The Current Account as A Dynamic Portfolio Choice Problem

Public Disclosure Auhorized Policy Research Working Paper 486 WPS486 Public Disclosure Auhorized Public Disclosure Auhorized The Curren Accoun as A Dynamic Porfolio Choice Problem Taiana Didier Alexandre Lowenkron Public Disclosure Auhorized The World Bank Lain America and he Caribbean Region Office of he Chief Economis March 29

Policy Research Working Paper 486 Absrac The curren accoun can be undersood as he oucome of invesmen decisions made by domesic and foreign invesors. These decisions can be decomposed ino a porfolio rebalancing and a porfolio growh componen. This paper provides empirical evidence of he imporance of porfolio rebalancing for he dynamics of he curren accoun. The auhors evaluae he predicions of a parial- equilibrium model of he curren accoun wih dynamic porfolio choices, in which porfolio rebalancing is driven by changes in invesmen opporuniies. Using daa for he Unied Saes and Japan, he auhors find evidence supporing innovaions in invesmen opporuniies as an imporan mechanism o explain inernaional capial flows. This paper a produc of he Office of he Chief Economis, Lain America and he Caribbean Region is par of a larger effor in he deparmen o undersand he driving forces of inernaional capial flows and heir implicaions for macroeconomic policies. Policy Research Working Papers are also posed on he Web a hp://econ.worldbank.org. The auhor may be conaced a didier@worldbank.org. The Policy Research Working Paper Series disseminaes he findings of work in progress o encourage he exchange of ideas abou developmen issues. An objecive of he series is o ge he findings ou quickly, even if he presenaions are less han fully polished. The papers carry he names of he auhors and should be cied accordingly. The findings, inerpreaions, and conclusions expressed in his paper are enirely hose of he auhors. They do no necessarily represen he views of he Inernaional Bank for Reconsrucion and Developmen/World Bank and is affiliaed organizaions, or hose of he Execuive Direcors of he World Bank or he governmens hey represen. Produced by he Research Suppor Team

THE CURRENT ACCOUNT AS A DYNAMIC PORTFOLIO CHOICE PROBLEM Taiana Didier The World Bank Alexandre Lowenkron Banco BBM JEL Classificaion: F2, F32, G Keywords: Inernaional Invesmen, Porfolio Choice, Curren Accoun Balance We are very graeful o Ricardo Caballero, Marcio Garcia, and Guido Lorenzoni for heir guidance and suppor. For valuable commens and suggesions, we hank Nicolas Arregui, Suman Basu, Olivier Blanchard, Marin Evans, Robero Rigobon, Ken Rogoff, Leandro Rohmuller, Carmen Taveras, Jose Tessada, Cedric Tille, and seminar paricipans a MIT, Bank of England, Federal Reserve Bank of New York, Federal Reserve Board, Georgeown Universiy, Graduae Insiue for Inernaional and Developmen Sudies (Universiy of Geneva), Universiy of California a San Diego, Universiy of California a Sana Cruz, Universiy of Houson, and he World Bank. The findings, inerpreaions, and conclusions expressed in his paper are enirely hose of he auhors and do no necessarily represen he views of he World Bank. All errors are our own. Email: didier@worldbank.org and alexandrelowenkron@bancobbm.com.br.

Inroducion In heory, he curren accoun can be undersood as he oucome of invesmen decisions made by domesic and foreign invesors. Empirically, we can sudy he oucome of hese decisions by analyzing a counry s gross foreign asse posiions. Therefore, facors ha affec hese gross posiions, such as asse reurns and exchange raes, also have an impac on he curren accoun. Furhermore, he recen empirical lieraure on he dynamics of counries porfolios highlighs he imporance of heir variaions over ime. Thus, in his paper, we analyze he implicaions of changes in a counry s opimal porfolio allocaion for he dynamics of he curren accoun. In paricular, we focus on ime-varying opimal porfolio shares caused by innovaions in he invesmen opporuniy se. 2 Mos significanly, we empirically evaluae he relevance of hese variaions o explain movemens in he curren accoun. We focus on he predicions of a parial-equilibrium model of he curren accoun wih dynamic porfolio choice. In his model, ime-varying invesmen opporuniies, capured by he dynamics of asse reurns, are he main mechanism behind porfolio rebalances. In sum, our approach highlighs changes in expeced asse reurns as an imporan facor o explain inernaional capial flows. Alhough he curren accoun is essenially an issue of porfolio allocaion, sandard macroeconomic models have no incorporaed his aspec of he problem unil very recenly. Even hen, commonly used models have saic-like soluions wih consan porfolios over ime. For example, Kraay and Venura (2) use Meron s (97) model of porfolio allocaion o analyze he curren accoun. This model assumes consan asse risk and reurn, and agens wih log-uiliies. The opimal porfolio allocaion is hus characerized by consan porfolio shares, implying ha a counry s ne foreign asse posiion is a consan fracion of is wealh. Therefore, See, for example, Lane and Milesi-Ferrei (26), and Gourinchas and Rey (26). 2 Oher reasons for opimal porfolio reallocaions have been suggesed, namely ime-varying preferences (e.g. risk aversion), parameer uncerainy, and financial consrains. 2

his model highlighs a porfolio growh componen as an explanaion for he dynamics of he curren accoun. On he oher hand, in his paper, we focus on porfolio rebalancing as he driving force behind he dynamics of he curren accoun. We exend exising srucural models of he curren accoun in order o incorporae he growing empirical evidence on he dynamics of counries porfolios. More specifically, we emphasize he imporance of innovaions in invesmen opporuniies, capured by changes in expeced asse reurns, as he main mechanism behind variaions in counries porfolios. Meron s (97) porfolio model is he foundaion of our heoreical framework. By changing wo cenral assumpions of Meron s model, we are able o obain a srucural model of he curren accoun wih dynamic porfolio choice. Firs, we assume ha asse reurns are non-i.i.d. and exploi heir predicabiliy. Second, we depar from he assumpion of a log uiliy funcion. To separae he elasiciy of ineremporal subsiuion from he relaive risk aversion parameer, and herefore o model savings and invesmen decisions separaely, we assume agens wih Epsein-Zin uiliy funcion. We also assume a relaive risk aversion parameer greaer han one. Boh assumpions are imporan in obaining ime-varying opimal porfolio shares for invesors wih long-erm invesmen horizons. As already discussed, he main mechanism behind opimal porfolio reallocaions in our model is ime-varying invesmen opporuniies, characerized by he dynamics of expeced asse reurns. The model allows us o obain clear predicions of his mechanism for he curren accoun balance. Nex, we empirically analyze he model s implicaions for he curren accoun. Due mosly o daa availabiliy, we focus on wo counries and heir bilaeral curren accoun. Campbell, Chan and Viceira s (23) mehod is used o solve and esimae he model for U.S. and Japanese invesors. We presen robus empirical evidence ha ime-varying invesmen opporuniies are 3

