 Hello and welcome to this session. This is Professor Farhad in which we would look at covariance and correlation coefficient as they apply in a portfolio of two risky assets. These topics are covered on the CPA as well as the CFA exam, 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 1800 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 might benefit other people, share the wealth, connect with me on Instagram. On my website, farhadlectures.com, you will find additional resources to supplement this course as well as your other courses. I strongly suggest you check out my website, especially if you are studying for your CPA, CFA or CMA exam. So let's take a look at a portfolio of two risky assets. Now, the question is why did I choose or why do we choose two risky asset to illustrate this portfolio? Well, because it's relatively easy to analyze. You only have two assets or two securities and they illustrate the principles. It doesn't matter in consideration that apply the portfolio of many. So whether you have two, five or 50, the concept is the same. So we're going to be assuming two assets will be a bond fund and a stock fund. So think about it, a bond and a stock, but we're going to be using a fund. Once we understand the portfolio of two risky assets, we will reintroduce the choice in the next session of the risk-free assets. So we'll add one more component to this portfolio. Structing an efficient portfolio of many risky securities is a straightforward extension of this stripped-down asset allocation exercise. So simply put, we're going to take this two risky asset portfolio and project to other efficient portfolio. So to optimally construct the portfolio of risky assets, we need to understand how the uncertainties of the asset returns interact. How do they interact with each other? Okay, so a key to this interaction of the portfolio risk is the extent to which the return of the two assets vary either in tandem or in opposition. What does that mean? It means when you are since we're on selecting two risky portfolio and you said we're going to have stocks and bonds. Now, why did we say stock and bonds? Because from history, when stocks go up, bonds go down. Or when stocks go down, bonds go up. So there's kind of an inverse relationship or they work. It doesn't have to be inverse, but they don't work the same way. They work in opposition. Now, how much that opposition is or that inverse relationship we're going to look at it shortly. But the point is you don't want two securities that work together. For example, if you buy PepsiCo and you buy Coke, guess what? To a great degree, PepsiCo and Coke work together. When people are consuming soda product, they either buy PepsiCo or Coke. So if you have a portfolio with both PepsiCo and Coke, well, those two stocks, they should work the same way. So maybe you want to have PepsiCo and have something else than maybe a software company, maybe Apple computers. Now, the consumption of PepsiCo and Apple product may not have a relationship. So people could consume more PepsiCo product, could consume less of Apple, or they could consume more of Apple product and less of PepsiCo. So they don't work the same way. And that's what you want. You want to have portfolios. They don't work in the same direction. And we're going to explain this little bit further using numbers. But that's the basic idea. So portfolio risk depend on the covariance and we're going to learn about this term between the returns of the asset and the portfolio. So we're going to look at their covariance. How do they vary among each other? And we will illustrate this concept using a simple scenario analysis. And this simple scenario analysis, as long as you understand it, we're going to generate some rules from it and apply it to any other portfolio. So the portfolio we're going to be using will be using this data. So it's very important to understand what you are giving and how we are computing these numbers. So I'm going to show you only a computation of one sample that you should be able to do the rest. Here what we have, if we have a stock fund and we have a bond fund, those are the two assets. And we have scenarios. We have severe recession, mild recession, normal growth and boom. And we have the probability of those recession occurring. And remember, your probability should add up to 100%. This should be all a review for you because we already did this. So the stock fund in a severe recession, we expect negative 37% return. Well, if we take 0.5 times negative 37, it will give us an expected return under this scenario of negative 1.9. And we'll do the same thing for the mild recession, normal growth and boom. And when we add them all up, the expected return of the stock portfolio alone is 10%. We'll do the same thing for the bond portfolio. In a severe recession, in a severe recession, which is a 5% chance, we could earn negative 9%. We'll take 0.05% times negative 9%. We'll give us negative 0.45%. Then we'll do the same thing for the other scenarios and the expected return of the portfolio alone of bonds is 5%. And hopefully this makes sense. The stock portfolio will have, usually the stock portfolio will have a higher risk. Therefore, we have a higher return than the