CF 12-3A – M&M Proposition II – Cost of Equity With and Without Tax – Corporate Finance

It M and M proposition two, we'll do one version without any tax. That's easier. We'll do a slightly harder version with tax. Uh, but let's get down to business. Flyer company has zero debt and can borrow at 4%. The company's weighted average cost of capital is 8% scenario one, assuming no taxes answer the required below. Uh, so required a what's the cost of equity. Okay. Well, if the company is zero debt and the weighted average cost of capital is 8%, the cost of equity is 8%, but let's prove it with a little bit of math. So there's our, um, assuming no taxes, uh, M and M proposition two formula that we introduced in previous videos and showed you how to derive this in previous videos as well. I wonder if anybody sat through that explanation. It was pretty long. Uh, but I would expect you're either in a test situation going to be given this on a formula sheet or being allowed to prepare a formula sheet or being expected to memorize this. Uh, I would not expect anybody to be deriving this formula from the weighted average cost of capital for me. Anyway, let's get down to business. We're given the formula here. Um, okay. So our cost of equity is the, uh, cost of capital 8% plus 8% minus 4%. Eight minus four is four, uh, times debt divided by equity. Well, here we go. Debt is zero equities, a hundred percent. So zero divided by a hundred is a hundred. So. Eight minus four is four, four times zero is zero. Our cost of equity here is 8%, right? Eight plus zero is eight. So, uh, in scenario one, the cost of equity is indeed 8%. Um, uh, okay, let's continue. And so what we're going to do is, uh, what's the cost of equity? 8% let's do B B says if the company changes its capital structure to be 25% debt, what's the cost of equity here? Okay. So scenario, uh, part B, right? So there's a, here's B. Our cost of equity here is the 8% plus eight minus four. I'm just going to put four here. Eight minus four is four times. And what was the borrowing here? 25% debt. So the debt to equity ratio is if it's 25% debt, it's 75% equity. Uh, and there we have it. So we've just got to do the math for actually 25 divided by 75.33333 times four, 1.33333. So 1.33333 is on the right. Eight is on the left. It's 9.33333%. Let's look at C C says now we're going to 75% debt. What's the cost of equity. I haven't forgotten. We're also asked to calculate the weighted average cost of capital at these various debt levels. So we'll do that momentarily. Our cost of equity here though, is 8% plus 4% times. Now what's the ratio. What it's flipped. Now it's 75 debt, 25 equity. And so the math here is 75 divided by 25. It's three times four is 12, 12 plus eight is 20%. Okay. So our cost of equity is rising as our debt levels are rising. That should hit your intuition is right. Because of course, more debt means more risk. More risk means the shareholders are going to require more returns. So cost of equity will of course rise. Now let's look at the weighted average cost of capital. Cause again, this is our cost of equity eight. So again, this is our, our E right. Our cost of equity, but we want to know what is the weighted average cost of capital. And let's do some math. In scenario part a, if we have 8% is our equity and equity is a hundred percent of the portfolio, then it doesn't really matter that we can borrow at 4% and that's 0% of our portfolio, just eight times a hundred plus four times zero. It's 8%. That's our weighted average cost of capital. In the second scenario, 9.33, 3, 3, 3% times, uh, in the second scenario, it was 75% debt or no, sorry. 25% debt, 75% equity. So 9.333, it's 75% equity and four times 25% debt. Uh, let's do the math here. 0.9, 3, 3, 3, 3, 3, 3, 3, 3, 3 times seven. Oh, I screwed that up actually. Let's do, let's do it this way. 9.33, 3, 3, 3, 3, 3 times 0.75. We get seven, right? 