imporan deerminans of he dynamics of he bilaeral curren accoun. We show ha variaions in expeced asse reurns change agens opimal porfolios in a direcion consisen wih he acual bilaeral curren accoun movemens. We also find ha posiive changes in he prediced bilaeral curren accoun are significanly associaed wih improvemens in he acual bilaeral curren accoun. Furhermore, we provide robus evidence ha prediced porfolio shares, combined wih acual daa on savings and consumpion insead of he model s predicions, can explain he dynamics of he bilaeral curren accoun. Our empirical resuls hus provide suppor for he main mechanism highlighed in his paper. Alhough our model effecively capures he dynamics of he bilaeral curren accoun, i does no successfully explain he level of he bilaeral curren accoun. There are wo reasons for ha. Firs, we do no impose eiher borrowing or shor-selling consrains in he model, and as a resul, we allow leveraged porfolios. We are hus bound o obain larger and more volaile capial flows han acual ones. This problem is ypical of models which assume perfec mobiliy of capial flows. Similar implicaions have been repored in porfolio allocaion models by Campbell, Chan, and Viceira (23), Evans and Hnakovska (25), and Mendoza, Quadrini, and Rios-Rull (27) for example. Correcing his issue in a porfolio model wih several asses is no simple, so we share his problem wih he res of he lieraure. Furhermore, similar models have been widely used o analyze issues of opimal porfolio allocaion. For example, Campbell and Viceira (999), Wacher (22), Normandin and S- Amour (25), and Sangvinasos and Wacher (25) highligh hese models success in explaining opimal porfolio choice in differen conexs. However, hey acknowledge ha hey are no well-suied o capure he dynamics of agens wealh. The models predic rapidly growing wealh, low consumpion-wealh raios, and relaively low consumpion volailiy. 4

Noneheless, he prediced dynamics of consumpion is reasonable: an invesor wans more wealh in saes when he marginal uiliy is higher. Therefore, given our focus on changes in opimal porfolio allocaion as an explanaion for he dynamics of he curren accoun, we believe ha our model is appropriae. I effecively capures variaions in he opimal porfolio allocaion caused by changes in he invesmen opporuniy se. To he bes of our knowledge, his is he firs paper o empirically analyze he relevance of porfolio rebalancing caused by changes in invesmen opporuniies for he dynamics of he curren accoun. Alhough we develop our own heoreical framework, a few heoreical papers should be menioned. 3 Devereux and Suherland (27) and Tille and van Wincoop (27) highligh he imporance of ime-varying porfolio shares in a dynamic sochasic general equilibrium (DSGE) model wih porfolio choice. In heoreical erms, hey show he imporance of porfolio rebalancing for ne and gross capial flows. Also from a heoreical perspecive, Evans and Hnakovska (25) and Hnakovska (26) use a general equilibrium model wih porfolio choice o discuss he size and volailiy of capial flows and heir deerminans. The res of he paper is srucured as follows. In Secion 2, we presen preliminary evidence o illusrae he empirical relevance of our argumen. Secion 3 presens our model of he curren accoun wih dynamic asse allocaion. In Secion 4, we furher develop our empirical analysis. We esimae he model for he U.S. and Japan and compare is predicions wih he acual bilaeral curren accoun daa. Secion 5 concludes and is followed by he appendices. 3 From a differen perspecive, he Inernaional RBC lieraure has incorporaed he effecs of changes in he produciviy of physical capial on invesmen decisions. See, for example, Backus, Kehoe, and Kydland (992). 5

2 Preliminary Empirical Evidence As a saring poin o illusrae he empirical imporance of porfolio rebalancing as opposed o he porfolio growh componen, a simple accouning exercise is helpful. A counry s wealh can be decomposed in o he sum of is ne foreign asse posiions and domesic asses: W NFA Dom. Asses () We can hen define he porfolio share in ne foreign asses as: NFA W (2) By differeniaing his equaion, we obain he sandard definiion of he curren accoun: NFA W Curren Accoun Porfolio Rebalancing Componen W Porfolio Growh Componen (3) Lasly, we perform a variance-decomposiion analysis based on equaion (3): var( W W ) var( W ) var( W ) 2cov( W, W ) (4) The resuls are shown in Table I. High-income counries are shown in he op panel, middleincome and low-income counries in he boom panel. 4 The firs column on his able, R- squared, repors how much of he ime-series variaion of he curren accoun can be explained by porfolio growh and porfolio rebalancing. The nex hree columns repor he hree RHS variables on equaion (4), scaled by he LHS variable. Lasly, he oher wo columns repor he relaive size of he porfolio rebalancing and he porfolio growh componens. The porfolio rebalancing componen is on average hree-imes as large as he porfolio growh componen. This is indeed he case for boh he U.S. and Japan for example. Alhough hese average effecs are large, here is heerogeneiy across counries. The porfolio growh 4 A deailed descripion of he daa is presened in Secion 4. 6

