[MUSIC] In the calculations that we've done in this course so far, it always seemed that we knew the right forecast, right? When we were computing that present value for example, we were analyzing investment decisions, we were starting from cash flows, right? And in our calculations, it always seemed that we knew what the cash flows were going to be in the future, right. When we were analyzing acquisitions, it seems that we knew what the cost savings would be for example. When a company acquired another company, we computed the net present value of the synergies assuming that we knew what the cost savings would be in future years, okay. The truth is that, in the real world, we don't really know, right. There is uncertainty. There is fundamental uncertainty about what the cash flow forecast are going to be. Cost savings, revenue increases, right. All of these numbers are going to be base on forecast. So a very important part of evaluation analysis is to learn how we deal with uncertainty, okay. What do we do when we recognize that we actually don't know what the forecasts are, okay? So that's our topic now. It is a very important, very fundamental topic in corporate finance. The truth is that we know the assumption we've been working so far is not really that we knew the forecasts, that's what we're going to learn now. Really, what we were doing is we were using the expected value, okay? All the numbers in our timelines, in our evaluations, in our acquisition analysis, they represented expected value, okay? So in this lecture, we are going to learn what an expected value means, okay? The idea is that we were using averages, we were trying to work with our best guess of what that number would be in the future, okay. This is gonna be true for all the model parameters we worked with. Cost savings, revenue increases, future investments, right. We were always trying to forecast, to calculate this average value, okay. So, our goal now is to learn how we deal with uncertainties. The uncertainty is going to become more explicit when we think about investment, and we're gonna talk about how we incorporate these uncertainties into our evaluation exercise, okay? So, the way I want to start is by actually going back to an example we talked about in module three, okay? The easiest one of the project evaluation analysis that we did. We had an example where there was an initial investment. The project was going to increase revenues and increase costs as well, right? There is some depreciation. All the numbers that you need to calculate the net present value of the project, okay. So in particular, we have forecasts about how much revenue the project is going to add in the next ten years, okay. So now what I want you to recognized is that every time a company is doing this type of calculations, the company really doesn't know for sure that the cash flow at time nine, for example, is going to be exactly $6.3 million, okay. We have to think about uncertainty, which is exactly what we're going to do now. The first thing we need to do is to figure out what we mean by these forecasts. What we mean by the numbers that we had in our spreadsheets, okay. And as we discussed already in the introduction, these forecasts should reflect what we think of as the project's most likely scenario, okay? So for example, when the company was trying to analyze this investment, right, trying to think about this product, the marketing department may have forecasted the added revenues as the product of the market price times the additional sales, okay? So the added revenue worth $12 million because, sorry, $12,000, right, because the price was $4 and you're selling 3000 units, so $4 * 3,000 equals $12,000, okay. So in the production department was probably responsible for forecasting the cost, right. And the way that the production department probably forecasted the cost is by adding variable cost to fixed cost, right. So the cost per unit times the additional sales plus an amount of fixed costs, right? Every project is going to have some costs that don't depend on the amount that you sell, we call those fixed costs, okay? So this is just to go a little bit deeper on the numbers, right? So now let's think about uncertainty, right? The truth is that the marketing department is making a guess, right? The marketing department is trying to estimate what the most likely value is going to be for the sales, for example, okay? Let's take sales as an example here, right. To make this more explicit, let's work with probabilities, okay? The good thing is that, at this point, we already worked with probabilities in this course, right. So you already know how to do calculations with probabilities, here is another example, right. Suppose that the chance that sales are actually $3,000 is 20%, okay. But there is also 20% chance that sales are gonna be equal to $1,000, $2,000, $4,000, or $5,000. Okay, so in this case, 1,000 would be the worst case scenario, right. Of course, having lower sales is not gonna be good for this project. 5,000 is going to be the best case, okay? And all the values are equally likely. Okay, so if you think about this problem, okay, the value that we used, right, in our evaluation, we used the value of 3,000. Okay, if you do a math here, which should be quite obvious, right, cuz everything is equally likely. You can compute the average value by multiplying the probabilities times the different possibilities. So 20% times 1,000 + 20% times 2,000 + 20% times 3,000 + 20% times 4,000 + 20% times 5,000. If you do the math, if it's not obvious to you which it could, you're gonna get exactly 3,000, okay? So 3,000 is the average of all these different possibilities. There are five different possibilities, okay, but the one that we input in our evaluation is always the expected value. The number you're gonna see in your timeline is always going to be the expected value, okay? So, the first tool we're going to learn to deal with uncertainties, what we think of as sensitivity analysis. We call this sensitivity analysis. The idea is to try to figure out how sensitive the net present value is to fluctuations in this key parameters, okay? Now we know