## _wp07284

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---

### I. Introduction — motivation and problem statement
- Open economy macro models typically represent international financial linkages via net foreign assets and the current account.
- Recent evidence documents large cross-country gross asset and liability positions (Lane and Milesi-Ferretti (2001, 2006)); raises questions:
  - Can international macro models explain observed structure of portfolio holdings?
  - What macro determinants drive the size and composition of gross portfolio positions?
  - Gross positions imply nominal exchange rate or equity price changes produce capital gains and losses with large effects on net foreign assets.

### Limitations of existing methods
- Standard general-equilibrium models with complex asset markets are difficult to solve for portfolio choice.
- Complete-markets approaches permit deriving implied country portfolios but are questioned by empirical evidence on failure of risk-sharing.
- In incomplete-markets models:
  - Equilibrium portfolio allocation and macro outcomes are mutually dependent and generally intractable except in restricted cases.
  - Standard first-order and second-order approximations fail: optimal portfolio allocation is generally indeterminate in a first-order approximation and indeterminate in the non-stochastic steady state.

### Proposed solution method — overview and key insight
- Method applicable to any standard open economy model with any number of assets, any number of state variables, and either complete or incomplete markets.
- Key innovation: at the level of approximation normally used in multi-country DSGE work, only a solution for steady-state asset holdings is required; time variation in portfolio shares is irrelevant for first-order responses of macro variables (consumption, output).
- Methodological approach:
  - Characterize optimal portfolio using a second-order approximation of the portfolio selection condition combined with a first-order approximation of the remaining parts of the model.
  - Endogenous portfolio weights depend on the variance-covariance matrix of excess returns produced by the general-equilibrium model; solve the simultaneous system to obtain a closed-form analytical solution for portfolio weights.
- Practical advantages:
  - Fits into standard DSGE solution approach; can be applied using a standard first-order accurate solution.
  - Simple to implement; analytical results for small models and numerical results for larger models.
  - Execution time comparable to obtaining a standard log-linear solution.

### Relation to literature
- Alternative numerical approaches: Judd et al. (2002), Evans and Hnatkovska (2005), Devereux and Saito (2005) — handle time-varying portfolios but are complex and depart from standard DSGE methods.
- Close relation to Samuelson (1970), Judd (1998), Judd and Guu (2001):
  - Samuelson: mean-variance approximation identifies optimal portfolio in a near-non-stochastic world.
  - Judd and Guu: use bifurcation point as approximation to overcome indeterminacy.
- The steady-state gross portfolio holdings derived correspond to the bifurcation (approximation) point identified by Judd and Guu.

### Scope and structure
- Presents general portfolio choice within a generic open economy model.
- Develops solution method that yields steady-state portfolio holdings and embeds in standard DSGE techniques.
- Higher-order aspects and time-variation require extensions; companion work addresses higher-order/iterative numerical solutions.

---

### Model setup and definitions (generic)
- Two-country world: home and foreign. Home produces YH at price PH; foreign produces YF at price P∗F.
- Home preferences: U = Σ_{τ=t}^∞ β^{τ−t} [u(C_τ) + v(·)] (equation (1)); u(·) twice continuously differentiable; v(·) captures non-portfolio-relevant components.
- n assets with return vector r_t = [r_{1,t}, r_{2,t}, ..., r_{n,t}]′ measured in units of home consumption good C.
- Home-period budget constraint: equation (2). Net wealth: W_{t−1} = Σ_i α_{i,t−1} (equation (3)).
- Numeraire: nth asset; define excess returns r_{x,t} on other 1n− assets and vector α′_{t−1} excluding α_{n,t−1} (equation (4)).

### Portfolio first-order conditions and equilibrium
- Home FOCs for portfolio choice: set (5). Foreign analogous: set (7).
- Market clearing: α_t + α∗_t = 0 (i.e., α_t = −α∗_t).
- Exact solutions infeasible due to nonlinearity and expectations; replace FOCs with second-order approximations.

### Approximation strategy and formal properties
- Approximation point: non-portfolio variables X at a non-stochastic steady state; portfolio vector α treated as endogenous and solved so second-order approximations of FOCs hold in neighborhood of (X, α).
- Use second-order Taylor expansion to capture second moments (risk characteristics).
- Key properties (as stated):
  - Property 1: In first-order approximation of non-portfolio parts, only α (portfolio at the approximation point) appears; deviations ˆα do not affect first-order macro dynamics.
  - Property 2: Solution of second-order approximation for portfolio requires only first-order accuracy of non-portfolio parts.
  - Property 3: Expected excess returns E_t[ ,1 x r + ] are zero in all time periods in a first-order approximation.
  - Property 4: Conditional one-period-ahead second moments from first-order approximation are non-time-varying if Σ (covariance matrix of innovations) is non-time-varying.
  - Property 5: α% enters first-order non-portfolio equations only via budget constraints; by Walras' Law only one budget constraint matters (home budget constraint linearised as 1ˆˆˆˆ tt t xt WWYC rβα′ − = + − + %).
  - Property 6: Realised excess asset returns ˆx_t r are zero-mean i.i.d. up to first-order accuracy.