bond portfolio. So this is something you should be familiar with. Let's compute the variants and the standard deviation of these two investments. Again, this should be a review. So how do we compute the standard deviation and the variance for the portfolio? Basically, what we do is we first compute the range. So how do we compute the range? Remember, the stock portfolio has an expected return of 10%. Under an expected return of 10%, if it's a severe scenario, the expected return is negative 37. So what's the range? What's the range? The range is we go from, let me go back, we go from 10 to negative 37. The range is negative 47. Notice it's negative 47. And under a mild scenario, we have 10% and the expected return is negative 11. Well, the difference between the range from the expected is negative 21. So this is negative 21. And this is how we compute it for the normal and the boom. And this is how we compute the deviation or the range for the bond. Same thing. Remember, the bond has an expected return of 5%. So you want to go from 5 to negative 9. From 5 to negative 9 is negative 14. Now, what we do then, and this is again a review, if you're not sure how I'm doing this, go to my previous recording or go look into the playlist for the variance and the standard deviation. To find the variance, you will square negative 47. So to find this number, you'll take 47 times 47, which is 47 square. Then you multiply this by the probability 0.05 for the severe recession and you will come up with 110.45. And you will do the same thing for the other scenarios. Then you compute everything and the variance for the stock portfolio is 347. Remember, we don't care about the variance. We want to compute the standard deviation. You'll take the variance and you compute the standard deviation and it is 18.63. You will do the same thing for the bond. You compute the variance. Then you compute the standard deviation. The standard deviation for the bond is 8.27. And I hope this makes sense. The stocks are riskier because they have a higher standard deviation. And bond is less risky, bond less risk relative to stocks. And this should make sense. And this is represented showing in the standard deviation. So this makes sense. So remember, the expected return here is 10%. The expected return here is 5%. You expect less return. You should expect less risk. If you expect more return, you should expect more risk. Hopefully, we know the basic concept. So let's form a portfolio. And here what we're going to do, we're going to look at a portfolio. We're going to mix, rather than have a stock alone, rather than have a bonds alone, we're going to have a portfolio 40%, 40% in stocks and 60% in bond. So simply put, we're going to take 0.04. This is our weight in stocks multiplied by the return plus this is for stocks. We're going to take the weight and bonds and the weight and bonds is 60% multiplied by the return of the bond for each scenario. And remember, we're going to multiply 0.4%. This is the weight multiplied by the return and a severe recession and a severe recession for the stock. We're going to get negative 37. This is multiplied by negative 0.37. And for a severe recession for a bond, we're going to get negative 9. We're going to get negative 9. So what we want to find out first is this rate of return. When we mix things up 40% stocks, 60% bonds. Let me get my calculator and just show you how we compute this because it's very important that you know where the numbers are coming from. So we're going to take 0.4, the weight, the weight of the portfolio, 0.4 times 0.37. It's negative 37. So that's negative 0.148 plus. We're going to take the weight of the stocks 0.6 times 0.09. It's 9%. That's 0.09. And that's going to give us 0.054. That's also negative because that's a negative return. When we add the two negatives together, we're going to get negative 20.2. So the expected return under a severe recession is negative 20.2. We'll do the same thing for the mild recession. We'll do the same thing for the normal growth. We'll do the same thing for the boom. Now, when we take these returns, multiply them by the probability, we're going to get the expected return for each scenario than the expected return of the whole portfolio. So the expected return of the whole portfolio, 40% stocks, 60% bond is 7%. So remember, the stocks had an expected rate of 10. The bonds had an expected rate of 5. When we combine them 40% to 60% together, we're going to give us an expected return of 7. It's less than 10, but more than 5, 7%. Now, what we're going to find out is the variance and the standard deviation for the portfolio. Remember, the expected return now is 7. Now, we compute the variance. We're going to go from 7 to negative 20.2. So remember, the expected return is 7. And negative 0.2 is one of the scenarios under the severe recession. Therefore, the variance, the range is negative 27.2. And we'll do the same thing for the mild recession, normal growth and boom. Then we're going to take those deviation from the mean, and we're going to square them. So we're going to take negative 0.27 times 2 squared, and that's going to give us 739. We're going to take 2.4 squared, 3.4 squared. And we're going to multiply them by the probability 