(Convexity theorem ) Suppose that an obligation occurring at a single time period is immunized against interest rate changes with bonds that have only nonnegative cash flows (as in the X Corporation example). Let P(X) be the value of the resulting portfolio, including the obligation, when the interest rate is r +2 and r is the current interest rate. By construction P(0) = 0 and P'(0) = 0. In this exercise we show that P(0) is a local minimum; that is, P" (0) > 0. (This property is exhibited by Example 3.10.) Assume a yearly compounding convention. The discount factor for time t is d: (a)= (1+r+2). Let dt = d; (0). For convenience assume that the obligation has magnitude 1 and is due at time i. The conditions for immunization are then P(O) = g;d d;=0 P'(O)(1+r) = {texd id; = 0. (a) Show that for all values of a and B there holds p"CO)(1 + r)2 = (2+at+B)cd (72+ ai +B)d;. (6) Show that a and can be selected so that the function t2 +at+ has a minimum at i and has a value of 1 there. Use these values to conclude that P" (0) 0. (Convexity theorem ) Suppose that an obligation occurring at a single time period is immunized against interest rate changes with bonds that have only nonnegative cash flows (as in the X Corporation example). Let P(X) be the value of the resulting portfolio, including the obligation, when the interest rate is r +2 and r is the current interest rate. By construction P(0) = 0 and P'(0) = 0. In this exercise we show that P(0) is a local minimum; that is, P" (0) > 0. (This property is exhibited by Example 3.10.) Assume a yearly compounding convention. The discount factor for time t is d: (a)= (1+r+2). Let dt = d; (0). For convenience assume that the obligation has magnitude 1 and is due at time i. The conditions for immunization are then P(O) = g;d d;=0 P'(O)(1+r) = {texd id; = 0. (a) Show that for all values of a and B there holds p"CO)(1 + r)2 = (2+at+B)cd (72+ ai +B)d;. (6) Show that a and can be selected so that the function t2 +at+ has a minimum at i and has a value of 1 there. Use these values to conclude that P" (0) 0