### Second-order portfolio optimality approximations (selected relations)
- Second-order expansion of home FOCs yields vector form (9) and foreign analogue (10).
- Combined equilibrium expressed by (11) and (12):
  - (11) and (12) bind equilibrium portfolio holdings and expected excess returns respectively.
- At employed approximation level portfolios are non-time-varying (time subscripts can be omitted) because expected excess returns are zero at first order and second moments non-time-varying when Σ non-time-varying.

---

### Partial-equilibrium insight and mean-variance form
- Substitute first-order linearised budget constraints (15) and (16) into (11):
  - (15): ()2 1,111 1 ˆ ˆˆ ˆ ˆˆ , txt ttt Cr WWYOαε β ′ ++++ = + − ++ %
  - (16): ()2 1,111 1 ˆ ˆˆ ˆ ˆ txt ttt Cr WWYOαε ∗ ′ ∗ ++ ++ = − − + + +
- Partial-equilibrium portfolio allocation α% from (17):
  - (17) ()1 1 2 xxx D V V O α ε − = − + %
  - V_{xx}, V_{xD}, V_{xA} depend on α% via impact of gross holdings on net wealth.
- Partial-equilibrium expected excess returns (18):
  - (18) [] ()23 11 ˆˆ 22 xxx A Er E r V O ρε [ ] = − + ++
- Forms resemble mean-variance solutions; full general equilibrium needed because V matrices depend on α% through general-equilibrium channels.

### General-equilibrium derivation and closed-form solution for α%
- Represent first-order non-portfolio system with net-wealth innovation treated as exogenous t ξ (zero-mean i.i.d.) to obtain state-space (20) with solution (21).
- First-order realised excess returns linear in exogenous shocks (22).
- Relations:
  - Realised excess return on portfolio: 1 ˆ txt r ξ α ′ + + = % (equation (23)).
  - Solve for 1 t ξ + and 1 ˆ x t r + in terms of exogenous shocks: (24) and (25).
- Key equilibrium expression (30):
  - (30) ()3 111 ,1 ˆˆˆ ˆ (/) tt t txt ECCQ r R D O ρε ∗ ′ +++ + [ ] − − = Σ + %%
- Equilibrium α% solves matrix equation (31): 0 RD ′ Σ = %%
- Closed-form equilibrium (35):
  - (35) ()1 221 12222 R D R D R R RD O α ε −′′′′ [ ] = Σ − Σ Σ + %
- Residual in (35) is first-order; actual portfolio α related to α% via α = Y α α β = % (notation as in source).
- Interpretation: α% is the equilibrium portfolio allocation in a world arbitrarily close to non-stochastic (Σ = ζ Σ_0, ζ → 0); α% corresponds to the bifurcation point in the set of non-stochastic equilibria.

### Practical implementation algorithm (outline)
- Equilibrium α% requires only first-order information from the non-portfolio model to construct D and R matrices and then apply (35).
- Implementation step (first listed in source):
  1. Solve the non-portfolio equations of the model in the form of (20) to yield a solution in the form of (21).
- (Further steps follow by extracting appropriate rows to form R1, R2, D1, D2 and computing α% via (35).)

