I'm going to talk about a really important concept called net present value which is really one of the key concepts that you take away from studying introductory finance. So when we talk about net present value what it may mean here? What we are talking about from a conceptual point of view? Well, we're talking about taking a project or something and saying how can we value it in terms of the cash outflows and the terms the cash inflows.
We've got some project at our firm that we're trying to make a decision. Do we accept this project or reject it. We only have so much money at the firm so we have to be judicious about how we spend it. We need to know this is something that we think we should do or not and so how do we go about making that decision?
Well, we can look at the cash flows associated with this project. Let's look at in terms of time periods
Let's say this is time period zero we got time period one that's the end of year one, end of year two, end of year three, four, and five. So let's say this project ends in year five. It's a five-year project we're debating whether we want to take it or not. Now what we want to do is want to map out these cash flows so let's say that the project is going to cost us initially $10,000 that we have to invest to get this project going, we have to buy a machine or something or whatever let's not really get into the details. With $10,000 we have to lay out in order to start this project. So that's a cash outflow, this is a negative number.
So $10,000 it's an outflow but now we want to think about the inflows of this. So let's go ahead and we'll say that we'll have in year one let's say that we'll have inflows of $2,500 now that's positive that's going to get added ultimately. Then we have in year two this project if we accept it will generate $4,000 cash and then $5,000 in period 3, $3,000 in period 4 and then the final period it will generate $1000 of cash.
So we want to look at the cash flows and we want to say let's find a way that we can somehow account for the time value of money because we know that $1,000 in period 5 is not worth $1,000 today and we want to somehow net these cash inflows with the cash outflow. So how do we go about doing that? Well, we are going to have to calculate the net of the present value of each cash inflow we can't just use an annuity formula because the cash inflows are a different number in each period so what we'll do is we'll treat each one as a single cash flow and then go ahead and discount it back to what its value is at period zero (0) and we don't have to discount the negative 10,000 because we know that negative 10,000 is today is 10,000 today so we don't have to worry about that one. But we have to discount the other cash flows here. So what we're going to ultimately have is an equation that is going to look like this. 
So our net present value is going to be equal to negative 10,000 again we do not need to discount that because 10,000 today is 10,000 today. So we don't have to worry about that but then we're going to add to that the cash inflows but discount it. So we've got 2500 that we get at the end of period 1 but we have to discount that so let me just rewrite the formula for the present value of a single cash flow.
Let's see if the cash flow over (1+r) to the t which means time and in period one of course t is one so we don't even concern ourselves with that and just a side note we're going to need a discount rate here and we'll say that the discount rate we could have earned let's say that we could have earned 6% on an investment with a similar risk that's our opportunity cost we're going to need that to calculate this formula here. So we're going to use this formula and now we're going to calculate the present value of that $2,500 at the end of year 1 so we've got 2500 in the numerator and then in the denominator, we're going to have (1+r) and we know they are being 6%, (1+r) is going to be 1.06. Now the year 2 sketch flows of $4,000 we're going to have to treat that as a separate cash flow. Now that's going to be discounted back to periods because that's two years from now so it's to the second power that t the time period.
Now for the third period we've got 5,000 in terms of the cash flow but we're going to have to discount that back three periods and then we've got at the end of year four we get 3,000 and we'll discount that back four periods and then finally at the end of the project we get a cash inflow of 1,000 which will discount back five years so ultimately this formula here is going to yield our net present value.
Now and then we'll talk about how to interpret that and what we're going to do with it in a moment but let's just go ahead and finish out these calculations. So this net present value is going to be equal to that negative 10,000 that we incur upfront to start this project and then all these cash flows when we add that all up it's going to be $13,239.
So now this where the net part comes in we net these together we're netting the cash outflow this is going out at the beginning of the project and then we've got the inflows this is the discounted value of the inflows we take it together and that's going to give us $3,239 is the net present value of this project.
Well what does that mean and how to interpret it? In general, assuming that the company has enough cash to do the project, what you want to do is you want to use a rule hereof if the NPV is greater than zero then accept the project.
So then of course 3239 is greater than zero so we would accept this project. Now let's explain that this rule here in a little more detail. What is this saying if the net present value is greater than zero well remember the discount rate that we chose of 6% well ultimately we can view that as an opportunity cost, this 6% if we don't invest the money in this project it could be somewhere else maybe you could have invested in stocks or bonds or something that would have earned us this 6% return so the capital is not free.
The money to do this project is not free it could have been deployed elsewhere to earn a return and so when the NPV is greater than zero what this is basically saying is that we have added value to the firm above and beyond the 6% that we otherwise could have done with that money. So by accepting this project we're actually earning a return to the firm that is higher than this discount rate that we were thinking about, higher than the opportunity cost. So we could have made 6% elsewhere, did we do better than? That's what we're trying to figure the NPV is greater than zero then the answer is "yes" we did better than that and so we need to accept the project.
