Hi. Welcome back. Hopefully, you've had time to think about two assets in a portfolio. What are the characteristics? Just quickly, you have to worry about the proportion of wealth put in both xa, xb. Very important. You have to worry about the individual risks measured by standard deviations or variances. But most importantly, some new animal has started appearing. That's the relationship between two things: Google and Yahoo. Remember, whenever you measure relationships, relationships will pick up something common, not something different. You see the awesomeness of finance to define risks into broad categories: market and specific. Let's see, revisit our two-asset choices and then move on to three and more. Was combining Google and Yahoo a good choice? I don't want you to do numbers. If you have the numbers, great, great. If you can figure out the standard deviation of Google and Yahoo and the relationship between the two, just substitute them in that formula. Take that opportunity to try to see what's going on. I'm asking you intuitively though, do you think was combining Google and Yahoo a good choice? Of course, your first question should be, ''Hey, bozo, what do you mean? Relative to what?'' Well, Mr. Bozo says the following, ''Just use your head, bozo, and try to think about it.'' The thing I'm asking is quite straightforward obviously, but kind of hidden is, could you have made a better choice given that you're investing? I think probably yes. The reason is, Google and Yahoo are roughly in the same space. Yahoo is much smaller, but let me say that Yahoo is probably competing with Google in the same space. What's likely to be the correlation? Pretty high positive. Are you likely to benefit from diversification much? Probably not. Would Google and Boeing be a better choice? Again, do the analysis. This is your week of running with all the examples you can. It's my week of showing you the beauty of what's going on. I'm not going to use too many examples, I'm going to use real data in real contexts. But for now, and I've already started real data, try to do this. But intuitively, what do you think? It's a better choice? Chances are yes. They're very different industries, they're doing very different things. Therefore, chances are they will have a lot of things specific to each other, which is the first sign of diversification. Common things? No. Specific? Yes. The risk of the portfolio is affected by how many factors? I'm going to start writing on this. Is it a good choice? I don't know. Certainly better. Intuitively do the numbers if you're there. The risk of the portfolio is affected by how many factors? Portfolio. It's gone from one to two assets, factors have gone to four. Why? Let's just wrap up. The four factors are: a, b, 2, a, b. There are four factors: one standard deviation of a, standard deviation of b, two standard deviations or two variances, but two relationships. Why are these different than the average risk of the two assets held in isolation? If I had only one, it would be this risk, the second this risk. What's the average of the two? Weights average. Very simple. Why are the average risk of the two held in isolation different? Because of this. Or if you like to think about it differently, because of Sigma a, b. Many people prefer to think about correlations rather than covariances. I think you know the relationship between the two. One is standardized and easy, intuitive, the other is not. I love covariances as much as I love correlations. Because which comes first? Correlation doesn't have a hope because covariance comes first, tells you the sign. It suffers from units and magnitude problems, so you scale it by what? How do you go from here to here? One more time. Just scale this by Sigma a, Sigma b. We have a very good sense, hopefully, of two assets. Let's keep moving and go on to three. Suppose you have three securities in your portfolio; Google and you dropped Yahoo!, nothing against Yahoo!, just from a diversification perspective. Google is in the internet domain and to be honest with you, I sometimes wonder what it does, because it's everywhere. But obviously, it's not the way you used to think in the past. It's not a physical commodity like Boeing. What does Boeing do? Man, huge physical thing, planes. Google? Huge, but I don't know what. That's what's fascinating and troublesome at the same time. Merck, well, it's a very different animal. What does Merck do? Merck produces medicines for people. Is it the same business as flying or being all over the internet? No. I'm picking three very different securities. What's happened? I've consciously started thinking about diversification, but I'm still small. Three, what did I say? You need to diversify it's more important, but you want to be large too. How large is a quick question? How would you measure the risk of each of the three stocks? Again, something for you to do and put in the table where they were not. Remember the table we started off this morning and left off last week? That table, try to substitute stuff as you calculate it. Okay? Go you may find it on the web. The standard deviations too. Let me assure you you can if you look hard enough, but try to calculate too. Turns out we'll see Yahoo! Finance gives you historical data. It may not be measured as well as say, for example, Morningstar, but it's free and you can do calculations, you can do covariances, correlations, and so on. How do you measure the risk of each? Sigma G, let me call it sigma Google, sigma Boeing. I don't remember, it's Merck, three separate entities. That's easy. How would you measure the risk of your portfolio? Now, this is a tough one. How many do I have? How many securities do I have now? Three and still, sorry, I jumped ahead. I don't want to show you the answer I want to talk about it before we go there. It's going to look nasty and nastier than what the two one was. Not a sum of clearly much more nasty than one asset. One asset is very simple standard deviation of Google or Boeing or Merck. Now we're combining the three. What are the first thing you have to think about when you have it? Not only that I've chosen Google, Boeing, and Merck, but what do I have to think about right away? I have to think about xa, xb, xc. What are these? These are the proportion of my investment going into the three. Look at the importance of this. Suppose I'm diversifying slowly but this is 0 and this is close to 0. Have I diversified? I haven't. The weights are very important. Having three securities superficially, in your portfolio, this is close to 0 and suppose this is 90 percent of your weight. You haven't really diversified. The proportion of wealth you are putting in each is very important. It's not the case that you put equal every time. I'm just giving you a sense of what's going on. How would you measure the risk of your portfolio? Think intuitively, how many, and we'll revisit this issue over another. What will matter? These three clearly will matter. But what will matter too? Sigma ab, which is what? Relationship between a and b and turns out in the data, it's also the relationship between b and a. The measurement of the two are likely the same thing within a certain set