---

### III. Applications — two worked examples

A. Example 1 — Two-country endowment model with nominal bonds
- Model primitives and shocks:
  - Home utility: equation (36) (textual form preserved).
  - Endowments and money supplies follow AR(1) in logs (37) and (42).
  - Shocks Yε, Yε∗, Mε, Mε∗ i.i.d. symmetric on [, ]εε− with 2[][ ] YY Y VarVarεεσ ∗ == and 2[][ ] MM M VarVarεεσ ∗ == .
  - Four shock processes independent; covariance matrix Σ composed of 2 Y σ, 2 Y σ, 2 M σ, 2 M σ entries as shown.
- Budget and returns:
  - Budget constraint (38); net wealth (39).
  - Real returns on bonds given by (40); law of one price PS = P∗S; quantity-theory relations (41).
- First-order approximate results and properties:
  - FOCs and resource constraint approximations yield (46)-(53) including:
    - Expected excess return zero up to first order: 2 ,1 ˆ (). txt ErOε + ⎡⎤ = ⎣⎦ .
    - Nominal interest differential equals expected inflation differential: (52).
    - Realised excess return expression (53).
  - State and control vectors used to form system (20): s, c, x as defined.
- Solution for bond holdings:
  - Resulting R1, R2, D1, D2 given by simplified relations:
    - [][] 12 0,1111RR==−−
    - [] 11 12 11 2(1)  ,0    0 YY DD ββ βζβζ β −− −− ⎡⎤ =−=   − ⎣⎦
  - Applying (121) yields bond holdings expression:
    - 2 22 2()(1) Y B B MYY σ αα σσβζ ∗ =−   =− +− %%
  - Qualitative conclusions:
    - Home consumers take a negative position in home-currency bonds, balanced by positive position in foreign-currency bonds.
    - Home price level is countercyclical; home currency bonds pay relatively more when home output is high → home bonds are a relatively bad hedge against home output risk; foreign bonds a relatively good hedge.
    - Home exchange rate appreciates with a positive home output shock, increasing return on home bonds relative to foreign bonds in that state.
    - Monetary policy volatility reduces gross holdings of bonds: monetary volatility reduces usefulness of nominal bonds as hedges even with flexible prices.

B. Example 2 — Two-country production model with equities
- Model extensions:
  - Endogenous production, labour supply, productivity and fiscal shocks; home utility in (56).
  - Consumption aggregator CES form (57); CPI (58).
  - Equity trade replaces bond trade; budget (59); net wealth (60).
  - Firms: linear labour production with productivity At (AR(1)) (61).
  - Government purchases G with AR(1) shocks; non-stochastic G consistent with yg ratio.
  - Equity payoff and return definitions as in text.
- First-order approximation and excess returns:
  - Excess return on foreign equity: expression (63).
  - System cast in form (20) and solved to state-space (21).
- Solution for equity holdings:
  - Additional assumption: AG ζζ = .
  - Computed R1, R2, D1, D2 (matrix forms given in text) with definition (1(1)). y gρθΘ = − + − .
  - Applying (35) yields equity holdings expression (64):
    - 2 2 2 1 1 2(1)(1)(1) y G E E yA g g ρ σ αα βθ    σ ∗ ⎛⎞ =−   =−− ⎜⎟ ⎜⎟ −−− ⎝⎠ %%
  - Qualitative conclusions:
    - In absence of government spending shocks, home households hold a perfectly pooled portfolio of home and foreign equity; given budget constraint (59) (home households receive all home profits), this requires E α % negative.
    - When 0 y g > , home equity is a good hedge against consumption risk from government spending shocks (home profits relatively high when government spending is high); households will hold less than a fully pooled equity portfolio.
    - Government spending shocks can explain partial or full home bias in equity portfolios.

---

### IV. Method implications, extensions, and practical notes
- The approximation method:
  - Is easy to implement and applicable in any model relying on first-order approximation methods.
  - Allows analysis of first-order properties of macro variables (GDP, consumption, real exchange rate) via impulse responses or second-moment calculations.
  - Can be used to study macro determinants of optimal steady-state portfolio holdings for any asset, under complete or incomplete markets.
- Time-variation in portfolio holdings requires higher-order approximation; companion paper (Devereux and Sutherland 2006b) extends methods to higher-order aspects.
- Method applicable to sticky-price open economy models; Devereux and Sutherland (2006a) use it to analyze monetary policy rules and asset holdings in sticky-price contexts.

*Source: _wp07284 - Section and references as provided in the PDF content unit.*

### References..............................................................................................................

### _wp07284 - References..............................................................................................................

### I. INTRODUCTION: motivation and problem statement
- Open economy macroeconomic models typically represent international financial linkages in terms of net foreign assets and the current account.
- Recent data show large cross-country gross asset and liability positions; Lane and Milesi-Ferretti (2001, 2006) show that these gross portfolio holdings have grown rapidly, particularly in the last decade.
- Large gross positions raise questions:
  - Can international macroeconomic models explain the observed structure of portfolio holdings?
  - What macroeconomic determinants drive the size and composition of gross portfolio positions?
  - Gross asset and liability positions imply that nominal exchange rate or equity price changes produce capital gains and losses that can have very large effects on the value of net foreign assets.2

### Limitations of existing methods and need for new approach
- Standard general equilibrium macroeconomic models with complex asset markets are difficult to solve for portfolio choice.
- Complete markets approaches (e.g., Engel and Matsumoto (2005) and Kollmann (2006)) allow solving macro equilibria without first solving asset-market behavior and permit derivation of implied country portfolios.
- Empirical evidence on failure of risk-sharing across countries casts doubt on the completeness of international financial markets; complete markets may not approximate the true position for optimal portfolios.
- In incomplete-markets models:
  - Equilibrium portfolio allocation depends on macroeconomic outcomes, and macroeconomic outcomes depend on equilibrium portfolio allocation — making models generally intractable except in restricted cases.
  - Standard first-order and second-order approximation methods fail: the optimal portfolio allocation is generally indeterminate in a first-order approximation and indeterminate in the non-stochastic steady state.

