International capital flows

Existing models of capital flows sit at two unsatisfying extremes: single-bond models that can only speak to net flows, and complete-markets models in which capital flows barely matter at all. “Most of what we know about capital flows is within settings where only one risk-free bond is traded. These models only have implications for net capital flows, not gross flows… At the other extreme are models where financial markets are complete. But capital flows do not really matter in these models and are rarely ever computed as the real allocation is independent of the exact structure of asset markets” (Introduction). The authors position their contribution against a stated need in the field, quoting Gourinchas (2006) that “the next obvious step is to build general equilibrium models of international portfolio allocation with incomplete markets,” and Obstfeld (2004) that “at the moment we have no integrative general-equilibrium monetary model of international portfolio choice, although we need one” (Introduction, fn. 3).

The standard method solves the order-O component of model variables from the order-O component of model equations, sequentially from O=0 upward – but for the difference in Home and Foreign portfolio shares (k^D), neither direction of this logic holds. The order-O component of k^D never affects the order-O component of any model equation, because it always enters multiplied by an expected excess return or a cross-country wealth difference that is itself zero at that order (Section 3.4, equations 9-10); and conversely, the order-O component of the portfolio Euler equation differential does not depend on the order-O component of any variable, for the identical reason (equation 11). “While the order O component of k^D_i,t does not affect the order O component of model equations, the lower order components of k^D_i,t do affect the order O component of model equations” (Section 3.4) – which is precisely what makes the problem a higher-order fixed point rather than a simple recursion.

The algorithm solves, in sequence for O = 1, 2, …, the order-(O-1) component of the portfolio share difference jointly with the order-O components of all “other” model variables, using the order-(O+1) component of the portfolio Euler equation differentials together with the order-O component of the “other” model equations (Section 3.5, “Solution Algorithm”). For O=1 this means: solve the first-order component of all non-portfolio variables as a function of the unknown zero-order portfolio difference k^D(0) using the first-order “other” equations, then pin down k^D(0) itself from the second-order component of the portfolio Euler equation differentials, which is “consistent with the intuition discussed in the introduction as k^D(0) depends on second moments that show up in the second-order components of the portfolio Euler equations” (Section 3.5).

Steady-state home bias (equation 30) is the sum of three ratios of second-order terms: a positive contribution from the cost of investing abroad, a contribution from the covariance between the real exchange rate and the excess return that is positive when relative risk aversion exceeds one, and a hedging contribution tied to future expected portfolio returns. “The first reflects the cost of investing abroad, tau, with a higher cost making investing in domestic equity more attractive. The second reflects the co-movements of the real exchange rate and excess return. Assuming gamma > 1, it is attractive for Home investors to invest in the Home equity if the excess return on Home equity is high in states where the Home price index is relatively high… [The third] is attractive for Home investors to invest in Home equity when the excess return on Home equity is high in states where expected future portfolio returns are low… This source is positive when there is consumption home bias” (Section 5.2). In the paper’s calibration these three forces combine to a home-equity share of 0.8, which the authors decompose as “a bias of +67% invested in the domestic country due to the financial friction tau, a negative bias of -40% due to a negative correlation between the real exchange rate and excess return… and a positive home bias of +3% due to the hedge” (Section 6.1, fn. 23).

A persistent, one-standard-deviation increase in Home productivity produces an immediate 2.6% real depreciation (rise in the relative price of the Foreign good), a 4.7% jump in the Home equity price, and a smaller 2.9% rise in the Foreign equity price (measured in Home goods), with both equity prices subsequently drifting back toward steady state. “The persistent increase in Home productivity boosts the supply of the Home good, leading to an immediate 2.6% increase in the relative price of the Foreign good (a Home real depreciation)… The persistent Home productivity shock immediately raises the Home equity price by 4.7%. The Foreign equity price rises by a small 0.3% [in Foreign goods]… While the increase in Foreign equity prices is larger when expressed in Home goods (2.9%), Home equity prices still increase by more on impact” (Section 6.2).

