 Hello and welcome to the session. This is Professor Farhad in which we would look at the normal distribution for a portfolio and the concept of value at risk. This concept is usually not explained when in college so hopefully I will do a good job explaining value at risk because I believe once you understand normal distribution you should be able to easily understand value of risk and that's why I bunched those two topics together. These topics are covered on the CPA as well as the CFA exam. Also in an essentials of investments or or principles of investments whether undergraduate or graduate as always I'm going 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 tutorial. If you like my lectures please like them, share them, put them in playlists. If they benefit you it means they may benefit other people. Connect with me on Instagram. On my website farhadlectures.com you will find additional resources to complement and supplement this course as well as your other accounting and finance courses. I strongly suggest you check out my website. In the prior session we looked at the scenario where we had four scenarios, basically scenario analysis, the probability for each scenario, the holding period return, the expected return of the portfolio, the deviation or the range from the expected return. We computed the variance and and from the variance we square root the variance to come up with the standard deviation. So the focus on this session is about standard deviation and how the standard deviation fits in a normal distribution. So let's go ahead and make sure we understand what is what is a standard deviation. A standard deviation is the amount of volatility, risk, uncertainty, fluctuation in your portfolio. So somehow we want to measure how risky is your portfolio. How is it measured? It's measured through the spread of the numbers. How spread are the returns or how spread are they? And it's represented by the letter by the by the letter Greek Sigma looks something like this. And what does what does the standard deviation tells us? Generally speaking, if we have a large standard deviation suggests that the data is more dispersed, more volatile, more risky. So the average is less representative. If the standard deviation is smaller, if we have a small standard deviation, suggests that the data is less dispersed, less volatile, and the average will say it's generally more representative. Now we're going to look at the standard deviation in a normal distribution graph, but just want to let you know these are the rules kind of to keep in mind. Now we're going to be working with a normal distribution. We're going to assume it's a normal distribution. And if what is a normal distribution, it's the bell curve shape, which we're going to see in a moment, we're going to assume we're working with this. What does the normal distribution tells us in conjunction with the standard deviation? Well, what we say is this a lot of the data. So 68% of your data of your return falls within one standard deviation. I'm going to show you real quick what does that mean and we're going to look at another graph. So if this is the mean and this is one standard deviation, this is plus one minus plus one minus one standard deviation. The majority of the return approximately 68% of the return falls within plus and minus one standard deviation. Most of the data falls between two standard deviations. So if we go to minus two plus two, I'm going to change colors here. And now 95% of the expected return falls within two standard deviation because now we are increasing the standard deviation. And almost all the data, 99% of the data falls within, let me change the colors again, three minus three plus three standard deviation. Basically, once we increase the standard deviation, it's going to encompass everything. It's going to, this orange will approximately include 99% of the returns within three standard deviation. Now, why it's not 100% because you could always have something outside the normal, which is called the black swan. But within three standard deviation, all the returns should fall within three standard deviation. The majority of it should fall within one standard deviation. 68% of it should fall within one standard deviation. Let's take a look at this actual example to see exactly how it works. Here we are working with a normal distribution, normal distribution bell curve. The expected return is 10% and the standard deviation is 20. Let's now use numbers and explain what I just told you a second ago. Well, if it's 10, within one standard plus one standard deviation, minus one standard deviation, it means it's going to be, we're going to, so it's 10 plus 20, we're going to be at 30. So notice 30% and 10 minus 20, it's going to give us minus 10%, minus 10%. So within one standard deviation, notice we have approximately 68% of the return. So 68% of the return, it's going to fall within one standard deviation. Now, what happened if we go to two standard deviation, negative 2 plus 2. Negative 2 and plus 2, the standard deviation is 20. 