### Proposed solution method: overview and key insight
- The paper develops a solution method applicable to any standard open economy model with any number of assets, any number of state variables, and either complete or incomplete markets.
- Key innovation: at the level of approximation normally used by open-economy macroeconomists in multi-country DSGE models, one only requires a solution for steady-state asset holdings; time variation in portfolio shares is irrelevant for first-order responses of macro variables like consumption and output.
- Methodological approach:
  - Characterize the optimal portfolio using a combination of a second-order approximation of the portfolio selection condition with a first-order approximation to the remaining parts of the model.4
  - The endogenous portfolio weights depend on the variance-covariance matrix of excess returns produced by the general equilibrium model, which itself depends on the portfolio positions — the simultaneous system is solved to give a simple closed-form analytical solution for portfolio weights.
- Practical advantages:
  - The solution formula fits into the standard solution approach for DSGE models and can be applied directly using a standard first-order accurate solution.
  - The technique is simple to implement and can produce analytical results for small models or numerical results for larger models.
  - Execution time for numerical solutions is no longer than required to obtain a standard log-linear solution.

### Relation to existing literature and methods
- Alternative approaches to incomplete-markets models:
  - Judd et al. (2002): numerical algorithm based on spline collocation.
  - Evans and Hnatkovska (2005): numerical approach combining perturbation, projection and continuous-time approximation.
  - Devereux and Saito (2005): continuous time framework allowing some analytical solutions and time-varying portfolios, but cannot handle general international macroeconomic models with diminishing-returns technology or sticky nominal goods prices.
- Comparison:
  - Methods by Judd et al. and Evans and Hnatkovska handle dynamic general equilibrium models and time variation in portfolios but are complex and depart from standard DSGE methods.
  - The present method is most closely related to Samuelson (1970), Judd (1998) and Judd and Guu (2001):
    - Samuelson: mean-variance approximation sufficient to identify optimal portfolio in a near-non-stochastic world.
    - Judd and Guu: use a Bifurcation theorem with the Implicit Function Theorem to overcome portfolio indeterminacy in the non-stochastic steady state and identify a bifurcation point as an appropriate approximation point.
  - The steady-state gross portfolio holdings derived here correspond to the approximation (bifurcation) point identified by the Judd and Guu method; the equilibrium portfolio can be rationalised as a bifurcation point in the set of non-stochastic equilibria.5

### Scope and structure of the paper
- This paper:
  - Presents the general portfolio choice problem within a generic open economy model.
  - Develops and describes the solution method that yields steady-state portfolio holdings and embeds naturally in standard DSGE solution techniques.
- Extensions:
  - Higher-order aspects of portfolio behaviour can be derived by considering higher-order approximations; a companion paper (Devereux and Sutherland (2006b)) shows how the method can be extended to higher-order aspects of portfolio behaviour. Tille and van Wincoop (2006) show how higher order solutions can be obtained numerically via an iterative algorithm.

*Source: _wp07284 - References..............................................................................................................*

### Section 4 presents two examples of how our technique can be use to solve for bond and equity

### _wp07284 - Section 4 presents two examples of how our technique can be use to solve for bond and equity holdings in simple two-country models. Section 5 concludes the paper.

### Model setup and definitions
- World: two countries (home and foreign). Home produces aggregate quantity YH at price PH; foreign produces YF at price P∗F. Foreign currency prices denoted with an asterisk.
- Home-agent preferences: U = Σ_{τ=t}^∞ β^{τ−t} [u(C_τ) + v(·)] (equation (1)); u(·) twice continuously differentiable; v(·) captures non-portfolio-relevant preference components.
- n assets with return vector r_t = [r_{1,t}, r_{2,t}, ..., r_{n,t}]′ measured in units of the home consumption good C.
- Home-period budget constraint (end-of-period holdings purchased at end of period t−1 to hold into t): equation (2). Net wealth relation: W_{t−1} = Σ_i α_{i,t−1} (equation (3)).
- Numeraire: nth asset; define excess returns r_{x,t} on other 1n− assets and vector α′_{t−1} excluding α_{n,t−1} (equation (4) representation).

