Exchange Rate Economics: Theories and Evidence

Spot and forward exchange rates 379
If R 0 t+1 is the return on an asset that has a zero conditional covariance with Q t+1,
then
E t (R 0 t+1 ) = 1
E t (Q t+1 ) ,
and
E t (R j
t+1 − R0 t+1 ) = −Cov t(Q t+1 , R j
E t (Q t+1 )
t+1 )
. (15.24)
If it is assumed that agents can trade an asset whose return is the minimum second
moment return (see Hodrick 1987) – Rt+1 m = Q t+1/E t (Q t+1 ) 2 – then Hansen and
Richard (1984) demonstrate that any return, Rt+k b ,on the mean-variance frontier
can be written as a weighted average of Rt+1 m and R0 t+1 :
R b t+1 = σ tR m t+1 + (1 − σ t)R 0 t+1 . (15.25)
With these relations it is possible to rewrite equation (15.14) in CAPM form as:
where
E t (R j
t+1 − R0 t+1 ) = β j
t (R b t+1 − R0 t+1 ),(15.26)
t = Cov t(Rt+1 b , R j
t+1 )
Var t (Rt+1 b ) .
β j
An alternative way of demonstrating this result (see Lewis 1995) is as follows. Since
relationship (15.14) holds for any asset with return j,it must also hold for the risk
free rate and we can write (15.15) as:
E t {Q t+1 (R j
t+1 − R f
t+1 )}=E t{Q t+1 ex j
t+1 }=0,(15.27)
where ex j
t+1 ≡ R j
t+1 − R f
t+1 and R f
t+1
is the risk free rate. Using the definition of
covariances and (15.14),equation (15.27) can be rewritten as:
E t (ex j
t+1 ) =−Cov t(R j
t+1 , Q t+1)R f
t+1 . (15.28)
and since this holds for any asset,such as the benchmark return:
E t (ex b t+1 ) =−Cov t(R b t+1 , Q t+1)R f
t+1 . (15.29)
And by substituting out for the risk free rate we get a similar expression to (15.26),
namely:
E t (ex j
t+1 ) =[Cov t(R j
t+1 , Q t+1)/Cov t (R b t+1 , Q t+1)]E t (ex b t+1 ),(15.30)
where R b is the benchmark return and R is the risk free rate.