that we have five different possibilities for the sales and we know how to compute NPV. So it should be fairly straight forward to answer the following question, right. How does the NPV change under these different assumptions. And really, that is the goal of sensitivity analysis. What I will do is I will provide the spreadsheet, okay, that you can use, an Excel spreadsheet that you can use to do the sensitivity analysis. Right, we're gonna have to change assumptions and it's really not that convenient to do this by hand or on a pencil and paper, you can try. But the best way to do sensitivity analysis is by using a computer and a spreadsheet, okay? And so for example, we can figure out that if sales are equal to 1,000, NPV is -16,900, if sales are equal to 5,000, the NPV is equal to 43,214. So consistent with our intuition, the higher the sales are, the higher the NPV is going to be, okay. Here is the snapshot from the spreadsheet, just for you to see. Like I said, you can get that on the website and use it yourself. Right, for example here, what I'm doing is experimenting with the worst case scenario, right? What happens if our additional sales are equal to 1,000 units, this is highlighted in green. And then here in the bottom, you have the NPV and the IRR, the internal rate of return of the project, okay. The spreadsheet computes all the cash flows automatically and then at the bottom you get the NPV and the IRR, okay. The spreadsheet might be useful for you as well as kind of a guideline for how you would do project evaluation, how would you compute net present value using Excel, okay. So it should be very straight forward to use this. Let us think now about a very important question, a fundamental question really, okay. How should you use the sensitivity analysis, right. So now, rather than having one number for the NPV, we seem to have five numbers, right, so it complicates our life. We learned in module three that we should take a project if the net present value is positive. So, now we have five different NPVs. What should we do? Okay? Let me make this question specific. Should you reject the project because the NPV is negative under the worst possible scenario? The answer is no, okay. So you have to be really careful about how you use the sensitivity analysis, right. You cannot reject the project because the net present value would be negative, is the worst possible thing happened, right. This idea sounds wrong, okay? And really it's, there is a very fundamental concept behind this, which is the idea that to create value, a company needs to take risk. I think it will be impossible to find an investment that creates true value that generates shareholder value, but that exposes the company to no risk, okay. Everything that creates value is going to be risky, okay. Value creation equals risk. That's the way to think about it, okay. In terms of net present value, I bet if you're doing this calculation correctly, net present value will always be negative under the most pessimistic assumption, okay. So if something goes wrong, your NPV will turn negative, right. The NPV of the project is still the number we computed in module three. There is only one NPV, okay. So remember this, there is only one NPV. We do not have five NPVs. NPV is based on the best possible guess. NPV is based on the expected cash flow. So the NPV is still 13,152, nothing changes, okay. So how, if the sensitivity analysis is not changing the NPV, what are we going to use it for, right? That's the question you may have in your mind now, so why are we doing this, right? Okay? One way that you can use the sensitivity analysis is simply by getting more information. More information is better than less information as long as you know how to use it, okay. It's very useful to know which variables are likely to affect the NPV, right? So for example, we've just learned that sales are a very important variable, okay. So you have to make sure the marketing department didn't do a crappy job of forecasting sales, right. It's crucial for you to know how much you're going to sell. It's kind of an obvious statement, but this is, we just learned the NPV equivalent of that, right? If sales turned out to be low, the NPV is going to be negative, okay. So one way that you can use the sensitivity analysis is to show it to the marketing department. And then tell them, look, we have to make sure we get this number right because if the sales is lower than a certain value, our project should not be started. We should not be selling this, okay. Sensitivity analysis can also be useful for planning purpose, right? We can figure out, for example, what is the minimum level of sales that are required for the project to create value, okay. And now you have the spread sheet that I provided so you can do it yourself. In our case, it turns out that sales have to be larger than 2,125 units. So that is your break even level of sales. If you sell less than that, right, then the project is no longer positive NPV, right. So think about that, that is a very useful number for the company to know, right. So now you can get your sales team, you can have a target for your sales team, right. You can tell the marketing department, look, this is the number that we have to hit. Are we sure we're gonna hit that? I mean sure, we're never gonna be sure. You see how dangerous it is? We always, we always want to be sure, right? We don't like risk, but the truth is that every time a company takes a project, you are taking risk, right. So anyway, going back here, okay. We know now that our sales have to be larger than 2,125 units, okay? So we can use that for planning purpose. If we find out two years ahead that our sales are not 3,000, that they are more likely to be 2,000 units. At this point, you might want to consider the option to abandon, for example, which is something that we have talked about already, right. It might be the correct decision at this point to abandon the project. So you may want to redo the net present value calculation and try to estimate whether you should continue this project or not, okay? So sensitivity analysis is very useful as long as you know how to use it.