### Portfolio first-order conditions and equilibrium conditions
- Home first-order conditions for portfolio choice: set (5) (matrixed form of Euler conditions for each asset).
- Foreign first-order conditions analogous: set (7).
- Market clearing: α_t + α∗_t = 0 (i.e., α_t = −α∗_t) implies 3(1)n− equations in unknowns α_t, α∗_t and E_t[ ,1 x r + ].
- Exact solutions infeasible generally due to nonlinearity and expectations; method replaces first-order conditions with second-order approximations.

### Approximation strategy and key properties
- Approximation point: non-portfolio variables X chosen at a non-stochastic steady state; portfolio vector α treated as endogenous and solved for so second-order approximations of first-order optimality conditions hold in a neighborhood of (X, α).
- Use second-order Taylor expansion to capture second moments (risk characteristics).
- Important formal properties:
  - Property 1: In a first-order approximation of non-portfolio parts, only α (portfolio at the approximation point) appears; deviations ˆα do not affect first-order macro dynamics.
  - Property 2: Solution of second-order approximation for the portfolio problem requires only first-order accuracy of non-portfolio parts (second moments obtainable from first-order realised values).
  - Property 3: Expected excess returns E_t[ ,1 x r + ] are zero in all time periods in a first-order approximation.
  - Property 4: Conditional one-period-ahead second moments from the first-order approximation are non-time-varying if Σ (covariance matrix of innovations) is non-time-varying.
  - Property 5: α% (portfolio at approximation point) enters first-order non-portfolio equations only via budget constraints; due to Walras' Law only one budget constraint matters (home budget constraint linearised as 1ˆˆˆˆ tt t xt WWYC rβα′ − = + − + %).
  - Property 6: Realised excess asset returns ˆx_t r are zero-mean i.i.d. up to first-order accuracy.

### Second-order portfolio optimality approximations (selected equations)
- Second-order expansion of home first-order conditions yields vector form (9):
  - ()23 ,1,11 ,1 1 ˆ ˆˆˆ 2 txtxttxt ErrCrOρε ++++ [ ... ] + − = (residual of order O(ε^3))
- Foreign analogous expansion yields (10).
- Combined, equilibrium conditions expressed as:
  - (11) ()3 111 ,1 ˆˆˆ ˆ (/)0 tt t txt ECCQ r Oρε ∗ +++ + [ ] − − = +
  - (12) [] ()23 111 ,1 11 ˆˆˆˆ ˆˆˆ (/) 22 xxttttxt ErErECCQ r Oρρε ∗ +++ + + [ ] = − + + ++
- These two equations: (11) binds equilibrium portfolio holdings; (12) gives equilibrium expected excess returns.

### Time variation and stationarity implications
- At the employed level of approximation, portfolio problem yields non-time-varying equilibrium: time subscripts can be omitted.
- Rationale: Properties 2, 3, and 4 imply expected excess returns zero at first order and second moments non-time-varying when Σ non-time-varying; therefore terms in (11) and (12) are time-invariant.

### Partial-equilibrium solution (insight and form)
- Substitute first-order linearised budget constraints (15) and (16) for 1ˆ t C + and 1ˆ t C ∗ + into (11):
  - (15) ()2 1,111 1 ˆ ˆˆ ˆ ˆ txt ttt Cr WWYOαε β ′ ++++ = + − ++ %
  - (16) ()2 1,111 1 ˆ ˆˆ ˆ ˆ txt ttt Cr WWYOαε ∗ ′ ∗ ++ ++ = − − + + +
- Partial-equilibrium portfolio allocation (α%) from (17):
  - (17) ()1 1 2 xxx D V V O α ε − = − + %
  - where V_{xx}, V_{xD}, V_{xA} depend on α% via impact of gross holdings on net wealth.
- Partial-equilibrium expected excess returns (18):
  - (18) [] ()23 11 ˆˆ 22 xxx A Er E r V O ρε [ ] = − + ++
- Note: forms resemble mean-variance solutions; full general equilibrium needed because V matrices depend on α% through general-equilibrium channels.

