4. Suppose that an obligation occurring at a single time period is immunized against interest rate changes with bonds that have only nonnegative cash flows. Let P()) be the value of the resulting portfolio, including the obligation, when the interest rate is r + A and r is the current interest rate. Here, A represents the change in the interest rate. By immunization construction, we have set P(0) = 0 and P' (0) = 0. In this problem, we would like to show that P(0) is a local minimum; that is, P"(0) 2 0. Assume a yearly compounding convention. The discount factor at time t is dt (A) = (1 +r + )). Let dt = dt(0). For convenience, we assume that the obligation has magnitude 1 and is due at time t. The conditions for immunization are then given by P(0) = _ adt - di = 0 P'(0)(1 +r) = > taedt - tdi = 0