(Asymmetric) Trade Costs, Real Exchange Rate Hedging, and Equity Home Bias in a Multi-Country Model *

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1 (Asymmetric) Trade Costs, Real Exchange Rate Hedging, and Equity Home Bias in a Multi-Country Model * Ju Hyun Pyun Korea University Business School September 15, 017 Abstract There has been controversy between (two-country) theory and the empirics about whether hedging against real exchange rate fluctuations in the goods market influences foreign equity holdings. This study reconciles the theory with the empirics by introducing a multi-country framework with asymmetric trade costs. We find that the incentive to hold foreign equities to hedge real exchange rate risk is negligible because multiple trade partners act as a hedging channel for real exchange rate fluctuations. Further, our theory calls for a country s covariance-variance ratio to be constructed as the sum of the bilateral covariance-variance ratios of the multiple partners. The empirical analysis of 4 advanced countries confirms the theoretical prediction. Keywords: Multi-country model, International portfolio allocation, Real exchange rate hedging, Equity home bias, Trade costs, Non-tradable goods JEL Code : F30, F36, F41, G11 * I gratefully acknowledge Paul Bergin and two anonymous referees for constructive suggestions. This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-015S1A5A801550). All remaining errors are my own. Korea University Business School, 145, Anam-Ro, Seongbuk-Gu, Seoul , Tel: , jhpyun@korea.ac.kr 1

2 1. Introduction A stream of general equilibrium (GE) research in international macro-finance has explained portfolio home bias by invoking frictions in international goods markets, such as trade costs and the existence of non-tradable goods. The portfolio decision drawn from this line of research is determined by the equilibrium-hedging properties of relative equity returns against the consumption risk that comes from the relative price difference between home and foreign goods owing to trade costs. In this model, technically, home equity holdings can be presented as a proportion of the covariance of the home real exchange rate (with the convention that a decrease in the home real exchange rate denotes a depreciation) and the home excess equity return divided by the variance of the home excess return (hereafter, the covariance-variance ratio). For instance, a positive covariance-variance ratio implies that home assets are a better hedge than foreign assets because the real exchange rate and home excess return move in the same direction. Previous studies using two-country GE models have debated whether the motive for hedging real exchange rate risk generates equity home bias in the model. This debate is closely related to the value of the abovementioned covariance-variance ratio. Obstfeld and Rogoff (000) derive a positive covariance-variance ratio from their parametric assumption that relative risk aversion is equal to the inverse of the elasticity of substitution and demonstrate that home bias in equities increases with trade costs. By contrast, using a more realistic parameterization, Coeurdacier (009) shows that the covariance-variance ratio is negative. He suggests that in his model, home consumers are better insured against real exchange rate fluctuations when they hold more foreign equities than home equities (i.e., they show bias toward foreign equities). 1 1 Obstfeld (007) allows separated equity claims on traded and non-traded industries and shows possible home bias in both tradable and non-tradable goods equities.

3 However, the subsequent empirical finding by van Wincoop and Warnock (010) pose a challenge to these theoretical studies. They compute the covariance-variance ratio for the United States and other developed countries based on the two-country model and find that the computed ratio is close to zero. A zero covariance-variance ratio means that demand for holding foreign equities to hedge against real exchange rate fluctuations hardly exists, and thus goods market frictions, which affect real exchange rates, do not influence the equity position. Some theoretical studies have begun to address this zero covariance-variance ratio by introducing an additional assumption that real exchange rate risk is fully hedged by either the forward market or the bond market (e.g., Engel and Matsumoto, 009; Coeurdacier, Kollmann, & Martin, 010; Coeurdacier & Gourinchas, 016). 3 Further, the literature focuses on other sources of shocks to explain a reasonable degree of equity home bias. For example, Berriel and Bhattarai (013) introduce a government spending shock and Coeurdacier and Gourinchas (016) examine redistributive shocks to the share of financial income relative to total output. This study bridges the gap between the theory and empirics by extending a two-country model to a multi-country model with asymmetric trade costs. 4 By doing so, we can derive a close to zero covariance-variance ratio in the model with reasonable trade cost parameters, which nullifies both the home bias and the foreign bias driven by trade costs. Our theory also calls for a country s covariance-variance ratio to be constructed by the sum of the bilateral covariancevariance ratios with multiple partners. Furthermore, we show that the computed covariance- This is an equity market capitalization-weighted average of 1 other industrialized countries. 3 Engel and Matsumoto (009) introduce nominal rigidities so that the real exchange rate can be hedged by forward foreign exchange markets, and Coeurdacier et al. (010) and Coeurdacier and Gourinchas (016) allow trade in real bonds denominated in goods to hedge real exchange rate risk. 4 Previous works such as Berg and Mark (015) and Bergin and Pyun (016) explain some of the anomalies in international macro-finance in a two-country setting by introducing a multi-country setting. Dedola, Lombardo, & Straub (011) introduce a multi-country model to allow for differences in financial development between advanced and emerging markets to investigate the impact of financial development on portfolio allocation. 3

