CF 10-2A – Expected Returns and Deviations – Corporate Finance

Let's take a look at problem 10-2a. This one's a challenging problem. I think it'll take us a while here. If you look at the bottom of the scroll bar, I bet you I'm over 20 minutes, maybe 21 minutes and 18 seconds. That's my prediction. Wouldn't that be amazing if I hit the post, hit the nail on the head, but we'll see. The reason it's challenging is calculating variances and variances are required to calculate standard deviations. Calculating variances and standard deviations by hand is time consuming and you just got to be slow and careful and that's what we will be. Let's jump into the problem. Boa's information about two stocks and various possible economic situations. So there's a 20% chance we'll be in a bust. And if we're in a bust, stock X gives us 10%, stock Y gives us negative 20%. There's a 70% chance this economics will be normal. And those are the expected returns in that situation. And there's a 10% chance of a boom cycle. So the first question is compute the expected returns for the individual stocks. I've just moved this to where I can scribble all over the screen, but it's all the same information. So what's the expected return of stock X? Well, the way you would do this, you'd say, well, there's a 10% chance or rather there's a 20% chance that we're going to get a 10% return. You just multiply those two probabilities, 20% times 10%. So 20% times 0.1 and that equals two. Now the two doesn't mean much, but what we do is we take the sum of these things and we get a weighted average, 70% chance of a 15% return. So, uh, 70 times 0.15 or 0.7 25, anyway, it's 10.5 and 10% chance of a 20 is two again. So I get 14.5%. That is my expected expector, no expected return of stock X. So 14 and a half percent is, that is, that is hideous expected return of stock X. My expected return of stock Y I go, okay, well, there is a 20% chance. We're going to have a negative 20% return. So negative 20, 20% times negative 20% is negative 4% 70% chance of a 20% return. That's positive 14 and 10% chance of a 55% return. That's 5.5 just doing some math here. So let's add that up for stock Y four negative plus 14 positive plus 5.5, 15.5%. So our expected return for stock Y is 15 and a half percent. Now you might look at this and say, well, I want to get more return. Therefore I want stock Y, but of course not every investor would agree with you. Lots of investors would say, I'd rather have stock X, which seems like way more stable returns. You know, uh, at worst case scenario of a 10% return is really, really good. So, uh, that's, uh, I I'd sign up for that. Uh, but you're, you, you may sign up for stock Y, but there's a reason that investors aren't just piling into one stock or another, and it's the variability in those returns. So variability in returns is a useful information to have. And that's where this question leads us. It, it asks us for the variability of the return and it asks us to calculate the variation and standard deviation in the return. So that is a, an economics or statistics concept. So let's do it. We're going to calculate the standard deviations and to do it first, we calculate the variance of the return. So stock X had an expected return of 14 and a half percent. So we deduct our expected return from the, uh, projected return in these various scenarios, and we get a difference. So 10 minus 14 and a half is negative 4.5%. 15 minus 14 and a half is 0.5%. 20 minus 14 and a half is positive 5.5%. I'd like these all to be decimals when I put them into my calculator. So negative 0.045. This is 0.005. And this is positive, uh, 0.055. Okay. So we've converted our, uh, percentages to decimals. Now calculating variance, it's the sum of the squared differences. So we, we took the difference between our, uh, uh, projected return on stock X and the, uh, average or mean return, the average return, and we got a difference of four and a half percent. Now we square those differences. So 0.045 negative, uh, squared. And we're going to get very small numbers here. 0.002025. This is going to be even smaller than 0.005 squared. 0.0123425. And now 0.055 squared. 