssectonal varaton, we use the average nterest rate dfference wth the Unted States as an nstrument. As s clear from 13 Ths table also shows that our asset prcng results are entrely drven by how exchange rates respond to consumpton growth shocks n the Unted States, not by soveregn rsk. the uncondtonal Euler equaton, ths s equvalent to testng the uncondtonal moments of managed portfolo returns:,e (7) E M t 1 R tr t 1 0, where R t s the average nterest rate dfference on portfolos 1 7 and ( R tr,e t 1 ) are the managed portfolo returns. We normalzed R t to be postve. 14 Instead of the varaton n average portfolo returns, we check whether the model explans the crosssectonal varaton n average excess returns on managed portfolos that lever up when the nterest rate gap wth the Unted States s large. In addton, we also use the nterest rate dfference for each portfolo as an nstrument for that asset s Euler equaton. Table 8 reports the FamaMacBeth estmates of the factor prces and preference parameters for our benchmark model. In the frst column, 14 We add mn( R t ) to the nterest rate dfferental.
15 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 103 TABLE 8 ESTIMATION OF LINEAR FACTOR MODELS WITH EIGHT MANAGED CURRENCY PORTFOLIOS SORTED ON INTEREST RATES Factor Prce Average R Portfolo R Nondurables [0.757] [0.830] Durables [0.974] [1.150] Market [9.012] [9.008] Parameters [5.069] [5.616] [0.018] [0.009] [0.022] [0.030] Stats R p value Notes: Ths table reports the FamaMacBeth estmates of the factor prces (n percentage ponts) for the EZDCAPM usng eght annually rebalanced managed currency portfolos as test assets. The sample s (annual data). In column 1, we use the average nterest rate dfference wth the US on portfolos 1 7 as an nstrument. In column 2, we use the nterest rate dfference on portfolo as the nstrument for the th moment. The standard errors are reported between brackets. The factors are demeaned. The last two rows report the R 2 and the pvalue for a 2 test. we use the average nterest rate dfference wth the Unted States as an nstrument. In the second column, we use the nterest rate dfference for portfolo as an nstrument for the th moment. The consumpton rsk prce estmates are very close to those we obtaned off the uncondtonal moments of currency returns, and, more mportantly, the benchmark model cannot be rejected n ether case. Consumptonbased models do a remarkable job n explanng the crosssectonal varaton as well as the tme varaton n returns, albet at the cost of a very hgh mpled prce of aggregate consumpton rsk. In Secton IV, we contrast ths model s performance wth that of the workhorse of modern fnance, the Captal Asset Prcng Model. As we show, there s not enough varaton n market betas to explan currency returns, but there s enough varaton n consumpton betas. We conclude that consumpton growth rsk seems to play a key role n explanng currency rsk prema. The next secton lnks our fndngs about rsk prema back to changes n the exchange rates. III. Mechansm We have shown that predcted currency excess returns lne up wth realzed ones when prcng factors take nto account consumpton growth rsk. Ths s not mere luck on our part. The next secton provdes many robustness checks. Ths secton sheds some lght on the underlyng mechansm: where do these currency betas come from? We frst show that the log of the condtonal expected return on foregn currency can be restated n terms of the condtonal consumpton growth betas of exchange rate changes. We then nterpret these betas as restrctons on the jont dstrbuton of consumpton growth n hgh and low nterest rate currences. A. Consumpton Growth Betas of Exchange Rates If we assume that M t 1 and R t 1 are jontly, condtonally lognormal, then the Euler equaton can be restated n terms of the real currency rsk premum (see proof n Appendx B): log E t R t 1 r f t cov t m t 1, r t 1 p t 1, where lower cases denote logs. We refer to ths log currency premum as crp t 1. It s determned by the covarance between the log of the SDF m and the real return on nvestment n the foregn TBll. Substtutng the defnton of ths return nto ths equaton produces the followng expresson for the log currency rsk premum: (8) crp t 1 cov t m t 1, e t 1 p t 1. Note that the nterest rates play no role for condtonal rsk prema; only changes n the deflated exchange rate matter. Usng ths expresson, we examne what restrctons are mpled on the jont dstrbuton of consumpton growth and exchange rates by the ncreasng pattern of currency rsk prema n nterest rates, and we test these restrctons n the data. Consumpton Growth and Exchange Rates. From our lnear factor model, t mmedately follows that the log currency rsk premum can