5%, 25% by these probabilities. Then we add them all up, and we're going to come up with the variance. Again, the variance, we don't use the variance. We use the standard deviation. We're going to take the variance and square it, and we're going to come up to the standard deviation. Now, what does that tell us? Well, it tells us the standard deviation of 6.65 is lower than the previous standard deviation. Remember, the standard deviation was 8.27 for the bond and 18.63. When we build a portfolio 60 to 40, the standard deviation is lower. That's good. Here we see the benefit of diversification. It means we lowered our risk. Yes, we did lower our return a little bit if we invested 100% of stocks, but we also lowered our risk. So let's just kind of learn what we have on the screen so we could move on. Again, this should be computing the variance and the standard deviation should be a review. So the lower risk portfolio is due to the inverse relationship between the performance of the bonds and the stocks. So here what we're starting to find out, it seems they don't work in tandem. They don't work together. They work inversely. And we actually, we can clearly see that. Now, in a mild recession, stocks fair poorly, but this offset by a large positive in the fund. In a mild recession, the stocks don't do well, but the bond will do well. So that's good. Conversely, in a boom scenario, when you have a good time, bond prices fall, stock prices do very well. So notice here, this is the benefit of diversification in a stock that has different type of securities that are inversely related. So portfolio risk is reduced because variation in return of the two assets are generally upsetting. And this is the benefit of diversification when the two securities didn't work in the same way. Now, we want to know how to measure that relationship. Okay, now we know that somehow they benefit when we combine them, the risk goes down. But how can we measure this, this relationship? Well, we're going to look at something called covariance and correlation, a correlation coefficient. So how can, how can, how one can measure the tendency of the return on two assets to vary in tandem or in opposition? We use two statistical measures. One is covariance and one is the correlation coefficient. So the covariance, look, look, it's coming from the word variance, okay? But this is covariance. The covariance is computed similar to the variance. How do we compute the variance? The variance, we take the range, which is the difference, the mean to the expected return and we square it. Here, what we're going to do instead of multiplying the difference of asset return from its expected value, we multiply it by the deviation of other assets returned from its expectation. And we'll see how, in a moment, the sine and magnitude of this product are determined whether the deviation from the mean moved together and whether they are small or large. And by doing so, we're going to find out how far, how much is the covariance? What's the magnitude of this multiplication? So let's take a look at the example here. So here's how we compute the covariance and the correlation coefficient. Here's what we do. First, we're going to find the deviation, which is what we already did, for the stock fund as well as for the bond fund. Remember how we did for the stock fund? We said the expected return is plus 10% for the stocks. And under a severe recession, we are at the expected return is negative 37. To go from 10% to the negative 37, we have the range of 47 or negative 47. Same thing with the bond. We said the expected return is five under a severe scenario. They're going to get negative nine. Five to negative nine is negative 14. So these are the ranges and we'll do the same thing for the other scenarios. Now, we're going to take the ranges and multiply them. We're going to multiply negative 47 times negative 14. It's going to give us positive 685. We're going to do the same thing here. But notice here, we have negative and positive. It's going to give us a negative number. It's good when we have a negative product because it means they are working in the opposite direction. A normal growth, they're positive, boom, one positive, one negative. Then what we do is we take those, multiply them by the probability, 0.05, 0.25, 0.4, and 0.3. Then we're going to get to these figures. We add up all these figures and we'll come up with the covariance of negative 47. Now, the covariance, that's fine, but we really want something other than the covariance. It's like the variance. So what we do, we want to find the correlation coefficient and we're going to explain what the correlation coefficient is. It's like the standard deviation. It makes more sense when you interpret the covariance in the correlation coefficient context. What we do is we'll take the covariance and we'll divide it by the standard deviation of the stocks times the standard deviation of the bonds or whatever standard deviation funds we have. And the correlation coefficient is negative 0.49. And this will make more sense. We'll discuss this in a moment because we need to learn this. But first, you need to learn how to compute it. So in column E, we multiply the stock fund deviation from its mean