Country Portfolios with Heterogeneous Pledgeability Separate Technical Appendix Tommaso Trani The Graduate Institute j Geneva

Contents 1 Model

1

2 Approximated Non-Portfolio Equations

2

3 Approximated Portfolio Equations 3.1 Homogeneous Pleadgeability Case . . . . . 3.2 Heterogeneous Pledgeability Case . . . . . . 3.2.1 Assuming Zero Order Heterogeneity 3.2.2 Assuming First Order Heterogeneity

1

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

3 3 3 4 5

Model

In the main text, I work with a model featuring heterogeneously pledgeable assets and make comparisons with a model where assets are homogeneously pledgeable. Furthermore, some assumptions are progressively made through the paper, in order to develop a solution strategy for country portfolios in the heterogeneously pledgeable asset case. Here, I list the equations that form the model used under the two alternative assumptions. For the model I build (heterogeneous pledgeability case), the set of equations involve: Equilibrium conditions: equations in appendix A.1 of the paper and their foreign counterparts. Processes for the exogenous states: equation (31) and its foreign counterpart, equations (32). For the model I used as a term of comparison (homogenous pledgeability case), the set of equations involves Equilibrium conditions: the same equations as before, but (A.1)-(A.2) and their foreign counterparts are replaced either with equations (18.a) and their foreign counterparts or with equations (18.b) and their foreign counterparts. Note that only the speci…cation of the latter relations allows to study …nancial shocks under homogeneously pledgeable assets. Processes for the exogenous states: the same equation (31) and counterpart as before, but equations (32) are replaced with (33) and its foreign counterpart. 1

2

Approximated Non-Portfolio Equations

Model equations are approximated up to the …rst order. Only the budget and leverage constraints are directly a¤ected by the equity portfolio share, ! t . To recover this e¤ect, in the main text I rewrite (1) and (2) as (1’) and (2’), respectively. The …rst order components of these equations (included the counterpart of (2’)) are: w (0) d (0) (0) (d (0) + q e (0)) (0) w (1) + d (1) + t Ht Ht (1) cI (0) cI (0) cI (0) bI (0) I q b (0) bI (0) b q (0) (0) I q (1) + b (1) (1) + b (1) Ht 1 t t cI (0) cI (0) cI (0) t 1 +~ ! (0) rxt (1) + r (0) N F At 1 (1) (AA.1) I I (0) c (0) (0) c (0) e ! ~ (0) xt (1) + N F At (1) + Ht (1) + qHt (1) + Ht (1)(AA.2) bIt (1) = r (0) bI (0) bI (0) (0) cI (0) (0) cI (0) bt I (1) = ! ~ (0) (1) N F At (1) + F t (1) + qFe t (1) + F t (1)(AA.3) xt r (0) bI (0) bI (0)

cIt (1) + N F At (1)

=

\ where ! ~ (0) = r (0) ! (0) =cI (0), N F At = N F At =cI (0) and N F At (0) = N F At (0). Only the budget constraint of home investors is strictly needed: avoiding redundancy, the other is satis…ed by Warlas’Law. In comparison, ! t does not a¤ect the leverage constraints if the model features homogeneously pledgeable assets. In such a case, (AA.1) is still valid, but (AA.2)-(AA.3) are substituted with bIt (1)

=

bt I (1)

=

(0) cI (0) e N F At (1) + qHt (1) + Ht (1) bI (0) (0) cI (0) N F At (1) + qFe t (1) + F t (1) bI (0)

(AA.4.a) (AA.5.a)

if the debt-to-asset ratio is a constant parameter, or with bIt (1)

=

bt I (1)

=

(0) cI (0) e N F At (1) + t (1) + qHt (1) + Ht (1) bI (0) (0) cI (0) N F At (1) + t (1) + qFe t (1) + F t (1) bI (0)

(AA.4.b) (AA.5.b)

if the debt-to-asset ratio is time-varying and speci…c for each investor. The …rst order components of home investors’budget constraint and - depending on how asset pledgeability is modeled - of both investors’ leverage constraints can be added to the other linearized model equations. The resulting system is standard: AEt Yt+1 (1) = BYt (1) + Cxt (1) 0

where A; B; C are matrices of coe¢ cients, Yt+1 (1) = [st+1 (1) ct (1)] are the endogeneous state and control variables and xt (1) is the vector of exogenous states. The solution of this system takes the form below:

where

1;

2;

1;

2

st+1 (1)

=

1 xt

(1) +

2 st

(1)

(AA.6)

ct (1)

=

1 xt

(1) +

2 st

(1)

(AA.7)

are matrices of numbers. 2

3 3.1

Approximated Portfolio Equations Homogeneous Pleadgeability Case

In a model where collateralizable assets are homogeneously pledgeable, home and foreign investors’portfolio Euler equations are I t+1 I Et t+1