4 variance ratios in the multi-country setting using price and equity return data for 4 advanced countries are consistent with the theoretical prediction. In the two-country model, the covariance-variance ratio is intuitively negative, as pointed out by Coeurdacier (009). When output in the home country is high, the home real exchange rate depreciates owing to the scarcity of foreign goods. Conversely, home equity returns are higher because home production is higher. Thus, home equity returns and the home real exchange rate move in opposite directions. However, in the multi-country framework, the motive for hedging home exchange rate fluctuation is weakened by the presence of multiple partner countries. Consider the following three-country example. Suppose there is a rise in country 1 s output (equity return), which leads to a decrease in the price of country 1 s goods. Now, if there is a third country (country 3), which substitutes country s goods with country 1 s goods, both of which are imported and require the payment of possibly different trade costs, then the demand for country s goods in the international market lowers, which results in an additional decrease in the price of country s goods. Thus, the relative price of goods in countries 1 and would not decrease as much as in the two-country setting. 5 The theoretical contribution of this study is that while we start building upon the twocountry work of Coeurdacier (009), we allow for multilateral trade partners and asymmetric trade costs among them. We show that the tremendous foreign equity bias disappears with reasonable trade cost parameters in a multi-country setting. Following Coeurdacier (009), we further include non-tradable goods into the model. While this generates slight equity home bias, we still preserve the property of the covariance-variance ratio driven by the multi-country setting. We also 5 Relative demand for home (country 1) goods compared with a bilateral partner s (country ) goods depends not only on the relative prices of both goods but also on those for home goods and the third country s (country 3) goods. Thus, the existence of a third country moderates the negative relationship between relative demand and price for the goods of countries 1 and. 4

5 demonstrate that when considering a higher-order approximation (over the first-order approximation), the symmetric two-country model and symmetric multi-country model provide slightly different implications because of the covariance structure at a risky steady state (Coeurdacier, Rey, & Winant, 011). 6 In addition, while van Wincoop and Warnock (010) calculate the covariance-variance ratio by using a two-country framework, our theory calls for a country s covariance-variance ratio to be constructed by the sum of the bilateral covariancevariance ratios. Lastly, we show that our empirical results are consistent with the theoretical prediction from the model. The remainder of this paper is organized as follows. Section develops the two-country model of Coeurdacier (009) into a three-country model and then discusses the implications. In Section 3, we show the simulation results with the calibrated parameters. Section 4 presents the empirical results for the covariance-variance ratios. Concluding remarks follow in Section 5.. A Three-Country Model.1. Goods market and preference We begin with a simple extension of Coeurdacier s (009) model to a multi-country framework. The model is a two-period (t = 0,1) endowment economy, as in Coeurdacier (009), with three symmetric countries: country 1 (home), country, and country 3. Each country produces one tradable good, and the representative agent in each country consumes all goods; goods can be shipped from one country to another with trade costs. In period t = 0, no consumption takes place, but agents can trade equities, which create claims on the country s future endowment. The 6 One may argue that along with the assumption that all countries are symmetric, when the number of countries increases, the importance of home-produced goods in the consumption of a representative agent decreases. Thus, the decrease in the importance of hedging the price risk of domestically produced goods is purely mechanical in this symmetric model. 5

6 representative household of country i has a standard CRRA preference, with the coefficient of relative risk aversion, γ, 1 C i U i E0 where 1 C i /( 1) 3 ( 1)/ cij for i = 1,,3, (1) j 1 where cij is the total consumption of goods from country j by the representative agent in country i. The parameter ϕ is the elasticity of substitution between home and foreign goods, which is assumed to be ϕ >1. Denote iceberg-type trade costs, τij. When a unit of goods is shipped across countries, 1/(1 ij ) goods arrive at the destination i. pi denotes the price of one unit of output in country i, and the price of one unit of country j s output sold in country i is p ( 1 ) p. The terms of i j ij j trade between countries is defined as the relative price of each country s tradable goods. Hence, the terms of trade between countries 1 and are q 1 p1 p. The consumption price index (CPI) of country i (i = 1,,3) is P 3 {(1 ) p } i ij j j1 1 1/(1 ), where τii=0. For simplicity, assume this trade cost is destination specific, τij = τi (j i). is defined as Based on the price index of each country, the real exchange rate between countries 1 and RER P p ((1 ) p ) ((1 ) p ) P ((1 ) p1) p ((1 ) p3) 1/(1 ) () where RER 1 p1 p 1, p p3 and (1 1) x (1 1) y ( x, y) (1 ) x (1 ) y 1/(1 ). A decrease in RER1 means the depreciation of the real exchange rate between country 1 and country. When 6

7 the terms of trade, q1, decrease, RER1 also depreciates... Financial markets Financial markets work in the same manner as described by Coeurdacier (009). Each country has an equity that creates claims on future endowment. The supply of each country s equity is normalized to unity. Each country s representative household has its own equity at birth and then trades equities in period 0. Country i has the following budget constraint at t = 0: 3 p for i = 1,,3, (3) S p S j1 where μij is the share of stock j held by country i at the end of period 0. ps is the stock share price, which is identical for all three countries. ij.3. Market equilibrium.3.1. Goods market conditions In period 1, a consumer in country i maximizes his/her utility under a budget constraint: for i = 1,,3, i j. (4) PC p c (1 ) p c p y p y i i i ii i j ij ii i i ij j j ji ji The budget constraint equalizes consumption expenditure to aggregate the (financial) incomes of the representative agent in country i, where financial incomes depend on the portfolio share {μij}. At this point, we take the portfolios chosen in period 0 as given. From the first-order conditions, we derive the following allocation across goods: c ii p C i i Pi, c ij (1 i ) p j Pi C i for i,j = 1,,3, i j. (5) The endowment constraints are given by 7

