Country Portfolios in Open Economy Macro-Models

Existing open-economy macro models mostly ignore portfolio composition – tracking only net foreign assets – even though gross cross-country asset and liability positions are, empirically, “multiples of GDP” for countries such as the UK, and this matters because valuation effects from exchange-rate and asset-price movements on such large gross positions “are the same order of magnitude as annual current accounts” (Introduction, p. 1, citing Lane and Milesi-Ferretti 2001, 2006). The authors argue this raises first-order questions – what determines the size and composition of gross portfolios, how portfolio composition affects the international transmission of shocks, and how it should shape optimal monetary and fiscal policy design – that “current theoretical models and solution methods cannot answer… in any very systematic way” (p. 1).

Because “optimal portfolios are not uniquely defined in a non-stochastic steady state” and are “also not defined in a first-order approximation to a DSGE model, since such an approximation satisfies certainty equivalence, so that all assets are perfect substitutes” (Introduction, p. 2). The paper demonstrates this concretely in its two-asset example: the non-stochastic version of the household’s portfolio first-order conditions implies “both assets pay the same rate of return,” and a first-order approximation implies “both assets have the same expected rate of return” – in each case “any value for α is consistent with equilibrium,” because “neither the non-stochastic steady state nor a first-order approximation capture the different risk characteristics of assets” (Sec. 2.2, pp. 8-9).

Since “the risk characteristics of assets only show up in the second-moments of model variables,” the authors show that a second-order approximation of the portfolio optimality (Euler) conditions – while the rest of the model is still solved only to first order – is sufficient to tie down the zero-order (steady-state) component of the portfolio, building on a principle “formalised by Samuelson (1970), who established that, in order to derive the zero-order component of the portfolio, it is necessary to approximate the portfolio problem up to the second order” (Sec. 2.2, p. 9). The paper’s stated “main contribution… is to show how this solution can easily be derived in standard DSGE models” (p. 9), producing closed-form analytical results rather than the numerical algorithms of related methods (Judd et al., 2002; Evans and Hnatkovska, 2005; Tille and Van Wincoop, 2007).

Because of “Property 2”: only the zero-order (constant) component of the portfolio, ᾱ, affects the first-order-accurate behavior of consumption, output, exchange rates and other macro variables – the first-order (time-varying) portfolio component does not, since the portfolio excess-return term in the budget constraint reduces to ᾱr̂x,t once the steady-state excess return is zero (Sec. 2.3, “Property 2,” p. 12-13). The authors conclude that “time variation in portfolios is irrelevant for all questions regarding the first-order responses of macroeconomic variables like consumption, output, real exchange rates, etc.,” so “the solution we derive exhausts all the macroeconomic implications of portfolio choice at this level of approximation” (Introduction, p. 2) – though they note that welfare analysis, which requires a second-order approximation of the model as a whole, does require the (time-varying) first-order portfolio component, addressed only briefly in Section 4.

The equilibrium home share of home equity holdings depends on δ, the share of capital (versus labor) income in total output, and on σKL, the covariance between home capital-income and home labor-income shocks (eq. 24-25, p. 14-15); when capital and labor income are uncorrelated (σKL = 0) “agents continue to hold a balanced portfolio of home and foreign equity,” but “the equilibrium portfolio deviates from an equal balance of home and foreign equity when there is some correlation between capital and labour income,” and “when there is a negative correlation, i.e. σKL < 0, there will be home bias in equity holdings” (Sec. 2.3, p. 15). The intuition is that home equity, as a claim on home capital income, is a poor hedge for home labor income when the two are negatively correlated, so households would ideally use equity to hedge labor-income risk, and this hedging demand – alongside the paper’s other parameters – shapes the tilt away from a naive 50/50 split.

Only in special, knife-edge cases – the general result is that risk sharing is imperfect. With only equities traded and no labor income (δ = 1), the model has a known exact solution in which “home and foreign agents hold a balanced portfolio of home and foreign equities,” which “implies perfect consumption risk sharing,” and the paper’s zero-order method exactly reproduces this benchmark (Sec. 2.3, p. 14-15, fn. 8). But “more generally, in cases where this is labour income risk… there is no exact solution to the model, but our zero-order solution provides an approximate solution” (p. 15) that in general falls short of full risk sharing.

Section 3 shows the same second-order-approximation logic extends to an n-asset economy with a general DSGE structure – arbitrary specifications for labor supply, sticky or flexible prices, and traded goods – where “only those parts of the model directly necessary for understanding the portfolio selection problem need to be explicitly described” (Sec. 3.1, p. 17). The home and foreign portfolio first-order conditions, together with market clearing, again reduce (via a second-order approximation) to a system of equations in the vector of excess returns and the covariance of those returns with relevant marginal-utility differentials, yielding “a general expression for the zero-order portfolio” (Sec. 3.2, p. 19) that specializes to the closed-form two-asset formula as a special case.

Adding home- and foreign-currency nominal bonds to the traded-equity (“EQ”) economy, producing the “EB economy,” changes the hedging problem because “monetary shocks now have an impact on equilibrium portfolio holdings,” since such shocks “have an impact on the real returns on nominal bonds and thus affect their hedging properties” (Sec. 3.3.2, p. 27). The paper finds that “in general, risk sharing is not perfect in the EB economy” despite the extra assets: “nominal bonds provide an extra means for hedging government spending and endowment shocks,” but “their efficacy as hedging instruments is partly undermined by monetary shocks,” with perfect risk sharing recovered only in special cases such as the absence of monetary shocks (σM = 0), where “nominal bonds offer a good hedge for government spending shocks, while equities provide a good hedge for endowment shocks” (p. 27-28).

Following the same Samuelson-style logic one order further, the paper states that tying down the first-order (time-varying) component of the portfolio requires a third-order approximation of the portfolio optimality conditions, combined with a second-order approximation of the rest of the model – “in order to derive the Nth-order component of the portfolio, it is necessary to approximate the portfolio problem up to order N + 2” (Sec. 4, p. 29, citing Samuelson 1970). The paper sketches the strategy – postulating a linear law of motion for the portfolio deviation in the model’s state variables, then solving for the coefficient vector that satisfies the third-order condition – but states that “the details of the solution procedure… are presented in Devereux and Sutherland (2007), where we derive a closed-form solution which is applicable to a wide class of models” (p. 30), i.e., the full first-order portfolio-dynamics solution is developed in a companion paper rather than here.

The authors position their method as resting on established mathematical foundations – Samuelson’s (1970) mean-variance approximation result and Judd and Guu’s (2001) characterization of the zero-order portfolio as a bifurcation point in the set of non-stochastic equilibria – while offering, as their “main contribution,” a way to derive this solution “easily” within standard DSGE perturbation methods (Sec. 2.2, p. 10-11). They contrast their closed-form analytical approach with contemporaneous numerical methods – Tille and Van Wincoop’s (2007) iterative algorithm, Judd et al.’s (2002) spline-collocation approach, and Evans and Hnatkovska’s (2005) perturbation/continuous-time hybrid method – describing the latter two as “very complex compared to our approach” and “a significant departure from standard DSGE solution methods” (Introduction, p. 4), and noting that the method, though motivated by open-economy applications, “can be applied to any heterogeneous agent DSGE model, whether in a closed or open economy context” (p. 3).