CF 9-3A – Variance Calculations – Corporate Finance

Let's take a look at problem nine, three, a, this problem has us looking at variance calculations and variance is a measure of variability. So we're going to look at a stock price and how it's changed over time. And we're going to calculate the variance of that stock price. And this is a statistical concept. If you've taken a stats class, you've probably calculated variance and standard deviation. This is just that. So this should be review for anyone that's done statistics. Now, why does that play into finance? Well, first, if you're an investor, the variability in stock prices is of intense interest to you. Uh, but investors look at a concept called beta, uh, when they measure variability as compared with the rest of the market, so we'll look at beta next chapter, and it's based on. This variance calculation. So this variance calculation is core to beta and beta is a core concept in corporate finance. So anyway, let's get going. And when we do this, it's very much like a stats class. So here we go. The percentage returns of XYZ companies stock over the last five years are shown in the table below. And there's XYZ stock and you can see it's, you know, had some good years and had some bad years. So it says during that time, the average T-bill rate was 4.2%. That is the risk free rate. In other words, without putting our money into some stock that could go negative, which it did, and we could just give it to the government, the U S government, and they'll give us 4.2%. And that's as close to a guarantee as exists in the world. So calculate the arithmetic average return of the stock. We'll ignore that risk free rate for the moment. The arithmetic average, I think of it as just the average. We're going to learn in a future question, something called the geometric average. That's a little more complicated. Arithmetic average. So you just add the numbers up and divide by however many you just added up. So we're going to go 20% minus 8% plus 17% plus 14%. Minus 3%. We're going to divide by there's 1, 2, 3, 4, 5, 5, 5 years, and we'll get an average, right? What's the average return in any given year? So 20 minus 8 plus 17 plus 14 minus 3. It's 40. So 40 divided by 5 is 8. Our average here is 8%. The arithmetic average return on our stock was 8%. What's the risk premium? Well, the risk premium is you invested in a stock that fluctuates in value. You expect a higher return than, because otherwise you just give it to the government and they give you 4.2% on your money, you'd lend it to them. So we got paid 3.8% more than what we could have got if we bought T-bills, right? 8 minus 4.2 is 3.8. That's the risk premium of this stock. Okay. So now it asks us for the variance of the returns. Here's how we calculate the variance. It's the sum of the squared differences, such an awkward turn of phrase here, and I'm sorry to say it in this way. The sum of the squared differences divided by N minus 1. It'll become easier when we do it a few times, but that's an awkward turn of phrase. So when I say the sum of the squared differences, I mean the difference between the given year and the average. So the average was 8%. So I just take 20% minus 8%, negative 8% minus 8%, 17% minus 8%, 14% minus 8%, and negative 3% minus 8%. Okay. So let's take the difference between the year and the average. So 20 minus 8 is 12. Negative 8 minus 8 is actually negative 16, right? Minus 8 minus 8. 17 minus 8 is 9. 14 minus 8 is 6. And negative 3 minus 8 is negative 11. And again, I'll just show you that in my calculator, make sure we're on the same page here. Negative 3 minus 8, negative 11. Okay. So that's the difference between our given year and the average. And you know, I think about the variance. The variance is like how high or below average you are. If we're eight every year, then our average is eight and there's no variability. It's just the same every year, but here there's variability. So this already is a measure of variability. How different are we in one year compared to the average? Uh, now we square those differences. So actually I'm going to restate these as decimals because punch them in the calculator as a decimal. So 0.12, negative 0.16, positive 0.9, positive 0.6, and negative 0.11. And now we square those. So to the power of two. So negative, or not negative, just 0.12 to the power of two, 0.12 squared. 0.0144. We have big, long decimals. Try to keep as many decimal points as you can. 0.16 negative squared is 0.0256 positive. When you square a negative number, it of course goes positive, 0.9 squared. Oh, I'm glad I looked here. This is not 0.9, it's 0.09 squared. This is 0.06 squared. I'm happy I double checked that. 0.09 squared is 0.0081, 0.0081, 0.06 squared. I can already guess 0.0036, but let's crunch it, 0.06 squared. Yeah, 0.0036 and 0.11 negative squared, 0.11 negative squared is 0.0121. Let's sum the squared differences here. So sum it up, 0.0144 plus 0.0256 plus 0.0081 plus 0.0036 plus 0.00121. And I get 0.0638. I was just crunching the number in my head saying, does that make sense that that's adding up? Because sometimes you miss a 0.0, you know, you miss one of the zeros there and end up with a number that's off. Okay. We divide this by n, the number of years or the number of items we're looking at, minus one. So we, and you might think, well, why don't I just divide by five, right? I took an average where I divide by five. Why don't I just divide this by five? And the answer is somewhat statistical. The idea here is we're looking at the percentage of returns of the company's stock over the last five years. Well, the company's had more years than five years, presumably that it's existed. So because we're not looking at the entire population, we go, we divide by n minus one. If we're looking at a population, you just divide by n. Why? I watched a statistics video for 10 minutes on this concept. Why is it n and n minus one? And they're talking about degrees of freedom. And I got to tell you, I lost the plot at about minute two of a 12 minute video. I couldn't, I was like, oh, I want to explain this in my video. I couldn't. So Google it. You can say why n or n minus one when doing a variance calculations, and you'll get an explanation. For me, it's just, if it's a sample of years, not the whole population, n minus one. In our class, it's going to be n minus one. Okay. So 0.0638 divided by n, the number of years, five minus one, five minus one is four, right? Five minus one is four. So 0.638 divided by four. It's a long-winded way of saying that. And I get 0.01595 or 1.595%. So what's the variance? It's 1.595%. Is that big? Is that small? What does that mean? The only way to do this would be to compare to other similar companies or companies in the industry. You can say, oh, that variance is very high for this industry, or that's very low for companies of this sort. We don't know is the reality. We don't know what it means unless we can compare to something else. So we'll just take it as a number at this point. So the variance is annotated as this. This, I think it's a sigma symbol. I could be wrong. I don't know my Greek alphabet very well. This symbol squared. And the next part asks us for standard deviation. Well, standard deviation is just that symbol. So to get from this symbol squared to this symbol, I just take the square root of whatever my variance was, and that'll give me my standard deviation. So the square root of 0.01595, take the square root, and I get 0.1263 or 12.63%. That's my standard deviation. Now, again, is this big? Is this small? It's going to depend on the company, the industry, what's going on in the market at that time to know if that's big or small. The real key calculation here is variance. Once you get variance, standard deviation is easy. The reason variance is key, though, is because it's very useful in computing beta, and beta is a concept we explore in some detail next chapter. Let's stop 9.3a there. This is enough statistics for me. I hope you liked it, though, and I hope you'll give me a thumbs up. Thanks so much for watching. See you in the next video. Bye-bye.