The term premium at time t for the future period [t , T] is the current
Question:
The term premium at time t for the future period [t
, T] is the current forward rate for that period minus the expected spot rate, that is f t
,T t − Et[yT t]. This exercise will give a link between the term premium and a state-price deflator ζ = (ζt).
(a) Show that BT t = Bt
t Et
BT t
+ Covt
ζt
ζt
,
ζT
ζt
for any t ≤ t
≤ T.
(b) Using the above result, show that Et
e
−yT t(T−t
)
− e
−f t
,T t (T−t
) = − 1 Bt
t Covt
ζt
ζt
,
ζT
ζt
.
Using the previous result and the approximation ex ≈ 1 + x, show that f
t
,T t − Et[yT t] ≈ − 1
(T − t
)Bt
t Covt
ζt
ζt
,
ζT
ζt
.
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