20 times 2 is plus 40%, minus 40%. Now if we go plus 40%, so if we have 10% the mean plus 40, that's going to give us 50%. We're going to be at 50% and 10% plus minus 40, it's going to give us at 30%. So here we are dealing with here. So plus two standard deviation, plus two standard deviation, minus two standard deviation, within those two standard deviation, we're going to have 95.44, let's just say 95% of the return. And within three standard deviation, within three standard deviation, what's three standard deviation here? Here the standard deviation is quite large, 3 times 20 is 60%. So 10 plus 60, we are at 70%. Within three standard deviation, we could have a return of 70%. But that's going to encompass 99% of the return and minus what's going to give us 10 minus 60, it's going to give us negative 50, negative 50. It's going to encompass this. So within three standard deviation, basically we're going to have all the possible returns. So this is what we are saying. Another way to look at this, let me show you another picture of this. I'm going to show it to you in another format, another pictorial. Hopefully it will make more sense because it's very important that you understand what are we doing here. Here's what we're saying. What we are saying is this. We have a portfolio and the mean is the expected return of this portfolio is 10%. The standard deviation is 20. So the standard deviation equal to 20. It means within one standard deviation, we could have a return of 30% and we could have a return of negative 10. But the majority of the return, if it's fall within this one standard deviation, if we go to standard deviation, well, we're going to add, now we're going to add 40%. So we're going to be at 50% and minus 30. Now we're going to have less here because the majority, it's going to be here 68. Now you might be saying, hold on, isn't 95 more than 68? Yes, but the 95 is a cumulative with the 68. So really the difference between 68, 95 and 68 is what's in the second degree standard deviation. Then the third degree standard deviation, it's 60. As we said, it's going to be 70% and minus 50. Let's have minus 50 here, minus 50. So within three standard deviation, the majority of return will fall here and this is 99%. Again, the majority falls the 68%. You might be saying, but isn't 95 greater? It's 95 minus 68. Isn't 99 greater than 95? It's only an additional 4% here if we're talking about the cumulative probability. So this is what we are saying here. So now let's take a look and kind of think out loud. What do we mean by far outcome from the mean? For example, a return of 15% below the mean would hardly be noteworthy if the typical volatility were high. So would we say a return less than 15% of the mean, would that be an issue in our example? And the answer is no. A return of less than 15% would still fall within this group. So if we have 10 minus 15 is minus 5. Okay, minus 5 is right here. So we are still within one standard deviation. So we would say a portfolio like this, if we are 15% below the mean, that's not a big deal. But the same outcome would be highly unusual if the standard deviation is 5. Let's see, let's change this example and assume the standard deviation is 5. Let's assume we have a portfolio. Now we're going to be working with the standard deviation of 5. So here we go. It's a 10% expected return and the standard deviation equal to 5%. Now what would things look like now? Now plus 5 is 15, minus 5 is 5. This is within one standard deviation, everything should fall within 66% of the expected return should fall within one standard deviation. If we go to standard deviation, we're going to add 5. It's going to be 20%, 20% and 0 because you know, 5 minus 5 is 0. This is within two standard deviation, two standard deviation and within three standard deviation, three standard deviation, we're going to be at 25% and negative 5%, negative 5%. Now in this scenario, if we say, if we say we are 15% below the mean, well in this scenario, 15% below the mean, we are 10%, 15% below the mean. We're at three standard deviation. So this is an, should be an unusual outcome because we are three standard deviation from the mean. So what does that mean? Although both portfolios have the same expected return, the difference between them is this one is less risky. We should not see, we should not see a negative 15%. It doesn't mean we will not see it, but for that to happen based on the portfolio volatility, it's unlikely, it's unlikely outcome here, it's normal. If we deviate 15% from the mean, we are still within one standard deviation. In this portfolio of the standard deviation as 5 and we deviate 15% from the mean, either positive or negative within three standard deviation. So it's very important we understand this concept that the standard deviation measures the volatility, the risk, and it tells us how risky. Basically, can you stomach this risk? That's the question. Would