# Math Help - Interest rate that gives a net present value of zero.

1. ## Interest rate that gives a net present value of zero.

A project has cash flows of -$12,000 in Year 1, +$5000 in Years 2 and 3, -$2000 in Year 4, and +$6000 in Years 5 and 6. Find the interest rate that gives a net present value of zero?

The answer is provided to me, it is 19.2%.
I would like to see what the process was to get that answer

2. wrong forum i think.

You did not say whether the casflows happen at the start or the end of each year. I will assume they happen at the start of each year.

Using standard notation
i = annual interest rate
$v=1/(1+i)$

you want to solve PV = 0
$-12000 + 5000v + 5000v^2 -2000v^3 + 6000v^4 + 6000v^5 = 0$

Use trial and error. For a first guess i will use 10%
@ 10%: PV = 2099
@ 20%: PV = -213

So the answer is between 10% and 20%. it looks closer to 20%

@18%: 328
@19%: 51.98

So the answer is between 19% and 20%

Keep going and you will get 19.2%

3. If 19.2%, then flows are at end, so:
-12000v + 5000v^2 + 5000v^3 - 2000/v^4 +6000/v^5 + 6000/v^6 = 0

As SFan told you, can't be calculated directly: so "hit and miss!".

4. If 19.2%, then flows are at end,
in fact, it doesn't matter

for cashflows at the start of each period you solve:
$(A)~~-12000 + 5000v + 5000v^2 -2000v^3 + 6000v^4 + 6000v^5 = 0$

For cashflows at the end of each period you solve

$(B)~~-12000v + 5000v^2 + 5000v^3 -2000v^4 + 6000v^5 + 6000v^6 = 0$
But, this factorises to (A)
$(B)~~~v(-12000 + 5000v+ 5000v^2 -2000v^3 + 6000v^4 + 6000v^5) = 0$

We know v is not 0, so any solution of B is also a solution of A. You can actually shift the payments by any (constant) length of time and it still works, provided that all payments are shifted by the same amount.

5. Agree; shudda known!