[MUSIC] We just talked about net present value, which is the concept, the calculation we're going to use to make corporate decisions, right? The net present value is a number. In our accounts receivable example, we computed $118,000,000. And we also talked about the fact that there is a direct equivalence between net present value and shareholder wealth so shareholder wealth would go down by $118,000,000. If the management decided to speed up the collection of accounts receivable. The problem is that when we think about investments, we like to think of investment returns in percentages. So when we think of the stock market return, we're not measuring that in dollars. Usually we're measuring it in percentages. What's the return on your portfolio? What's the return on Pepsi Co. stock. We are always thinking of percentages. So, that is why we need this additional concept, which is the rate of the return on an investment. It's very important for us to be able to compute the rate of return on any investment that we are given. And the idea is that's going to give you a percentage measure of how much you're getting out of that investment, okay? And here is the definition and I know that the first time people see this definition it sounds a little bit confusing, okay? The rate of return of any investment can be defined as the discount rate that makes the net present value of the investment equal to zero, okay? That is the most general definition of a rate of return that we can come up with, okay? And by the way we have this alternative name, just like NPV, we have this alternative name for the rate of return, which is the IRR, okay? That stands for Internal Rate of Return, okay? So let us see why, let us try to understand why the rate of return is the discount rate that sets the NPV to zero. Okay, so let me give you an example. A very simple one and my experience teaching this is that, most people that have done some math, you know basic algebra can figure this out, okay? So suppose that you have an investment that requires $10,000 today. And that's going to pay back $11,000 in a years time. And ask yourself, what is the rate of return on this investment? Let me go over here. So you invest $10,000 today, let's do this in a timeline. Okay? And we're gonna use later. And then you'll get $11,000 a year from now. Each should be obvious to most people that their rate of return is 10%. Okay? So I'm pretty sure that the moment I gave you this problem, you'll say yeah, the answer is 10%. What kind of problem is that? Right? Too easy. Okay? But I want to use that problem to show you that this concept works. Okay? So think about, let's go inside your brain, okay? So this is fun, we're actually going to try to think about, what is the equation that you're solving in your brain when you got that 10%. And I'm pretty sure that this is the equation you're solving the first one here. What you did is you deducted 10 from 11, so I'm making $1,000. And then you divided 1,000 by 10,000. Okay. So that's how you get the return of 10%. Okay. So if you've done basic algebra you can try to manipulate this equation a little bit. Just change the numbers around. Okay. So, our equation what you can do here is you can divide the 10,000 by 10,000. So you'd have 11,000 divided by 10,000 minus 1. We know that that's equal to 10%. And then you can start rewriting this. So you can send 1 to the other side, you get 1 plus 10%. And now you can divide everything, first actually you send the 10,000 to the other side, multiplying the 1 plus 10%. And then you divide the 11,000 by 1 plus 10%, okay? So, this can be written as follows, right? If you deduct 10,000 from the right-hand side here, this can be written as 0 = 11,000 / (1 + 10% ) Minus 10,000. So think about this, we did the timeline. So you have 10,000, 11,000. So this term here that we just derived, let me highlight this. This term here that we just derived is nothing more than the net present value of an investment. That pays off $11,000 in a year, and requires an investment of $10,000. Right? When the discount rate is 10%, we are discounting by 10%. Highlighted here for you, okay. What is this NPV? The NPV is equal to zero. Okay? So the equation you're solving here in your brain is mathematically equivalent to writing down the NPV and setting the NPV to zero. Finding the discount rate that sets the NPV to zero, okay. If you don't believe me we can do it in another way, right. Write down the NPV of that problem. It's very simple, right. We learned NPV already. So the NPV = -10,000 + 11,000 / ( 1 + Discounted rate). And then do step number two. Find the discount rate that makes the NPV equal to zero. So set this equation to zero, you will see, it's very easy to see that the value that you have to plug in here to set the NPV to zero is 10%. Okay? So if the discount rate is equal to 10% the NPV is going to be exactly equal to zero. Okay? So this is why the definition of rate of return is the discount of NPV rate to zero, okay? Of course, you could have solved this problem another way. You could have used Excel, for example. Excel also has an IRR function. And this one actually works fine [LAUGH] as opposed to the NPV function that we just talked about. All you have to do is to input the cells. Okay. So is the IRR of cell one and cell two. Right. And then Excel would tell you that the IRR is 10%. Okay. Of course, for this problem you probably don't need Excel. For another problem you may wanna use Excel. So I wanted to talk about this as well. Okay. Let us now see if we can apply what we just learn. Okay. Let me give you this problem to work on. We have an investment that requires 10,000 today, and now it produces a cash flow in perpetuity. So the cash flow is expected to grow at 4% a year. And the question I have for you is, what is the rate of return of this investment? So every time we solve a problem like this, what I want you to try to do is to use what we learned. Set the NPV to zero, okay? This turns out to be a very powerful idea that helps us solve many problems in finance. Okay? So what is the NPV? Right? We learned the growing perpetuity formula, right? It's minus 10,000 the investment today, then the present value of the cash flow is going to be 500 divided by the discount rate minus four percent. Right? And I didn't give you the discount rate? Right? You might be like well, but how am I gonna solve this problem? What is the discount rate? Remember, that's not what we are trying to do here. In this problem, what we are looking for is the rate of return. We are looking for the IRR. And step number two here is the definition of the IRR. What the IRR is is the discount rate that makes the NPV equal to zero. The discount rate that makes the NPV equal to zero. And if you solve this equation, mathematically what you're going to see is that the IRR is 9%. If you put 9% here you're gonna get 500 divided by 5%, which is gonna be equal to 10,000. So you get an IRR of 9%. So this is really cool. I gave you the schedules. It was probably not obvious what the rate of return is. But I can tell you, with certainty, that the IRR, the rate of return on this investment, is 9%. Okay? Let us try to use Excel in this case. Okay? And here you are going to see that there is going to be a problem, right? You can do it, but it's going to take a lot of boring work. Okay, let me show you this. Suppose we were doing this with 30 years. Right. So we have an investment of 10,000 and then we have the cash flows. 500 in year one, 520 in year two and it's growing, right, the cash flow continues to grow. When you get to year 30 the cash flow is 1559, okay, I just did this manually. And then you can ask Excel compute the IRR for me, Excel will tell you in this case that the IRR is just 6.65%. The problem is that here we have growth. The cash flow is growing, so if even if you do it for 30 years, the value is still too high, this is still 1,559. You know, its still matters, right. So the rate of return of this investment really is higher because this cash flow is going to continue to grow, right. So, if you just ask Excel to do it before 30 years, you'd get an answer that is a considerable under estimate. Of the true rate of return on the investment, okay? So in this case, you actually do have to use the little trick that we learned, right? Instead of using Excel, you have to write down the NPV and then set the NPV equal to zero.