of assumptions we have made. What else? Sigma ac and sigma ca. Finally what? Sigma bc, sigma cb. I've shown you all the elements. The weights are extremely important. The personalities called standard deviations are very important but what is also important is the relationships. You'll see a pattern in a second. I'm going to put up these generic equations every time I do equations there will be a list of equations for risk and return like I did for time value of money will be on the website. But just stare at this. I know it's a little bit intense, there are lot of terms going on. But let's start off and let's just walk through this a little bit slowly. Is this any surprise? Remember the squares are simply because variances are squared, no. When would I have only this as my risk? When x_a is one, all my wealth in one. This is the second personality, second variance, third variance. I told you these will appear. But now how many relationships are there? Two, between a and b. They are the same because sigma ab is equal to sigma ba. Two between a and c, and two between b and c. What have I done going from equation one to the next? The first term is the same, the second term is the same, the third term is the same. Actually it turns out all terms are the same except starting with these covariances, I'm replacing covariance by correlation. Remember again, what is row ab? Correlation is equal to sigma ab over sigma a times sigma b, and I'm using this to substitute. It's simply because people prefer to look at rows rather than covariances. Some do, I actually am so used to both I don't particularly care in this context. When you think about it, the important thing is, the intuition we derive. We start off by the intuition, we end with the intuition. How many unique variances? We'll take a break again after this because after this specific, we'll jump to a lot of assets and then take a break because that's a good time to get a sense of what's been happening. How many unique variances. Since we've done this, I'm going to go a little bit faster, but I'll do it with you, and this is a good thing to guess. Three. Don't forget weights. Because if one of them weight is close to zero, it's two so it matters that you're investing. How many relationships? Actually, six. I'll explain in a second. How many unique? Three. How many total factors affecting the portfolio? By that I mean, how many terms going on? Nine. This step is just clarification, so let me ask you the following, three unique variances, what are they? Sigma a squared, sigma b squared, sigma c squared, or the standard deviations? How many relationships? Why did I say six? Because think like this, ab, ba, two, ac, ca, four, bc, cb, six. In a room with three people, how many relationships? Just do this, call them a, b, c, you will have six relationships. Turns out in the data in between, under our assumptions, what happens is there are three unique. Why? Because sigma ab is equal to sigma ba, these are the first two. Then sigma ac is equal to sigma ca and sigma bc equals cb. Therefore the six and the three. But this part, if it confuses you, just ignore it for the time being. Everybody is okay with 3 plus 6 is 9? How did I get nine? Remember this, nine is three squared. It's very important to understand how the pattern is emerging. Why three squares, three people in a thing squared is nine. Three of them are what? Variances, unique? Six, are relationships. Let me go one step further before taking a break and show you the most fascinating thing ever. Suppose you have 500 assets. Which portfolio has 500 assets? S&P 500, can you believe one day I walked into class and said, "How many things are in S&P 500?" Rightly enough the class thought I was an idiot, which wasn't a surprise to me, but anyway. 500. How many total factors? Remember, in what? You should ask, in what? In the variance. Total factors, 500 squared. It's mind boggling, how many. 250,000, there should be four zeros, and 5 times 5 is 25. Two hundred and fifty thousand things happening in your portfolio. Isn't it mind boggling? Why did I pick 500? Because there's simply 500 heads. But why did I just jump from three to five? Because it would take two years to finish this class if I went from three to four to five. But the other reason I did is because portfolios already exists, so why create their own? And that's something I want to talk about for a second. You buy mutual funds which already exist. Vanguard is one of the companies which has taken portfolio theory literally by saying that if you buy existing portfolios, you're capturing the whole market. How much you put in the market and then in the risk-free asset, as it turns out, will capture pretty much anything you can do. Finally, why buy individual securities and put them in a portfolio? Because you would incur huge amounts of transaction costs. How many unique variances? How many guys in the room and girls? How many unique personalities? You should know the answer to that? It's linear. Five hundred. Yes. You have googled, you have moogled , you've doogled. All of them in a room, 500, how many personalities? Five hundred. You don't even need to do the math. But remember, weights are very important. Let's assume they're roughly equally weighted, what is the weight going to be? One over 500. In other words, you're just buying a portfolio. Let's assume it's equally weighted, it may or may not be. It's equally weighted is that you don't want have to worry about how much do I put in this asset and and all that. You see, in some sense now, in large portfolio, the weight is not that important. What's more important? Diversification. Therefore you spread, so don't buy 500 in technology, buy 500 across. How many relationships? This is going to blow your mind, you should know the answer. Two hundred and forty nine thousand, five hundred relationships going on in your portfolio. Why? How did I get that? Very easy. Total relationships is squared 250, subtract the unique, what am I left with? You can actually do this math separately, directly, but why do it when you know the answers in a simple fashion? But what will those be? One to two, two, one, one to three, three, two and all of those. Just think through this. But what's the bottom line? How many unique relationships? That's not the bottom line issue. This unique relationship is roughly half of these. Why? Because they come in pairs. The bottom line of what we have learnt right now before we take a break is, very quickly bottom line, and I'll repeat this after the break. Think about it. I'll quickly go through this, take a break. Variance of an asset determines risk faced by you only if you hold one or very few. Variance of any asset is unimportant when held in large diversified portfolio. Specific risks gets canceled out. An assets contribution to the portfolio's risk is determined largely by its relationships with the other assets. All relationships between people or assets is due to common rather than specific things. Let's take a break now. We'll pick up with this slide. I'll go through this very seriously and slowly, and we will think through each one to come up with the bottom line. Let's take a break. See you soon. Bye.