Suppose that a bond pays a cash flow C_i at time T_i for i = 1, ..., N. Then the net present value (NPV) of cash flow C_i is NPV_i = C_i exp (-T_i y_Ti). Define the weights w_i = NPV_i/sigma^N_j = 1 NPV_j and define the duration of the bond to be DUR = sigma^N_i=1 omega_i T_i, which is the weighted average of the times of the cash flows. Show that d/d delta sigma^N_i=1 C_i exp {-T_i (yT_i + delta)}|delta = 0 = - DUR sigma^N_i=1 C_i exp{-T_i yT_i and use this result to verify change bound price/bond price approximatelyequalto - DUR times delta