0

= Et

rHt+1

rF t+1 e

0

=

rHt+1 e

(AA.8)

rF t+1

(AA.9)

where > 0 is a second order transaction cost, which erodes part of the returns earned by every investor on her holdings of foreign assets. Both equations imply rH (0) = rF (0) = r (0). A second order approximation of (AA.8)-(AA.9) yields 1 2 = Et rxt+1 (1) + rxt+1 (2) + + rxt+1 (1) + 2 1 2 0 = Et rxt+1 (1) + rxt+1 (2) + rxt+1 (1) + 2

0

I t+1

(1) rxt+1 (1)

I t+1

(1) rxt+1 (1)

Combining them, I obtain the condition to determine ! (0), 0 = 2 + Et

I t+1

(1)

I t+1

(1) rxt+1 (1)

(AA.10)

and the condition for the corresponding excess return on home versus foreign equities: Et rxt+1 (1) =

1 1 2 Et rxt+1 (2) + rxt+1 (1) + 2 2

I t+1

(1) +

I t+1

(1) rxt+1 (1)

(AA.11)

The equilibrium portfolio share of a model with homogeneously pledgeable assets - denoted ! E ho (0) - can thus be determined extracting from (AA.6)-(AA.7) the rows needed to compute (AA.10). The details of this computation are in appendix A.3, inside the paper.

3.2

Heterogeneous Pledgeability Case

In the model I build, collateralizable assets are heterogeneously pledgeable. In this case, home and foreign investors’portfolio Euler equations are I t+1 I Et t+1

0

= Et

rHt+1

0

=

rHt+1 e

rF t+1 e

+ Mt (

rF t+1 + Mt (

Ht Ht

F t) F t)

(AA.12) (AA.13)

These portfolio selection conditions imply either rH (0) rF (0) = GP (0) ( H (0) F (0)) or rH (0) = rF (0) = r (0), H (0) = F (0) = (0). It depends on the assumption one makes about the pledgeability of collateral assets. According to the …rst assumption, home and foreign collateral have di¤erent debt-to-asset ratios already in the steady state; according to the second, the steady state is fully symmetric (i.e., symmetry is satis…ed also for pledgeability and return on equities). Approximation and properties of equilibrium portfolios are sensitive to the assumption we choose. 3

3.2.1

Assuming Zero Order Heterogeneity

Under the assumption that assets are heterogeneously pledgeable already in the zero order, a second order approximation of (AA.12) yields 0

= Et rH (0) rHt+1 (1) rF (0) rF t+1 (1) + rx (0) It+1 (1) h +GP (0) ( H (0) Ht (1) F (0) F t (1)) + x (0) Mt (1)

Et

I t+1

(1)

i

1 1 2 2 +Et rH (0) rHt+1 (2) + rHt+1 (1) rF (0) rF t+1 (2) + rF t+1 (1) 2 2 1 I 2 +Et rx (0) It+1 (2) + (1) + It+1 (1) (rH (0) rHt+1 (1) rF (0) rF t+1 (1)) 2 t+1 1 1 2 2 +GP (0) H (0) Ht (2) + Ht (1) F (0) F t (2) + F t (1) 2 2 1 1 I 2 2 I +GP (0) x (0) Mt (2) + Mt (1) (1) Et t+1 (2) + 2 2 t+1 +GP (0) M (1) ( H (0) Ht (1) F (0) F t (1))

and, for (AA.13), I similarly get 0

I = Et rH (0) rHt+1 (1) rF (0) rF t+1 (1) + rx (0) t+1 (1) h +GP (0) ( H (0) Ht (1) F (0) F t (1)) + x (0) Mt (1)

I t+1

+Et rx (0) +GP (0)

H

(0)

(2) + Ht

1 2

2

I t+1

(1)

1 2

Ht

(2) +

+

I t+1

(1) (rH (0) rHt+1 (1)

2

(1)

F

1 2 (0) Mt (2) + Mt (1) Et 2 +GP (0) M (1) ( H (0) Ht (1) F (0) +GP (0)

I t+1

(1)

i

1 2 rF (0) rF t+1 (2) + rF t+1 (1) 2

1 2 + rHt+1 (1) 2

+Et rH (0) rHt+1 (2)

Et

(0)

I t+1

x

Ft

Ft

(2) +

(2) +

1 2

I t+1

1 2

rF (0) rF t+1 (1)) 2

Ft

(1)

2

(1)

(1))

Combining these equations, I discover that ! (0) must now satisfy 0

=

I (rx (0) GP (0) x (0)) Et It+1 (1) t+1 (1) + GP (0) x (0) (Mt (1) h 2 I +Et rx (0) It+1 (2) GP (0) x (0)) It+1 (1) t+1 (2) + (rx (0)

+ (rF (0) +GP (0)

rH (0)) + Et

x

I t+1

(1)

(1) (rH (0) rHt+1 (1)