8 y c (1 ) c 3 i ii i ji ji for i = 1,,3, i j. (6) By using equations (5) and (6), we derive the relative outputs of countries 1 and : y y P P1 C P 3 P1 1 3 q1, C1 C1 where ( x, y) is a continuous function of x and y such that C (7), 1 x(1 ) y(1 ) ( xy, ). x (1 ) (1 ) y Note that in equation (7), the relative outputs of countries 1 and depend on country 3 s information, which is different from the two-country model in Coeurdacier (009)..3.. Financial market conditions In period 0, equity trade is allowed. The marginal cost of buying an additional unit of stock i in period 0 is equal to the expected marginal gain from the endowment in period 1. 7 We rewrite the first-order conditions in relative terms for countries 1 and by using the shadow endowment price as follows: C1 C E0 ( )( R1 R ) P1 P 0, where p j y j R j for i = 1,. (8) p s Further, market clearing in asset markets requires the sum of the portfolio shares of the three countries and the net supply of equities to be equal to 1: 1 i 1 i i3 for i = 1,,3, (9) 7 i,0 ps E0 p y j j C i P for j = 1,,3, where λi,0 is the Lagrange multiplier of the period 0 budget constraint (3). i 8

9 1 for i = 1,,3. (9 ) 1i i 3i.3.3. Competitive equilibrium A competitive equilibrium is described as a set of endogenous variables ФN that satisfies the consumer utility maximization problem, market-clearing conditions, initial endowment, and exogenous trade costs. The endogenous variable set ФN comprises all consumption allocations, asset allocations, prices, and the terms of trade, that is, ФN ={cij, μij, pj, ps, qij} (for i,j=1,,3). The exogenous variable set includes output (endowment) and trade costs, as follows: ФX ={yi, τi} (for i=1,,3)..4. Log-linearization around the equilibrium Denote the deviations of the variables from the equilibrium (in percent) by (^): x x xˆ, where x x is the steady-state value at the equilibrium. We follow the higher-order approximation techniques of Devereux and Sutherland (011) and Tille and van Wincoop (010). We take an approximation around the equilibrium where all three countries have the same endowment yi (for i=1,,3). At the steady-state equilibrium, the prices of goods are the same and the zero components of prices are assumed to be p p p First-order approximation The first-order component of the real exchange rate is ^ RER P ˆ P ˆ q ˆ q ˆ, (10)

10 (1 1) where 1 (1 (1 1) ) (1 1) (1 (1 ) ) 1 1 1, 1(1 ) (1(1 ) ) (1 ) (1 (1 ) ) 1 (1 ) (1 ) (1 (1 ) ) (1 ) (1 (1 ) ). ^ RER P ˆ P ˆ q ˆ q ˆ (10 ) (1 1) (1(1 1) ) (1 1) (1(1 3) ) where3 1 1, 1(1 3) (1(1 3) ) (1 1) (1 3) (1(1 3) ) (1 ) (1 (1 ) ) (1 ) (1 (1 ) ). Take the first-order approximation of equation (7), the relative outputs of countries 1 and, as a benchmark, ˆ ˆ ˆ y1 y q1 1 PC1 ( 1) RER1 PC13( 1) RER13 (7 ), where 1 PC 1 1 P C PC denotes the relative consumption expenditure. The excess stock returns between countries 1 and can be derived from (7 ), R p y p y qˆ1 1 3 qˆ13 1 PC1 PC13 (1 )(1 ) ( 1)( ) (11) R p y p y qˆ qˆ1 4 PC1 3 PC13 (1 )(1 ) ( 1)( ). (11 ) In a multi-country setting, if there is a rise in the output of country 1 (home), the relative price of home goods (the terms of trade) will not fall by as much as it did in a two-country setting, 10

11 because the relative outputs of countries 1 and are positively related to the relative prices of goods from countries 1 and 3. Moreover, the period 1 budget constraint, equation (4), is loglinearized. We express the period 1 budget constraint in relative terms by using the portfolio holdings in equations (9) and (9 ). Note that in this exercise, we assume that the trade costs are source-specific (μ11 = μ1) as well as symmetric trade costs for countries and 3. Hence, the equity share solution is refined as follows. The home equity shares of countries and 3 are pinned down as the same (μ = μ = μ33), and country 1 would hold the same share of foreign equities as countries and 3 (μ1 = μ13 = (1 μ1)/): 1 1 PC1 ( ) R1 ( 1 ) R13, (1) 1 1 PC13 ( 1 ) R13( ) R1. (1 ).4.. Second-order approximation The second-order approximation of the portfolio equation (8) is We also derive this for countries 1 and 3: cov( PC 1, R 1 ) (1 1/ ) cov( RER 1, R 1 ). (8 ) cov( PC13, R13) (1 1/ ) cov( RER13, R13). (8 ) Equations (8 ) and (8 ) are rewritten by using equations (1) and (1 ): 1 1 ( ) var( R1) (1 1/ ) 1cov( q1, R1) cov( q13, R1), (13) ( 1) var( R13) (1 1/ ) 3cov( q13, R13 ) 4cov( q1, R13), (13 ) where var( R 1 ) denotes the variance of excess returns of country 1 s stock over country s and 11

12 cov( q, R 1 ) 1 denotes the covariance between the terms of trade and excess stock returns for countries 1 and..5. Solution for the equilibrium portfolio From equations (13) and (13 ), we derive an equilibrium expression for the home portfolio share for country 1, μ1, which is similar to that in Coeurdacier (009) and van Wincoop and Warnock (010): 1 1 cov( RER1, R1 ) cov( RER13, R13) var( R1) var( R13) 1 1 cov( q, R ) cov( q, R ) cov( q, R ) cov( q, R ) 1. (14) var( R1) var( R13) In equation (14), home equity holdings, μ1, depend on two terms: (i) the market portfolio (1/3) due to the diversification motive and (ii) the hedging component due to real exchange rate fluctuations, leading to 1 cov( q, R ) cov( q, R ) cov( q, R ) cov( q, R ) var( R1) var( R13) This term is also called the home bias measure in van Wincoop and Warnock (010). The equity solution for the home country in a three-country model is similar to that in a two-country model. However, when computing the covariance-variance term in the hedging component, the three-country model differs from the two-country model. First, in this three-country framework, country 1 s excess stock returns over country s depend not only on the terms of trade between countries 1 and but also on those between countries 1 and 3 (see equation (11)). Thus, the 1

