15. Suppose that a bond pays a cash flow Ci at time Ti for i = 1,...,N. Then the net present value (NPV) of cash flow Ci is NPVi = Ci exp(−Ti yTi ).

Define the weights

ωi = NPVi

N j=1 NPVj and define the duration of the bond to be DUR =

N i=1

ωiTi, which is the weighted average of the times of the cash flows. Show that d

dδ

N i=1 Ci exp{−Ti(yTi + δ)}









δ=0

= −DUR N

i=1 Ci exp{−Ti yTi }

and use this result to verify Eq. (3.31).