16 104 THE AMERICAN ECONOMIC REVIEW MARCH 2007 be restated n terms of the condtonal factor betas: crp t 1 b 1 cov t c t 1, e t 1 p t 1 b 2 cov t d t 1, e t 1 p t 1 b 3 cov t r m t 1, e t 1 p t 1. Ths equaton uncovers the key mechansm that explans the forward premum puzzle. We recall that, n the data, the rsk premum (crp t 1 )s postvely correlated wth foregn nterest rates R, t : low nterest rate currences earn negatve rsk prema and hgh nterest rate currences earn postve rsk prema. To match these facts, n the smplest case of the CCAPM, the followng necessary condton needs to be satsfed by the condtonal consumpton covarances: cov t c t 1, e t 1 cov t c t 1, e t 1 small/negatve when R t, s low, large/postve when R t, s hgh. The same condton apples to durable consumpton growth d t 1 and the market return r w t 1 n our benchmark, threefactor model. Ths s exactly what we see n the consumpton betas of currency, reported n Fgure 3. Both n the tme seres (comparng the bar n the left panels and the rght panels) and n the cross secton (gong from portfolo 1 to 7), low foregn nterest rates mean small/negatve consumpton betas. On the one hand, currences that apprecate on average when US consumpton growth s hgh and deprecate when US consumpton growth s low earn postve condtonal rsk prema. On the other hand, currences that apprecate when US consumpton growth s low and deprecate when t s hgh earn negatve rsk prema. These currences provde a hedge for US nvestors. Gven the pattern of excess return varaton across dfferent currency portfolos, the covarance of changes n the exchange rate wth US consumpton growth term needs to swtch sgns over tme for a gven currency, dependng on the portfolo t has been allocated to (or, ts nterest rate). There s a substantal amount of tme varaton n the consumpton betas of currences. Ths reflects the tme varaton n nterest rates and expected returns wthn each portfolo over tme. Yet, most of our results can be understood n terms of the average consumpton betas: on average, hgh nterest rate currences expose US nvestors to more consumpton growth rsk, whle low nterest rate currences provde a hedge. The next subsecton explans where these betas come from and why they are correlated wth nterest rates. B. Where Do Consumpton Betas of Currences Come From? The answer s tme varaton n the condtonal dstrbuton of the foregn stochastc dscount factor m. Investng n foregn currency s lke bettng on the dfference between your own and your neghbor s IMRS. These bets are very rsky f your IMRS s not correlated wth that of your neghbor, but they provde a hedge when her IMRS s hghly correlated and more volatle. We dentfy two potental mechansms to explan the consumpton betas of currences. Low foregn nterest rates sgnal ether (a) an ncrease n the volatlty of the foregn stochastc dscount factors; or (b) an ncrease n the correlaton of the foregn stochastc dscount factor wth the domestc one. To obtan these results, we assume that markets are complete and that the SDF are lognormal. Essentally, we renterpret an exstng dervaton by Davd Backus, Slvero Fores, and Chrs Telmer (2001), and we explore ts emprcal mplcatons. Currency Rsk Prema and the SDF. In the case of complete markets, nvestng n foregn currency amounts to shortng a clam that pays off your SDF and gong long n a clam that pays off the foregn SDF. The net payoff of ths bet depends on the correlaton and volatlty of these SDFs. Assumng that the nflaton betas are small enough and that markets are complete, the sze of the log currency rsk premum crp t 1 s gven by See Appendx B for a proof.