### General-equilibrium derivation and closed-form for α%
- Represent first-order non-portfolio system with net-wealth innovation treated as exogenous t ξ (zero-mean i.i.d.) to obtain state-space (20) and solution form (21).
- First-order realized excess returns linear in exogenous shocks (22): ()2 11121 ˆ xt t t r R R O x ε ε + + + = + +
- Realized excess return on portfolio relates via (23): 1 ˆ txt r ξ α ′ + + = %
- Solve for 1 t ξ + and 1 ˆ x t r + in terms of exogenous shocks: (24) 11 t H x ε + + = % ; (25) ()2 11 ˆ xt t r R O ε ε + + = + %
- Using relationships between these and the expression for 111 ˆˆˆ(/) t t t CCQρ ∗ + ++ − − yields (30):
  - (30) ()3 111 ,1 ˆˆˆ ˆ (/) tt t txt ECCQ r R D O ρε ∗ ′ +++ + [ ] − − = Σ + %%
  - where Σ is covariance matrix of ε.
- Equilibrium α% solves matrix equation (31): 0 RD ′ Σ = %%
- Expanding and simplifying yields closed-form equilibrium (35):
  - (35) ()1 221 12222 R D R D R R RD O α ε −′′′′ [ ] = Σ − Σ Σ + %
- Residual in (35) is first-order; α (actual portfolio) = Y α α β = % (as noted: α = Y α% β).

### Interpretation: α% as limit in a near-non-stochastic world
- Let Σ = ζ Σ_0 with scalar ζ > 0 and Σ_0 valid covariance matrix. Solution α% in (35) is independent of ζ, so it equals the equilibrium portfolio allocation in a world arbitrarily close to non-stochastic (ζ → 0).
- As ε → 0 (equivalently ζ → 0) the residual in (35) → 0, so α% → true portfolio holdings in the non-approximated model.
- Thus α% corresponds to the bifurcation point in the set of non-stochastic equilibria (consistent with Judd and Guu (2001) discussion).

### Practical implementation summary (algorithmic outline)
- The solution for equilibrium α% requires only first-order information from the non-portfolio model to construct D and R matrices and then apply (35).
- Implementation steps (start of list provided in source):
  1. Solve the non-portfolio equations of the model in the form of (20) to yield a solution in the form of (21).

*Source: _wp07284 — Section 4 presents two examples of how our technique can be use to solve for bond and equity holdings in simple two-country models.*

### 2. Extract the appropriate rows from this solution to form

### _wp07284 - 2. Extract the appropriate rows from this solution to form

### III. Application of the method — overview
- The solution technique is applied to two example open-economy models to illustrate how to compute portfolio holdings in first-order approximations.
- Key steps: log-linearize first-order conditions and market clearing conditions; collect into a first-order linear system in the form of (20); apply any standard linear solution algorithm to obtain state-space solution in the form of (21); extract matrices R1, R2, D1, D2 and compute portfolio shares via expression (35).

### A. Example 1: A two-country endowment model with trade in nominal bonds
- Model setup:
  - One-good, two-country endowment economy.
  - Home utility: 1
    1
    t
    t
    tt
    t
    C
    UE
    ρ
    τ
    τ
    β
    ρ
    −
    ∞
    −
    =
    =
    −
    ∑                                                    (36)
  - Home and foreign endowments follow AR(1) in logs:
    1,1
    ,
    loglog,loglog
    tY   t  Ytt  Y   t
    Yt
    YYY Yζεζε
    ∗
    ∗∗
    −−
    =+  =+                          (37)
  - Shocks: Yε and Yε∗ i.i.d. symmetric on [, ]εε− with 2[][ ] YY Y VarVarεεσ ∗ == .
  - Asset trade: home and foreign nominal bonds. Budget constraint (home):
    ,1 ,
    ,1   ,
    tBtBttt
    Bt   Bt
    Wr   rYCαα
    ∗∗
    −
    −
    =++−                                          (38)
  - Net wealth: ,
    ,
    tBt
    Bt
    Wαα
    ∗
    =+                                                          (39)
  - Real returns on bonds:
    11
    ,,
    ,,
    tt
    BtBt
    BtBt
    tt
    PP
    rRr R
    PP
    ∗∗
    ∗
    −−
    ∗
    ==                                               (40)
  - Law of one price: PS P∗ = where S is nominal exchange rate.
  - Quantity theory price relations:
    ,
    tttt tt
    MPYMP Y
    ∗∗∗
    ==                                              (41)
  - Money supplies follow AR(1) in logs:
    1,1
    ,
    loglog,loglog
    tMt   Mtt Mt
    Mt
    MMM Mζεζε
    ∗
    ∗∗
    −−
    =+  =+                           (42)
  - Money shocks Mε and Mε∗ i.i.d. symmetric on [, ]εε− with 2[][ ] MM M VarVarεεσ ∗ == .
  - Four shock processes assumed independent; covariance matrix Σ is
    2
    2
    2
    2
    00 0
    000
    000
    00 0
    Y
    Y
    M
    M
    σ
    σ
    σ
    σ
    ⎡⎤
    ⎢⎥
    ⎢⎥
    Σ=
    ⎢⎥
    ⎢⎥
    ⎢⎥
    ⎣⎦