componen can be as large as 98% in China and 92% in he U.K. and Malaysia or as low as 37% in Argenina. Bu in eiher scenario, he porfolio rebalancing componen is imporan o explain he dynamics of he curren accoun. Nex, we focus on he empirical relevance of changes in expeced asse reurns as he main mechanism behind he porfolio rebalancing componen o explain he dynamics of he curren accoun. We propose a simple reduced-form model of he bilaeral curren accoun. We assume ha ex-ane domesic and foreign expeced asse reurns (boh in levels and in differences), and domesic and foreign savings can explain he dynamics of he bilaeral curren accoun, from he perspecive of a domesic invesor. Expeced asse reurns are capuring changes in invesmen opporuniies as a driving force behind porfolio reallocaions. For example, as domesic expeced asse reurns increase, ceeris paribus, here is an incenive for boh domesic and foreign invesors o rebalance heir porfolios oward domesic asses. Therefore, a decrease in foreign invesmens by domesic invesors and an increase in domesic invesmens by foreign invesors imply a deerioraion of he curren accoun. The opposie effec happens if foreign expeced asse reurns increase. Domesic and foreign savings capure a porfolio growh effec. Larger domesic savings should imply larger holdings of foreign asses, being hus associaed wih an improvemen on he bilaeral curren accoun. Conversely, if foreign savings increase, we should observe a negaive effec on he bilaeral curren accoun. Equaion (5) summarizes his reduced-form model: BCA ( E R ) 2 ( E R ) 3( E R E R ) 4 ( E R E R ) S 5 S 6 (5) 7

We esimae equaion (5) for he bilaeral curren accoun beween he U.S. (domesic) and Japan (foreign) from 96 o 25. 5 The following asses are considered: shor-erm and longerm governmen bonds and equiies. 6 The esimaed coefficiens are correcly signed and saisically significan in boh annual and quarerly samples. 7 Figure I plos he fied values of his model. For comparison purposes, we also show he resuls for a model wih only a porfolio growh componen. This laer model is a varian of Kraay and Venura s (2) model of he curren accoun. As can be seen in Figure I, our simple seup, wih porfolio rebalancing effecs, explains he dynamics of he bilaeral curren accoun remarkably well. I provides us preliminary evidence ha porfolio rebalancing induced by changes in expeced asse reurns can be empirically imporan in explaining he dynamics of he curren accoun. In he nex secions, we furher develop he argumen behind he preliminary evidence shown here. We exend exising srucural models of he curren accoun in order o allow opimal ime-varying porfolio shares caused by changes in invesmen opporuniies. We hen empirically evaluae he predicions of our model of he curren accoun. 3 A Dynamic Porfolio Allocaion Model of he Curren Accoun In his secion, we presen a srucural model of he curren accoun, wih dynamic porfolio choice. The main mechanism behind porfolio reallocaions in our model is ime-varying invesmen opporuniies, characerized by he dynamics of expeced asse reurns. Therefore, his model provides a heoreical framework o furher analyze he empirical evidence shown in 5 As will become clear in Secion 4, he empirical analysis of his paper focuses on he bilaeral curren accoun beween he U.S. and Japan, as opposed o he oal U.S. curren accoun. 6 Expeced reurns are calculaed using a vecor auoregression sysem (VAR) wih pas reurns and oher predicive variables idenified in he finance lieraure. Our mehodology is explained in Secion 4. 7 See Appendix A for a deailed descripion of he esimaed regressions. 8

Secion 2. I allows us o obain clear predicions of he effecs of changes in expeced reurns on he curren accoun balance. Our model is an exension of Meron s (97) model o examine he dynamics of he curren accoun. Meron s model assumes agens wih logarihmic uiliy funcions and i.i.d. asse reurns. I hus implies ha long-erm invesors behave as mean-variance opimizers, choosing he same porfolio as a shor-erm invesor. Given he assumpions, he opimal porfolio allocaion is characerized by consan porfolio shares. To obain a model wih dynamic porfolio choice, we change wo cenral assumpions of Meron s model. We assume agens wih an Epsein-Zin uiliy funcion and non-i.i.d. asse reurns. 3. The Environmen The model is se in discree ime. We consider a parial equilibrium analysis in which agens face exogenous asse reurns. There is an arbirary se of raded asses. We also assume ha all individuals are idenical and have access o he same informaion se regarding he curren sae of he world. This common-knowledge assumpion is sandard in inernaional macroeconomic models and i implies ha capial flows in our model do no resul from differences of opinion on fuure asse reurns or risks. 3.2 A Represenaive Counry Consider a counry populaed by idenical and infiniely lived individuals whose preferences are represened by Epsein-Zin (99) recursive preferences defined over heir consumpion sream: ( ) C E U, U ( C, E ( U )) (6) 9

where, C is consumpion a ime, < δ < is he ime discoun facor, γ > is he relaive risk aversion coefficien, and ψ > is he elasiciy of ineremporal subsiuion. This uiliy funcion ness as special cases he power uiliy specificaion, in which he relaive risk aversion (RRA) coefficien is he reciprocal of he elasiciy of ineremporal subsiuion (EIS), and he log-uiliy specificaion, in which boh parameers are equal o one. Therefore, hese preferences have he flexibiliy of modeling he EIS and he RRA parameers separaely. The former has firs order effecs on savings versus consumpion decisions and only secondary effecs on invesmen decisions. In conras, he RRA parameer is essenial o porfolio allocaion. Hence, his funcional form disenangles savings and porfolio allocaion decisions. 8 We assume ha individuals can inves in domesic and foreign asses. There are n securiies available for invesmen a home and n securiies available abroad, so ha 2n is he oal number of available securiies. Therefore, he inra-emporal budge consrain can be defined as: W n C A A, (7) i i, n j j, where W is oal wealh a ime, A i, is he amoun invesed in domesic asse i a ime, and A j, is he amoun invesed in foreign asse j a ime. The wealh accumulaion equaion can hen be defined as: n n i, i, i j W R A R A, (8) where R i, + is he gross real reurn on domesic asse i from ime o ime +, and R j, + is he gross real reurn on foreign asse j from ime o ime +. 8 The reason for his paricular uiliy funcion as opposed o a more sandard power uiliy funcion will become clear in Secion 3.4. j, j,

