CF 10-7A – CAPM Calculations – Corporate Finance

Let's take a look at problem 10-7a. We're doing lots of calculations. These calculations have us doing weighted averages. There's a little bit of algebra here, but I think it's an interesting problem. Let's jump into it. So 50% of an investor's portfolio consists of one stock with the following characteristics, the expected return of the stock 17%, the risk-free rate 4%, and the stock has a beta of 1.4. That means if the market moves by 10%, we would expect our stock to move by 14%. We've got a swingier and riskier stock than the average in the market. The other 50% of the portfolio is invested in U.S. government bonds, which are in 4%, and we all know the beta of these risk-free bonds is zero, and the reason is there's no swinginess to them, right? The market swings. They're still going to return 4%. Okay, so what is the expected return of this portfolio? We've got our stock, and we've got our government bonds, and our stocks are expected to return 17%, our government bonds are expected to return 4%, and we got half stock, half government bonds, so our return is going to be just in the middle. Now, we'll do this as a weighted average calculation, but you could do a simple average, 17 plus 4 divided by 2. It's 10 and a half. I did it in my head, but this is how I'd advise you to do these types of calculations, because the weight's not always going to be equal, so you take the weight, which is 50% in our portfolio. It said where 50% of the portfolio is stock, 50% is the government bonds, you multiply them through, 17% times 50% is 8.5, 4 times 50%, half of 4% is 2%, then you add them together, 8.5 plus 2, 10 and a half percent. We got it. We got part A in the bag. Let's move on to part B. I'm actually going to scroll down just to clear off some space. If an investor desired a portfolio beta of 9, what would the relative weights of the stock and government bonds need to be? Another weighted average calculation this time, we'll compare our stock and our government bonds, but not along the expected return line, no, no, along the lines of betas. So the beta of the stock is 1.4, and the beta of the government bond is zero. Now, had we been 50-50, let's just estimate that. So if the weights, now this is not going to give us the right answer. Don't write this down. We'll write down the right answer in a minute. This is just to illustrate a point. Had we been 50-50, 1.4 times 50 is 0.7. Zero times 50 is zero. So 0.7 plus zero is 0.7. The weighted average beta of our portfolio is 0.7. And it's saying, no, no, no, we want a beta of 0.9. We're going to have to take on more stock, right? Because adding government bonds to this portfolio doesn't add risk. So we're going to have to have more than 50% in stock. But what's the number? So here's how you do it. I'll change my ink here, maybe. Let's change this, let's change this, and let's change this. And we can erase that too. Okay, so we're sort of starting from scratch again. We are desiring a beta of 0.9. We know whatever number goes in here, and we don't know the numbers yet, but we know whatever number goes here in government bonds, zero times whatever is going to be zero. That's one of the cool things about multiplying by zero. So we know 1.4 times this number, it's got to be 0.9, right? It's got to work out to 0.9 to make our portfolio beta 0.9. So how do I figure that out? Well, I just go 0.9 divided by 1.4. And we'll get there. 0.9 divided by 1.4 equals 0.642857. Oh, nice. Even number. I like when I write a question that's beautiful, even numbers like this. Okay. Well, 64, let's call it 64.3%. Okay. Well, if we've got only two items in our portfolio stock at 64%, the government bonds got to be the rest. So I just take away one plus minus. It's 35.7%. And there we have it. We've solved part B. Uh, if an investor wants a portfolio of, uh, with the beta of 0.9, what's the relative weight of the stock and government bonds, we'd have 64% stock, 36% government bonds, and we'd be right on our 0.9, our desired beta, beta, if you're in the UK, I think, what do you say? Beta or beta? I wonder they talk about software betas. When I listened to podcasts with UK hosts, I call them betas though. Okay. Let's continue. I could get very sidetracked there. If an investor desired a beta of three, what would the portfolio weights need to be? Oh, we need way more stock. So it's exact same question. I'm going to just erase what I've got here in blue. And I'm going to say, okay, we want to have a three