0.003025. Okay. So again, very small numbers. That's life. Now making matters a little more difficult for us is we want to apply the probabilities of the situation. And so again, our numbers are going to get microscopically small to the point of not meaning much to me. When we work back to standard deviation, that number means something to me. Even the variance number in my heart, I don't have a strong feeling when I see a variance. I want to see the standard deviation that gives you better information. But in any event, let's let's we're getting to the variance first, uh, we apply our probability. So we multiply this one by 20%, multiply this one by 70%, multiply this one by 10%. And again, we get very small numbers. I'll start with the bottom cause it's already in my calculator. Multiply that by 0.1 and we're at like 0.1233025. Okay. Uh, next one 0.0, 1, 2, 3, 4, 2, 5 times 0.7, 0.1, 2, 3, 4, 1, 7, 5. My Lord in 0, 0, 2, 0, 2, 5 times 0.2. Oops. Missed the button 0.2. There we go. Uh, 0.1, 2, 3, 4, 0, 5. And now we just add these up and this is our variance. So that number plus 0.1, 2, 3, 4, 1, 7, 5. Just got to be careful. We just go slowly here. 0.0, 1, 2, 3, 3, 0, 2, 5 equals. Holy cow. We got 0.000725. That is our variance. Again, that number looks to me like a very small variance and that's what we expect. There wasn't a huge variance between the options, especially if 70% of the time you're going to land on that. That's not a huge variance. Even when we hit our outliers, it's not a huge variance. So, uh, standard deviation gives us this number as a percent. I take the standard deviation by just taking the square root of the number. And I get 0.0269, which I'll just convert to a percentage right now. I'll call it 2.7%. So our V our standard deviation here, 2.7%. Uh, uh, so again, we're expecting a return of. Uh, 14 and a half percent plus or minus 2.7% is the standard deviation. Okay. So in any event, it's a measure of the variability here, uh, of our returns. Okay. Uh, let's move along and do the same thing for stock Y. So stock Y's expected return was 15.5%. And what we're going to find is look at these returns, way more variance, way higher standard deviation. So I subtract my expected return for that stock negative 20 minus 15.5 negative 35.5%, 20 minus 15.5 is 4.5%. And 55 minus 15.5 is 39.5%. Just doing the math in my head. Uh, we square the differences. So I'll state these as decimals, uh, negative 0.355, negative 0.045, actually positive doesn't really matter. And positive 0.395, the reason I say it doesn't matter is because we're going to square them. And so negative numbers become positive one squared. So 0.355 negative squared 0.12. I'm going to stop here. Uh, 0.1260. I'll just stop after 4. Uh, 0.045 squared, uh, 0.0020, uh, and 0.395 squared 0.1560, 0.1560. Okay. So we've squared the differences. Now we apply the probability of a situation, 20%, 70%, and 10%. And we will sum that up to get our variance. So let's go to it. Uh, 0.126 times 0.2, 0.0252, 0.002 times 0.17. No, not 0.17, 0.7. My goodness. Uh, 0.0014. And finally 0.156 times 0.10, 0.0156. We're going to sum those up. So we'll add that to 0.14 and add that to. Oh, that, that seems off. Hold on. Let me just cool my jets. The number seemed weird. 0.156 times 0.1, 0.0156. I'm going to add, I think this is where I screwed up. 0.0014. There we go. And I'm going to add 0.0252. So I get 0.0422. And now I square root it to get my standard deviation, which again, our last standard deviation was 2.7%. This is going to be way bigger than that. Let's do a square root and figure it out. 20.5%. It's like 10 times bigger. So yeah, much bigger standard deviation here with stock Y than stock X. Now, what do we use this for? Well, it's used a lot in, you know, the variability of a stock is, is like useful information for an investor to know. And so these are the statistics behind this, right? This is the math behind this. Okay. So we've computed the standard deviation. We've solved part B. Moving on to C and this is another sort of challenging part. Uh, there's actually two pieces. The first part's easy. The second part's harder. Um, it says, assume your portfolio is weighted 70% to stock X, 30% to stock Y compute the expected return for the portfolio. So we've said stock X is going to give us 14.5% and 70% of our money is going into stock X, 30% is going into stock Y. So you just multiply 70% times our expected return. 