you prefer a portfolio with a standard deviation of 15% or would you prefer, sorry, of 20% or would you like a standard deviation of 5%? Okay. Now, you know, if you want more risk, you will take this portfolio. Why? Because this portfolio, you may hit 50%, you may hit 70% return, but you also you may hit negative 30 and negative 50. In this portfolio, based on the standard deviation, the maximum you will get is 25% and the worst it's going to get is negative 5%. So it's up to you. More risk, more return. That's basically the concept. And how do we measure this? Standard deviation. I just showed you when we compute the standard deviation. Now, we're going to look at how we can compute the standard deviation, giving a normal distribution. So it's very important to kind of just be comfortable with the math. We can transform any normally distributed return into a standard deviation score by first subtracting the mean return, okay, then divide by the standard deviation. Basically, what are we saying? We're looking at this. We're going to take the return minus the expected return divided by the standard deviation. That's going to give us the standard deviation score. So let's take a look at an example to see how this works. So we want to find, let's assume we want to find what is the standard deviation if we are dealing with positive one, let's assume 30%. If we are dealing with a 30% return, 30% minus the expected return, 10% minus the expected return, and that's the range in the numerator, divided by the standard deviation 0.2. So 0.2 divided by 0.2 equal to 1. So notice at 30%, we have standard deviation of 1. Now, we don't have to stick with this. We can try 33%. We can try 25% and do the same thing to find out where do we stand for the standard deviation, or we can try minus 5% to find out what's the standard deviation for minus 5%. So make sure you are comfortable with this formula. Once we have the return, once we have the expected return, once we have the standard deviation, we can find the standard deviation for that return specifically. Conversely, we can rearrange the formula and find, given the standard deviation, we can find the return. So let's do this, just make sure we are comfortable with this. Using the same example, here what we do, we'll take the expected return of the portfolio, the expected return of the portfolio is 10% 0.10 plus the 10%, and we're going to use in the same example, we're going to use 30%. So the standard deviation of 30% is one standard deviation times the standard deviation of the portfolio 0.2. So if we do this computation, 1.2, 1 times 0.2 plus 0.1, which is 0.2 times plus 0.1, which is 0.3. So we can find 30%. So basically make sure you are comfortable if you want to compute the percent, giving standard deviation expected return, or if you want to compute the standard deviation, giving the expected return and a particular return for the portfolio. So it's just want to make sure you're comfortable with those formulas. Now, what happened when the investor is a little bit concerned with risk? Because the standard deviation measured the dispersion of possible asset return. That's true, but sometimes you want to know what's the worst case scenario, like what could happen if we really have a worst case scenario. So suppose you are worried more specifically about large investment losses in a worst case scenario for your portfolio. So basically, the investor might ask, how much would I lose in a fairly extreme outcome? For example, in a 5% tile of the distribution, basically there's a 5% chance that something that could happen, there's always that chance. What is my possible losses at 5%? So simply put, what you are saying is, you want to know the 5% is a possibility, 5% something could happen. If that 5% happened, what would be, what do I expect my investment to experience? What would be my return? So in the investment language, this cutoff is called value at risk, donated by V small a and capital R to distinguish it from the variance. So it's a V, a, R value at risk, value at risk. So a loss of verse investor might desire to limit portfolio value at risk that is limit the loss corresponding to a particular threshold probability such as 5%. So you want to know at what's the possibility of 5% that happening, that worst case scenario happening? How much could I lose? You could also do it at 1%, you could do it at 5% or at 1%. So let's take a look at value at risk. So this value at risk compute the 5th percentile of a normal distribution with a mean of zero, mean of zero, it means we have the mean and a variance of one. So under those circumstances, we're looking at standard deviation at negative because it's going to be a loss, negative 1.6464485. So simply put, what is our return? Basically, what is our return at this standard deviation? Remember, it's negative 1.64485. In other words, that is 1.64485 standard deviation below the mean would be the 5th percentile of the distribution and therefore correspond to a, to