Can nontradables generate
substantial home bias?
Technical Appendix
Paolo Pesenti and Eric van Wincoop
Federal Reserve Bank of New York
November 1999
1
A model of international portfolio choice
with nontradables: extensions
In this Appendix we focus on the technical aspects of the model presented
in the main text.
Domestic residents receive an endowment stream of nontradables S(t),
whose stochastic rate of growth is given by
dS(t)
= ¹dt + ¾X d»(t)
S(t)
(A.1)
where ¹ and ¾X are constant parameters and » is a standard Wiener process.
The instantaneously stochastic returns on domestic and foreign assets are:
dR(t) = ´dt + ¾R d!(t)
dR¤ (t) = ´dt + ¾R¤ d! ¤(t)
(A.2)
(A.3)
Here R is the cumulative return on the domestic asset, R¤ the cumulative
return on the foreign asset, and ´ the expected return per unit time, assumed
to be equal across countries and constant over time. The instantaneous correlation between the returns is denoted ½ ´ [E (d!d!¤)] =dt. The correlations
between the growth rate of nontradables consumption and the two asset returns are denoted ½RX ´ [E (d!d»)] =dt and ½R¤ X ´ [E (d!¤d»)] =dt.
The stochastic process for domestic ¯nancial wealth W (measured in
terms of traded goods) is
dW (t) = [n (t) dR (t) + (1 ¡ n (t)) dR¤ (t)] W (t)
+p (t) S (t) dt ¡ C (t) dt ¡ p (t) X (t) dt
(A.4)
where C(t) is domestic consumption of tradables, X (t) consumption of nontradables, n(t) is the fraction of wealth invested at home and p (t) is the
relative price of nontradables.
The value function V [W (t); X(t)] is de¯ned as:
V [W (t); X(t)] ´ max Et
C(t);n(t)
Z 1
t
e¡±(s¡t)U[C (s); X(s)]ds
where ± is the rate of time preference. In equilibrium we have:
S (t) = X (t) ;
dS (t)
dX (t)
=
S (t)
X (t)
1
(A.5)
and the relative price of nontradables p(t) is determined as the ratio of the
marginal utilities UX =UC . Optimal consumption/portfolio choice C(t) and
n(t) solves the Bellman equation:
±V [W (t); X(t)] = max U [C (t) ; X (t)] + (Et dV [W (t) ; X (t)]) =dt (A.6)
C(t);n(t)
subject to (A.1)-(A.4) and the appropriate boundary conditions.
Applying Ito's Lemma to the value function, we obtain
1
1
dV = VX dX + VW dW + VXX dX 2 + VWW dW 2 + VXW dXdW:
2
2
(A.7)
Substituting dX and dW with their expressions (A.1) and (A.4), after taking
the conditional expectation of (A.7) we can rewrite (A.6) as follows:
1
±V [W; X] = max U[C; X] + VX X¹ + VW (W ´ ¡ C) + V XXX 2¾2X
C;n
2
h
i
1
2 2
2
2 2
+ V WW W n ¾R + (1 ¡ n) ¾R¤ + 2n (1 ¡ n) ½¾R¾R¤
2
+V WX WX ¾X [n½RX ¾R + (1 ¡ n) ½R¤ X ¾R¤ ]
Optimal portfolio shares are obtained by taking the ¯rst-order condition
for a maximum with respect to n. Di®erentiating the previous expression
yields:
h
0 = VWW W n¾R2 ¡ (1 ¡ n) ¾R2 ¤ + (1 ¡ 2n) ½¾R¾R¤
+VWX X¾X [½RX¾R ¡ ½R¤X ¾R¤ ]
i
(A.8)
from which we derive the expressions for n and bias in the main text.
The model can be generalized in several respects. Consider ¯rst the case
in which expected returns are not constant over time, but rather exhibit
mean reversion around a constant steady-state mean. Speci¯cally, replace
(A.2) and (A.3) with
dR(t) = ´ (t) dt + ¾Rd!(t)
dR¤(t) = ´ ¤ (t) dt + ¾R¤ d!¤ (t)
where
d´ (t) = ¡k (´ (t) ¡ ¹́) dt + Ád!(t)
d´¤ (t) = ¡k (´¤ (t) ¡ ¹́) dt + Á¤d!¤ (t)
2
(A.9)
(A.10)
are two Ornstein-Uhlenbeck processes, with ¹́, Á and Á¤ constant positive
parameters. Intuitively, a shock d! pushes upward the current return and
the future expected return on domestic assets. If the expected return ´ is
currently above its long-run average ¹́, it is expected to decline; if ´ < ¹́
instead, ´ is expected to raise. In the long run, the distribution for the
expected return ´ is stationary, and ´ is normally distributed with asymptotic
mean ¹́ and asymptotic variance Á 2=2k. Similar considerations hold for ¹́¤.