13 additional variability due to the existence of country 3 makes the covariance-variance ratio greater than that in the two-country setting. Note that if we assume symmetric trade costs in all countries here (τ1 = τ = τ3), then our home bias measure collapses into a simpler form as follows (where θ1 = θ3 =θ and θ = θ4 = 0): 1 cov(, ) 1 3 q1 R1 cov( q13, R13) var( R1) var( R13)..6. An N-country model with trade costs Now, we extend the three-country model to an N-country model. First, we conjecture the equity solution for the N-country case with trade costs: N 1 N 1 1 cov( RERi, j, Ri, j) i 1 N N. (15) j1, ji var( Ri, j) The home equity share (µ) in the N-country model in equation (15) resembles that of the threecountry model in equation (14). Again, it comprises (i) the market portfolio (which is 1/N) due to the diversification motive and (ii) the home bias measure, which includes the hedging component due to real exchange rate fluctuations, namely the covariance-variance ratio. Here, the latter home bias measure can be divided into three terms. The first term (N-1)/N is related to the number of countries, which contributes to increasing the home bias measure when N goes from two to infinity. The second term includes risk aversion (γ) with which home bias increases. The last term is the sum of the covariance-variance terms. The home country s real exchange rate risk can be diversified if there are more trade partners in the international market. Hence, the covariancevariance ratio decreases with the number of countries. Note that when trade costs are symmetric 13

14 across countries, N j1, ji cov( RER, R ) cov( q, R ) cov( q, R ) cov( q, R ). i, j i, j i1 i1 i i in in i... var( Ri, j) var( Ri1) var( Ri) var( RiN).7. Adding non-tradable goods into the three-country model Coeurdacier (009) also extends his model by allowing countries to receive endowments of nontradable goods along with tradable goods. Here, we closely follow Coeurdacier (009). The hedging portfolio now depends on the covariance between the relative equity returns and relative price of tradeable to non-tradable goods Model set-up Each country now produces two goods, a tradable (T) and a non-tradable good (NT). In period 0, agents trade home and foreign equities, which are claims to aggregate output of individual countries, but they cannot trade separate claims on tradable and non-tradable output in each country. 8 The equity returns of country i are a weighted share of the returns in both sectors. In period 1, country i receives an exogenous endowment NT exogenous endowment yi T yi of the tradable good of country i and an of the non-tradable good. The stochastic properties of endowment shocks are assumed to be symmetric: ( T ) T E0 yi yi ; ( T Var ) 0 yi T; ( NT E ) 0 yi yi for i=1,,3. NT, and ( NT Var ) 0 yi A household in country i has the same CRRA preferences over the aggregate consumption index. Here, the aggregate consumption index is a bundle of tradable and non-tradable goods as follows: NT 8 See Coeurdacier (009) for a detailed justification of this assumption. 14

15 1 C i U i E0 where 1 /( 1) 1/ T ( 1)/ 1/ NT ( 1)/ Ci ( ci ) (1 ) ( ci ), (16) where T c i is the consumption of a composite tradable good using tradable goods and NT c i is the consumption of non-tradable goods. ε is the elasticity of substitution between tradable and nontradable goods. η is the steady-state share of spending devoted to tradable goods. The tradable consumption index for country i is c T i 3 j1 T ( c ) ij ( 1)/ /( 1), where T c ij is country i's consumption of the tradable good from country j. The consumer price index is as follows: 1/(1 ) T 1 NT 1 Pi ( Pi ) (1 )( Pi ), (17) where T P i is the price index over tradable goods in country i and NT Pi is the price of non-tradable goods. The tradable goods price index is defined by P p p 3 T T 1 T 1 i ( i ) {(1 ij) j} ji 1/(1 ), where T pi is the price of the tradable goods in country i. The terms of trade between countries i and j are T T defined by q p / p. ij i j The goods market constraints for the tradable and non-tradable goods in country i=1,,3 are given by 3 T T T i ii (1 ij) ji ji y c c for i = 1,,3, i j (18) y NT i c for i = 1,,3, (18 ) NT i y y y for i = 1,,3. (18 ) T NT i i i The set-up of financial markets is the same as the case without non-tradable goods. However, note that here with non-tradable goods, a stock gives investors a claim on the aggregate output of the 15

16 country at the market value (where aggregate output at the market value of country i is the sum of sales in the tradable and non-tradable sectors). A household in country i owns the local stock and faces the budget constraint shown in equation (3). We also recall the market-clearing conditions in asset markets and those of net equity supply (equations (9) and (9 ))..7.. Solution methods and equilibrium portfolios As in the benchmark model, we solve the equilibrium equity portfolios μ by using the first-order approximation of the non-portfolio equations and second-order approximation of the Euler equations. We do not describe the resolution and approximation of the model (as the steps are similar to those of the benchmark model) and simply present some of the key equations (see also Coeurdacier, 009). The first-order approximation of the real exchange rate equations of countries 1 and is ^ ^ ˆ ˆ NT 1 1 ˆ 1 1 ˆ 13 (1 ) 1 RER P P q q P, (19) where ^ ^ ^ NT NT NT 1 1 P P P is the (first-order approximation of) the relative price of the non-tradable (1 1) goods of countries 1 and. 1 (1 (1 1) ) (1 1) (1 (1 ) ) (1 ) (1(1 ) ) (1 1) and 1 (1 ) (1 (1 1) ) (1 1) (1 (1 ) ) (1 ) (1 (1 ) ) Importantly, the real exchange rate now depends positively on the terms of trade (q) as well as on the relative price of the non-tradable goods (P NT ). The intertemporal allocation across goods from the consumer s utility maximization problem (16) and market-clearing conditions for tradable and non-tradable goods (equations (4), (18), (18 ), and (18 )) implies the following 16

