Exercise 5 (Forward interest rates). A forward rate at current time t for period (T1, T2), t s T, is the rate agreed at t at which one can lend money during (T1, T2]. Denote this forward rate by f12. Suppose that the current zero rates (continuously compounded) during (t, Ta] and [t, T2] are r and r2, respectively. (i) Consider two portfolios: (A) Deposit x during [t, T2] at rate r2; (B) Deposit y during [t, T) at rate r, and go short one forward contract to lend yen(1,-1) at rate f12 during [T1, T2]. Show that at time T2, their values are xera(T3-1) and yeri (T1-1) efiz(T2-T1), respectively. (ii) For the choice y=e="(T;-1)- fiz(T2-T1), (8) show that the two portfolios in (i) have the same value 1 at time T2. Conclude that x = y, and deduce f12 = r2(T2 t) r(T1 t). (9) T2 - T6 (iii) Show the conclusion of (ii) using a no-arbitrage argument. (Hint: Compare the two portfolios in (i) with x = y = 1. If one always has higher value than the other at time T2, one can create an arbitrage opportunity.) x = erz(12-1)