a var of 5%. So if you want to see it on a graph, here's what we're looking at. We're looking at this 5%, let me clear all of this. We're looking at 1.66, so negative 1.66, so someplace here and this someplace here, that's, that encompass 95%, which is what's left is 5%. This is what we mean by this. So this is 1., negative 1.644, which is 95%. Okay, so let's see how we can do this, how we could, how we could perform this computation, but now since you saw it on a graph. So the formula to find the, the value at risk at 5%, we're going to expect, we're going to take the expected return and we're going to do plus minus, basically it's minus, minus negative 1.64485 times the standard deviation. So we are away from the mean 1.64485 times the standard deviation and this represent 95% of the occurrence. So what's left is the 5%, okay. We could also use excel sheet and I'm going to show you this, non-standard normal distribution function, which is norm inverse and I'm going to show it to you in an excel. Okay, so on a prior session we looked at this scenario and we computed the expected return of this portfolio. We computed the expected return and we computed the, we computed the, the standard deviation. If you're not sure how we come up with this, please see the, please see the prior recording. I'll also put this in the description, this example, this way you can see it. So we already computed the expected return, we already computed the standard deviation. So let's compute, first of all, before we compute the value at risk, let's see what the value at risk tells us about this portfolio. The value at risk is right here. Look, they already, they already kind of did this computation for us and at 5%, they expect losses of negative 56.5. They already told us this. They already know, based on estimate, this is an estimate. Based on the estimate, it's negative 0.65. Now assuming this is a normal portfolio, normally distributed portfolio, which is, we assume that's the case, let's compute value at risk, which is, we're going to take the, we're going to take the expected return plus minus negative 1.64485 times the standard deviation. Let's take a look at these figures and start to do the computation. Okay, what is the expected return? The expected return is 0.303074. The expected return was already computed, 0.3074 from a prior session. Now we need to compute the second part of the formula, which is taking 1.64485 times the standard deviation. Given those two together, even those two together, we get a value at risk when we sum these up of 29.62 or basically 29.62%. So based on a normally distributed portfolio, given this expected return and a 5% probability, we can experience this as negative. We can experience negative 29.62. Well, guess what? That's based on the computation, but based on our estimate, it could be up to 56.5. What does that mean? It means this portfolio, this portfolio could have a worse prediction than what we are computing in a normal distribution. It means there is a tail risk. It means, let me just show you kind of what is a tail risk is. Again, this is assumed this is a normal portfolio. We think at 1.644, we could have negative 29.62% negative return, but based on our knowledge, we think this return is way here. It's like 50, 56.5%. So what does that mean? It means this portfolio is quite risky, is quite risky. Although we think it's 0.29 based on our computation, but what they told us from the get-go at 5%, we could have a losses of 56.65. So it tells us a little bit more. As I told you, you could always compute this using Excel. So let's take a look at normal distribution using Excel. So if you want to use Excel, what you do is you put norm inverse. Let me just do this for you. So what you do is you go to the function and let's look at the function. It's easier. Norm inverse. And what they ask you to do once you put the formula, the probability, we're looking at 5% probability. The mean, which is the expected return of this portfolio, probability c6, which is 0.05. The mean of the portfolio is g7, which is 30.74. And the standard deviation is 36.67. If we compute this, and again, voila, it's the same number, 29, you know, rounding, it's fine, 29.59. Again, this is based on our computation. But when we created the scenario, we think at 5%, we could experience a loss of 56%. Risky portfolio, because if that 5% happened, our losses could be way more than 29, although we measure it at 29, but it's going to be way more than 29%. In the next session, we'd look at the topic that's similar to value at risk, which is called kurtosis and skewness. As always, I'm going to ask you to like this recording. If you like it, share it, put it in playlists. And obviously, if you are this far listening, it means you like that it benefit you. It means it might benefit other people. As always, also, I'm going to remind you to visit my website forhatlectures.com to supplement and compliment your accounting and finance courses. Good luck and study hard.