[MUSIC] In order to master net present value and IRR calculations, You're going to need to practice them. There is no other way. If you wanna master anything like playing guitar for example or computing IRR, the only way to do it is to exercise after exercise. So make sure you work your way through all the quizzes, the practice quizzes and the final quiz. Do a lot of problems. There's gonna be an assignment as well, that you can use to practice, to try your hand at doing these calculations. That is really the only way to master this. What I wanna do, is go through a few examples with you though. So I want to go through three examples here to illustrate a few additional issues that we have to be aware when computing NPV and IRR. The first one here is a project analysis. So we have a project that is a moralistic example of a project where you have an initial investment. And then the project is going to add both revenues and cost to their company. So you have added revenue and added expenses, so obviously this is a profitable project. It's adding a profit of $9,000 per year, let's say this is measured in thousand. You have some depreciation. Depreciation is naturally going to come from the initial investment. The investment has to be depreciated. We're going to consider a planning horizon of ten years. And as we discussed already, having a fixed horizon is not natural. The way we handle this is by thinking about the salvage value. At the end of the ten years, we have to think about how much are we going to get for that project. If we sell that project for example, how much is it worth? So we're going to call that the salvage value. What we want to do is to compute the NPV and the IRR of this project. So let's go here and check the cash flow. This table is building the cash flow for you. At year zero we have the minus 40,000 investment. And then notice that I'm dividing this in two periods, years six to ten and years one to five in the left corner. So you have the profits and notice the role of depreciation here. The important thing of this calculation is the role of depreciation. Depreciation has to be deducted from profits before you compute taxes. Why is that? Because depreciation is tax deductible. If a company has a higher depreciation, you're gonna pay less taxes to the government. This is what we're doing here by deducting the $8,000 in appreciation from your taxable income. So your actual taxes in years one through five is just 300, your after tax income is 700. But we are not done with depreciation at the bottom here in the cash flow before we can add cash flow we have to add appreciation back. Why? Because depreciation is a known cash expense. Depreciation doesn't mean that it's actual cash coming out of the company. Nothing is happening in terms of cash flow. The way we address that is by adding depreciation back to compute the cash flow. So the cash flow in years one through five is 8.7 thousand. The calculation is exactly the same in years six to ten. The only difference is that the depreciation is zero. So now, you'll pay taxes on your entire profit. Meaning that you're gonna get a lower cash flow. Your taxes are gonna be higher. Taxes now are 2.7 thousand, your cash flow's gonna go down to 6.3 thousand. So cash flow is higher when depreciation was higher. Salvage value, at the end that's very simple, it's just the 4,000 that was our assumption minus the tax which is 30% of 4,000. In this case, I told that the salvage value was before tax value. So just to recap this, since the depreciation is tax deductible we have to deduct it from profit before we compute taxes but then we have to add it back to compute cash flow. So to make sure you understood this, ask yourself why is the cash flow lower in years five to ten. When depreciation is lower, the cash flow is lower, make sure you understand why is this the case. Given the cash flows of course we can compute NPV. So the way we do it is by applying the NPV formulas that we talked about. You can do it in Excel of course. A problem like this, if you wanna do it by hand, you literally have to write down all the cash flows and discount all of them. So an easy way of doing it is just to write down the timeline in Excel and then use the NPV function, but you can do it by hand as well. And I recommend that you do a couple of problems both ways to make sure that you are using Excel correctly. The NPV here should be $13,152. And now, let's think about IRR. We just talked about IRR. Let's make sure you understand IRR. A couple questions for you before we talk about them. First one is this a case where we can compute IRR? Are we allowed to compute an IRR? Second is before you compute it, can you guess a range for what it should be? So can you make a guess on what the IRR should be. The answers. Number one, yes. This is a scale with standard cash flow pattern. You are investing in the project today and then the project is producing cash flow in the future. So yes, it has a well defined rate of return. If you have to guess, we learned that if the NVP is positive, the IRR should be bigger than the discount rate so a reasonable guess here would be that the IRR has to be lower than 8%. The IRR should be lower than the discount rate. If you do the calculation in Excel again, Excel will be the best way of doing this, using the Excel function IRR. Writing down the cash flows use the Excel function, you will see that the IRR is 15.25%. So 15.25 bigger than the discount rate same thing as a positive NPV. This is a good project that would increase shareholder value. Example number two. Let's recap the concept of incremental cash flows, and really make sure we understand how to use it. This example is one where computing incremental cash flows is a little bit more complicated, similar to the accounts receivable problem. So we already did a lot of work to get our incremental cash flows in that problem. Now, this is another example where you have to do a little bit of work before you figure out what are the right cash flows to discount. So here you have a machine that the company is considering to invest in. These are the only cash flows that you have. Very simplified situation. You might have an investment of minus 70,000 and then the machine produces profits or cash flow of 25,000 every year for a period of four years. So what do we do to make a decision on this, we need the discount rate. And we have to compute NPV, and additional assumption here that the machine is gonna be discontinued in 2014 and has zero salvage value. So there is no other cash flow as I said before. So should be very easy to compute the NPV of this cash flows. Using Excel, using the formula, you should get an NPV of 12,800 as of 2010. What should the company have done at that point the company should have invested in the project. It has positive