 Hello and welcome to the session, this is Professor Farhad and which would look at an example that deals with risk and risk premium and specifically how to compute the standard deviation as well as the variance. These topics are covered on the CPA as well as the CFA exam and also covered in an essentials or principles of investment course. As always, I would like to remind you to connect with me on LinkedIn if you haven't done so. YouTube is where you would need to subscribe. I have 1,700 plus accounting, auditing, tax, finance as well as Excel tutorials. If you like my lectures, please like them, share them, put them in playlists. If they benefit you, it means they might benefit other people. Connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources to complement and supplement your resource, your finance as well as your accounting education. When I mentioned the word example, it means there was an explanation before I work this example. Therefore, if you are interested in kind of learning about the topics, standard deviation, variance, expected return holding period, you want to make sure you see the prior recording. Don't forget to check out my website. So this is the problem that we are going to be working with. The current value of stock portfolio is 23 million. A financial analyst summarizes the uncertainty about the next year holding period return using scenario analysis and the following spreadsheet. So we have four scenarios. High growth, normal growth, no growth and a recession. What are the holding period return of the portfolio in each scenario? Compute the expected holding period return and the standard deviation of returns. So we are giving the scenarios for scenarios. We are giving the probabilities. They always at up to 100%. We are giving the expected year end value of the portfolio under different scenarios. We are giving also the annual dividend yield in million for each scenario. So to work this problem, I'm going to be using an Excel sheet. So I'm going to switch to the Excel sheet so it's easier to show you the computation as we go through this exercise. So on the Excel sheet, we're going to work this problem. The first thing is we want to know is the holding period return for each scenario. How do we compute the holding period return? Remember, the ending period of the portfolio is given. The beginning period is 23. So if we're going to end up with 35 minus 23. That's going to be the capital gain plus the dividend 4.4 divided by the original value of 23. That's going to give us the holding period return to simply put what I'm going to do is I'm going to show you the formula. How we did this, then copy the formula down for all the resources for all the other scenarios. So if we take D3 minus 23, the beginning portfolio plus the dividend plus E3 divided by the original value of the portfolio of 23. It's going to give us 71.3 return. What I'm going to do, I'm going to take this and copy the formula for the other returns. So under the normal growth, we expect 34.8%. Under no growth, we expect negative 17.4. And under a recession, we expect negative 56%. The next thing we're going to do is we're going to take the probability of that happening 30% times the holding period return to find the expected value of the portfolio under different scenarios. So if we take 30% times 0.713, we'll give us 21.39. Now we can reduce it to do or three decimal point. I'm going to go ahead and do the same thing for the other scenarios and the expected return of the portfolio, which is an important number, 30.73%. So the expected return of the portfolio is 30.74%. So you're asked to compute the expected return and this is the expected return. Now, that's fine. That's the expected return. I want to see the variability. How much this portfolio is going to vary? Why is that important? So real quick, I'm going to tell you why that's important. Let's assume you are choosing between two stocks, stock A and stock B. And the expected return over 30 years or 20 years or for any particular period, but it's a meaningful period for stock A is 10%. For stock B is 10%. Well, what's the difference between the two? Well, the difference is there's no difference from the expected return over the same period. They both earned 10%. But what we want to know is the variability. How much did stock A over the period of time varies? So in other words, we want to know the standard deviation. How much it deviated from the mean, stock A versus stock B? So if the deviation for stock A is 15 and the deviation for stock B is 25, we say that stock B is riskier. Why is it riskier? Because in terms of deviation, stock B deviated more from the mean. This is to measure the variability of the stock. Why is that important? Because it depends on your comfort. How comfortable are you? Because you want to quantify this uncertainty. How much did stock B deviated? Maybe stock B looks like this and stock A looks like this. So you're more comfortable with a stock that doesn't deviate a lot from the mean versus a stock that goes up and down much more. How do we do this? We compute the standard deviation. How do we compute the standard deviation? First, we have to find how much does each scenario deviate from the mean. So let's find scenario 1. Scenario 1 deviates from the means. Let's see. It deviates from the mean 0.40. How do I know this? Well, the expected return is 0.74, but the return is 0.70. So I'm going to take the difference between those two. So it deviates 40 points. Let's look at B. B, we're going to take the difference between this. Notice B doesn't deviate a lot. Only 4%. We're going to do C, not C, the third scenario, no growth. The third scenario, you deviate negatively. Because we don't care deviation. It doesn't matter whether it's negative or positive. But that's how much it deviates. And under the recession, it deviates a lot, 87%. So basically here, we're looking at the deviation at the range. And let me show you what do I mean by the deviation or the range. I like to show it to you. So this way, you can see it. So this is the expected return, 30.74. Let me change my color because I'm going to be using this here. Now for the high growth scenario, it's 70.13. So this is the range. This is how much it deviates. For the second scenario, for the normal growth, it is right here. So notice it's only 4%, which is 4%. So this is what we mean by the range. So first, we compute the range. Notice the range has negatives. So we want to eliminate the negatives. We don't care about negatives. We don't care about the negatives. What we're going to do, we're going to take the range and we're going to square it so it becomes positive and multiply it by the probability to find the variance. So now we want to compute the variance. So to compute the variance, we're going to take the probability, which is 30%, times the deviation raised to the second power. And we're going to do so for all four scenarios. Then we're going to add them up and we're going to find the variance. And the variance is, let me just show you the variance, the sum of all those three. The variance is 0.13405. Now from the variance, we could compute the standard deviation. And basically the standard deviation is the square root of the variance, which is 36.67. Now the question is what does that mean? What does the standard deviation mean? It means within one standard deviation, within one standard deviation, we vary 36.67. That's all what we need to know for now. Now what we're going to do in the next session, I'm going to look a little bit more about this is one standard deviation. So it varies 36.67. So basically you will see this in the normal distribution later on. If this is 30.74 within one standard deviation, within one standard deviation, which is, so we're going to be, let's just add 30.74 plus 36.67. So within one standard deviation, 67.41, and we're going to take the difference now between 30.67 minus 36.67. Equal, let's just get the function. And that's equal to negative six, negative six. So within one standard deviation, we could go up to 67.41 and negative six. Again, we're going to talk about one, two, and three standard deviation when we talk about the normal distribution. I just wanted to show you this. Now why is standard deviation is important? Because we could have another stock or we could have another portfolio. With the same expected return, but with a different standard deviation. So let's assume we have another portfolio and the standard deviation happens to be 10%. Well, what does that mean? It means within one standard deviation, we could be at 40.74 or 20.74. It means this portfolio, it deviates less. Deviate less means it's less risky. And that's why we have to compute the standard deviation. Don't worry, we're going to talk about this in the next session, where I discuss the standard, the normal distribution within the portfolio and explain a little bit more about the importance of the standard deviation. All what I wanted to show you here is how to compute the standard deviation in an example. In the next session, as I said, we'll look at this. As always, I'm going to remind you to like this recording, share it, put it in playlist, and check out my website for additional resources. Good luck and study hard.