Under these new assumptions, the value function includes now ´ and ´¤
as arguments, and the modi¯ed Bellman equation is now:
±V [W; X; ´; ´¤] = max U[C; X] + VX X¹ + VW [W [n´ + (1 ¡ n) ´¤] ¡ C]
C;n
1
¡V´ k (´ ¡ ¹́) ¡ V´¤ k (´¤ ¡ ¹́) + VXX X 2 ¾2X
2
h
i
1
+ VWW W 2 n2 ¾R2 + (1 ¡ n)2 ¾R2 ¤ + 2n (1 ¡ n) ½¾R¾R¤
2
1
1
+ V´´Á 2 + V´¤ ´¤ (Á¤ )2 + VW XW X ¾X [n½RX ¾R + (1 ¡ n) ½R¤ X ¾R¤ ]
2
2
+VW´ WÁ [n¾R + (1 ¡ n) ½¾R¤ ] + VW ´¤ W Á¤ [n½¾R + (1 ¡ n) ¾R¤ ]
+VX´X ¾XÁ½RX + VX´¤ X¾X Á¤½R¤ X + V´´¤ ÁÁ¤½
The ¯rst order condition with respect to n now yields
h
0 = VW (´ ¡ ´ ¤) + VW W W n¾2R ¡ (1 ¡ n) ¾2R¤ + (1 ¡ 2n) ½¾R¾R¤
i
+VWX X¾X [½RX¾R ¡ ½R¤X ¾R¤ ] + VW´ Á [¾R ¡ ½¾R¤ ] + VW´¤ Á¤ [½¾R ¡ ¾R¤ ]
so that the portfolio share invested in domestic assets is equal to
n =
¾2R¤ ¡ ½¾R¾R¤
VW
´ ¡ ´¤
¡
¾R2 ¤ + ¾R2 ¡ 2½¾R ¾R¤ VW W W ¾R2 ¤ + ¾2R ¡ 2½¾R¾R¤
V
Á [¾R ¡ ½¾R¤ ]
VW´¤
Á ¤ [½¾R ¡ ¾R¤ ]
¡ W´
¡
+ bias,
V WW W ¾2R¤ + ¾2R ¡ 2½¾R¾R¤ VWW W ¾R2 ¤ + ¾R2 ¡ 2½¾R¾R¤
where bias is the same formula given in the main text.
Interpreting the formula above, the share of wealth invested in domestic
assets is equal to the minimum-variance portfolio share (¯rst term), corrected
to account for discrepancies in expected returns (second term), to hedge
3
against uncertainty in the expected returns themselves (third and fourth
term), and to hedge against nontradables uncertainty (the bias term). Only
the last term is associated with hedging against nontradables. The ¯rst four
terms are present even in the absence of nontradables. The key point is
that a higher expected return of one country's assets leads investors of all
countries to shift their portfolio toward that country's assets, and therefore
leads neither to a systematic di®erence in portfolio shares across investors
in di®erent countries, nor to a systematic tendency to skew investors' portfolios toward the assets issued in their countries of residence. Such systematic tendency is still captured by the term bias, which measures the impact
of country-speci¯c variables on wealth allocation, although theoretically the
presence of nontradables may also have a small secondary impact on the
composition of investors' portfolios through their e®ects on the preference
terms VW´ =VWW W and V W´¤ =VWW W .
Similar considerations apply in the case of stochastic volatility. Assuming
for simplicity of exposition that only domestic asset returns are heteroskedastic, the processes for asset returns are now
dR(t) = ´dt + ¾R (t) d!(t)
dR¤(t) = ´dt + ¾R¤ d!¤(t)
(A.11)
(A.12)
where the law of motion for ¾R (t) is given by:
q
d¾R (t) = ¡k [¾R (t) ¡ ¾¹R] dt + Á ¾R (t)dz(t);
In other words, ¾R follows an autoregressive process similar to the one considered above for expected returns: the conditional volatility of domestic
returns is subject to shocks dz and is time-varying around a steady-state
average ¾¹R . Deviations from the steady-state average decay at the rate k.
