Net Present Value (NPV)

The Net Present Value rule is a method used to understand whether a project is worth undertaking, by expressing future cash flows in terms of cash today. It can be used to compare to project (the one with the higher cash flow will win) or to make a decision whether to accept or reject a project (accept those ones with a positive NPV). The Net Present Value of a project is the difference between the present value of the benefits and the present value of its costs, or simpler, the present value of all its future cash flows. NPV = PV (Inflows) - PV (Outflows) = PV (All cash flows - with respective sign)

What is the Present Value?
To understand the implications of the Net Present Value rule, we must introduce the concept of Time Value of Money, i.e. the difference in value between the money today and the money in the future. Therefore, we need a methodology that allows to move all the future cash flows to today (t=0) or to an arbitrary future date.
In order to compare or combine cash flows, they need to be at the same point in time. Therefore, if we want to move cash flows backwards (i.e. bring them to t=0 or today) they must be discounted at a certain interest rate:
PV : Present Value
Cn : Cash Flow at time t = n
rn : Market Interest Rate for t = n
PV = Σ Cn ÷ (1 + rn)n
In the example below, we compute the PV as follow (assume that r is constant interest rate and equal to 3%):
PV = $ 1000 ÷ (1 + 0.03)0  +  $ 1000 ÷ (1 + 0.03)1  +  $ 1000 ÷ (1 + 0.03)2  +  $ 1000 ÷ (1 + 0.03)3  +  $ 1000 ÷ (1 + 0.03)4   +  $ 1000 ÷ (1 + 0.03)…  +  $ 1000 ÷ (1 + 0.03)N

What’s the different between present value and future value?
The future value is the result obtained by the doing the opposite operation of discounting, which is called compounding. That is necessary when we want to move our cash flows to time t=N:
FV : Future Value
Cn : Cash Flow at time t = n
rn : Market Interest Rate for t = n
FVN = C0  (1 + r)N
That is:
FVN = $ 1000 (1 + 0.03)N