As can be seen from equaions (7) and (8), we do no model labor income. The income available for consumpion a ime is given by he reurns on porfolio holdings and by he sales of hese asses (shor sales are allowed). This counry s GDP can be inerpreed as he oal real reurn on domesic asses, independenly of who owns hem. From equaion (7), we can define porfolio shares A /( W C ) : i [ A i, /( W C )] j i A [ A i, j, i i, /( W j C j A j, )] j, W, C where i, is he proporion of a counry s wealh, ne of consumpion, invesed in a domesic asse i from o +, and is he proporion of a counry s available wealh invesed in foreign j, asse j from o +. The real porfolio reurn, R p, +, is hus given by: R n n p, i, Ri, i j j, R j,. (9) Finally, equaions (7), (8), and (9) can be combined in order o obain he ineremporal budge consrain: W R p, ( W C ). () In summary, he problem faced by individuals in his represenaive counry is o choose consumpion (C ) and porfolio shares ( ) ha maximize (6) subjec o (), given an iniial level of wealh W. We hus allow counries o differ in heir size, i.e., invesors from differen counries can sar wih differen levels of wealh, W. In his seup, Epsein and Zin (99) show ha invesors opimal consumpion decision mus saisfy he following Euler equaion:

E ( ) ( C / C ) R p, Rk,, k, () where R k, is he gross real reurn on any asse, including he porfolio iself. When invesmen opporuniies are consan, porfolio shares are also consan, implying ha R p,+ is ime-invarian. Thus, he opimal consumpion policy, characerized by equaion (), implies a consan consumpion-wealh raio. I also enails a consan porfolio share in all available asses. In oher words, agens behave as shor-erm invesors and opimally choose a myopic porfolio allocaion. To obain dynamic porfolio choices, we relax he hypohesis of consan invesmen opporuniies over ime. In our seup, if asse reurns are non-i.i.d., a relaive risk aversion parameer greaer han one is a sufficien condiion for he opimal porfolio allocaion o be dynamic (non-myopic). 9 Thus, we assume a relaive risk aversion parameer greaer han one. Therefore, o model ime-varying invesmen opporuniies, we explore he empirical evidence ha financial asse reurns are predicable o some exen. For example, Amromin and Sharpe (26) provide empirical evidence based on survey daa suggesing, for example, ha expeced sock reurns are exrapolaed from acual reurns. We hus assume ha asse reurns follow a firs-order vecor auoregression (VAR). This assumpion capures he hisorydependence of expeced reurns. The empirical finance lieraure has idenified several predicive variables, besides he hisorical values of asse reurns hemselves. Nominal yield on shor-erm bonds, he erm spread, and earnings-o-price raio have been documened o forecas asse reurns for many asse classes. Thus, we also use hese variables (s ) o esimae expeced reurns. 9 See Campbell and Viceira (22) for an exensive sudy on sraegic asse allocaion. See Campbell (987), Campbell and Shiller (988), Fama and French (988, 992), and more recenly, Campbell and Yogo (26) and Wacher and Warusawiharana (27), among many ohers. A similar specificaion can be found in Campbell, Chan, and Viceira (23) and Barberis (2), for example. 2

Define a vecor z + conaining he log real reurn of a benchmark asse (r,+ ), log excess reurns of domesic and foreign asses, i.e. (r i,+ - r,+ ) and (r j,+ - r,+ ), and oher sae variables (s + ) used o predic asse reurns: ri, r, r,... z x, where x, r j, r, s... (2) and where r,+ is he log real reurn on he benchmark asse, an d x + is he vecor of log excess reurns, measured as excess reurns over his benchmark asse. As menioned above, we assume ha z + follows a VAR() process: z z v 2 ' '... ' x s i i d where v ~ N (, v ) and v x xx xs. s xs ss (3) We allow shocks o be cross-secionally correlaed, bu homoskedasic and independenly disribued over ime. In oher words, we assume ha sae variables are no able o predic changes in asse risk. Therefore, only changes in expeced asse reurns affec porfolio choices. Even hough his assumpion may be unrealisic, i is no resricive from he perspecive of longerm porfolio allocaion. The empirical evidence suggess ha changes in risk are no persisen enough o have large effecs on porfolio choices. 2 3.3 The Curren Accoun A every period, agens decide how o allocae heir wealh, ne of consumpion, among available financial asses. By analyzing his porfolio choice, i is possible o deermine he oal wealh allocaed o domesic and foreign asses a each poin in ime. More specifically, we can 2 See Campbell (987), Harvey (99), and Chacko and Viceira (25). 3

de ic and foreign invesmens made by domesic agens. Hence, dom ermine he opimal porfolio allocaion. Therefore, obaining an expression for he curren accoun balance is sraighforward. The curren accoun balance of he Home counry (H) can be defined as domesic savings minus invesmen in domesic asses: CA = S - I. Using equaions (7) and (8), i is clear ha domesic savings mus equal domes esic savings are given by: ). ( ) (,,,,,, H j H j H i H i A A A A S (4) j i Invesmen in domesic asses is given by he change in holdings of all domesic asses, aggregaed across all counries (c) ha have access o domesic asses: ). (,, c i c i c i A A I (5) Thus, he curren accoun balance of he Home counry is defined as (4) minus (5): )]. ( ) ( [ )] ( ) ( [ ) ( ) (,,,,,,,,,,,, H c i i i j H H H j H H H j H c i c i c i j H j H j C W C W A A A A CA c c c c c c C W C W (6) Similarly, he Home counry bilaeral curren accoun wih a Foreign counry (F) can be defined as: Equaions (6) and (7) clearly show how changes in wealh and opimal porfolio shares affec he bilaeral and he oal curren accoun balances, respec ely. Therefore order o explain he dynamics of he (bilaeral) curren accoun, we need an explici soluion for hese F F F i F F F i C W C W )] ( ) ( [,, (7) i F j H H H j H H H j HF C W C W BCA )] ( ) ( [,,,,, iv, in 4