by the bond fund deviation. We talked about this. The product will be positive if both are positive. If both are positive and it will be negative, if it will be positive also of both are negative too. It will be negative of one positive and one negative. This is basically, we talked about this. And if we look at row four, again, the stock fund returns false short of its expected value by 21% while the bond exceeds its expected value by 10%. Again, what's happening here, these two products are working in the opposite direction. When one goes up, the other one goes down. So let's take a look at the covariance computation first from a statistical perspective. The covariance is the probability of weighted average of the product. It's called covariance measured by the average tendency of the asset return to vary in tandem. And this is the kind of the statistical formula. And basically, each particular scenario in this equation is labeled as I. So this is the severe recession for the stocks, the return of the stocks minus expected return. And we sum them all up and we'll do the same thing for demand. In our scenario, we had 10, when our example, we had 10 scenarios. Okay. And the negative value for the covariance indicate that two assets on average vary inversely. And this is really what you want. Now let's talk about the correlation coefficient, because this is important. Like variance, the unit of covariance is a percent square, which is why it's difficult to interpret, because it's a percent square, that's not what we want. We want a correlation coefficient. For instance, how do we interpret negative 74.8? Okay, we really can do that. The best way to find out whether it's a strong or not strong relationship is to interpret this as a correlation coefficient. So how do we compute the correlation coefficient? It's the covariance. We're going to take the covariance, which is negative 74.8. And we're going to divide this by the product of the standard deviation of the return of each fund. So we're going to divide this by the standard deviation of stocks times the standard deviation of bonds. Okay. And it's represented by the Greek letter rho. So our correlation coefficient for our example is negative 0.49. Now what does negative 0.49 mean? Well, the correlation is a pure number. It can range from negative one to one. So notice the maximum is one, you know, the maximum is one, and the lowest one is negative one. How do we interpret this? If we have a correlation of negative one, it indicates that one asset return varies perfectly inversely with the other. That's perfect. If you have a correlation of negative one, that's perfect. It means if you know what's going to happen to one stock, you know what's going to happen to the other stock. It's the exact opposite. So here, if you could have a portfolio with a correlation of negative one, it means get those stocks and put them together because now diversification is the best one. One goes down, the other one goes up. You can predict 100% the variability of one asset return if you know the return of the other asset. So this is like perfect, perfect inverse correlation. If you have a positive one, it would indicate perfect positive correlation. Here, diversification is not useful. It's like have a PepsiCo and Coke. And here, I'm simplifying so you understand. So PepsiCo and Coke, you would assume that those two stocks work together. So if you have a portfolio of PepsiCo and Coke, you're not really diversified because they work together. You want a portfolio that when one stock does bad, the other one, the other investment does well. So they work the opposite or not in tandem of each other. So the diversification here is not affected. You don't gain anything from diversification. Here, if you compute the portfolio standard deviation and the mean, they are a weighted average of the component of the security because they work the same way. There's no benefit from having one stock over the other. So the choice amount, the choice amount portfolio depends on how risk aversion you are because the diversification is out of the question. Are you comfortable with this portfolio? A correlation of zero means the two assets are unrelated. So one goes south, one goes north. There's no relationship between them. There's no correlation between them. Historically, the correlation coefficient between stocks and bonds is 0.2, historically based on historical data. 0.2, which is good. It means they vary positively, but not in the same direction. That's fine. So the correlation coefficient of point row of 0.49 confirms the tendency of the return of the stock and the bond varies inversely. So here in our example, the stock fund and the bond fund that we have varies inversely. And this is the best. When one goes up, the other one goes down. So what did we learn so far about tourist asset portfolio? Here's the three rules that we learned. Rule number one, the rate of return on a portfolio is the weighted average return on the component securities with the portfolio proportion as weight. Simply put, the return of the portfolio, you will take the weight times the return, the weight of the stock times the return. And we did this mathematically. The expected return of the portfolio