The bene…ts of international diversi…cation have been known for many decades, but it is only recently that investors have started allocating a signi…cant portion of their portfolio holdings in foreign assets. To manage the risk of international portfolios, investors need to know the speculative and hedging demands in the cross-country variation in global return uncertainty.
This study investigates the international asset allocation for global investors, which in-corporates the hedge demands in controlling the stochastic variation due to PPP deviation.
The development of our approach adding to the previous works of Lioui and Poncet (2003) is that we compare the obtained optimal strategies with certain market structure in order to clarify the hedge e¤ects in …nancial decision allowing for global investors. Finally, hypothet-ical mutual funds are constructed in our work to ful…ll the proposed demands. The optimal investment strategies are a leveraged growth optimal portfolio, but with contingent leverages as time goes by.
Following the four-fund theorem stated in Rudof and Ziemba (2004), the optimal portfo-lio consists of into four components: the international myopic portfoportfo-lio, the domestic interest rate hedge portfolio, the cross country interest rate di¤erential hedge portfolio and the do-mestic riskless asset. With respect to the most common approach used in the literature, the market structure and the certain utility employed to describe the investor’s attitude toward risk allow us to …nd the general pattern of the optimal strategy for investors through dynamic
fund separation methodology.
7 Appendix
7.1 Evaluation of a Certain Conditional Expectation
Theorem 2 (Feynman-Kac Formula, c.f. Lamberton et al (1991), Theorem 5.1.7) Let u be a well-behaved function de…ned on [0; T ] Rn: If u satis…es
@u
@t + Atu ru = 0; 8(t; x) 2 [0; T ] Rn
and
u(T; x) = f (x);
then
u(t; x) = E f (XTt;x) exp
Z T t
r( ; Xt;x)d ;
where the At is the in…nitesimal operator of the n dimensional di¤ usion process dXt = b(t; Xt)dt+ (t; Xt)dWt. The conditional expectation is taken with respect to t; where Xt= x:
We consider a conditional expectation u(t; x) as the following
u(t; x)) = E exp k Z T
t
Z( )2d (9)
which is conditioned at t and Xt = Z (t) = x; where Z(t) is a standard one-dimensional Wiener process and k is a constant. Note that, the conditional expectation (9) is akin to (30) modulo a deterministic factor and the evaluation of the more general (30) may bene…t from the following approach. By Feynman-Kac formula, we immediately write down the
PDE satis…ed by u; which is
@u
@t = 1 2
@2u
@x2 kx2u and subject to the boundary condition
u(T; x) = 1:
Assume that u satis…es the form
u(t; x) = exp (t)x2 '(t) ; (10)
then the boundary condition becomes
(T ) = 0; '(T ) = 0:
After the separation of variables, we have
x2 0(t) + 2 (t)2 k = 0
and
(t) = '0(t):
Solution of the above two ODEs yields
(t) = rk
2tanhp
2k(T t) (11)
and
'(t) =1 2ln
h coshp
2k(T t) i
: (12)
7.2 Evaluation of Constant Parameter Models
The following list is the summary of the underlying dynamics in this constant case, and note that all coe¢ cients without argument notation are all constants.
de(t)
where
and
From Thus, by substituting (22) and (23) into (3), the de…nition of , we have
(t; T ) = exp
Upon inspection, only the …rst term in the last equality would generate stochastic components after taking conditional expectations. We proceed to carry out the calculation.
Applying the decomposition of (t) in (21) to the integral
exp
we have We neglect the integrals 12RT
t 1( )0 1( )d , RT
t 1( )0dZ( )on the right-hand side of (24) because of the deterministic contributions after taking the conditional expectations. There are three stochastic integrals left, namely
1
Applying (25), we have
d (rf( ) rd( )) = q ( )dt +
where q ( ) = dq( )=d : De…ne (t) such that
(t) = Z t
o
2( )0 2( )d : (27)
Integration by parts and the application of Itô’s lemma with (rf( ) rd( ))2render the integral
After substituting (26) into (28), it is clear that the only term we need to specify is Z T
through repeated integration by parts, where
(t) = Z t
0
( )q ( )d : (29)
Thus, we may summarize our results in the following lemmas
Lemma 3 With the assumptions of our …nancial model, there exist two deterministic func-tions (T ) in equation (27) and (T ) in equation (29) such that
1 2
Z T t
2( )0 2( ) (rf( ) rd( ))2d (30)
= 1
2(rf(T ) rd(T ))2 (T ) 1
2(rf(t) rd(t))2 (t) (rf(T ) rd(T )) (T ) + (rf(t) rd(t)) (t) +X
i
Z T t
( )dZi( ) + ( ):
Lemma 4 With the assumptions of our …nancial model, there exist two deterministic func-tions e (T ) such that the integral
Z T t
1( )0 2( ) (rf( ) rd( )) d
may be treated in a similar fashion. The …nal result is Z T
t
1( )0 2( ) (rf( ) rd( )) d (31)
= (rf(T ) rd(T )) e (T ) (rf(t) rd(t)) e (t) +X
i
Z T t
( )dZi( ) + ( );
where
e(t) =Z t 0
1( )0 2( )d : (32)
Lemma 5 With the assumptions of our …nancial model, substituting the expression (25) into the stochastic integral
Z T t
(rf( ) rd( )) 2( )0dZ( );
we have
Z T t
(rf( ) rd( )) 2( )0dZ( )
= Z T
t
q( ) + X5 i=1
liZi( )
!
2( )0dZ( )
= X
i
Z T t
( )dZi( ) +X
i;j
Z T t
li 2j( )Zi( )dZj( ); (33)
where 2j( ); 1 j 5 denotes the j th component of the 5 1 function 2( ):
Collecting all the results of (30),(31) and (33) obtained above, we compute J ( ; t; T ) as
J ( ; t; T ) = Et (t; T ) 1
= A( ; t; T ) exp
2 (1 )(rf(t) rd(t))2( (T ) (t))
exp 1 (rf(t) rd(t)) e (T ) e(t) (T ) + (t) ;
where A( ; t; T ) is a deterministic function. Here we utilize the independence property of (rf(T ) rd(T )) (rf(t) rd(t)) with respect to the conditional expectation operator Et[ ] because of the expression (25), and the fact such that the expressionRT
t li 2j( )Zi( )dZj( ) is independent with respect to Et[ ] is also used. Applying Itô’s lemma, we have
dJ ( ; t; T ) J ( ; t; T )
= 1
n
( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o d (rf(t) rd(t)) + ( ) dt
= 1
n
( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o X5
i=1
lidZi(t) + ( ) dt:
We immediately obtain the following proposition:
Proposition 6 The instantaneous conditional ( 1) moment of the Arrow-Debreu prices of the reference country bond of maturity T is given by
J ( ; t; T ) = Et (t; T ) 1
= A( ; t; T ) exp
2 (1 )(rf(t) rd(t))2( (T ) (t))
exp 1 (rf(t) rd(t)) e (T ) e(t) (T ) + (t) ;
where A( ; t; T ) is a deterministic function. The di¤ usion vector J( ; t; T )> of the process of dJ ( ;t;T )
J ( ;t;T ) is given by
J( ; t; T )0 (34)
= 1
n
( (T ) (t)) (rf(t) rd(t)) + e(T ) e(t) (T ) + (t) o l1 l2 l3 l4 l5 :
Substituting the expressions of (t); (t) and e (T ) in (27), (29) and (32), respectively and (5), (16), (21) and (34) into (6), we obtain the expression of optimal allocation strategy
(t) of risky assets.
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