ime-varying opimal porfolio shares and for he dynamics of wealh. More specifically, in he case of he bilaeral curren accoun, we need he soluions for boh Home and Foreign counries. 3.4 Model s Approximae Soluion Campbell, Chan, and Viceira (23) show ha here is no closed-form soluion for his mulivariae model of sraegic asse allocaion. However, hey propose an approximae soluion mehod. They show ha we can reduce his model o an approximae sysem of linear-quadraic equaions for porfolio weighs and consumpion as funcions of he sae variables. Therefore, we follow heir procedure in order o obain an approximae soluion o our model. The soluion o our model is characerized by hree equaions: he porfolio reurn, he ineremporal budge consrain of he represenaive counry, and he Euler equaion. We can rewrie equaion (9), which characerizes he gross porfolio real reurn, in he following way: R n ( R R ) ( R R ) R, (8) p, i, i,, j, j,,, i2 j where he firs asse, whose real reurn is given by R,, is a domesic shor-erm insrumen used as a benchmark asse. Even hough asse reurns are measured relaive o his benchmark asse, i is no assumed o be riskless. This benchmark asse is subjec o shor-erm inflaion risk. The log reurn on he porfolio can hen be approximaed as: x ( xx r n 2 r, ' x '( x xx ), (9) 2 p, 2 where diag ) is a vecor conaining he variance of excess asse reurns, and is a vecor of porfolio shares. This approximaion holds exacly in coninuous ime and i is highly accurae for shor-ime inervals. 3 3 This approximaion o he log reurn on he porfolio has he effec of ruling ou he possibiliy of bankrupcy. See Campbell and Viceira (22). 5

The nex equaion is he budge consrain, equaion (). Log-linearizing i around he uncondiional mean of he log consumpion-wealh raio, we obain he following expression for he wealh dynamics: w rp, ( c w ) k, (2) where exp( E[ c w ]) and k log( ) ( )log( ) /. This form of he budge consrain is exac if he elasiciy of ineremporal subsiuion ( ) is equal o, in which case and c w is consan. Lasly, we apply a second-order Taylor expansion o he Euler equaion () around he condiional means of c, rp,, rk, o obain: log Ec ( ) Erp, Er var c 2 ( ) r p, k, r k,, k. (2) This form of he Euler equaion is exac if consumpion and asse reurns are joinly lognormally disribued, e.g. when. In sum, he model s approximae soluion can be described by hese hree equaions, (9), (2), and (2). The opimal soluion is accurae for an elasiciy of ineremporal subsiuion around one, independen of he value of he relaive risk aversion parameer. 4 A model wih a disincion beween hese wo parameers is imporan o he empirical evidence presened in he nex secion. We evaluae he sensiiviy of he opimal porfolio allocaion o differen values of he relaive risk aversion parameer. In he power uiliy case, as we increase he relaive risk aversion parameer, he model soluion becomes inaccurae. However, wih he Epsein-Zin 4 This is consisen wih recen esimaes of he elasiciy of ineremporal subsiuion. See for example Vissing- Jorgensen (22) and Yogo (24). 6

uiliy funcion, we have he auonomy o do so wihou inerfering wih he accuracy of he soluion. This reason underpins our focus on an Epsein-Zin uiliy funcion as opposed o he more sandard power uiliy funcion. Campbell, Chan, and Viceira (23) show ha he opimal porfolio choice is linear in he VAR sae vecor. I is characerized by he following opimal porfolio allocaion: where 2 A xx H x x ( ) 2 A xx H x xx, and and are consans. A A z, (22) x xx, They also show ha he opimal consumpion rule is quadraic in his VAR sae vecor: c w log p, ( ) E( rp, ) k E( c w ), (23) where E ( r p, ) and p, are quadraic funcions of he VAR sae variables. A numerical recursive procedure, described in Campbell, Chan, and Viceira (23), is used o solve for he opimal consumpion and porfolio shares. Using equaion (7), we are hus able o consruc a measure of he prediced curren accoun balance based on his model of he curren accoun wih dynamic porfolio choice. 4 An Applicaion o he U.S. Bilaeral Curren Accoun wih Japan In his secion, we presen a quaniaive analysis of he framework developed in Secion 3. More specifically, we focus on he bilaeral curren accoun beween he U.S. and Japan. The model yields opimal porfolio rules ha are linear in he vecor of sae variables. Therefore, we empirically evaluae ime-varying porfolio shares, caused by changes in expeced asse reurns, 7

as an explanaion for he acual dynamics of he U.S. bilaeral curren accoun wih Japan. We esimae our model separaely for invesors in he U.S. (Home) and in Japan (Foreign) from 96 o 25. We hen consruc he ime series of porfolio weighs for each counry, i.e.,,h α j, F and α i,. Afer aggregaing foreign holdings for boh counries, we presen a firs round of empirical evidence. We analyze wheher variaions in expeced asse reurns change agens opimal porfolios in a direcion consisen wih he acual bilaeral curren accoun movemens. Nex, we combine hese opimal porfolio weighs, according o equaion (7), o obain our prediced measure of he bilaeral curren accoun. We ake ino consideraion differences in he counries sizes. We hen evaluae wheher our prediced measure can explain he dynamics of he acual bilaeral curren accoun daa. Finally, as a robusness exercise, we consruc a hybrid version of equaion (7): we use he opimal porfolio shares combined wih acual daa on wealh, savings, and consumpion. We hus obain anoher measure of he prediced curren accoun. We re-esimae he relaion beween he prediced and he acual bilaeral curren accouns. In summary, we provide srong empirical evidence ha changes in invesmen opporuniies are an imporan mechanism behind he dynamics of he bilaeral curren accoun beween he U.S. and Japan. 4. Why he Bilaeral Curren Accoun beween he U.S. and Japan? A large number of counries have significan exposure o U.S. asses. An empirical analysis of he mechanism highlighed in his paper for he oal U.S. curren accoun would hus require an esimaion of he model for all hese counries. Moreover, many asses would need o be considered in our quaniaive analysis. By focusing on wo counries and heir bilaeral curren accoun, we only need o analyze he behavior of wo invesors. Hence, we empirically sudy he 8

