Exchange Rate Economics: Theories and Evidence

376 Spot and forward exchange rates
foregone marginal utility from investing in this asset is Vt iu
c t
/P t and the expected
marginal utility of the payoff is E t [βu ct (Vt+1 i +Di t+1 )/P t+1]. Equating the marginal
benefit to the marginal cost produces an expression that must be satisfied by all
equilibrium returns defined in terms of the home currency:
E t (Q t+1 R j
t+1 ) = 1 ∀ j,(15.13)
where Rt+1 i ≡ (V t+1 i + Di t+1 )/V t
i and Q ≡[βu ct+1 P t /u ct P t+1 ] is the so-called
pricing kernel. This expression may be related to the forward rate in the following
way. If R f
t+1
is the nominal interest rate on a risk free discount bond paying one
unit of home money,M,in period t + 1 (= (1 + i)) (a certain payoff ) and so the
price of such an asset is simply:
or
1/R f
t+1 = E t(Q t+1 ),(15.14)
1 = E t (Q t+1 R f
t+1 ),
equally we can think of this condition holding for an interest differential:
or
1/R f
t+1 − 1/Rc t+1 = E t(Q t+1 ),(15.15)
0 = E t (Q t+1 (R f
t+1 − Rc t+1 )).
For the equivalent foreign position we have:
or
1/R f ∗
t+1 = E t{βu ∗ c t+1
P ∗
t /u ∗ c t
P ∗
t+1 }≡E t(Q ∗ t+1 ),
1/R f ∗
t+1 = E t(Q ∗ t+1 ). (15.16)
By exploiting the familiar covered interest parity (CIP) condition – (1 + i t ) =
(1 + i ∗ t )(F t/S t ) – (15.14) and (15.16) may be rewritten as:
F t = S t R f
t+1 /R f ∗
t+1 = S tE t (Q ∗ t+1 )/E t(Q t+1 ). (15.17)
As Lewis (1995) notes,this relationship between the spot and forward exchange
rate is quite general and to solve for the forward rate using the specific form of
the Lucas model requires substituting (15.12) or the monetary extension of (15.12),
derived in Chapter 4,which we use here:
[
F t = S t R f
t+1 /R f ∗ u
∗
t+1 = ct M t y ∗ ]
t
u ct Mt
∗ E t (Q ∗ t+1
y )/E t(Q t+1 ). (15.18)
t