- First-order conditions and approximations:
  - FOCs for consumption and bonds (home and foreign) given by (43) and (44).
  - Resource constraint (45): tt tt CC YY ∗∗ +=+.
  - First-order approximations yield:
    - (46): ()() 22 11 ,1,1 ˆˆˆ ˆ ˆˆ , tttt tt BtBt CE   C  rOC E   C  rOρρερρε ∗∗ ∗∗∗ ++ ++ ⎡⎤⎡⎤ −=− +  +− =− +  + ⎣⎦⎣⎦
    - (47): () 2 ,1 ,1 ˆˆ tBtt Bt ErErOε ∗ + + ⎡⎤ ⎡⎤ =+ ⎣⎦ ⎣⎦
    - Resource constraint approximation (48): () 2 ˆˆ ˆˆ tt tt CC YY Oε ∗∗ +=++
    - Budget constraint approximation (49): () 2 1, 1 ˆ ˆˆˆ ˆ ttttBxt WWYC rOαε β − =+−++ %
    - Quantity theory (50): ))ˆˆˆ ˆ , tt tttt MPY   M  P Y ∗ ∗∗ −=−=
    - Excess return on home bonds (51): () 2 , , ˆˆˆ xtBt Bt rrr Oε ∗ =− +
  - Property demonstrations:
    - Expected excess return zero up to first order: 2 ,1 ˆ (). txt ErOε + ⎡⎤ = ⎣⎦ .
    - Nominal interest differential equals expected inflation differential: (52) ()() () 2 ,1111 , ˆˆˆˆˆˆ Bttttttt Bt RR   EPP  EPP Oε ∗ ∗∗ −−− − ⎡⎤⎡ ⎤ −=    −−−+ ⎣⎦⎣ ⎦
    - Realised excess return (53): ()() () 2 ,11 ˆˆˆ ˆ ˆ xttttttt rEPPEPPOε ∗∗ −− ⎡⎤⎡ ⎤ =−− −+ ⎣⎦⎣ ⎦ — equals difference between home and foreign price surprises and negative unexpected change in exchange rate.
  - Definitions for expectations (54) and (55):
    - (54): 11 ˆˆˆ ˆ [],[ ] EE ttt t  tt PEP P EP ∗∗ −− == 
    - (55): ,1 ,1 ˆˆˆ E ttBtt Bt rErEr ∗ + + ⎡⎤ ⎡⎤ == ⎣⎦ ⎣⎦

- State vector definitions used to form system (20):
  - s = 11   1 ˆˆ ˆ EE tt t   t sPPW '∗ −− − ⎡⎤ = ⎣⎦
  - c = ,ˆˆ ˆˆ ˆ ˆˆ E ttttttDtxt cCCrPPYr '∗    ∗ ⎡⎤ = ⎣⎦
  - x = ˆˆ ˆ tt t t xYY M M '∗∗ ⎡⎤ = ⎣⎦

- Solution for bond holdings — matrices and portfolio:
  - The resulting expressions for the matrices R1, R2, D1 and D2 are given by
    - [][] 12 0,1111RR==−−
    - [] 11 12 11 2(1)  ,0    0 YY DD ββ βζβζ β −− −− ⎡⎤ =−=   − ⎣⎦
  - Applying (121) yields the following expression for bond holdings:
    - 2 22 2()(1) Y B B MYY σ αα σσβζ ∗ =−   =− +− %%
  - Qualitative findings (from the example text):
    - Home consumers take a negative position in home currency bonds, balanced by a positive position in foreign currency bonds.
    - The home price level is countercyclical, so home currency bonds have a relatively high payoff when home output is high—making home currency bonds a relatively bad hedge against home output risk, while foreign currency bonds are a relatively good hedge.
    - Equivalent: home exchange rate appreciates in response to a positive home output shock, increasing return on home bonds relative to foreign bonds in this state.
    - Monetary policy volatility reduces the gross holdings of bonds: although prices are fully flexible, monetary volatility is costly because it reduces the usefulness of nominal bonds as a risk-hedging instrument.