U.S. and Japan and heir bilaeral curren accoun. In his case, only U.S. holdings of Japanese asses and Japanese holdings of U.S. asses maer. There are many reasons for choosing he U.S. and Japan in our empirical exercise. The firs one is daa availabiliy. Bilaeral curren accoun daa beween he U.S. and Japan, Canada, or he U.K. is available since 96 on a quarerly basis. The daa for oher counries sars in he lae 97s, and herefore, has an insufficien ime span for our purposes. Furhermore, in our empirical exercise, we use asse reurns on sock markes, governmen bonds, and privae firm profis (reurn on equiy). This las variable is no available for he U.K. and Canada, alhough we could have excluded i from our analysis. Second, Japan was economically relevan for inernaional capial flows from 96 o 26. Boh he U.S. and Japan are represenaives of he so-called global imbalances. The curren accoun defici in he U.S. was soaring and reached 7% of is own GDP in 25 almos 2% of world GDP. Japan has long been he counry wih he larges curren accoun surplus. Furhermore, he U.S. and Japanese curren accoun balances were mirror images of each oher unil he lae 99s, suggesing ha hey could have had a large counerpar in each oher s balances. Third, he oal U.S. curren accoun and he U.S.-Japan bilaeral curren accoun have similar dynamics. This is highlighed in Figure II, which plos hese series as a percenage of U.S. GDP. Movemens in he oal U.S. curren accoun clearly resemble movemens in he U.S. bilaeral curren accoun wih Japan. Thus, deerminans of he bilaeral curren accoun can be indeed relevan o he undersanding of facors affecing he oal U.S. curren accoun. Lasly, i is well known ha counries porfolios are subjec o home bias ha is, porfolio composiion ends o be biased oward domesic asses. For example, insiuional invesors in he 9

U.S. held only % of heir porfolios in foreign equiy and bonds in 23. A similar paern is observed in Japan, where insiuional invesors held only 6% of heir porfolios abroad in 23. 5 Alhough domesic residens hold he majoriy of heir asses in heir own counries, a large number of foreign invesors, if allowed, end o hold hese foreign asses as well. Survey daa published by he U.S. Deparmen of he Treasury shows ha residens of Japan were he larges foreign porfolio invesors in U.S. securiies by a wide margin in 25. 6 They held U.S. $. rillion (or 6% of he oal holdings of U.S. securiies by foreign invesors), whereas residens of he U.K., he second major invesing counry, had holdings of U.S. $.56 rillion, only half he holdings of Japanese invesors. Previous surveys show ha his paern is sable over ime. For example, in 994, when he firs survey was conduced, Japan held 8% of he oal foreign holdings of U.S. securiies. A he same ime, Japan has consisenly been one of he main desinaions of foreign purchases of securiies by U.S. residens. U.S. invesors held around % of oal marke capializaion of equiy markes in Japan in 25. 7 Moreover, in 994 U.S. residens invesed 5% of heir foreign porfolio holdings in Japan he counry ha araced he larges share of U.S. porfolio invesmens abroad. A more recen survey shows ha Japan is sill a large desinaion for U.S. funds, aracing 2% of U.S. holdings of foreign securiies in 25. Alhough no daa is available on he holdings of oher foreign invesors in Japan (or even oher holdings of Japanese invesors), he evidence presened here suggess ha he U.S. has been a major paricipan in his marke. By focusing on he U.S. and Japan and heir bilaeral curren accoun, given he survey evidence presened above, we are analyzing he wo larges holders of U.S. securiies: U.S. invesors hemselves and Japanese invesors, he larges foreign holders. We also examine large 5 See IMF (25). 6 See Repor on U.S. Porfolio Holdings of Foreign Securiies, U.S. Deparmen of he Treasury. 7 See Repor on Foreign Porfolio Holdings of U.S. Securiies, U.S. Deparmen of he Treasury. 2

holders of Japanese securiies: Japanese invesors hemselves and U.S. invesors. In sum, U.S. and Japanese invesors ogeher are possibly he larges holders of U.S. and Japanese securiies. Moreover, his survey evidence also suggess ha U.S. and Japanese invesors hold he majoriy of heir porfolios in he U.S. and Japan hemselves. Therefore, his empirical evidence, combined wih he daa presened in Figure II, suggess ha U.S. and Japanese asses are he mos relevan asses affecing he bilaeral curren accoun beween he U.S. and Japan, and possibly he oal U.S. curren accoun. Therefore, we assume in our empirical exercise ha U.S. and Japanese invesors can only hold asses from eiher he U.S. or Japan. Because of he limied ime span of our sample, we do no consider oher asses; four decades of daa would no be enough for an esimaion of our VAR sysem. On he oher hand, we assume ha invesors from oher counries can hold asses anywhere, including he U.S. and Japan. Thus, if a Japanese invesor decides o sell some of her holdings, a U.S. invesor does no need o buy hem. In oher words, we are considering a parialequilibrium analysis. We are fully aware of he limiaions of his las assumpion. Including asses from oher counries in he analysis could significanly change he calculaed opimal porfolio allocaion among he asses acually considered here. For robusness purposes, we ried o include asses from a hird counry in our empirical analysis. According o he survey evidence repored by he U.S. Deparmen of he Treasury, he U.K. and he Euro-area as a whole are he relevan candidaes. Thus, we included in our exercise asses from eiher he U.K. or Germany, he laer as a represenaive of Euro-area asses. The resuls are qualiaively similar o he ones presened in his paper and, herefore, no repored. Moreover, if he inclusion of oher relaively large invesors does no qualiaively change our empirical analysis, he inclusion of oher smaller invesors is similarly unlikely o affec our resuls. 2

