(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(a) 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(O) = 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;()= (1+r+1)-. 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(0) = c d d=0 P'(O)(1+r) = {tcd id; = 0. (a) Show that for all values of a and B there holds p"(O)(1+r)2 = (12+at+B)cd (72 +ai+B)dz. (b) Show that a and can be selected so that the function 12 +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(a) 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(O) = 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;()= (1+r+1)-. 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(0) = c d d=0 P'(O)(1+r) = {tcd id; = 0. (a) Show that for all values of a and B there holds p"(O)(1+r)2 = (12+at+B)cd (72 +ai+B)dz. (b) Show that a and can be selected so that the function 12 +at+ has a minimum at i and has a value of 1 there. Use these values to conclude that P"(0) > 0