17 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 105 (9) std t m t 1 std t m t 1 corr t m t 1, m t 1 std t m t 1. Its sgn s determned by the standard devaton of the home SDF relatve to the one of the foregn SDF scaled by the correlaton between the two SDFs. What does ths equaton mply? Obvously, ether a hgher condtonal volatlty of the foregn SDF or a hgher correlaton of the SDFs n the case of lower nterest rate currences and the reverse for hgh nterest rates would generate the rght pattern n rsk prema. Example. In the case of the smple CCAPM, these two mechansms can be stated n terms of the jont dstrbuton of consumpton growth at home and abroad. Assume that the standn agents n both countres share the same coeffcent of relatve rsk averson. Then, abstractng agan from the nflaton betas, the sgn of the condtonal rsk premum s determned by std t c US t 1 corr t c US t 1, c t 1 std t c t 1. A low correlaton of foregn consumpton growth wth US consumpton growth for hgh nterest rate currences, and a hgh correlaton for low nterest rate currences, creates the rght varaton n currency rsk prema. More volatle consumpton growth for low nterest rate currences also delvers ths pattern. What s the economc ntuton behnd ths mechansm? In our benchmark representatve agent model wth complete markets, the foregn currency apprecates when foregn consumpton growth s lower than US aggregate consumpton growth and deprecates when t s hgher. When markets are complete, the value of a dollar delvered tomorrow n each state of the world, n terms of dollars today, equals the value of a unt of foregn currency tomorrow delvered n the same state, n unts of currency today: Q t 1 /Q t M t 1 /M t 1, where the exchange rate Q t s n unts of the US good per unt of the foregn good. Thus, n the case of a CRRA representatve agent n the Unted States, the percentage change n the real exchange rate equals the percentage change n consumpton growth tmes the coeffcent of relatve rsk averson: q t 1 ( c t 1 c t 1 ). If the foregn standn agent s consumpton growth s strongly correlated wth and more volatle than that of hs US counterpart, hs natonal currency provdes a hedge for the US representatve agent. For example, consder the case n whch foregn consumpton growth s twce as volatle as US consumpton growth and perfectly correlated wth US consumpton growth. In ths case, when consumpton growth s 2 percent below the mean n the Unted States, t s 4 percent below the mean abroad, and the real exchange rate apprecates by tmes 2 percent. When consumpton growth s 2 percent n the Unted States, t s twce as hgh abroad (4 percent), and the real exchange rate deprecates by tmes 2 percent. Ths currency s a perfect hedge aganst US aggregate consumpton growth rsk. Consequently, nvestng n ths currency should provde a low excess return. Thus, for ths heteroskedastcty mechansm to explan the pattern n currency excess returns, low nterest rate currences must have aggregate consumpton growth processes that are condtonally more volatle than US aggregate consumpton growth. Ths s n lne wth the theory. All else equal, n the case of power utlty, an ncrease n the condtonal volatlty of aggregate consumpton growth lowers the real nterest rate. 16 If real and nomnal nterest rates move n synchronzaton, a low nomnal nterest rate should predct a hgher condtonal volatlty of aggregate consumpton growth. Of course, f nflaton s very hgh and volatle, the nomnal and the real nterest rates effectvely are detached, and ths mechansm would dsappear, as t seems to n the data. Tme varaton n the correlaton between the domestc and the foregn SDF s the second mechansm. In the prevous example, f the consumpton growth of a hgh nterest rate country s perfectly negatvely correlated wth US consumpton growth, then a negatve consumpton shock of 2 percent n the Unted States leads to a deprecaton of the foregn currency by tmes 2 percent. Ths currency deprecates when US consumpton growth s low. Consequently, nvestng n ths currency should provde a hgh 16 Ths can be shown by startng from the Euler defnton of the real rskfree rate and by assumng that aggregate consumpton growth s lognormal.
