Exercise . 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 B T t = B t t Et

, B T 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

) = − 

B t t

Covt

(

ζt

ζt

,

ζT

ζt

)

.

Using the previous result and the approximation ex ≈  + x, show that f

t

,T t − Et[yT t

]≈−



(T − t

)B t t

Covt

(

ζt

ζt

,

ζT

ζt

)

.