4.2 Daa Descripion We use quarerly daa exending from he second quarer of 96 o he hird quarer of 25. As already discussed, we consider financial asses from he U.S. and Japan. The daa was obained from Global Financial Daabase, he financial saemens from he Minisry of Finance in Japan, and he U.S. Flow of Funds Accouns calculaed by he U.S. Federal Reserve. The following asse classes are considered in he analysis: socks, shor-erm governmen bonds, long-erm governmen bonds, and privae firms profis (ROE). U.S. sock reurns are calculaed as reurns on he S&P 5 index, and Japanese sock reurns are given by he reurns on he Tokyo Sock Exchange Topix All Shares Index. The U.S. and Japanese reurns on shor-erm ineres raes are he quarerly reurns implied by he Fed Funds rae and he Japanese Discoun rae, respecively. The reurn on long-erm governmen bonds is calculaed as he reurn on - year consan mauriy U.S. governmen bonds and as he reurn on 7-year Japanese governmen bonds. Governmen bonds of longer mauriy were no available for Japan. ROE is consruced as he oal operaional profis divided by capial (ne worh). Our model is wrien in real erms. Therefore, our benchmark asse is he ex-pos real reurn on shor-erm governmen bonds. More specifically, he benchmark asse is he real reurn on a U.S. shor-erm bond for a U.S. invesor and a Japanese shor-erm bond for a Japanese invesor. Real reurns are consruced as he difference beween he log reurn on an asse and he log of CPI-inflaion. In our heoreical framework, invesors analyze excess reurns over he benchmark asse. Thus, all excess reurns are calculaed as he log difference beween he real reurn on a specific asse and he real reurn on he appropriae benchmark asse, boh denominaed in he same currency. We use he log change of he real exchange rae o conver reurns o a common currency. We define he log real exchange rae as he sum of log nominal exchange rae and log 22

domesic CPI less he log foreign CPI. Lasly, we use variables known o predic asse reurns, such as nominal shor-erm yield (3-monh T-Bills), price-o-earnings raio, and he nominal erm spread in governmen bonds. Table II repors he summary saisics for real asse reurns denominaed in local currency. Daa is in annualized percenage unis. I shows he sample average and he sandard deviaions of he quarerly asse reurns used in he analysis. The able also repors hese sample saisics for he CPI-inflaion raes and he real exchange rae. Among he U.S. asses, he shor-erm governmen bond is he safes asse, wih an average real reurn of.8% p.a., and equiies are he riskies asse, wih larger real reurns, 7.% p.a. on average. A similar paern is observed in Japan. Socks are also he riskies asse class and shor-erm governmen bonds, he safes, wih average real reurns of 7.6% p.a. and.4% p.a., respecively. Average inflaion raes are smaller in Japan han in he U.S., bu more volaile. Lasly, he real exchange rae shows, on average, an appreciaion of he Japanese yen agains he U.S. dollar in our sample from 96 o 25. We have esed all series of asse reurns for uni roos using Augmened Dickey-Fuller ess. These ess srongly rejec uni roos in all daa series considered, excep for he reurns on U.S. long-erm governmen bonds. However, we recognize he low power of hese ess and he evidence in favor of mean-reversion in he long run, and assume U.S. long-erm bonds o be saionary. 8 Our VAR esimaions also include he CPI-inflaion raes, he nominal exchange rae, or he real exchange rae, depending on he specificaion considered. Boh CPI-inflaion raes are saionary according o Augmened Dickey-Fuller ess. These ess on he nominal exchange rae and he real exchange rae could no rejec he exisence of a uni roo. However, he empirical evidence on he saionariy of exchange raes is highly conroversial. 9 Therefore, 8 Excluding his variable from he analysis does no qualiaively change he resuls. 9 See for example Rogoff (996) and Imbs, Mumaz, Ravn, and Rey (25). 23

in order o show he robusness of our resuls, we presen hem considering exchange raes in levels or in differences, or no exchange rae a all. Besides asse reurns, our empirical analysis also uses daa on he bilaeral curren accoun beween he U.S. and Japan, oal wealh, naional savings, naional consumpion, and GNP. The bilaeral curren accoun daa is from he U.S. Bureau of Economic Analysis. We follow he mehodology described in Kraay e al. (25) o consruc measures of oal wealh. Naional savings, naional consumpion, and GNP are from IMF s Inernaional Financial Saisics. 4.3 VAR Esimaion Our empirical resuls depend on he esimaion of he sysem of equaions (3). We hus repor he esimaions based on five differen specificaions of his VAR sysem. 2 Through he res of he paper, we repor resuls for all hese specificaions. We show ha he expeced asse reurns obained from hese differen esimaions have similar dynamics. We will argue ha our empirical analysis is robus o hese differen esimaions. In oher words, he porfolio allocaions implied by hese differen VAR sysems are similar in heir composiion, and hus lead o similar predicions for he bilaeral curren accoun beween he U.S. and Japan. In our heoreical model, we have assumed ha invesors from boh counries have access o he same informaion se. They use he same model and know he curren sae of he world. Therefore, o characerize he dynamics of asse reurns, we esimae a single VAR ha reas Home and Foreign symmerically. 2 The single framework described in his secion summarizes concisely he informaion se available o boh invesors, alhough i is no in he forma of he 2 We have analyzed more han hese five repored specificaions, bu choose no o repor hem here. The resuls are qualiaively similar o hose shown in his paper. 2 One of he VAR specificaions considered here canno be esimaed by his unified framework; oher conrol variables are needed. Thus, in his paricular case, separae VARs are esimaed for U.S. and Japanese invesors. 24

sysem of equaions (3). In Appendix B, we show how o obain he parameers of he sysem of equaions (3) for each invesor from hese esimaed VARs. When esimaing hese VARs, we have imposed he following resricion: he uncondiional means of he variables implied by he esimaed coefficiens should be equal o heir full-sample arihmeic counerpars. Moreover, he esimaed sysems migh be subjec o finie sample bias. However, bias correcions are complex in a mulivariae sysem. Thus, no correcions were aemped here. Insead, he esimaed coefficiens are aken as given and known by invesors. As already menioned, we esimae five differen specificaions of his sysem of equaions. The following variables are considered: real asse reurns in local currency, predicive variables (nominal yield on T-Bills, price-earnings raio, and he erm spread), he nominal and real exchange raes, and he inflaion raes for boh counries. The firs esimaed sysem includes only real asse reurns and predicive variables for boh he U.S. and Japan. This is our basic VAR. 22 The oher esimaed VARs add conrol variables o his basic sysem. Our second specificaion includes he real exchange rae in levels. The hird one adds he real exchange rae in differences insead of levels o he basic sysem. Our fourh specificaion includes he nominal exchange rae and he CPI-inflaion raes. Lasly, our fifh specificaion expands he basic VAR sysem by including he nominal exchange rae in differences and he CPI-inflaion raes. We repor only he esimaion of he VAR sysem based on our second specificaion in order o save space. 23 The resuls are presened in Table III. The esimaed coefficiens are comparable o he ones idenified in he finance lieraure. 24 The coefficiens in all equaions are joinly 22 This sysem canno be esimaed by our single unified framework oher conrol variables are necessary. One VAR was esimaed for U.S. invesors wih all variables denominaed in U.S. dollars, and anoher VAR was esimaed for Japanese invesors wih all variables denominaed in Japanese yen. 23 The oher esimaed VARs are qualiaively similar and he resuls are available upon reques. 24 See Campbell and Shiller (988), Fama and French (988, 992), Hodrick (992), Leau and Ludvigson (2), Campbell and Yogo (26), Ang and Bekaer (27), among many ohers. 25