18 106 THE AMERICAN ECONOMIC REVIEW MARCH 2007 TABLE 9 CONSUMPTION GROWTH REGRESSIONS Country R 2 AUS CAN FR GER ITA JAP NE SWE SWI UK Pooled Notes: Ths table reports the results for the followng regresson: c t 1 2 (R, t R $ US t ) c t c US t 1 t 1. The last row reports the results from a pooled tme seres regresson. The sample s and the data are annual (for the Netherlands the sample s and for Swtzerland ). We used the optmal lag length to estmate the spectral densty matrx (Andrews 1991). 2 and 2 are, respectvely, one standard error below and above the pont estmate 2. excess return. Thus, for ths correlaton mechansm to explan the pattern n currency excess returns, the correlaton between domestc and foregn consumpton growth should decrease wth the nterest rate dfferental. Emprcally, we fnd strong evdence to support that mechansm: foregn consumpton growth s less correlated wth US consumpton growth when the foregn nterest rate s hgh. Evdence. The heteroskedastcty mechansm s also at the heart of the habtbased model of the exchange rate rsk premum n Verdelhan (2005). In hs model, the domestc nvestor receves a postve exchange rate rsk premum n tmes when he s more rsk averse than hs foregn counterpart. Tmes of hgh rsk averson correspond to low nterest rates. Thus, the domestc nvestor receves a postve rsk premum when nterest rates are lower at home than abroad. Test of the Correlaton Mechansm. In addton, we document some drect evdence n the data for the correlaton mechansm. For data reasons, we focus on nondurable consumpton growth only. Usng a sample of ten developed countres, we regressed a country s nondurable consumpton growth on US nondurable consumpton growth and US consumpton growth nteracted wth the lagged nterest rate dfferental: c t 1 US 0 1 c t 1 2 R, t R $ US t c t 1 t 1. The results obtaned over the post Bretton Woods perod on annual data are reported n Table 9. The coeffcents on the nteracton terms 2 are negatve for all countres, except for Japan. The table also reports 90percent confdence ntervals for these nteracton coeffcents. They show that the 2 coeffcents are sgnfcantly negatve for seven countres. The last row of each panel reports the pooled tme seres regresson results. The 90percent confdence nterval ncludes only negatve coeffcents. As s clear from the 2 estmates n column 3, the condtonal correlaton between foregn and US annual consumpton growth decreases wth the nterest rate gap for all countres except Japan. We also found the same pattern for Japanese and UK consumpton growth processes (not reported). IV. Robustness Ths secton goes through a number of robustness checks: (a) we look at other factor models, (b) we splt up the sample, (c) we ntroduce other test assets, (d) we reestmate the model on developed currency portfolos, and (e) we reestmate the model usng the GMM.
19 VOL. 97 NO. 1 LUSTIG AND VERDELHAN: CURRENCY RISK PREMIA 107 A. Factor Models The Captal Asset Prcng Model (CAPM), from Wllam Sharpe (1964) and Jack Treynor (1961), s a useful benchmark. In ths model, the excess return on the US total market portfolo s the only asset prcng factor. We use the Center for Research n Securty Prces (CRSP) valueweghted excess return, denoted R w, as a proxy for the market return: (10) M t 1 E M t 1 1 b w wr t 1. Of course, the same decomposton of the rsk premum n market prce of rsk ( w ) and betas ( w ) apples here. The model mples that the market prce of rsk w equals the expected excess return on the market, because the market has a beta of one. In addton, we consder the bond and equty factor models developed by Fama and French (1992). Fama and French add the return on a portfolo that goes long n small and short n bg frms (R SMB t 1 ) and the return on a portfolo that goes long n hgh booktomarket and short n low booktomarket stocks (R HML t 1 ) as addtonal equty prcng factors. 