Exchange Rate Economics: Theories and Evidence

Spot and forward exchange rates 377
The earlier relationships can be used to re-express the Fama result that the variance
of the risk premium is greater than the variance of the expected change in the
exchange rate – var(λ t ) is > var(E t s t+k − s t ) as:
Var{E t (Q ∗ t+1 /Q t+1) −[E t (Q ∗ t+1 )/E t(Q t+1 )]} > Var{E t (Q ∗ t+1 /Q t+1)}.
(15.19)
In words,expression (15.19) indicates that the risk premium is the difference
between the ratio of expected marginal rates of substitution in consumption and
the expectation of this ratio and the variance of this difference exceeds the variance
of the expected ratio of marginal rates of substitution.
From (15.15),and by exploiting the CIP condition,a further implication of
(15.15) is
(
)
S t+1 − F t
E t Q t = 0,(15.20)
S t
which may be rewritten to give an alternative interpretation of the forward
premium as:
( ) (
St+1 − F t
E t = Cov Q t ; S )
t+1 − F t
|I t [−E t (Q t )] −1 ,(15.21)
S t S t
or
( ) (
F t St+1
= E t − Cov Q t ; S )
t+1 − F t
|I t [−E t (Q t )] −1 ,
S t S t S t
where the risk premium is the second term on the right hand side and this clearly
implies that the forward rate need not be an unbiased predictor of the future spot
rate. Note that in this model this biasedness result can occur even with risk-neutral
agents,a situation characterised by the linearity of the utility function underlying
(15.21).
15.3.2 Testing the general equilibrium risk premium model
A number of researchers have empirically implemented the general equilibrium
approach to modelling the risk premium and we consider the various tests in this
subsection.
15.3.2.1 Direct (error orthogonality) tests
Mark (1985) considers a variant of equation (15.15),in which the return is defined
as the forward forecast error: 2
(
)
S t+1 − F t
E t Q t = 0. (15.22)
S t