6.99, 9, 9, 9, 7, 4 times 25%. That's a one seven plus one is eight. And what we find here is the way the average cost of capital is the same, no matter what. And we'll do it for the last one, just to prove that out. Um, our cost of equity is 20%. And in the third scenario, we have 25% equity. Our debt remains 4%, but in this scenario, we have 75% debt. 20 times 25% is five, four times 75% is three, five plus three. Can anyone on my fingers? Eight. So our weighted average cost of capital is eight. Our value firm value is the same here, no matter what, right? The way to average cost of capital sort of drives firm value and certainly a key number for any firm and our firm value is the same, no matter what. Uh, regardless of the debt to equity ratio. Now, what we're going to see is once we incorporate tax companies that take on debt end up being more valuable. And the reason is their weighted average cost of capital goes down. And the reason for that is debt causes interest and interest is tax deductible. And so the tax deductibility of interest causes the whack to go down. And for value to go up, let's, uh, let's go ahead and solve this in scenario two. So scenario two, exact same thing, but this time we assume a tax rate. So that brings into us to a new formula here and formula is pretty familiar. It's pretty similar to the previous one, just a couple of minor. Well, significant, but minor tweaks, uh, rather than RA here, you can see it says, are you, that is the, uh, cost of capital or cost of equity, uh, of an unlevered version of our company. In other words, a company with zero debt, what's the cost of equity or the cost of capital, same difference, uh, when there is zero debt. And in this case, it's 8% is our, our you, um, And then one minus the tax. And so tax rate is 20%, one minus the tax rate is going to be 80%, one minus 0.2 is 0.8, so we'll plug in 80% here. Let's just plug in our numbers. So what is the company's cost of equity at a time when we have zero debt? Let's run the numbers here. So, uh, our unlevered cost of capital is eight plus eight minus four. So eight minus four is cost of debt is four. Eight is R U four is R D eight minus four is four. I'm just going to put four here, uh, times the debt equity ratio. Well, if we have zero debt, our debt to equity is zero over a hundred times 80. So what we end up happening is the thing that I'm writing in brackets here, just multiply it to zero. Cause we're multiplying everything by a zero. So our cost of equity, uh, at a time when we have zero debt is going to be the same as our weighted average cost of capital. That should make intuitive sense, right? Uh, if you have no debt, your cost of equity is your cost of capital. Let's move on to part B. So that was part A part B, you know, it gets more interesting. Our cost of equity, uh, uh, remains this unlevered is 8% plus four is R U minus R D is four times and what's the ratio, 25% debt, 75% equity. So 25, 70 fifths times 80%, one minus the tax rate. So again, let's take the middle stuff. I always do the ratio first. I don't know why you don't have to, of course, 0.33333 times four. It's 1.33333, uh, times 0.8, 1.07. So 8% plus 1.07 is 9.07%. Okay. Let's move on to C, uh, C says, uh, same calculation. Just flip the debt and equity. Now we're 75% debt, 25% equity. So same type of math, eight plus four times 75, 25 times 80%. Let's run the numbers here. Uh, 75 divided by 25 is of course, three, uh, three times four is 12, 12 times 0.8 is 9.6, 9.6 plus eight. Our cost of equity here is 17.6%. Okay. So we've done our cost of equity and well, they were different from the ones up here, right? Eight, nine and 20, and this is eight, nine and 17, like slightly lower nine, 9.07 versus nine point. I think it was three, three. Um, so it's slightly less our cost of equity as we climb the ladder here. Um, but still the same story is told, right? You end up, uh, in a situation where you borrow more. Well, equity costs more because risk is higher. So there's a premium placed on that. And, uh, uh, shareholders will want a higher return given that they're taking on higher risk. Okay. So that's sort of the story there, but weighted average cost of capital tells a more interesting story. So let's figure out our weighted average cost of capital. So, uh, in scenario a, it's pretty simple 8% and we have a hundred percent, um, uh, equity in scenario one, we had zero debt. Again, I keep circling the same thing. Um, and our cost of debt or our debt cost is 4%. But remember, we want to know the after tax. Now we got a tax rate to deal with. So we actually do 4% times one minus the tax rate, which is 80%. One minus T one minus 20% is 80%. So I'm actually going to use 4% times 80%, four times 0.8. I'm just going to put it in 3.2 as our sort of after tax cost of debt. And then times the proportion of debt, which in scenario one is 0%. So I went through all that work for a zero at the end of the day, but I'll use 3.2 as our after tax cost of debt times the proportion of debt. So eight times a hundred plus 3.2 times zero is eight. Our weighted average cost of capital in the first default scenario is eight. Let's look at the part B scenario. So the part B scenario, our cost of equity is 9.07%. Uh, our ratio here is 25 debt, 75 equity. So times 75% or proportion of debt plus our after tax debt, uh, times the proportion of debt. So 3.2, uh, times, uh, 70, what is it? Seven, the 25% Jeepers creepers easy for me to say. Okay. So let's run the numbers. Nine point oops, 9.07. My goodness. 