I'm going to work this time with a hypothetical company, and you'll see in a minute why it simplifies a little bit what we have to do. Then we'll discuss what would be different if we were working with a real company. So let's think about it in this way, let's suppose that we start with an unlevered company. An unlevered company by definition, is a company fully financed by equity that doesn't have any debt in its capital structure. Now by definition, an unlevered company has a debt ratio equal to 0, because D divided by D plus E, if the D is zero, then the debt ratio will be zero. Remember that if we use only two sources of financing, debt and equity, then these two proportions must add up to one. If I want to know what is my proportion of equity, if your X_D plus X_E is equal to 1, then the X_E, which is the proportion of equity, would be 1 minus X_D. If I have no debt, then the X_D will be zero, and the proportion of equity will be 100 percent. So all I'm saying is that, remember that these two proportions must add up to one. Whatever is that the proportion of debt must be the proportion of equity. If I'm not using any debt, then I'm using all my financing comes from equity. Let's assume we're not going to use it now because we have no debt at this stage. But let's assume that, when you have very little debt, the cost of debt will be two percent. Let's take a statutory corporate tax rate of 21 percent. This may change dramatically across countries. That number may be very different across countries. As a matter of fact, it changed relatively recently in the US. It used to be 34 percent for many years, now it went down to 21 percent. What really matters is that whatever country you live in, then companies have to pay a proportion of their profits they have to pay to the government, and that is what the corporate tax rate actually measures. Let's suppose that is a 21 percent and in this case, the after-tax cost of debt will be 1.6 percent. We're going to assume the cost of equity, although we could calculate that with the CAPM at five percent, and that will give me a cost of capital, which is by definition equal to the cost of equity because I'm not using any debt, which is five percent. So just to run very quickly into the calculations, we start with the proportion of equity, which we know is 100 percent minus the proportion of debt. We know we're not using any debt, so all the financing comes from equity. X_D is equal to 0, X_E is equal to 1. We're assuming a cost of debt of two percent. We're calculating an after-tax cost of debt of 1.6 percent by using that 21 percent corporate tax rate. So that's a tax break. That means that paying interest on the debt cost you less than two percent because the government finance is part of that, and so you're real, your actual cost of debt is not two percent, but 1.6 percent. We're assuming five percent cost of equity, and if we throw everything into a formula, we have X_D, the proportion of debt, which is zero, we have 1.6, the after-tax cost of debt, which is 1.6 percent. But that doesn't play any role in this expression because all that is zero. Then we have 100 percent equity financing with a five percent cost. That gives me a cost of equity and the cost of capital of five percent. Now, I'm going to show you the same table, but for every increasing costs of debt. I've done all the calculations, but you may want to check that you know how to reproduce those calculations. As we move from left to right, you'll see that the proportion of debt is increasing. So we're looking at proportions between no use of data at all, which was the case that we've seen just a minute ago, all the way to 60 percent debt. Of course remember that the proportions of debt and equity must add to one. So whenever we increase the proportion of debt, we must be decreasing the proportion of equity. We also established before that the more debt that you use, everything else equal, the worst is going to be your rating and the more you're going to have to pay for that debt, and so as you move from left to right, the cost of debt is increasing. Of course, if we use a corporate tax rates that is fixed, in this case at 21 percent. If the cost of debt is increasing as we increase the proportion of debt, the after-tax cost of debt must be increasing too. Then we have the cost of equity. Here's what we said, when you increase the proportion of debt, you increase the beta, and therefore that increases the cost of equity, and that's what those numbers reflect. We could calculate those numbers, we could calculate the beta, we could put them into the CAPM and calculate those numbers. I'm just sweeping that calculation away. But we do see in that table, what we've said before, which is that the cost of equity is increasing in the debt ratio. At the end we have calculated the cost of capital. So if you go to the second column, to the one with 10 percent debt, just to make sure that you know how to calculate this. Although it would be a good exercise to calculate several of those columns. We have 10 percent of debt at the very top of the table. Then we have an after-tax cost of debt at 1.7 percent, which comes from debt that costs me now a little bit more than before, which is 2.1 percent and the corporate tax rate of 21 percent, that gives you the 1.7 after-tax cost of debt. I'm using 10 percent of debt. Therefore, I must be using 90 percent of equity. That equity now cost me 5.2 percent, and when I take that weighted average, I end up in 4.85 percent, which is the number that you see in the second numerical column. What is capital structure about? Is to do an exercise like this. This is sometimes called pro forma, it goes by different names. This is kind of a simulation if you will, of what would be your cost of debt, your cost of equity, and therefore your cost of capital and their potential capital structures. Remember, we're talking about an unlevered company. We don't have any debt right now, but we're thinking, how would our company look like if we had 10 percent, 20 percent, 30 percent, and whatever percent of debt. What is capital structure all about is looking at the very bottom line and finding what is the lowest number. The lowest number in this case is 4.8 percent. Because if you use a little bit less debt than 20 percent, then the cost of capital goes up to 485. If you use a little bit more than your cost of capital goes up to 496. So the minimum number, notice that U shape when I have no debt is up here, then it starts going down. We're reached the minimum at 480, and then it starts going up again. Well, that minimum is the cost of capital that we want to have. Of all the cost of capital we're examining, this is the lowest that we can have. Now we need to look at what is the proportion of debt that takes me to that particular cost of capital, and that is that 20 percent. So when we minimize the cost of capital, that is exactly where we find the proportions of debt and equity that we want to implement in a company. This is really what capital structure is all about. We could complicate some of the intermediate calculations quite a bit more. But at the end of the day, capital structure is about determining the ideal proportions of debt and equity that you want to have in the company, calling ideal where you minimize the cost of capital, which in this case again happens to be 20 percent debt and 80 percent equity. By the way, some companies actually find that lowest cost of capital is using all equity, and some companies find that using a lot more debt finds him the lowest cost of capital. So this depends typically on two things. Depends a lot on the type of sector that you're in. But it also depends, as it is the case right now when you have very low interest rates, sometimes it's difficult for a company to resist the temptation of issuing debt. So nowadays, with rock-bottom interest rates, you have companies that actually do not need any debt, but there's debt anyway because it's so cheap. The typical technology company go back 10, 15 years ago, most of them would have no debt where they would be fully financed by equity. But now most of the technology companies, including the very large ones that don't need any money, from Microsoft to Apple to Amazon, all of them have debt because it's so cheap. So you know how much you have to pay for that debt is going to be an input. One of the things that you need to consider when you're considering that trade-off between the cost of equity and minimizing the cost of capital.
