So let's talk about net present value analysis, and let's start to talk about projects. So when we calculate net present values and we do an analysis of different projects, what it allows us to do is to make comparisons among alternative investment projects when the cash flows vary through time and they occur at different points in time. So let's consider the two following projects, both of these projects require an initial investment of $2,000, and both projects are going to last for 3 years. But as we'll see, the cash flows are not the same through time. So here are the two projects. If you see project A, the initial investment is -2,000 and you can see that it doesn't return any cash to the company for the first two years. But then in the third year it returns a cash flow of $4,500. Project B also requires an initial investment of $2,000 and it returns $2,000 in the first year, it returns another $2,000 in the second year and it returns $100 in the third year. If you were to ask an accountant, which of these were has the higher net income? The accountant would simply sum up those cash flows, and the accountant would say, the net income associated with project A is higher than the net income associated with project B. But of course, what we've been learning is that money has a time dimension. And you'll see that the timing of the cash flows is very different between these two projects. So in order to actually figure out which of these projects is better, we need to compute what the present value of the cash flows is associated with these projects. So now let's calculate the net present value of these two projects because we saw that the money arrives at different points in time, and so it's going to be important to know what the value of these projects is based on the time value of money. We're going to assume a 10% interest rate. But now we're going to talk about projects, where not going to use the word interest rate any longer. When people talk about evaluating projects, the R in the formula, they usually talk about that R in the formula as either the discount rate, the hurdle rate or the company's cost of capital. The discount rate is the rate at which we're discounting the cash flows, that's where that word comes from. And in fact, this is often referred to as discounted cash flow analysis. So somebody talks about DCFs or discounted cash flow they're just talking about calculating net present values. The hurdle rate that term comes from, a project has to get over a hurdle in order to be accepted and that hurdle rate is the hurdle that the project has to get over. The R is also often referred to as a company's cost of capital and it is the cost to the company of obtaining the capital that it uses to finance itself. And we'll come back later and talk more about the cost of capital. If we were to calculate the net present value of project A, what would we have to do? Well, you would take -2,000, and remember that's the initial investment, and that's the amount we assume is being paid out today to invest in this project. So we don't discount that, because the present value of a dollar today is just a dollar. And so we're writing out a check for $2,000 today, or we have a cash-out flow of $2,000, and the present value of that is just minus 2,000. That, of course, in project A we received $4,500 at the end of year three, and so we're going to calculate the present value of that by taking $4,500 and dividing by 1.1 to the third power. When we do that we find out that $4,500 flow is worth 3,380.85 in present value terms. So the net present value of project A is 1,380.85. Project B we have a $2,000 outflow at time zero. So again, that's not discounted. We received $2,000 at the end of year one, so we calculate the present value of that by dividing the 2,000 by 1.1. We receive another $2,000 at the end of year two. We calculate the present value of that by dividing by 1.1 squared. And then we receive $100 in year three and we calculate the present value of that by dividing by 1.1 to the third. And when we calculate the net present value of project B, you can see that it has a present value of 1,546.1. Here is the project that we just looked at where the initial investment was -$2,000. We then were going to receive a positive cash flow of $2,000 at the end of year 1. Another $2,000 at the end of year 2. And then finally, $100 at the end of year 3. So let's think about calculating the present value of each of these cash flows. Now the $2,000 outflow is occurring at time zero. So in this particular case, that is already a present value. So we can just write equals B9 in this case, and we have the -$2,000 there. So now, we're going to discount the cash flow for year 1. The cash flow for year 1 is in C9, so we're going to take C9 and we're going to divide that by the 1.1, which is 1 plus the discount rate of 10%. We're then going to take the cash flow for year 2, which is in cell D9 and we're going to divide that by 1.1 and now we're going to have to use the caret again because we want to divide by 1.1 squared. And then finally, we're going to take the cash flow for year three which is in cell E9, so we're going to take E9 and we're going to divide that by 1.1 raised to the third power. Now that we know the present value of each of the cash flows, we can just calculate the net present value by summing up all those present values. So we're just going to calculate the sum of B10 to B10, and when we do that we're going to find out that the present value of the sum of the cash flows is 1,546.2. Now what if we wanted to use the NPV function? There's a common mistake that people make when they use the NPV function, and it would probably be useful for me to make that mistake so that you can see the mistake that sometimes people make. So if you remember when we want to put in the NPV function, the first thing that we have to put in is what the discount rate is, and in our case its 10%. And now what I'm going to do is for the range that the cash flows. I'm going to put in B9 through E9. So I'm going to pick up the -2,000, the +2,000, the +2,000, and the 100. So I'm going to enter B9 to E9. And you'll notice the answer is not the same. And