$15$ points, $5$ points each part. Consider two semi$-$annual coupon bonds $($CB$),$

each with maturity $T$ years. The first $CB$ has face value $F_1,$ annual coupon rate $q_1,$

and is currently trading at the price $p_1.$ The second $CB$ has face value $F_2,$ annual

coupon rate $q_2,$ and is currently trading at the price $p_2.$ Without loss of generality

you may assume $q_2>q_1.$ Determine the value of the following securities:

$(1)$ A $T$ year zero coupon bond $($ZCB$)$ with face value $F.$ What condition on the

current yields of the bonds is needed to ensure the ZCB price is strictly positive?

$(2)$ An annuity that will make payments of A twice per year for the next $T$ years.

$(3)$ A third semi$-$annual CB with maturity $T$ years, face value $F_3,$ and annual coupon

rate $q_3.$