Recall that in class duration was defined as the magnitude of the percentage change in the price (P) of a bond for a given percentage change in 1 + R, where R is the yield on the bond. (Also see notes N5 on BB.) That is, D = dP/dR[1 + R/P]. Also recall that the price of the bond is simply the present value of the stream of payments: P = C_1/(1 + R)^1 + C_2/(1 + R)^2 + C_3/(1 + R)^3 + ... + C_k/(1 + R)^k + ... + C_K - 1/(1 + R)^K - 1 + C_K/(1 + R)^K where C_k is the cash flow at period k, K is the total number of periods, and R is the periodic yield. By taking the derivative of P (as defined by the present value formula) with respect to R, show that the formula for calculating duration is simply the weighted average of the times (k) in which the payments are made, where the weight for each period is the fraction of the total present value of the bond that comes from that period's payment. That is, show that D = -dP/dR[1 + R/P] = (C_1/(1 + R)^1/P) * 1 + (C_2/(1 + R)^2/P * 2 + (C_3/(1 + R)^3/P) * 3 + ... +(C_k/(1 + R)^k/P) * k + ... + (C_K - 1/(1 + R)^K - 1/P) * (K - 1) + (C_K/(1 + R)^K/P) * K