17 equilibrium conditions for countries 1 and : T T ˆ ˆ ˆ y1 y ˆ ˆ q1 1 ( )( 1q1 q13) ( 1) RER PC 1 1 ( )( 3qˆ ˆ 13 4q1) ( 1) RER13 PC13, (0) NT NT NT (1) yˆ yˆ P ( 1) RER PC These two conditions are similarly derived for countries 1 and 3. The excess stock returns between countries 1 and can be derived from (7 ): T NT NT R1 qˆ ˆ ˆ 1 y1 P1 y1 ( ) (1 )( ). () We also derive the excess stock returns between countries 1 and 3 in the same way: T NT NT R13 qˆ ˆ ˆ 13 y13 P13 y13 ( ) (1 )( ). ( ) Again, we recall equations (1) and (1 ) of the first-order approximation of the budget constraints and the second-order approximation of the portfolio equations (8) and (8 ). We rewrite these second-order approximation equations as follows: NT ( ) var( R1) (1 ) 1cov( q1, R1) cov( q13, R1) (1 ) cov( P1, R1), (3) ( 1 ) var( 13) (1 1 R ) 4cov( q1, R13) 3cov( q13, R13) (1 ) cov( P NT 13, R13), (3 ) where var( R 1 ) denotes the variance of the excess returns of country 1 s stock over country s and cov( q, R 1 ) denotes the covariance between the terms of trade and excess stock returns for 1 NT countries 1 and. cov( P, R1) 1 indicates the covariance between the relative price of the nontradable goods and excess stock returns. Then, we solve μ1 by using these two equations, (3) and 17

18 (3 ), as follows: NT 1 q1 R1 q13 R1 P1 R1 var( 1) 1 1 cov(, ) cov(, ) (1 )cov(, ) 1 1 [ R NT 4 q1 R13 3 q13 R13 P13 R13 var( R13) cov(, ) cov(, ) (1 )cov(, ) ]. (4) Note that when assuming symmetric trade costs across countries, this formula collapses into NT NT 1 1 1cov( q1, R1) (1 )cov( P1, R1) 3cov( q13, R13) (1 )cov( P13, R13) var( R1) var( R13 ) 3. Simulation This section depicts the home equity solution and the covariance-variance ratio derived from the theory on trade costs. Here, we highlight the role of the multi-country setting in shaping the home bias measure and covariance-variance ratio. First, we assume symmetric trade costs and then allow for asymmetric trade costs Symmetric trade costs: Comparison with the two-country GE model Baseline simulation Figure 1 plots the home equity share (μ) and covariance-variance ratio with respect to trade costs from the two-country model to the multi-country model. Note that Coeurdacier (009) derives an analytical form of the home equity share (μ) from the two-country framework as follows: cov( RER 1 var( R 1 1, R ) 1 ) (1 )(1 ) (1 1 ), (5) 18

19 where, again, γ is relative risk aversion, is the elasticity of substitution between home and foreign goods, and the trade cost parameter, 1 (1 ) 1 (1 ) , is different from that in the multi-country model. We use the same parameter values relative risk aversion (γ) is and the elasticity of substitution (ϕ) is 5 as Coeurdacier (009) in his two-country model. In Panel A of Figure 1, when N= (for the two-country case), the blue line with a circle marker indicates foreign equity bias when trade costs are less than 14% (e.g., Anderson & van Wincoop, 004, estimated trade costs are in the range of 40% to 70%). In addition, an increase in trade costs (τ) rather increases foreign bias in the portfolio, in contrast to the argument that trade costs lead to home bias in equities. If trade costs are higher than 14%, the portfolio starts to be biased toward home equities; however, the model with unrealistically high trade costs seems to deliver too much home bias, which is not consistent with the real data. As the number of countries increases, the balanced portfolio share changes from 1/ (twocountry setting) to 1/N (N-country setting), which shifts the home equity share at the origin downward. However, interestingly, the home equity shares respond to trade costs differently as the number of countries increases. For a greater number of countries, the home equity share becomes closer to the balanced portfolio share while even increasing trade costs, and it begins to show foreign bias only for very large trade costs (i.e., for N=50, foreign bias starts to appear when trade costs are above 100%). When we allow for a multi-country setting, foreign bias starts to disappear with reasonable trade cost parameters in the range of 40% to 70%. In particular, when N=50, there is no longer foreign bias and home equity holdings are not sensitive to the size of trade costs. Panel B of Figure 1 plots the home country s covariance-variance ratio. First, the covariance-variance ratio starts from zero and decreases with trade costs. However, when the

