Well, let's study the general case. Again, this is a market building balance sheet. This is V but now, this is D, and this is equity. In the previous episode, we said that if this line goes up and down, then the behavior of left side and the right side is exactly the same because there was no debt. Here however, we can claim that in a modified way, we can say that whatever happens to this line, then the left side behaves exactly as does the portfolio of debt and equity. Therefore, if these changes are small, we can take these proportions as fixed. Well, clearly, if this is a significant move, let's say, if this move here, then all that accrues to equity because debt, in this case, is not affected at all. However, if this goes up this way, then the closer to this point arrives this line, then the riskier becomes the debt. But these are all minor things. So, again, this part is a portfolio. And, now, we can say that if these other fixed things then beta firm, and we said that to start with because this is the origin of risk is what? This is D over D plus E times beta debt plus E over D plus E times beta equity because the portfolio consists of the proportion of D over D plus E of debt and E over D plus E of equity. Well, from this we can rewrite this formula and we can say that now beta equity is what? It's beta Firm plus D over E. And, here, beta firm minus beta debt. You can see that, well, if there is little debt, then there is not that much risk with this debt. And then you can approximate to say that beta debt is equal to zero. So this then disappears and so does this. And you can see that beta equity is actually increased because of leverage. Well, it can be easily be seen that if this line changes, then like I said, if this V increases by one percent, then equity increases by more than one percent because debt doesn't change. And so all this piece accrues to equity. So there is no miracle here. And oftentimes, in this formula, people say that this is the sort of the asset risks or the business risk, while this is the risk of financing. Now, within this course so far, we have claimed that the financial decisions, they have to be taken sort of separately from investment decisions. And we said that their influence is not of the first order. Now, this is the only area in this sort of simple approach because the details of that, we will discuss in some greater detail later in our M&A course, but for now, the key story is that we see how leverage contributes to the higher risk of equity. And, well, by the same token, we can rewrite the formula for return, and we'll say r firm is equal to D over D plus E, r debt plus E over D plus E r equity. Again, if there was little risk then debt becomes riskless and debt becomes RF. Now, we are close to another very famous formula, but for that, I will flip it over and rewrite it. It can be shown that for debt, you have the debt, the interest tax shield and, therefore, the formula should be sort of a just a little bit, and I'll put it like this. D over D plus E rD times one minus T, where T is the tax rate, plus E over D plus E times rE and that all is called weighted average cost of capital. This is a famous and, oftentimes, it's called the textbook formula that everyone uses. And we will obviously use it in not only our problems but also in our all valuation procedures in the real projects and we will use that heavily in our final project for this course. And the only challenge to the use of this formula is that, oftentimes, people do not, quoted, waste their time on what we have done with you over these weeks. And they just take this formula as a given, and then they just feed it with what they take from the market. Most often here, they put RF. That's not so great. Well, clearly, there are some challenges or what you have to take as the tax rate because it changes from the marginal tax rate may not change, but this is a minor details. Now, we will take a look at that and try to see some practical challenges. Well, it might be even boring for you, but these challenges are very much the same that we dealt with before. First of all, this is data or inputs, but for what? What is for rE? How can we get this rE? Because, again, if the company is traded, then you can always say, "Well, I know exactly what rE is because I can watch the market serious." But at the same time, the whole valuation procedure is not of that much importance because I do know the market value of the company. I just take the price quote and multiply that by the number of shares outstanding and that's it. However, if you are dealing with a project that you have to value that is not traded, then you have to find a proxy for this rE. And that is somewhat a challenge, and we will, in the next episode, I will give you a detailed example. Then I will see how you can extract that from the market. Now, the next thing here is, again, with rD, well, it's okay because normally you have to take. Well, again, where do you take that? You find bonds of equivalent risk and then you see what's rD for them because not always you can use RF as a good proxy. So if you did all that correct, then you can easily proceed. And there are quite a few other special examples and special challenges that there are somewhat in our handouts and then in textbooks that are about this. My idea is just to warn you a little bit. So when you see the formula, it seems very easy. You almost reach the goal, but your job is to make sure that when you feed this formula, you do that with the right inputs. Again, what I am saying is boring. I perfectly recognize that, but in order to understand things in, order to remove or reduce the probability of making a huge mistake, this is this boring watching if you will, and that will really save you. Now, what else I would like to say here? Well, there are some other challenges, and they are not depicted by the formula, but they must be taken into account. Well, first of all, this is project length. Well, we've talked about that. We said that, oftentimes, people seem to be sort of shortsighted. They prefer to take a shorter view because this is, in the corporate world, that may influence your performance evaluation. Now, by the same token, we can say that there is also another thing that we talked about. This is forecast accuracy. Therefore, for example, if you dealt with this weight average cost of capital, and if you use that for analyzing long-term projects with some unknown forecast, then you can say, "Well, maybe these ratios of debt and equity will change fundamentally. And maybe it's not a good idea to use this weight average cost of capital as a discount rate for a long-term period of time. So that cannot be answered right now because it depends upon the company, upon the projects. And, remember, when we talked about company cost of capital, we saw that even there, it's sort of a challenge. And then the final thing that I would like to point out here is this idea of debt which is leverage and risk. I would put them all together. So what's the idea here? Oftentimes, people say if you looked at the way the weight average cost of capital formula, you can see that if you take on more debt then because the return on debt is generally significantly lower than the return on equity, it seems that your cost of capital is being reduced and, therefore, it's great if I took more debt and then the NPV of my new projects will be higher. Well, the main problem here again is that, first of all, if you take on a project, it may not represent the company's growth proportionately in all directions if you will. So that may be a special project. Also, when you would like to take on more debt for financing this project, then people have a tendency to use some collateral from other projects when securing this debt. And, basically, this is very much like allocating overhead. So when you take on debt to finance a new project within the company, you have to properly allocate all costs and the risks associated with this debt in this project evaluation and, oftentimes, people ignore that. So all these traps, they, by no means, prevent us from using this. Instead, I'm just sort of trying to come up with these red flags to show., well, before you use the very simple taxable formula, please take a more thorough look at what you are about to feed it with. And now, the best way to have some idea about that is to study an example, and that's exactly what you will do in the next coming episode.
