There was a two part question assigned to my class for homework. It's already been turned in, but I couldn't get the 2nd part, and am curious how it is done:
suppose there is a k between 0 and n such that either:
show there is a unique i > -1 for which the net present value of this transaction is 0.
this is the part I got. I don't need help on it, but am just introducing it, because I'm sure it somehow is used to prove the next part.
Let be an arbitrary sequence of net cashflows, and let
Suppose both and are non-zero, and that the sequence has exactly one change of sign.
Show there is a unique i > 0 such that the net present value of these cash flows is 0 (although there may be one or more negative roots).
This is the part I need help on. I can show what I've done so far, but it really isn't anything except showing there is a root i > 0. I haven't shown that it is unique.
we need to show that the equation has a unique solution 0 < v < 1.
if v = 0,
if v = 1,
we are given F_0 and F_n are non-zero and opposite signs of each other, so by the intermediate value theorem there exists a root v, 0 < v < 1.
This is all I have come up with, I don't know if that is a good starting point for the rest of the solution or not. any help would be appreciated! Thanks in advance.