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



.