is the weighted average of the expected return of the component, which is, again, the expected return of the portfolio is the weight of the bonds times the expected return of the bond plus the weight of the stock. You remember, we said 60%, 40%, and we had different scenarios for the return, then we add them up, then we find the expected return. And our example was 7%, if you remember this. So far, so good. So far, what we learned is the rule one and rule two says that the portfolio actual return and its mean are linear function of the component security. Simply put, if you have more bonds, well, you're going to have the return closer to the bond value. If you have more stocks, it's based on the weight. You're going to have more return closer to the stock. It's based on the weight. This is not true when we compute the portfolio variance. The portfolio variance does not care about the weight. The portfolio variance on a risky portfolio is a little bit different. So this is the formula for it. This is the formula for it. And what's going to affect the variance of the portfolio is not the weight. So the weight does not really matter. It's the nature of the stock, not the weight of the stock. I'm sorry, not the nature of the stock, the nature of the relationship between the two stocks. So the weight is irrelevant. What we, the variance tells us how do they interact? And based on the interaction, if the interaction is negative, that's a good covariate, good in a sense that we want it. If we want to build a portfolio, we don't want them to work together. We want the stocks to work differently. So when some do well, the others don't, and the opposite is true. So we can assess the benefit from the diversification from rules two and three to compare the risk and the expected return of a better diversified scenario benchmark. So let's take a look at, illustrate this concept again. Let's assume we have the expected return of the bond is 5%. The standard deviation of the bond is 8%. We have stocks, the expected return of the stock is 10%. The standard deviation is 19. And the correlation coefficient is 0.2. This is what we have. This is the, this is what we have. So if we have a stock that's 100% and bond, if we have stock that 100% and bond easy, we expect 5% return with a standard deviation of 8, 100% of bonds. Let's assume we want to invest 40% in stocks, 60% in bond. Here's what's going to happen. The expected return is 7%. No, the 7% is greater than investing only in bonds. Now, that's good, which represent a 2% increase in bond. Now, using rule number three, let's compute the standard deviation. When we compute the standard deviation of this 40 and 60%, we find that the standard deviation is 9.76, which is good. It means it's way lower than 19 and just a little bit higher than 8. But let's, let's do a quick computation to show you why rule number three doesn't go with the weight. If we compute the weighted average component of the standard deviation, it will give us 12.4. This is not how we compute it. But what I'm saying is the standard deviation doesn't care about the weight. That's what we're trying to say. Unless we're going to see later, yes, I mean, unless the correlation is perfect, is plus one, then you know, then it's going to be, this is how you do it. But this is not how it works. So because we combine those two stocks together, we combine them. We were able to lower the risk to 9.76, not 12.4. Because if the risk is based on the weight, the standard deviation should be 12.4. So the difference is 2.64. And this reflects the benefit of diversification because we diversify. Okay, so this benefit is cost free in a sense that diversification would allow us to experience full contribution of the stock, stocks higher expected return while keeping the portfolio below the average of the component of the standard deviation. And this is the beauty of diversification. You can lower your risk. You could lower your risk and still have a decent return. So your return is not based on your weight. The benefit of the diversification will give you a decent return with lower risk. And that's what you want. You always want. Remember, when I started this, and we're going to see this again, you have risk and you have return. And here's what's going to happen. The more risk you take, the more expected return you take. But if somehow, if somehow you can, rather than this is being a linear somehow, if you can take less risk, so basically what you really want, and we're going to see this in a moment, you want to take less risk and higher return. You want something like this taken less risk, but the return goes up very quickly. Like it's going to be like little bit banded. That's what you want. You want less risk and higher return. So here's what we can say. What analysts can and must do is show the investor and an entire investment opportunity. So what we did is we said 40, 60. This is the scenario that we kind of looked at. 40, 60. Let me highlight it for you. This is the scenario. So here we have stocks. Here we have bonds. The expected return is seven. The standard deviation is 9.77. This is the portfolio that we have. And initially we said, let's assume everything is in bonds. When assume everything is in bonds, we had an expected return of five and standard deviation of eight. But here what we did, we're going to build an investment portfolio set, which is set of all attainable combination of risk and return offering by the portfolio formed using the available asset in different proportions. Here we're looking at different proportion. Now we can play with this little bit more. So can you find out what is the lowest risk? Well, I'm going to look here and find out what's the lowest standard deviation. Which portfolio gives you, if you're looking for the portfolio with the lowest standard deviation, you would need to look here. So let's look. The lowest standard deviation is right here, 7.80. 