So a famous study asked corporate executives what is your preferred method for making capital budgeting decisions? And they were allowed to list more than one. Seventy five percent of these corporate executives listed net present value, which is somewhat comforting, it would've been better if it would have been a 100 percent, but 75 percent is still okay, especially considering that the survey included some small, very small firms. However, the net present value was not the most favored measure. That honor belonged to the internal rate of return. Seventy six percent of the executives used the internal rate of return. So what is the internal rate of return? That's what I'm going to explain in this set of lectures. So you'll see what it is and why perhaps it is so popular. One thing that you'll see about the internal rate of return that I believe makes it so popular is that, it is a single number. Given the cash flows of a investment most of the time, but not all of the time, you can boil down all of these cash flows in the initial costs to one number, the internal rate of return, whereas the net present value gives you a number, yes, but it depends on your assumption of the discount rate. This may be why people like the internal rate of return. However, there are some very significant costs with using the internal rate of return. After these lectures, you will be able to weigh these costs against this benefit of having only one number and decide which is your preferred method. The internal rate of return, which everybody calls the IRR, is the discount rate that makes the net present value, namely NPV, equal to zero. So the IRR rule says except if the IRR on the project is greater than the discount rate otherwise reject. Let's contrast this with the net present value rule using an example. So we've got a really simple project, the cost is a $100 so our cash flow at time zero is negative a 100 and in one year it will pay off a $110. So the cash flow at time one is a $110. So the NPV, that's easy to calculate, it's minus 100 plus 110 over one plus r. So the IRR solves the following equation, zero equals minus 100 plus 110 over one plus r. So the IRR is the discount rate r that makes NPV equal to zero. So if we go ahead and solve this equation, we see that r is equal to a 110 over a 100 minus one or 10 percent. So let's contrast the IRR rule and the NPV rule in this example. So we need to set a value for the discount rate. Let's say the discount rate is eight percent. So substituting this into the formula for the net present value of our investment, we get NPV equals minus 100 plus 110 will now we have divided by one plus r, so it's 1.08, and that is equal to 1.85 greater than zero. So NPV rule says accept. What about the IRR rule? Well remember the IRR is equal to 10 percent, that's the discount rate that makes the NPV equal to zero. So notice r is less than the IRR. The IRR says, except if the IRR is greater than r, which is the case in it here. So the IRR rule says accept. So in other words, they agree, and that's not just because of the values that we picked. So let's consider what the net present value looks like as a function of r. So this is r on the x-axis, NPV on the y axis and these graphs are actually kind of helpful in understanding the relation between NPV and IRR. So notice that NPV is a decreasing function in r. If r were to equal zero, which would be a rather low r, we would know that the NPV would equal 10, then it's decreasing, it's not linear, actually, it crosses zero at the IRR, which in this case is equal to 0.10. But what I want you to realize from this graph is that if the NPV is greater than zero for this case, namely, when the NPV is given by this function, the IRR is above the discount rate. So if we think about this part of the graph over here, what's true? What's true is that NPV is greater than zero, what's also true is that r is less than the IRR, so the IRR is greater than r. If we look at this part of the graph, then the opposite is true, right? NPV is less than zero and IRR lies below r. For this project, we can conclude IRR and NPV rules agree. Now that's for this project, but it's actually kind of general reasoning, anytime where you have a project where the NPV as a function of r looks like this, something like this, then the IRR and NPV rules will agree by the same reasoning as I just told you. So this point here is the IRR so if NPV looks like this, namely is characterized by this graph, IRR and NPV rules agree. Now, when might it look like that graph? When does NPV look like this? Namely, what we're really using here is that NPV is a decreasing function of r. Well, this is true if you have a project where first you invest money and then you get money out. So that's true for what we call a normal investing project. So it's true if NPV equals C_0 plus C_1 over one plus r plus C_2 over one plus r, squared plus dot dot dot, and C_0 is less than 0, C_1, C_2, et cetera, these are all greater than 0, then we're fine. You can see that in that case, the NPV is going to be a decreasing function of r. Now that's not always going to be the case. So next we're going to discuss the cases where the NPV rule and the IRR rule do not agree.