Gross capital flows are dominated by active portfolio reallocation rather than by the mechanical growth of existing portfolios with national savings. “A first step towards understanding the drivers of capital flows is to break them down into portfolio growth and portfolio reallocation components… While this [portfolio growth] channel is not negligible under our parameterization, [the results] show that the portfolio reallocation effect dominates the overall dynamics of gross capital flows” (Section 6.4). On impact there is a “retrenchment” in which both Home and Foreign investors shift toward their own domestic assets (driven mainly by the rise in the variance of the excess return and in its covariance with the real exchange rate and hedging term), while in later periods both reallocate toward foreign equity as the expected excess return channel takes over (Section 6.4).

Most of the change in the expected excess return following the shock is not associated with any capital flows at all; only one of three components of that change actually drives reallocation. The paper decomposes the change in the expected excess return into three parts – one from the change in relative equity supply, one from changes in second moments (variance/covariance) affecting the average portfolio share, and one from the rise in Home savings interacting with home bias – and finds that “the first two aspects therefore illustrate the need to be careful when linking capital flows with changes in expected returns. Most of the changes in the expected excess return are not related to capital flows at all,” while the third component “moves in opposite direction from the overall expected excess return, which rises after the shock, again illustrating the pitfalls in empirically linking capital flows to changes in expected returns” (Section 6.4).

Because expected excess returns are zero to the first order (a standard no-arbitrage condition), the model’s net external debt is financed entirely by the present value of expected future trade surpluses, not by favorable expected returns on external assets relative to liabilities – a result the authors explicitly note cannot match some empirical findings. “As expected future excess returns are zero to the first-order, the net external debt is simply equal to the present value of expected trade surpluses. The model can therefore not account for empirical findings by Gourinchas and Rey (2006) that net external debt is to some extent financed by differences in expected returns… [this] is a standard arbitrage condition found in virtually any asset pricing model, and can only be relaxed by introducing elements that break the arbitrage across various assets” (Section 6.5). In the numerical example, the initial 6.2%-of-GDP net external debt is financed by future trade surpluses, with the individual components of expected returns (dividend yield, price changes, exchange rate changes) nonetheless shown to be economically large even though they net to zero as a first-order determinant of debt (Section 6.5).

Higher costs of investing abroad reduce welfare, with the loss rising to about 1.2% of wealth at the benchmark friction of 0.4%, and the loss function is concave – extra frictions matter less once investors are already close to full home bias. “The welfare loss rises to about 1.2% when tau = 0.4% as in the benchmark parameterization. In addition, the loss is concave in tau. When tau gets close to 0.5%, the portfolio approaches full home bias with investors holding only domestic equity. With little exposure to foreign equity, investors are little affected by further changes in the financial friction” (Section 6.6). The authors note that welfare analysis, like the computation of gross flows, requires combining the third-order component of the portfolio Euler equations with the second-order component of the other model equations – the same higher-order machinery developed for gross flows (Section 6.6).

The solution method is, by the authors’ own account, “essentially the same” as one developed independently and simultaneously by Devereux and Sutherland (2006a,c), but the two papers differ in emphasis: Devereux and Sutherland focus on the most efficient way to solve the portfolio fixed-point problem and provide an analytical solution for a broad model class, while this paper instead emphasizes the general reason standard methods break down and works through the implied dynamics of gross and net capital flows in a specific model. “Our focus is different in two ways. First, we characterize at a general level why standard solution methods for DSGE models break down with portfolio choice, and present an iterative solution method to solve for portfolio choice that applies to any order of approximation. Second, we illustrate the implications of this method for the dynamics of gross and net international capital flows in a simple model” (Section 2). The paper also distinguishes its approach from the “unusual hybrid” continuous-time/discrete-time methods of Evans and Hnatkovska (2005, 2006a,b), noting that its own method “stays much closer to… existing methods, modifying them in a way that accommodates portfolio choice” (Section 2).