7.80, right here, let me highlight this line, gives you the lowest standard deviation, which would give you a return of 5.46. It means if you invest your money, 90.8, almost let's say 90 and 10% in bonds, 10% in stocks, which is 90.8 and 0.092. Let's say 90 and 10, no, let's say not 90, because we have 90 here. If you invest 90.8, let's see, if you invest 90.8 and 9.2, 9.2, you will get the lowest risk, the lowest risk of the portfolio. So if you are risk averse, 100% risk averse, you would tell your advisor, I want to invest in this portfolio, put 90.8 and bonds, and 9.2 in stocks, which would give you the lowest. With this proportion, the portfolio standard deviation will be 7.8, and portfolio expected return 5.46. Is this your preferred portfolio? Well, it all depends on your risk. That depends on your risk aversion. So I'm going to take this data, and we're going to show you a graph. And notice the graph, because of the diversification, it bends. And we're going to look at this later on. So here's what we mean. This is the bond portfolio. If you are asked, a bond portfolio versus a portfolio here, which one will you choose? Notice here, they both have the same or close standard deviation, same risk. But this one here, let's call this 1A, this one here has a higher return. So you would always take, if you are choosing between portfolio A and portfolio B, you always take portfolio A. Why? Because look, this is the risk, and this is the return. You are taking the same risk, but getting more return. Wouldn't you take this? I'm taking the same risk, but I can earn 1% more. Of course, I will take this. In this relationship, there's a name for it. It's called the mean variance, mean variance. And look, I mean, it sounds very complicated, but all what it means, mean means return, and variance means risk. So you're comparing the risk to the return. I like to call it risk return relationship, but it's a mean variance. And here's what's going to happen. When we have all this, when we graph this investment opportunity set, again, it would look, it would look a little bit bended. Now, would you choose this stock or would you choose this stock here? Now, obviously, those are basically the same. What do I mean by this? Here, you might earn 8.5, but you're taking 15% risk. Here, you are earning a little bit more, but you are taking more risk. Now, it all depends on your risk tolerance. Would you prefer to take more risk or less risk? For example, you would never choose portfolio Z over this portfolio. Why? Because look at portfolio Z. Portfolio Z, compared to the stock portfolio, the risk is you are taking a standard deviation of 21%, and your earning is a little bit above 8. Here, the stock portfolio, your risk is 19, and your return is closer to 10. Obviously, I would never choose portfolio Z because this will not make any sense. Why will I take more risk with less return? So basically any portfolio in this area, you really don't want it. You really want this portfolio here, here, and here. Why? These portfolio, they're going to give you a higher return with less risk. For example, if you're going to compare this ABC portfolio to this portfolio, you will take ABC. Why? Because they're giving you the same return, but this portfolio ABC has a 14-point standard deviation. So hopefully, this will start to make sense. And we're going to carry on with this in the next session. But notice here, it bends a little. And the reason it bends because of the benefit of diversification. And here, at this point here, we have minimum variance portfolio. And this is this portfolio right here. Minimum variance portfolio is this portfolio right here where the return is 5.46. And the standard deviation is 7.8. The minimum variance portfolio, then it will start to go up. Notice it will start to go up. Now in the next session, we're going to do, we're going to look at the optimal risky portfolio with risk-free assets. So we're going to take everything that we learned. The first thing I'm going to start this session next session, I'll say, okay, make sure you view the prior session, because in addition to the two risky portfolio or many risky portfolios, I'm going to be adding the risk-free asset. As always, I'm going to ask you to please like this recording if you like it, share it, put it in playlist. If it benefited you, it means it might benefit other people. Connect with me on Instagram. Don't forget to visit my website for additional resources for this course as well as your other courses. Stay safe, study hard, and good luck.