### B. Example 2: A Two-Country Production Model with Trade in Equities
- Model setup extensions relative to Example 1:
  - Endogenous production, labour supply, productivity and fiscal shocks.
  - Home utility:
    1
    1
    t
    t
    ttt
    t
    CEKL
    ρ
    τ
    τ
    β
    ρ
    −
    ∞
    −
    =
    ⎛⎞
    =
    −
    ⎜⎟
    −
    ⎝⎠
    ∑                                                  (56)
  - Composite consumption C as CES aggregation of home and foreign goods:
    11
    1
    11
    11
    22
    HF
    CC C
    θ
    θ
    θθ
    θθ
    θθ
    −
    −−
    ⎡⎤
    =+
    ⎢⎥
    ⎢⎥
    ⎣⎦
                                                        (57)
  - Consumer price index (58): 1 1 11 11 22 HF PP  P θ θθ − − − ⎡⎤ =+ ⎢⎥ ⎣⎦
  - Equity trade replaces bond trade: budget constraint (59):
    ,1 ,
    ,1   ,
    tEtEtttttt
    Et   Et
    Wr   rwLTCααπ
    ∗∗
    −
    −
    =+ ++−−                                    (59)
  - Net wealth: , , tEt Et Wαα ∗ =+ (60)
  - Firms: linear labour production with productivity At; At follows AR(1) in logs (61).
  - Government purchases G with same composition as C; government spending follows (62) with AR(1) and symmetric shocks; non-stochastic equilibrium level G matches yg (ratio of government spending to GDP).
  - Home equity payoff real πt/Pt = Πt; real price of home equity denoted 1Et Z− ; gross real rate of return on home equity: ,,,1 () . EttEtEt rZZπ − =+
  - First-order conditions similar to previous example with equities replacing bonds; labour supply condition tt Cw K ρ− = ; profit maximisation condition ,1 Ht t t P w P φ φ = − .
  - Market clearing:
    - HtHtHtHtHt CCGGY ∗∗ +++=
    - FtFtFtFtFt CCGGY ∗∗∗ ++++=

- First-order approximation and excess return:
  - Excess return on foreign equity (63):
    - )))) () 2 ,, ,1 ,,1 , ˆˆˆ [(1)]   [(1)] EtEtEtEt xttt rZZ    ZZOβπββπβε ∗∗ ∗ −− =−+−−−+−+
  - System collected in form (20); solved to state-space (21).

- Solution for equity holdings — matrices and portfolio:
  - Additional assumption: AG ζζ = .
  - Computed expressions for matrices R1, R2, D1 and D2:
    - 1 2(   1)(1   )Rρθβ= ⎡⎤ −− ⎢⎥ Θ ⎣⎦
    - (1)(1)(1)(1)(1)(1)(1)(1)(1)(1) 2 (1)(1)(1)(1) yyyy AA    AA R ggggβθβθρ βθρ βθ βζβζβζβζ = −−−  −−−−−−− Θ−Θ−Θ−Θ− ⎡⎤ −− ⎣⎦
    - 1 2(1) D β = − ⎡⎤ ⎢⎥ Θ ⎣⎦
    - (1)(1) (1)(1)(1)(1) 2 (1)(1)(1)(1) yy AAAA D gggββ βθβθ βζβζβζβζ = −− −−−−− Θ−Θ−Θ−Θ− ⎡⎤ −− ⎣⎦
    - where (1(1)). y gρθΘ = −  +   −
  - Applying (35) yields equity holdings expression (64):
    - 2 2 2 1 1 2(1)(1)(1) y G E E yA g g ρ σ αα βθ    σ ∗ ⎛⎞ =−   =−− ⎜⎟ ⎜⎟ −−− ⎝⎠ %%
  - Qualitative findings:
    - In absence of government spending shocks, home households hold a perfectly pooled portfolio of home and foreign equity; given budget constraint (59) (home households receive all home profits), this requires that E α % is negative.
    - When 0 y g > , home equity is a good hedge against consumption risk from government spending shocks (home profits relatively high when government spending is high); households will hold less than a fully pooled equity portfolio.
    - Presence of government spending shocks can explain partial or full home bias in equity portfolios.

### IV. Method implications and extensions
- The paper develops a simple approximation method for portfolio choice problems in open economy models that:
  - Is easy to implement and applicable in any model relying on first-order approximation methods.
  - Allows analysis of first-order properties of macro variables (GDP, consumption, real exchange rate) via impulse responses or second-moment computations (volatility and comovement).
  - Can be used to study macroeconomic determinants of optimal steady-state portfolio holdings for any asset or combination of assets, under complete or incomplete markets.
- Time-variation in portfolio holdings requires higher-order approximation; a companion paper (Devereux and Sutherland 2006b) extends methods to higher-order approximations to analyze time-variation in portfolio behaviour.
- The method can be applied to sticky-price open economy models (e.g., Obstfeld and Rogoff (1995), Benigno and Benigno (2003), Devereux and Engel (2003), Corsetti and Pesenti (2005)); Devereux and Sutherland (2006a) use it to analyze monetary policy rules and asset holdings in sticky-price models.

*Source: _wp07284 - 2. Extract the appropriate rows from this solution to form (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07284.pdf_