1 2 2 Mt (1) Mt (1) 2 Mt (1)) ( H (0) Ht (1) F (0) F t (1))

(0) Mt (2)

+GP (0) (Mt (1)

I t+1

Mt (1)) I t+1

2

(1)

i

rF (0) rF t+1 (1))

Mt (2) +

4

(AA.14)

and that the corresponding expected excess return must be as follows: Et (rH (0) rHt (1) =

rF (0) rF t (1))

GP (0) ( H (0) Ht (1) F (0) F t (1)) 1 I (rx (0) GP (0) x (0)) Et It+1 (1) + t+1 (1) 2 1 [GP (0) x (0) (Mt (1) + Mt (1)) (rF (0) rH (0)) ] 2 1 1 I 2 2 I I I (1) + t+1 (1) (rx (0) GP (0) x (0)) Et t+1 (2) + t+1 (2) + 2 2 t+1 1 I Et It+1 (1) + t+1 (1) (rH (0) rHt (1) rF (0) rF t (1)) 2 1 1 1 2 2 GP (0) H (0) Ht (2) + Ht (1) F t (1) F (0) F t (2) + 2 2 2 1 1 2 2 GP (0) x (0) Mt (2) + Mt (2) + Mt (1) + Mt (1) 2 2 1 GP (0) (Mt (1) + Mt (1)) ( H (0) Ht (1) F (0) F t (1)) 2

(AA.15)

Conditions (AA.14) and (AA.15) are both a¤ected by …rst and second order components of model variables in such a way that a …rst order solution of the model such as (AA.6)-(AA.7) is not su¢ cient to determine the zero order portfolio and to guarantee that the expected excess return is zero. 3.2.2

Assuming First Order Heterogeneity

Under the alternative assumption that in the zero order of the model assets are symmetrically pledgeable, which is the key assumption I make in the text, the second order components of (AA.12)-(AA.13) are

0

0

1 2 = Et rxt+1 (1) + rxt+1 (2) + + rxt+1 (1) + It+1 (1) rxt+1 (1) 2 GP (0) (0) 1 2 + xt (1) + xt (2) + xt (1) + Mt (1) xt (1) r (0) 2 1 2 I = Et rxt+1 (1) + rxt+1 (2) + rxt+1 (1) + t+1 (1) rxt+1 (1) 2 GP (0) (0) 1 2 + xt (1) + xt (2) + xt (1) + Mt (1) xt (1) r (0) 2

Again, these two equations can be combined to yield the condition that ! (0) should satisfy, 0 = 2 + Et

I t+1

(1)

I t+1

(1) rxt+1 (1) +

GP (0) (0) (Mt (1) r (0)

5

Mt (1))

xt

(1)

(AA.16)

as well as the condition for the expected excess return: Et rxt+1 (1)

GP (0) (0) r (0)

=

xt

(1)

GP (0) (0) 1 2 xt (2) + xt (1) r (0) 2 GP (0) (0) I I (Mt (1) + Mt (1)) t+1 (1) + t+1 (1) rxt+1 (1) + r (0)

1 2 Et rxt+1 (2) + rxt+1 (1) 2 1 Et 2

+

xt

(1) (AA.17)

De…ning GP (0) (0) xt (1) r (0) as an "overall excess yield", (AA.16)-(AA.17) can clearly be computed using the same strategy as for (AA.10)(AA.11). Note in fact that (AA.16)-(AA.17) are close in essence to the conditions used by Devereux and Sutherland (2011) and Tille and Van Wincoop (2010), so the determination of portfolios is surely less cumbersome than what it would be attempting to satisfy (AA.14)-(AA.15). Hence, the equilibrium portfolio share of a model with heterogeneously pledgeable assets - denoted ! E he (0) can be determined, as above, extracting from (AA.6)-(AA.7) the rows needed to compute (AA.16). In the paper, I perform this computation adopting the iterative procedure detailed in appendix A.3 there. Figure 1 shows an example of how this simple algorithm works for the satisfaction of (AA.16). Given the way the portfolio share is de…ned, ! he (0) < 0 when there is home equity bias. The graph is showing a case like this. t+1

(1) = rxt+1 (1) +

Figure 1. Computing (AA.16) as an Implicit Function of ! he (0) -5

3

x 10

Cross-Border Portfolio Choice Condition Lower Bound of Min(distance) Upper Bound of Min(distance)

2

approximation to zero

1 0 -1 -2 -3 -4 -5 -8

-6

-4 -2 0 equilibrium portfolio holdings

6

2

4

References [1] Devereux, M. B. and A. Sutherland (2011) "Country Portfolios in Open Economy macro models", Journal of European Economic Association, 9(2), 337-369. [2] Judd, K. (1998) Numerical Methods in Economics, MIT Press. [3] Tille, C. and E. Van Wincoop (2010) "International Capital Flows", Journal of International Economics, 80(2), 157-175.

7