over here. Well, okay. Well, we're solving for this again. The weights, uh, zero times, whatever that weight is is going to be zero. 1.4 times something is it's got to be three down here. So 1.4 times what is three? I go three divided by 1.4 to 1.14289, maybe 2.14. I'm going to say 2.143, 2.143. Now what is that as a percentage? It's 214.3%. Okay. This is confusing. And look at the question says, if investors are made of three, what would the portfolio weights need to be? This produces a funny looking result. Yeah, it is funny looking to have 200% of my portfolio in stock. How can you have more than a hundred percent of your portfolio in anything? And in fact, if I want this to work, I need to have negative 114.3% in government bonds, because of course my portfolio has got to add up to a hundred percent, you know, if I had 30% stock, I would need 70% bonds because 30 plus 70 is a hundred well, if I have 200% stock and I have, uh, well, I know my, this plus this has got to equal a hundred percent. So I know this one's got to be negative 114.3. It's a crazy number, right? It's a strange number to see. What does it mean? It means to get this much beta and this is a lot of beta, right? This is high volatility, high risk. We need to borrow and we need to borrow at that government rate, the 4% rate. We need to borrow money to invest more money in stock than we even have. So it means you need to lever up, right? You need to get leverage and buy more stock. You know, maybe you're buying options or you're borrowing money to buy the stock, but it means you need to be taking even more risk on it. If this is what you want to do. So, so the implication here is we're borrowing, you know, if you had, let's just pretend these are thousands of dollars, right? You have a hundred thousand dollars in your portfolio. You got to invest all a hundred thousand dollars. Plus you need to borrow another $114,000 to invest in that stock to get sufficient risk and sufficient variants, uh, to get a beta of three. Okay. That's what produces that funny looking answer. Let's move on to D D says what relative weights between the stock and government bonds would be needed to produce an expected return of 9%. Okay. This kind of goes back to the first, uh, calculation. Maybe I'll scroll up there and do it on top of that. Is this the one? Yeah. This is the one. So what this one is saying is, you know, okay. The stock has returned of 17%. The bonds have returned a 4%. Uh, and our overall, if they're 50 50, our overall return is 10 and a half. Now it's saying, well, I want a return of nine. What relative proportions do I need to get a return of nine? I'm going to write this out slightly different. So I'm going to say, okay, our stock is 17% times its proportion, which we're going to call, uh, P in this example, I called it X, but then I had a times X when I did this, uh, question previously, and I got confused, I'm going to call it prop proportion plus 4%, 4% times. One minus the proportion. In other words, if the proportion here is 70, the proportion here is one minus 70. It's 30, right? So we're only solving for one variable. Thankfully, right. It looks kind of like we might have two variables, but it's really only one. And that needs to equal 9%. Now we just solve for our proportion variable. So 17% times proportion. So I'm just going to get rid of the percent signs. Cause there's percent signs all through it. I'm multiply both sides by a hundred disorderly. It's the numbers are going to work out. So I'm going to have 17 P proportion plus four minus I'm multiplying the four in minus four P equals nine. Okay. So now I can really solve, uh, 17 minus four, right? 17 minus four is 13. P equals nine minus four is five. So five divided by 13, 38.5% P equals 38.5%. What does that mean? It means we want to have 38.5% of our money tied up in stock. The rest tied up in government bonds. Let's figure what that is. Minus one 61.5% of our money tied up in government bonds. Let's make sure we're right. So we can just calculate this through 17 times 0.385, 6.545. And, uh, four times 0.615, 2.46. All right. So it's 2.46 plus 6.545 equals 9.005. There's some rounding going on, right? We rounded a couple of numbers, but we got there. This, you know, if we had 38% of our portfolio in stock and 61% in bonds, we would get a 9% expected return. There we go. Uh, if you made it to the end, as always, I genuinely appreciate the thumbs up. If y'all loved it, maybe hit the subscribe button. Thanks for watching. Have a great day. Bye bye.