20 number of trade partners increases, the covariance-variance ratio approaches zero with reasonable trade cost parameters and its curvature flattens. This finding implies that the diversification benefits driven by the multiple partners help nullify the foreign equity bias observed in a twocountry setting. One may argue that this is unsurprising in that the multi-country model simply results from decreasing the size of each country relative to the world and, therefore, decreasing the importance of home-produced non-tradable goods in the optimal consumption basket. To address this concern, the present study could be developed in various directions. For example, as Dedola et al. (011) use the benefit of the multi-country setting in their analysis by allowing for asymmetry in the size (or development) of individual countries, it could be interesting to examine the consequence of a multi-country model with asymmetry assumptions. In the following subsection, we therefore relax the assumption of symmetric trade costs across countries and discuss the implication from the adoption of asymmetric trade costs. [Insert Figure 1] 3.. Asymmetric trade costs with a multi-country model In Figure, we allow for asymmetric trade costs and examine how these influence the equity positions from our baseline three-country model to an N-country model. The same parameter values are used as in Figure 1 (γ =, ϕ = 5). For simplicity, we assume that the trade costs for all countries except for country 1 (home) are 63%. We simulate the home equity share and covariance-variance ratio for country 1 with respect to country 1 s trade costs. Panel A of Figure shows the home bias measure that quantifies the degree to which home equity holdings deviate from the balanced portfolio, instead of depicting home equity holdings, for comparison purposes. There is home bias if this measure is greater than zero and foreign bias otherwise. 0

21 For all cases from N = 3 to N = 50, the home bias measure deviates little from zero in contrast to the cases with symmetric costs, even when trade costs vary from 0% to 300%. In addition, compared with Figure 1 with symmetric trade costs, the discontinuity of equity solutions disappears in all cases. Indeed, home equity holdings change little as trade costs move. For the three-country case, there is still foreign equity bias with reasonable trade costs ranging from 40% to 70%. However, the degree of foreign equity bias becomes much smaller (the home bias measure is at most 0. compared with 0.5 for the three-country case with symmetric trade costs). More importantly, when increasing the number of trade partners, foreign equity bias reduces. For N=50, the home bias measure turns out to be close to zero regardless of the trade costs for country 1. In Panel B of Figure, the covariance-variance ratios also display similar patterns to the home equity shares. When trade costs are 63% in the baseline three-country model, the covariancevariance ratio is around 0.37, while it is 0.5 in the two-country model. Furthermore, when increasing the number of countries, the value of the covariance-variance ratios turns out to be starkly different from that of the two-country model of Coeurdacier (009). For instance, if trade costs are 63%, the covariance-variance ratio is about zero in the multi-country model (when N=50 or greater). Thus, the simulation result from the multi-country model with asymmetric trade costs shows that the asymmetric structure of trade costs contributes to decreasing the magnitude of the large negative covariance-variance ratio observed in the two-country model. In particular, it helps obtain a covariance-variance ratio of almost zero. It is thus conjectured that combinations of asymmetric trade costs may abolish a negative covariance-variance ratio. [Insert Figure ] 3.3. Model with non-tradable goods 1

22 In this section, we simulate the model with non-tradable goods following Obstfeld (007) and Coeurdacier (009) to strengthen the robustness of the results. While our simulation results derived from using the multi-country model with trade costs indeed make the negative covariancevariance ratio close to zero and thereby eliminate foreign equity bias, we fail to generate equity home bias. Here, we include non-tradable goods to focus on the role of goods market frictions in generating equity home bias. This approach also allows us to check the interaction between trade costs and non-tradable goods (e.g., Collard, Dellas, Diba, & Stockmann, 007; Obstfeld, 007; Coeurdacier, 009). Note that previous studies of equity home bias introduce not only non-tradable goods but also various shocks to generate equity home bias in the model. Figure 3 plots the home equity share (μ) and covariance-variance ratio of the model including non-tradable goods with respect to trade costs. We use the same parameter values used in Coeurdacier (009): risk aversion (γ) is, the elasticity of substitution between home and foreign goods (ϕ) is 5, the elasticity of substitution between tradable and non-tradable goods (ε) is 0.5, and the share of tradable goods in consumption expenditure (η) is In particular, we follow Coeurdacier s (009) benchmark case on the exogenous income process. The volatility ratio of the exogenous endowment process ( / ) is 0.5 and the correlation of the shocks NT T between sectors within each country ( ( ˆ T NT Corr y, y ˆ )) is 0.3. [Insert Figure 3] First, Panel A of Figure 3 replicates the home equity share and corresponding covariancevariance ratio for the two-country model with symmetric trade costs as shown in Coeurdacier (009). The dashed line indicates the balanced portfolio share. The home equity share here shows similar patterns to that of the two-country model with only trade costs: as trade costs increase, the home equity share begins to exhibit foreign bias and then switches to home bias with extremely

23 high trade costs. However, unlike the model with only trade costs, non-tradable goods generate equity home bias when trade costs are less than about 50%, which also generates a positive covariance-variance ratio. Panel B of Figure 3 shows home equity holdings and the covariance-variance ratio for the three-country model with symmetric trade costs and Panel C illustrates those for the three-country model with asymmetric trade costs (we assume that the trade costs for countries and 3 are 63%). The non-tradable goods in both cases contribute to producing equity home bias. In addition, when extending the two-country model to the three-country model, we show that a negative covariancevariance ratio turns out be close to zero or even positive. In Panel B of the three-country model with symmetric costs, the covariance-variance ratio is slightly greater than zero when trade costs range from 40% to 80%. Then, the covariance-variance ratio becomes negative when the trade costs for all three countries are above 80%. Panel C of the three-country model with asymmetric trade costs shows that the home equity share and covariance-variance ratio are not sensitive to trade costs, as also shown in Figure. Here, the asymmetric structure of trade costs again eliminates the discontinuity of home equity holdings. Unlike trade costs, non-tradable goods play a role in generating home bias and making the covariance-variance ratio greater than zero for any trade costs parameters ranging from 0% to 300%. In particular, the covariance-variance ratio turns out to be 0.1~0. with trade costs from 40% to 80%. 4. Empirical Analysis In this section, we compute the covariance-variance ratios by using real data based on a multicountry theoretical setting. We set the United States as the reference country. Note that our multi- 3