the reason the answer's not the same is because the NPV function in Excel assumes that the first cash flow that you reference in the formula is a period away. In this case, we're doing year, so it's assuming it's a year away. So this calculation is actually assuming that that -2,000 is a year away, but it's not. It's a time zero, it's an investment that we're making today, so when I use the NPV function this actually was not right. So how would we actually use the NPV function in this particular case? Well, what we have to do is in the NPV function itself, we only want to reference C9 to E9. So I'm going to go back and I'm going to change the B9 to a C9. And then what this will have done is it will have calculated the present value of the year 1, the year 2, and the year 3 cash flow but you can see I haven't subtracted off the initial investment. So what I would have to then do is I would have to subtract off the 2,000. So, I'm going to just take the -2,000 off. And when I do that I get exactly the same answer. So even though the total sum of the cash flows from project A is higher at 2,500 than it is for project B at 2,100, at a 10% discount rate, this analysis will tell us that the net present value of project B is higher than the net present value of project A. These net present values that we've just calculated are the point estimates of the value created by taking the project. So the point estimate of the value of project A is 1,380.85. And the point estimated the value created from project B is 1,546.1. If we take positive net present value projects, we're increasing the value of the firm. And this gets back to the objective function we talked about, about maximizing the value of the firm. So taking positive net present value projects relates directly to that objective of increasing firm value. If we take negative net present value projects, we are actually destroying firm value. We're actually investing in projects that are going to lower the value of the firm. In the case of projects A and B, if we had to choose just one of the project because we couldn't take them both. Maybe there are two solutions to the same problem or maybe we only have $2,000 to invest this period. We would take project B over project A because it creates the most value of the two choices. Let's assume for a minute that the cash flows of projects A and B were the same as we showed before. What would you have to do to the discount rate to have project A have a higher net present value than project B? Because given that the sum of the cash flows for project A was higher than the sum of the cash flows for project B, there will be a discount rate at which the value of project A will begin to exceed the value of project B. And, as it turns out, what we would do is we would reduce the discount rate that we used. And as that discount rate goes down, both the value of project A and the value of project B will go up. But the value of project A will go up faster because it gets all of its cash flows in the third year. At a discount rate of 6.525%, the present values of the two projects would be the same. You would be indifferent between project A and project B if your cost of capital was 6.525%. If your cost of capital was lower than the 6.525% you actually would prefer project A over project B because at that point project A would have a higher net present value. Let's talk about zero net present value projects for a minute because there is something important to be learned here. So let's consider a project that had an initial investment of $2,000 at time zero, and there was a single cash flow that you got from that project at the end of year one that was $2,200. Let's suppose that the discount rate is 10%. Well, what's the net present value of this project? Well, as it turns out, the net present value of this project is 0. If you took the -2,000 and you added to it the 2,200 divided by the 1.1, you would get exactly 0. So notice that this project earns a rate or return of 10%, 200 divided by 2,000 and that is exactly equal to the discount rate or the cost of capital in this example. But notice even though it earned a rate of return of 10% because it just earned the cost of capital, it didn't create any value. So the key thing to see is that managers only create value when they earn a rate of return on their project that exceeds their company's cost of capital. If you invest in projects that earn a rate of return that's just equal to your cost of capital you're not adding any value. And if you're investing in projects that have a rate of return that's less than your cost of capital those are going to be negative net present value projects and those are actually going to destroy firm value. So one of the things that's useful to know about is something called the cash flow perpetuity model. And this is a short cut way of calculating the present value of an infinite stream of cash flows. When you are willing to assume that those cash flows are going to grow at a constant rate g and the discount rate is constant. And you can calculate the present value of that infinite stream of cash flows starting at t + 1 and going through infinity. Simply by taking that t + 1 cash flow and dividing by the discount rate minus the growth rate, or r- g. And that will give you the value of that infinite stream of cash flows that began at time t + 1, as of time t. Now in order for this calculation to make any sense, it has to be that r, the discount rate is strictly greater than the growth rate in the cash flows. And that cash flow has to be positive. Now you might say, well, what's the use of this calculation, we wouldn't expect that anything could generate cash flows in perpetuity or for an infinite period of time. Well as it turns out, in present value terms we're really only talking about cash flows for about 50 to 60 years. So while the calculation is based on or assumes an infinite stream of cash flows, in economic terms we're really only talking about 50 to 60 years. And as we'll see later, this shortcut for calculating the present value of an infinite steam of cash flows will prove helpful to us in certain circumstances. And we'll discuss that a little later.