16. (Convexity theorem o) 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(2) be the value of the resulting port- folio, including the obligation, when the interest rate is + and is the current in- terest 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 Exam- ple 3 10 ) Assume a yearly compounding convention. The discount factor for timer is

d, (): (+2) Let

d, d,(0). For convenience assume that the obligation has magnitude 1 and is due at time 7 The conditions for immunization are then P(0)=Gd, - d = 0 P'(0)(1+r) = tc,d, - id; =0 I

(a) Show that for all values of or and there holds P" (0)(1+r) = r+ ar + B)c,d, (1 + a + )d;

(b) Show that a and B can be selected so that the function 2+at+ has a minimum at 7 and has a value of I there. Use these values to conclude that P" (0) 0