17 For bond prcng, they use the slope of the yeld curve (R long t 1 ) and the default spread on corporate bonds (R corp t 1 ). These factors proxy for the underlyng undversfable macroeconomc rsk (Fama and French 1993). Table 10 lsts the results for the CAPM and the FamaFrench factor models. We start wth the CAPM n the frst column. The prce of market rsk w s estmated to be around 7 percent. Ths number s n lne wth the theory, whch prescrbes a market prce of rsk of 7 percent, the average excess return on the market. However, the CAPM explans only 4 percent of the varaton n returns over the entre sample. Introducng the FamaFrench bond and equty factors does not mprove the prcng much. The FamaFrench equty factors explan 8 percent, whle the bond factors explan 20 percent. The mean absolute prcng error does not drop below 200 bass ponts for any of these models, compared to 32 bass ponts for the EZDCAPM. The prcng errors for the frst and 17 SMB means smallmnusbg and HML means hghmnuslow. TABLE 10 ESTIMATION OF LINEAR FACTOR MODELS WITH EIGHT CURRENCY PORTFOLIOS SORTED ON INTEREST RATES Factor prce CAPM FFequty FFbonds Market [9.873] [10.569] SMB [5.782] HML [6.892] slope [6.446] default [3.170] Stats MAE R p value [0.000] [0.000] [0.000] Notes: Ths table reports the FamaMacBeth estmates of the factor prces (n percentage ponts) usng eght annually rebalanced currency portfolos as test assets. The sample s (annual data). The standard errors are reported between brackets. The last three rows report the mean absolute prcng error (n percentage ponts), the R 2, and the pvalue for a 2 test. the seventh portfolo are large, n excess of 100 bass ponts, n all three models. The factor models, whch work n equty and bond markets, break down n currency markets. Clearly, the currency excess returns are not spanned by FamaFrench equty or bond factors. Ths makes currency portfolos partcularly useful as test assets. Kent Danel and Sherdan Ttman (2005) argue that even factors that are loosely correlated wth HML and SMB wll appear successful n explanng the cross secton of asset returns, but our currency returns are not correlated wth these. In fact, our currency portfolos are outofsample test assets, as advocated by Jonathan Lewellen, Stefan Nagel, and Jay Shanken (2006). B. Post Bretton Woods Whle the same nvestor Euler equaton apples to fxed and floatng regmes, the jont dstrbuton of consumpton growth and foregn currency returns s affected by a change n the exchange rate regme, and ths may affect the estmaton. To address ths, we splt the sample. ConsumptonCAPM. The results for the subsample are reported n Ta
20 108 THE AMERICAN ECONOMIC REVIEW MARCH 2007 TABLE 11 ESTIMATION OF LINEAR FACTOR MODELS WITH EIGHT CURRENCY PORTFOLIOS SORTED ON INTEREST RATES Panel A. Consumpton models CCAPM DCAPM EZCCAPM EZDCAPM Nondurables [1.087] [1.095] [0.914] [0.914] Durables [0.959] [0.905] Market [7.804] [7.259] MAE R p value [0.312] [0.535] [0.222] [0.479] Panel B. Factor models CAPM FFequty FFbonds Market [8.443] [8.684] SMB [5.188] HML [5.965] Slope [9.628] Default [2.393] Stats MAE R p value [0.001] [0.001] [0.001] Notes: Ths table reports the FamaMacBeth estmates of the factor prces (n percentage ponts) usng eght annually rebalanced currency portfolos as test assets. The sample s (annual data). The standard errors are reported between brackets. The factors are demeaned. The last three rows report the mean absolute prcng error (n percentage ponts), the R 2, and the pvalue for a 2 test. ble 11. Panel A reports the Consumpton CAPM estmates, and Panel B reports the factor model estmates. The estmated prce of consumpton growth rsk s 2.4 percent n the benchmark model, and t s stll sgnfcant, whle the prce of durable consumpton growth rsk s around 3 percent. The mpled coeffcent of rsk averson s 98, close to our earler estmate of 114. Our benchmark model, the EZ DCAPM, explans 65 percent of the varaton over ths subsample, and the mean absolute prcng error ncreases to 128 bass ponts, substantally hgher than the number for the entre sample. Even though all four models pass the 2 test, only the models wth durable consumpton growth as a factor explan a large fracton of the crosssectonal varaton n returns. Factor Models. The results for the factor models are shown n the second panel of Table 11. In ths subsample, the CAPM explans none of the varaton, and the FamaFrench factor models explan less than 18 percent of the varaton n returns. The mean absolute prcng error does not decrease below 290 bass ponts. The prce of market rsk s not sgnfcantly dfferent from zero n any of the models. None of these factor models passes the 2 test. C. Other Test Assets As an addtonal test of the statstcal sgnfcance of our results, we examne whether the compensaton for aggregate rsk n currency
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