Let's try a practice problem that's specific to finance that also uses the idea of algebraic equations, and inverting those equations. Here we have an equation that identifies the relationship between a levered beta and an unlevered beta. We'll be calculating the beta a little bit later on in these videos. What you should know is that a beta is a statistic that captures the amount of variability in the return of a stock. Relative to the variability of the whole market. And so the levered beta captures some of the amount of the debt that is carried by an organization, you'll learn this in finance. So, what we have is we have an algebraic expression that says that the levered beta is a function of my unlevered beta. Now if I want to get my unlevered beta by its lonesome. Then what I have to do is, I have to get rid of all of these other components around the beta, right? So now, one of the things I can do over here on this side, and since I have a beta unlevered here and a beta unlevered here. Is I can factor out that beta, and so I'm left with this. Beta L equals beta U times all of this beastly stuff that's left, 1 + 1 minus taxes times my debt over my equity. Now what I can do is I can divide both sides by this whole beast of an expression here, 1 + 1 minus tax times my debt over my equity. But if I divide this side, I also have to divide that side by the same thing. 1 + 1 minus my tax, times delta, or my debt over my equity here, now all of this cancels out and I'm only left with my unlevered beta. And then this, this remains in the denominator underneath my beta levered. So I'm left with this equation right here, beta unlevered equals beta levered. Divided by this beast, 1 + 1 minus T times my debt divided by my equity. So the top, we'll call it equation 1, this is the expression of my levered beta. And equation 2, that's the expression of my unlevered beta. Now, we can engage in some mathematics. And actually calculate the unlevered beta relative to the levered beta just by plugging and chugging. So we say like suppose that a company, Smith Services, has $30 million in debt and $100 million in equity. Has an equity beta, a levered beta of 1.2, and pays a 35% tax rate, what is its unlevered beta? Well, I can take my levered beta and turn it into my un-levered beta by plugging in all the information here into this equation. So my unlevered beta, beta U is equal to my levered beta, 120, 1.20, divided by this beast, 1 + 1 minus 0.35, 35. There we go, times my debt, which is 30 million, divided by my equity, 100 million. And so this denominator, I get 30 million divided by 100 million gives me 0.3, and so I can multiply my 0.3 times my 0.65, so 1 minus 0.35. 1 minus 0.35 gives me 0.65, I multiply that by 0.3, and I'm left with 0.195, so this ends up being 1.2. Divided by 1.1, 95. So I take 1.2 divided by 1.195, and I get an unlevered beta of 1.004. Now you're going to find out in finance, that then we can use this to re-lever your beta, right? When you're going to re-lever your beta, now you can, let's say, retire your debt, or get a new debt to equity ratio. Because now you're taking your own levered beta and then you're going to re-calculate a levered beta. Given a new leverage rate, a new amount of debt, a new amount of equity, what have you. But essentially all we've done is taken our linear equation, our algebraic equation and inverted it. And once we can invert it, we can un-invert it as well. This is an example of using algebra, using inverted equations, solving for unknowns, filling in the information, and calculating them. Practicing these types of problems will make you much more efficient in your MBA studies or in other studies in general.