significan a he sandard significance level, as can be seen from he low p-values of he F- saisics. The U.S. shor-erm reurn is significanly explained by he shor-erm nominal yield and he erm spread wih a posiive coefficien, however, is own lagged value is no significan. The R-squared is similar o wha has been found in oher sudies. The same variables significanly explain U.S. long-erm governmen bond reurns. U.S. sock reurns are negaively relaed o price-earnings raio. No oher variable is significan in his equaion. Sock reurns have proven raher difficul o predic, and, as expeced, his equaion has he lowes R-squared. U.S. ROE, U.S. shor-erm bond yield, and he U.S. price-earnings raio are significanly explained by heir own lagged values, illusraing ha a univariae AR() process could describe hem reasonably well. The resuls for he Japanese real reurns are less ypical han he ones for U.S. asses. Mos of he predicive variables do no significanly explain asse reurns, which in urn can be explained mosly by heir own lagged values. However, empirical evidence on Japanese reurns is scarce. Therefore, we do no lenghen our discussion here. As already highlighed, he model uses he informaion on expeced asse reurns. Table IV repors summary saisics of expeced real asse reurns implied by he esimaed VARs. They are repored in local currency. Common across all specificaions, shor-erm governmen bonds are he safes asse. In boh U.S. and Japanese markes, socks are he riskies asse. Therefore, he basic mean-variance paern of acual reurns is refleced in hese expeced reurns. Furhermore, he sandard deviaion of expeced real reurns is sable across differen specificaions. Alhough hey consisenly increase when he nominal exchange rae and he CPIinflaion raes are included in he VAR (insead of he real exchange rae), hey do so by less han % p.a. Thus, hese measures of expeced reurns seem robus o differen sysem esimaions. The real expeced reurns are on average equal o acual reurns, given he in-sample predicions 26

considered here. Neverheless, hey are much less volaile han acual reurns (summary saisics repored in Table II). In oher words, hese real expeced reurns are more persisen han he acual real reurns. 4.4 Opimal Porfolio Choice and he Bilaeral Curren Accoun Using he esimaed VAR coefficiens, he model is calibraed using differen relaive risk aversion parameers ( ). As already menioned, he model s calibraion is accurae for elasiciies of ineremporal subsiuion around. Therefore, resuls are repored for differen risk aversion coefficiens, bu we assume ha. 99 and. 92 in annual erms. 25 We firs calculae each counry s opimal allocaion o foreign asses. For a U.S. invesor, his opimal allocaion is he sum of all holdings in Japanese asses. Similarly, for a Japanese invesor, i is he share of Japanese pos-consumpion wealh invesed in U.S. asses. Formally, we obain he ime, H F series of and :, H, H, H, H, H sb, lb, socks, ROE,, (24) F F sb, F lb, F socks, F ROE,. (25), H F Figure III plos he ime series of (- ), and he U.S. bilaeral curren accoun wih, Japan. The negaive of he opimal Japanese porfolio shares allocaed o U.S. asses is he variable relev an o he U.S. bilaeral curren accoun according o equaion (7). These opimal porfolio shares are calculaed based on he VAR specificaion wih real exchange raes. We plo hese series for relaive risk aversion parameers of and. 26 25 The resuls are robus o differen parameer values of he ime discoun facor and he elasiciy of ineremporal subsiuion, as long as hese values are close enough o. 26 The empirical evidence on he equiy premium puzzle suggess values beween and 6. See Ai-Sahalia and Lo (2). 27

Even hough his figure addresses only par of he sory, i sheds some ligh on agens behavior. The main mechanism behind he ime-varying porfolio shares in our model is he expeced changes in asse reurns across he differen asses. If we assume a permanen improvemen in he U.S. invesmen opporuniy se and everyhing else remains unchanged, hen, according o our model, an invesor should increase her porfolio share on U.S. asses. If his invesor is Japanese, she would increase her holdings of U.S. asses. If a U.S. invesor is considered, her holdings of Japanese asses should fall. This implies, ceeris paribus, ha he U.S. bilaeral curren accoun wih Japan should worsen. The repored resuls are robus o he exclusion of individual asses and o differen parameer values. Alhough he dynamics of opimal porfolio shares does no change considerably wih he relaive risk aversion parameer, he average values of, H and F are highly sensiive o his parameer. When a smaller value of he relaive risk aversion parameer is used, individual porfolio shares are exremely high because of leveraged porfolios. Reasonable values for hese shares are obained only when larger parameer values are considered. In our model wih exogenous asse reurns and endogenous porfolios, agens ake advanage of any small excess risk-adjused reurns. High levels of relaive risk a version are hus needed o discourage excessive porfolio leverage. This parameer migh be capuring he model s sensiiviy o he well-known equiy premium puzzle, exensively documened in he inernaional finance lieraure. Given he mechanism highlighed in his paper and he esimaed expeced asse reurns, he paern of increased weigh on U.S. asses repored in Figure III is sriking. The correlaion coefficien beween hese wo measures is around.75, varying lile wih he relaive risk aversion parameer. The fi of he graphs is remarkable, especially if one considers ha only informaion on asse reurns was used. Therefore, he main argumen in his paper relies on hese 28