Assuming that at some initial time t = 0 it is ¾R(0) > 0, the square root
term in the previous equation guarantees that ¾R is always positive at any
time t. We denote the correlations involving ¾R as ½¾R ´ [E (d!dz)] =dt,
½ ¾R¤ ´ [E (d! ¤dz)] =dt and ½¾X ´ [E (dzd»)] =dt.
The modi¯ed Bellman equation is now:
±V [W; X; ¾R ] = max U [C; X ] + VX X¹ + VW [W ´ ¡ C]
C;n
1
2
¡V¾R k (¾R ¡ ¾¹R) + VXX X 2¾X
2
4
h
i
1
+ VWW W 2 n2 ¾2R + (1 ¡ n)2 ¾2R¤ + 2n (1 ¡ n) ½¾R¾R¤
2
1
+ V¾R ¾R Á2¾R + V WX WX ¾X [n½RX ¾R + (1 ¡ n) ½R¤ X ¾R¤ ]
2
p
+VW¾R W Á ¾R [n¾R½¾R + (1 ¡ n) ¾R¤ ½¾R¤ ]
p
+VX¾R X¾X ½¾X Á ¾R
and taking the ¯rst-order condition, the portfolio share invested in domestic
assets is equal to
p
¾R2 ¤ ¡ ½¾R¾R¤
VW¾R Á ¾R [¾R½¾R ¡ ¾R¤ ½¾R¤ ]
n= 2
¡
+ bias.
¾R¤ + ¾2R ¡ 2½¾R¾R¤ VWW W ¾R2 ¤ + ¾2R ¡ 2½¾R¾R¤
As in the case with time-varying asset returns, we now have an additional
hedging component related to stochastic volatility. The ¯rst two terms are
present even in the absence of nontradables. All investors, no matter where
they are located, prefer to invest in assets whose return rises when volatility
increases. The term bias, on the other hand, captures systematic discrepancies in national portfolios as the result of optimal hedging behavior. We do
want to be somewhat cautionary though by again pointing out that nontradables may have a small secondary impact on portfolios through their e®ect
on the preference term V W¾R =VWW W .
2
The sign of ¿ 0 without balanced growth
In the context of a deterministic framework we want to show that the sign of
¿ 0 is opposite to that of the growth rate of the fraction spent on nontradables.
Without uncertainty, the di®erential equation for ¿, described in Appendix A, becomes:
(1 ¡
¿ 0z
1+m
)(¿ ¡ ´ + ¹) =
²(± + °¹ ¡ ´)
¿
1 + °²m
(B.1)
It follows that
dX
dC
d¿
dW
¿ 0z
¡
= ¹¡
¡
=¹¡
(¹ ¡ ´ + ¿) ¡ ´ + ¿ =
Xdt Cdt
¿ dt W dt
¿
¿ 0z
1+m
= (1 ¡
)(¹ ¡ ´ + ¿) =
²(± + °¹ ¡ ´) (B.2)
¿
1 + °²m
5
Therefore, dC=C > dX=X if and only if ´ > ± + °¹.
Assume that the latter condition holds. Since m = (1 ¡ ®) ®¡1(C=X)1¡1=² =
C=pX, it follows that the fraction spent on nontradables 1= (1 + m) rises
(falls) over time if ² < 1 (² > 1). Moreover, since
dC
1+m
= ¹¡
²(± + °¹ ¡ ´);
Cdt
1 + °²m
(B.3)
it follows that @ (dC=Cdt) =@m > 0 when the two consumption goods are
complementary in preferences (²° < 1). Therefore,
@ (dC=C dt)
@ (dC=Cdt) @m
=
=
@ (C=X)
@m
@ (C=X)
µ
1¡®
=¡
®
¶
µ
(1 ¡ °²) (± + °¹ ¡ ´) (² ¡ 1) C
X
(1 + °²m)2
¶ ¡ 1²
(B.4)
is negative (positive) when ² < 1 (² > 1).
