The purpose of this exercise is to show that the claim of the gross return pure expectation hypothesis is inconsistent with interest rate uncertainty. In the following we consider time points t0 < t1 < t2.

(a) Show that if the hypothesis holds, then 1

Bt1 t0

= 1 Bt2 t0 Et0



Bt2 t1



.

Hint: Compare two investment strategies over the period [t0, t1]. The first strategy is to buy at time t0 zero-coupon bonds maturing at time t1. The second strategy is to buy at time t0 zero-coupon bonds maturing at time t2 and to sell them again at time t1.

(b) Show that if the hypothesis holds, then 1

Bt2 t0

= 1 Bt1 t0 Et0 1

Bt2 t1



.

(c) Show from the two previous questions that the hypothesis implies that Et0 1

Bt2 t1



= 1 Et0



Bt2 t1

 . (*)

(d) Show that (*) can only hold under full certainty. Hint: Use Jensen’s inequality.