9.07 times 0.75, 6.8 plus a 0.32. No, let's do 3.2 times 0.25, 0.8, 6.8 plus 0.8 is 7.6%. So we're seeing a little difference here, right? Up above, it was 8% every time for our way to average cost of capital here, because we're taking tax rates into consideration and because we now have some debt, we'll guess what happens. Our way, the average cost of capital falls from eight to 7.6%. All right, let's do the last one. We'll see the way that our cost of capital will fall further still. Uh, so our cost of equity went way up 17.6%. Wow. But we only have a 25 equity and 75 that this time the cost of that remains just as cheap as ever. 3.2% times, uh, 75% in this center. So let's run the numbers here. Uh, 17.6 times 0.25, 4.4 plus while it's the same 3.2 times 0.75 is still 0. Oh no, what am I talking about? 3.2 times 0.75 is not the same. It's 2.4. So 4.4 plus 2.4 is 6.8%. Okay. So what is this telling us? What does this even mean? Uh, if our weighted average cost of capital is going down, the value of our firm is going up, right? We're able to, uh, take on more projects. It's a better for the firm. Uh, and just having a lower weighted average cost of capital implies a higher firm value. So the implication of this is clear firm should just take on as much debt as humanly possible. But wait a second. I'm an accountant. I'm a frugal guy by my nature. I think that is on a personal level. Why would I recommend this? And the answer here is, well, that's what the math will tell you to do and finances, math and money. And so if you're just going a pure math game, the math game says. My goodness, our weighted average cost of capital goes down. The value of our firm goes up. Let's take on more debt, but, and, and finance books will tell you this as well. Uh, it's kind of a fool's game if you take it to its logical end. And the reason is if you take it to its logical end, you will go bankrupt. The risk of bankruptcy becomes more and more significant the more debt you take on. And I always think of this like a role playing video game. If you're a video gamer like me, you might've played a role playing game and you might've even decided I want to be a magic user, a wizard in this role playing game. And so here's what happens to wizard characters that do the logical thing and take it to its logical end. Cause I'm a numbers guy and I go, all right, my wizard just leveled up. I let's see, I can add to his or her strength. I can add to their dexterity. I can add to their magic power and I will always add to their magic power. Every time I level up more magic power, more magic power, more magic power, more magic power. And whatever happens to my characters is I just start killing all the bad guys cause I'm so powerful. My magic spells are spectacularly strong. And then they come up with a bad guy who can withstand one of my spells and they go up to me and they go. And they kill me in like one shot because I have zero hit points, right? I've invested everything in magic power, nothing in hit points. And I go, they go think, and I'm dead. That's what happens to companies that take this to its logical end. They go, we're going to lever up to the extreme degree, right? We're going to take on more and more and more debt, as much debt as humanly possible. And then one little problem, think, and they're bankrupt. So lenders know this. They're not going to lend to the logical degree and investors and even owners of the company know this, but it's very tempting to look at this formula and say, I should borrow to the hilt because it maximizes firm value and I should be trying to maximize firm value bankruptcies. The thing that sort of prevents this from being the greatest infinite money strategy you've ever heard of. All right. I hope this video was helpful. I hope, uh, you know, you don't take too much risk as you're headed towards your, uh, midterm or final exams. And I hope you'll hit one of those buttons for me. Thanks for watching. Have a great day. See you in the next video. Bye bye. Bye.