Exchange Rate Economics: Theories and Evidence

Spot and forward exchange rates 383
where it is assumed that the only form of uncertainty comes from the value of the
exchange rate in the next period. The first-order condition for this model can be
written as:
(1 + i ∗ t )E t(S t+1 /S t ) − (1 + i t ) = φω t (1 + i ∗ t )2 Var t (S t+1 /S t ). (15.41)
Asssuming that ln(s t+1 ) ≡ s t+1 is conditionally normally distributed,expression
(15.41) can be written as:
E t (s t+1 ) − s t + i ∗ t − i t + 0.5 · Var t (s t+1 ) = φω t Var t (s t+1 ). (15.42)
An analogous expression to (15.42) can be derived for investors in the foreign
country who maximise a function of the mean and variance of wealth in foreign
terms:
−E t (s t+1 ) + s t − i ∗ t + i t + 0.5 · Var t (s t+1 ) = φ(1 − ω ∗ t )Var t(s t+1 ),(15.43)
where ω ∗ t represents the fraction of wealth that foreigners invest in their own
bonds. If µ is the share of total wealth held by domestic residents,then by multiplying
equation (15.42) by ω t and equation (15.43) by ωt ∗ and adding the resulting
expressions together,the following expression may be obtained:
E t (s t+1 ) − s t + i ∗ t − i t =[−0.5 + (1−φ)(1−µ t ) + φ(µ t ω t + (1−µ t )ω ∗ t )]
× Var t (s t+1 ),(15.44)
where the term µ t ω t + (1 − µ t )ω ∗ t is the value of foreign bonds held in the world
as a fraction of world wealth which we define as ¯ω t and so (15.44) may be written
more compactly as:
E t (s t+1 )−s t + i ∗ t −i t = Var t (s t+1 )[(1−φ)(1−µ t )−0.5]+φVar t (s t+1 ) ¯ω t ,
(15.45)
where the home–foreign bond differential is related to the share of foreign bonds in
world wealth,to the share of wealth held by domestic residents and to the variance
of the exchange rate. As Engel (1996) points out equation (15.45) can be thought
of as a restricted version of equation (15.38). This can be seen more clearly by
rewriting (15.38) using the terminology underlying (15.45):
E t (s t+1 ) − s t + i ∗ t − i t = α t + δ t (1 − µ t ) + γ t ¯ω t . (15.46)
In the standard portfolio-balance model considered in Chapter 7 and represented
by (15.36) and (15.46) the time-varying parameters α t , δ t and γ t are
not restricted. The key distinguishing feature of the mean-variance version