It is now easy to check that the sign of ¿ 0 is the opposite of that of the
growth rate of the fraction spent on nontradables. We show this for ² < 1; so
that the growth rate of tradables consumption is a negative function of the
ratio C=X. Consider the impact of an unanticipated permanent increase in
X occurring at time 0. Conjecture ¯rst that ¿ 0 = 0. In this case, tradables
consumption at time 0 does not react to the change in X. However, the
fall in the C=X ratio instantaneously translates into a higher growth rate of
tradables consumption.
Although the growth rate of tradables consumption decelerates in the
future, the level of tradables consumption is always higher than before the
shock to X: In fact, suppose that at some future time ~t the after-shock path
of C crosses the before-shock path. Necessarily, at this future date C=X is
lower than it would have been before the shock. But this implies that at ~t the
growth rate of tradables consumption must be higher than the growth rate
before the shock, so that the two consumption paths can never intersect.
As a result, tradables consumption is always higher after the shock to X:
Since ¯nancial wealth is not a®ected by the shock, the budget constraint is
violated.
Suppose now that ¿ 0 > 0: At time 0, tradables consumption jumps upward in response to the rise in X: The logic is the same as above: if there
6
existed a time t~ at which the old and new path of C met, at that time the
consumption ratio C=X would be lower than before the shock, and dC=C
would be paradoxically higher. As C is permanently above the old path, once
again the budget constraint is violated. Only the case ¿ 0 < 0 does not entail
logical inconsistencies or violations of the transversality condition. One can
use similar arguments for ² > 1 and for ´ < ± + °¹, to show that the sign of
¿ 0 is always the opposite of that of the growth rate of the fraction spent on
nontradables.
3
Small levels of risk
We want to show that under balanced growth, a marginal increase in the
standard deviation of X from zero has a negligible e®ect on ¿ and ¿ 0 as they
depend on the variance of X rather than the standard deviation.
Proposition. Let ¹ = ° ¡1(´¡±) and ±¡(1¡°)¹ > 0. Holding ½, ½RX, ½R¤ X ,
¾X =¾R, and ¾X =¾R¤ constant, @¿ =@¾2X and @¿ 0 =@¾2X, both evaluated at
2
¾X
= 0, are ¯nite functions of X and W .
^ ´ §=¾x2,
Proof. Recalling the de¯nition of § and ¾ in Appendix A, de¯ne §
^ and ^
¾^ ´ ¾=¾2x; and hold §
¾ constant with respect to changes in
¾X . After substitution of n from equation (A.8) back into the Bellman
equation, the latter becomes:
2
±V = U(C; X ) + VW W ´ ¡ VW C + VX X¹ + A(W; X; ¾X
)¾2X
where
^ ¡1i 1
1 VWW W 2 V WX WX ^
¾0 §
+
+ VXX X 2
0
^
¡1
0
^
¡1
2 i§ i
2
i§ i
³
´2
0
¡1
^ i
¾^ §
1 V 2 X 2 0 ^ ¡1
¡ WX [^
¾§ ^
¾ ¡ 0 ¡1 ]:
^ i
2 VW W
i§
A(W; X; ¾2X ) ´
Now take the right hand side di®erential of the Bellman equation with
2
2
respect to ¾X
; evaluated at ¾X
= 0. Using (i) the envelope condition
UC = VW , (ii) the homogeneous function property VW W + VXX =
7
(1 ¡ °)V , (iii) the balanced growth condition ¹ = 1° (´ ¡ ±); and (iv)
the fact that without uncertainty ¿ = 1° ± + (1 ¡ 1° )´ = ´ ¡ ¹ , we get:
V¾X2 = a 0A(W; X)
where a0 = (± ¡ (1 ¡ °)¹)¡1 and A(W; X) is shorthand for A(W; X; 0).
Di®erentiating the equation above with respect to W, and using the
envelope condition VW = UC , we get:
aAW = VW¾X2 = UCC
which implies
@¿
W
@¾2X
@¿
aAW
:
2 =
@ ¾X
UCC W
Similarly, di®erentiating with respect to both X and W yields
aAWX = VWX¾2X =
@ @UC
=
2 @X
@¾X
@
@¿ 0
@¿
0
(U
¿
+
U
)
=
U
CC
CX
CC
2
2 + U CCX W
2
@¾X
@¾X
@ ¾X
which implies
@¿ 0
aAWX aAW UCCX
=
¡
2
2
@¾X
UCC
UCC
2
This proves that both @¿ =@¾X
and @¿ 0 =@¾2X; evaluated at ¾2X = 0; are
¯nite functions of X and W .
8