24 country setting does not require an equity market capitalization-weighted average of the individual countries to represent the rest of the world (as the counterpart of the United States), as in the twocountry model of van Wincoop and Warnock (010). Instead, we calculate the covariance-variance ratio of the real exchange rate and excess stock returns between the United States and individual countries j and then sum them according to the formula N j1, jus cov( RERUS, j, RUS, j ) shown in var( RUS, j ) equation (15). We introduce our data and present the results in the following subsection Data We collect monthly equity (stock) market price index data for 4 advanced countries, including the United States, from DataStream for comparison purposes with van Wincoop and Warnock (010). Note that data for some emerging markets are available. However, owing to possible heterogeneity in this group, we exclude emerging countries from this analysis. Most advanced countries stock index data are available from 1961 (see Table 1 for the list of countries and data availability). We limit data up to 007 to avoid abnormal influence from the global financial crisis on equity returns and prices. For the relative prices of countries that measure the real exchange rate index in the model, we collect four kinds of price index data to calculate the real exchange rate. First, the price measure is the price level of GDP (production price index (PPP)/exchange rate) from Penn World Table 8.0 ( An advantage of using these data is that they allow for the wider coverage of countries (up to 4 countries including the United States). Second, we use the countries general price index, the Consumption Price Index (CPI) from the World Bank, and the Production Price Index (PPI) from OECD statistics for robustness checks. 4

25 Lastly, we introduce the deflators of countries value-added over all industries from the OECD Structural Analysis (STAN) database. [Insert Table 1] 4.. Calculation of the covariance-variance ratio Figure 4, using the GDP deflator from Penn World Table 8.0, shows our baseline result for the bilateral covariance-variance ratio between the United States and other individual countries with respect to distance from the United States. Here, we use geographical distance as a proxy for trade costs. The bilateral covariance-variance ratios are scattered in the range between 0.5 and 0.3 with respect to trade costs. Interestingly, most of them are close to zero except for Iceland (ISL), the Netherlands (NLD), and Switzerland (CHE) for which the ratios are somewhat large or small (0.5 for ISL, 0.43 for NLD, and 0.41 for CHE). Among the 1 countries except for the above three, 1 show negative covariance-variance ratios but the magnitude is small and close to zero. Nine countries exhibit ratios greater than zero but less than 0.1. However, our theory calls for the US covariance-variance ratio not to depend on specific partners but rather be computed by summing the bilateral covariance-variance ratios of all partners. When summing all these bilateral ratios, we find that the US covariance-variance ratio, zero. N j1, jus cov( RERUS, j, RUS, j ) var( US, j ) R 0.05, which is close to [Insert Figure 4] We use the alternative measures for countries price levels as robustness checks. Figures 5 and 6 show the covariance-variance ratios by using the CPI and PPI, respectively. The covariancevariance ratios calculated by using the CPI in Figure 5 range from 0.11 to The US bilateral 5

26 covariance-variance ratios against most countries are close to zero. Figure 6 shows that the covariance-variance ratios by using PPI are between 0.18 and 0.1. Finally, we sum all the bilateral ratios over all partner countries in both figures and construct the US covariance-variance ratio ( and , respectively). Again, the US covariance-variance ratios are close to zero, which is consistent with our theoretical prediction. Lastly, Figure 7 uses the deflator of the value-added of all sectors (industries) in a country. Here, we present interesting findings: While the bilateral covariance-variance ratios are again between 0.16 and 0.1, the US covariance-variance ratio that comprises all the bilateral ratios turns out to be 0.87, which is slightly greater than zero. This result echoes our simulation result for the model with non-tradable goods. Note that when using real data for the United States and 1 industrialized countries, van Wincoop and Warnock (010) compute their benchmark covariancevariance ratios based on the two-country framework; these ratios are close to zero but slightly positive (0.16 when relative risk aversion (γ) is 5 and 0.3 when γ goes to infinity). However, the covariance-variance ratios calculated by van Wincoop and Warnock (010) are not exactly comparable with those from our theoretical results in the three-country model because they impose a two-country setting. [Insert Figure 5] [Insert Figure 6] [Insert Figure 7] 5. Conclusion In this study, we shed light on the role of equity holdings on real exchange rate hedging and examine theoretically whether trade costs play a role in generating bias in equity positions. 6

27 We extend the two-country GE model to a multi-country model and allow for asymmetric trade costs. While previous studies built upon two-country models show the gap between the theory and empirics, we reconcile the theory with the empirics simply by introducing a multi-country framework. Then, we show that the calculated covariance-variance ratios for excess stock returns and real exchange rates using data for 4 advanced countries are consistent with our theoretical prediction in the multi-country setting. An implication from this study is that our theoretical and empirical finding supports those of Berriel and Bhattarai (013) and Coeurdacier and Gourinchas (016), who maintain that equity home bias does not occur from real exchange rate hedging but rather from the hedging properties of stock returns against other sources of risk. Indeed, the presence of (asymmetric) trade costs in a multi-country framework supports that trade costs play a role in generating neither equity home bias nor foreign bias. Here, the existence of only trade costs (even very high trade costs of about 300%) is unlikely to explain equity home bias portfolios. However, non-tradable goods as an extreme friction in goods markets can work to generate home bias and render the covariancevariance ratio positive. Our theory-based covariance-variance ratios between the real exchange rate and equity returns in a multi-country setting help decompose the covariance-variance ratio into trade costs, the covariance between the terms of trade change and equity return, and that between the price of non-tradable goods and equity returns. Future research could aim to examine individual margins to explain the covariance-variance ratio more in detail. Lastly, the results from the multi-country model suggest that future GE model research should consider the multi-country framework and explore various sources of risk across countries with asymmetry assumptions to understand home bias in the portfolios in the international market. 7