14.5, 14.5 times 70% gives us a number 10.15. And we're going to add that to 15.5 times 30% and we have 4.65. We add them together. 10.15 plus 4.65 and I get 14.8%. That's the weighted average return in the portfolio. So that answers part C, the first part, compute the expected return. It's 14.8%. Now on to, uh, the expected return and standard deviation. Okay. This one, it's a multi-step process, but we'll, we'll get there. Uh, 70% of my stock is stock X, 30% of my stock is stock Y. So let's just go through the economic situation of a bust. Let's say again, I've got 70% of my money in stock X, 30% of my money in stock Y, and there's a bust cycle. What happens to me? Well, uh, stock X 70% goes up by 10%. So 70% of my stock goes up by 10%. That means it's a plus 7% to me again. Uh, 70% goes up by 10%. 7, 30% goes down by 20%. That's a negative six. So overall, plus seven minus six, I'm plus 1%. My portfolio, my, my wealth is going up by 1% in a bust. What's happening in my wealth in a normal situation, 70% of my stocks going up by 15%, 0.7 times 15. That's a plus 10.5 to my wealth. And, uh, uh, 30% of my stock is going up by 20%. I think I was pointing at the wrong 70, 30% of my stocks going up by 20%. That's a plus six. So in the event of a normal cycle, I'm expecting my portfolio to go up 16 and a half percent. What about a boom cycle? Well, 70% of my stuff's going up by 20%. That's plus 14 and 30% is going up by 55%, 0.3 times 55. That's 16 and a half 14 plus 16 and a half is 30 and a half. And that's another plus. So good news for my portfolio. It looks like it's going up no matter the cycle. Um, but what is the, uh, sorry, I have to now apply just hold on. I need to think for a half second. I've got it. Sorry. I was like, okay. So these are the states of that might be, I might go up 1%. I might go up 16 and a half percent, or I might go up 30%. So all I do is I take away my expected return to figure out the variance from expectation. And it's just a pretty straightforward variance problem from here. So 1% minus 14.8% is negative 13.8%. Uh, 16 and a half minus 14.8 is 1.7 positive. Uh, 30 and a half minus 14.8 is 15.7. All right, let's state these as decimals. Then we'll square them. So this is negative 0.138. This one is positive 0.017. And this one is positive, uh, 0.157. We square those. So 0.138 negative squared, uh, is 0.190, 0.0190, 0.017. We square 0.0003. And the last one, uh, 0.157 squared 0.0247. Okay. So we've squared those. We also need to apply the probabilities here. So we multiply here by 20%. By 70%. And by 10%. We'll answer that. We'll sum those all up and that's going to calculate our, uh, variance. Okay, let's do it. And again, we're going to get very small numbers here. Uh, uh, but, uh, that's what we're after 0.019 times 0.2 is 0.0038. 0.0003 times 0.7, oops, missed the times, times 0.7 is 0.0002. And we'll leave off the one point. I'm going to four decimal places here. 0.0247 times 0.1 is 0.0025. Okay. Let's add up our list to get our standard or our variance 0.0038 plus 0.0002 plus 0.0025. So the sum there is 0.0065. That is my variance square root to get the standard deviation square root of that number. Is, uh, let's see, 8.06%. So our standard deviation of these returns, sorry, these ones is 8.06%. Uh, to answer the question again, uh, what's the expected return for, oops, I'm looking at the wrong thing here. What's the expected return 14.8%. What's the standard deviation 8.06%. The last piece, actually a very short one, mercifully short, given how long that took. Assume a risk-free rate of two and a half percent, the poor, uh, for the portfolio above same portfolio. Uh, uh, what is the expected risk premium? Okay. Well, our expected return. So the risk premium is just the expected return minus the risk free. Right. That gives us our risk premium. So this one's easy. Our expected return of that portfolio is 14.8%. That actually wasn't that hard to get our risk-free rate two and a half percent. So what's the premium for taking on risk? It's 12.3%, just the difference between the two. And there we have it folks. We've solved the big one. Uh, I hope this problem helped. I know it's, it's difficult. Some of these variants and standard deviation things can be tough, but I hope I'm helping you get through it. Thanks so much for watching and I'll see you in our next video. Bye-bye.