Net Present Value (NPV)

We're 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, well, what do we mean here? What are we 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 cash inflows. So let's use an example here. So we've got some project at our firm that we're trying to make a decision. Do we accept this project or reject it? Okay, we only have so much money at the firm, so we have to be judicious about how we spend it. We need to know, okay, is this 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. So let's look at in terms of time periods. Let's say this is time period zero. We've 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 at 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 we want to map out these cash flows. So let's say that the project is going to cost us initially in period zero. So this is up front. Period zero is right now. Let's say it's going to cost us ten thousand dollars that we're going to 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, but ten thousand dollars we have to lay out in order to start this project. So that goes right here. So that's negative. That's a cash outflow. This is a negative number. Okay, so ten thousand dollars is 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 twenty five hundred dollars. Okay, now that's that's positive. That's going to get, this is going to get added ultimately. And then we have in year two, this project if we accept it will generate four thousand dollars cash and then five thousand in period three, three thousand in period four, and then the final period it'll generate a thousand dollars of cash. So what we want to get is we want to look at the cash flows and we want to say, okay, let's find a way that we can somehow account for the time value of money because we know that a thousand dollars in period five is not worth a thousand dollars 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 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. And we don't have to discount the negative ten thousand because we know that negative ten thousand, that's negative ten thousand today. So we don't have to worry about that one, but we have to, we're going to 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 ten thousand. Again, we do not need to discount that because ten thousand today is ten thousand 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 the twenty five hundred that we get at the end of period one, but we have to discount that. So let me just go ahead and rewrite the formula for present value of a single cash flow. That's C being the cash flow over one plus R to the T. And in period one, of course, T is one, so we don't even concern ourselves with that. And then also, 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 six percent 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 twenty five hundred dollars at the end of year one. So we've got the twenty five hundred in the numerator and then in the denominator, we're going to have one plus R and we know the R being six percent, one plus R is going to be one point zero six. OK, now the year two's cash flows of four thousand dollars, we're going to have to treat that as a separate cash flow. OK, now that's going to be discounted back two 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 five thousand 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 three thousand dollars and we'll discount that back four periods. And then finally, at the end of the project, we get a cash inflow of a thousand dollars, which will discount back five years. So ultimately, this formula here is going to yield our net present value. 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 ten thousand that we incur up front to start this project. And then all these cash flows here, when we add that all up, it's going to add up to thirteen thousand two hundred and thirty nine dollars. So now we put now this where the net part comes in. We net these together. We're netting the cash outflow. This is 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 three thousand three thousand two hundred and thirty nine dollars is the net present value of this project. Well, what does that mean and how to interpret it? Well, in general, assuming that the company has enough cash to do the project, what you want to do is you want to use a kind of rough rule here of if the MPV is greater than zero, then accept the project. Accept. And then, of course, three thousand two hundred and thirty nine is greater than zero. So we would accept by this rule. Now let's explain this rule here in a little more detail. What is this saying if the net present value is greater than zero? Well, remember, remember our our discount rate that we chose of six percent? Well, ultimately, we can kind of view that as an opportunity cost. This six percent. If we don't invest the money in this this project here, if we don't spend it there, well, it could be somewhere else. Maybe you could have invested in stocks or bonds or something that would have earned us this six percent return. So 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 MPV is greater than zero, what this is basically saying is that we have added value to the firm above and beyond the six percent that we otherwise could have done with that money. So by 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 six percent elsewhere. Did we do better than that? That's what we're trying to figure. And if the MPV is greater than zero, then the answer is yes, we did do better than that. And so we need to accept the project.