28 References Anderson, J., & van Wincoop, E. (004). Trade costs. Journal of Economic Literature, 4, Berg, K. A., & Mark, N. C. (015). Third-country effects on the exchange rate. Journal of International Economics, 96(), Bergin, P. R., & Pyun, J. H. (016). International portfolio diversification and multilateral effects of correlations. Journal of International Money and Finance, 6, Berriel, T. C., & Bhattarai, S. (013). Hedging against the government: A solution to the home asset bias puzzle. American Economic Journal: Macroeconomics, 5(1), Coeurdacier, N. (009). Do trade costs in goods markets lead to home bias in equities? Journal of International Economics, 77, Coeurdacier, N., & Gourinchas, P.-O. (016). When bonds matter: Home bias in goods and assets. Journal of Monetary Economics, xx, xx xx. Coeurdacier, N., Kollmann, R., & Martin, P. (010). International portfolios, capital accumulation and foreign assets dynamics. Journal of International Economics, 80, Coeurdacier, N., Rey, H., & Winant, P. (011). The risky steady state. American Economic Review, 101(3), Collard, F., Dellas, H., Diba, H., & Stockmann, A. (007). Goods trade and international equity portfolios. NBER Working Papers Dedola, L., Lombardo, G., & Straub, R., 011. Home bias and portfolio dynamics in a multicountry model, mimeo. Devereux, M. B., & Sutherland, A. (011). Country portfolios in open economy macro models. Journal of the European Economic Association, 9, Engel, C., & Matsumoto, A. (009). The international diversification puzzle when goods prices are sticky: It s really about exchange-rate hedging, not equity portfolios. American Economic Journal: Macroeconomics, 1, Obstfeld, M. (007). International risk sharing and the costs of trade. Ohlin Lecture, Stockholm School of Economics. Obstfeld, M., & Rogoff, K. (000). The six major puzzles in international macroeconomics: Is there a common cause? NBER Macroeconomics Annual, Tille, C., & van Wincoop, E. (010). International capital flows. Journal of International 8

29 Economics, 80, van Wincoop, E., & Warnock, F. (010). Is home bias in assets related to home bias in goods? Journal of International Money and Finance, 9,

30 Figure 1. Symmetric trade costs and equity home bias in a multi-country model (comparison with Coeurdacier, 009) Home equity holdings Covariance-variance ratio 1.9 Panel A. Home equity share N= N=3 N=10 N= τ1 (trade cost) Panel B. Covariance-variance ratio N= N= N=10 N= τ1 (trade cost) 30

31 Figure. Asymmetric trade costs and equity home bias in a multi-country model (γ =, ϕ = 5, tau(-i)=0.63) Home bias measure Covariance-variance ratio Panel A. Home bias measure N=3 N=5 N=10 N= τ1 (trade cost) Panel B. Covariance-variance ratio N=3 N=5 N=10 N= τ1 (trade cost) 31

32 Figure 3. Model including non-tradable goods Home equity holdings & Cov-var ratio Panel A. country model with symmetric trade cost Home equity share Covariance-variance ratio Balanced portfolio line τ 1 (trade cost) Home equity holdings & Cov-var ratio 3

33 Home equity holdings & Cov-var ratio Panel C. 3 country model with asymmetric trade cost Home equity share Covariance-variance ratio Balanced portfolio line τ1 (trade cost) 33

34 Figure 4. Covariance-variance ratio for the United States, price of GDP, PWT 8.0 Cov_var ratio (GDP deflator, PWT 8.0) CAN ISL IRL ESP FRA NOR ITA GBR BELFIN DNK PRT DEU SWE NLD CHE GRC ISR N j1, jus cov( RERUS, j, RUS, j ) JPN KOR var( RUS, j ) HKG 0.05 NZL AUS simple distance (most populated cities, km) Figure 5. Covariance-variance ratio for the United States, CPI, WDI Cov_var ratio (Consumer price index, WDI) CAN ISL BEL IRLGBR DEU DNK CHE ESP FRA NLD SWE NOR FIN ITA PRT GRC ISR N j1, jus cov( RERUS, j, RUS, j ) JPN KOR var( RUS, j ) HKG NZL AUS simple distance (most populated cities, km) 34

35 Figure 6. Covariance-variance ratio for the United States, PPI, OECD Cov_var ratio (Product price index, OECD) IRL NLD BEL DEU DNK FIN FRA NOR SWE GBR ESP ITA GRC N j1, jus cov( RERUS, j, RUS, j ) var( US, j ) R JPN AUS simple distance (most populated cities, km) Figure 7. Covariance-variance ratios for the United States, Value-added deflator, OECD Cov_var ratio (Value added deflator, OECD) GBR NOR BEL ESP FRA DNK CHE FIN ITA PRT NLD DEU SWE GRC N j1, jus JPN cov( RERUS, j, RUS, j ) var( RUS, j ) simple distance (most populated cities, km) 35