NPV, investing in the project. Buy the machine, put the machine in operation. However, let's say that now it's one year, it's just one year after. But technology moves very quickly and the company actually has a pleasant surprise one of the engineers develop the better machine that is going to reduce cost. So it reduces cost, it increases cash flows. And it's gonna be discontinuing the same era. Again, let's assume there's no salvage value for simplicity. Of course, in our real world problem will be a bit more complicated than that. But that's what the new machine looks like. It requires a bigger investment of $90,000 to put it in operation. It's a more complex machine. But then it's going to increase your profits significantly in the last three years. The question now is should the company replace the machine or with the new one or should it keep the old machine? So you can see that this has the same flavor of the accounts receivable problem that we worked with. So depreciation, let's give some data on depreciation. The old machine has been depreciated to 50,000 and it can be sold for the 39,000 in 2011. Should we sell the old machine and replace it with the new one? That's the question that we have to answer, okay? And here the key idea that we're going to use is the idea of incremental cash flow. New minus old. We have to think about which cash flows are a direct consequence of replacing a machine. What is going to happen if you replace the machine? So first we are gonna talk about things that don't matter. Before we actually derive the incremental cash flow, let's talk about two numbers that are irrelevant. So we are in 2011. And you invested 70,000 in the machine in 2010. Does that matter? No, the answer is no. Whatever you do in 2011, you can not recover that 70,000 anymore. It's gone. The 70,000 are gone, it's not gonna show in our calculation at all. The 70,000 is what we call a sunk cost. The cost that is already gone. You paid for it in the past. Whatever happened in the past should not affect your NPV anymore. We get some data on the depreciation, but I also told you that there are no taxes. So in this case, the depreciation actually doesn't matter. Depreciation in this case is irrelevant. Because there are no taxes so we can ignore that number as well. Every time you do a project analysis there is gonna be some information that is gonna turn out to be irrelevant. And one of the tasks that you have is to try to figure out which information is relevant and which information is irrelevant. So let's think about the relevant information. And I'm actually gonna ask you to do some of these calculations. So we are here in 2011, and we have to think about the idea of incremental cash flow. So what is old? Old is to keep the old machine. If you keep the old machine, your cash flow is zero. But if you decide to put the new machine in place, then what is gonna happen is that you are gonna sell the old machine for 39,000 and buy the new one for 90,000. So ask yourself what is incremental cash flow? And then from 2012 to 2014, the old machine produces 25,000, the new one produces 45,000. Again ask yourself, what is the incremental cash flow in that case? The answer should be very simple. Using the new minus old idea. What happens is that again, every time there is the word incremental cash flow, every time there is the word cash flow actually, what we have to do is to think about new minus old. So in 2011, the incremental cash flow would be minus 51,000. Because you can sell the machine for 39,000, and you have to buy the new one for 90. So the incremental cash flow is the difference. And then from 2012 to 2014 the incremental cash flow is 20,000, the difference between new minus old. So if you use the new minus old idea, you should have been able to obtain those. What is the NPV? After you figure out the cash flows in this problem, computing the NPV should be really easy. I write down the timeline. Here you have the timeline with all the cash flows. And use the Excel NPV function or do it by hand. You should find an NPV of $541.94. Now ask yourself, should we replace the machine? The answer of course, is yes. It's a small NPV cuz the machine, you have to invest 90,000 on it or 51 if you deduct the 39,000 you can sell the old one for. You're only gonna get an NPV of $500. But that doesn't matter. If the NPV is positive, this is like the one cent, two cent example. As long as the NPV is positive we should go ahead and replace the machine. But of course everybody at this point is probably thinking, if I'm only gonna make $500 why do I care? Maybe I shouldn't do it. I agree, this small NPV is gonna give us reason to think. And really, what we are concerned about is uncertainty. Investment decisions have a lot of uncertainty associated with them. And we have to consider are we really sure about our numbers. And since the NPV is so small, it's very easy to make a mistake in this case. In module four which is coming up, we will think about how to address uncertainty in investment decisions. So this is a preamble to the discussion we're going to have in module four. Final example, this is a fun one. You actually don't have to do any calculations here. Consider this, now we have a real estate developer who is considering whether to build a mall in a piece of land that the developer already owns. And they estimate construction costs 20 million. And they estimate the present value of all of the future rental income. So this is a smart manager. They learned NPV. They know they have to include all the future rents. And the value is 25 million. Question for you is, is the mall a positive NPV project? It might seem that it is, cuz 25 is bigger than 20. The right answer is actually that you do not have enough information to answer this. Think about it. The notion of incremental cash flow requires you to think about all the consequences of taking a decision. So if the real estate developer decides to build the mall in this case, then you cannot use the land for anything else. The land is yours, yes. But you can sell it. If you don't build the mall, you can sell the land. So if the land could be sold for example, for more than 5 million then the project is gonna be negative NPV cuz the profit you make, the NPV of operating the mall minus the construction cost is 5. But you could sell the land for more than five, so the project would not be positive NPV anymore. The value of the land in this case is what we think of as an opportunity cost. Every time you make a decision you have to consider all the relevant opportunity costs and include them in your valuation if you want to make the right decision. So what the real state developer would need to do here is before making a decision on the mall you have to do some research and try to figure out how much you can sell the land for.