We are back. I have encouraged you to stare at this equation because it's very important, even though there are only 2 assets. And to answer the following, first understand what's going on. You've gone from investment in one Google alone into investment into Google and Yahoo. But I've asked you a specific 2 sets of things. One, a realization that we talked about is that as soon as you go from one asset in your portfolio to two, you create a dynamic but there are variances of both in there or some deviations. Plus you create dual relationships. Makes a lot of sense, right? Okay. So the question I left with you which is more specific, is when do you think you will not benefit from diversification? Or, another way to say that is, when will sigma p be equal to the average (sigma i's)? How many sigma is there? Two. When will that happen? In other words, there's no point diversifying. Tell me the intuition and then we'll be done. The math, I'll let you do, the intuition is very straightforward. If Rho a b = 1.0, what does that mean? That if Yahoo and Google move perfectly together, all the time. Yahoo goes up by 1%, Google goes up by 1%. Yahoo goes down by 1%, Google goes down by 1%. 2%, 2%. Sign and magnitude are all going together. That will make sense. Why divide up your money between two things which have a different name but are essentially the same? How likely is this to happen? Low, right? There's no such thing as two perfectly same companies, okay? So put in 1.0 here. Substitute 1.0 here and try to show that this is true regardless of magnitude. So I'm now pushing you to do this, right. You should be able to show it. It's simple algebra and I'm not going to use numbers because I want you to run with this and do it. But the second question is the following, before we move on. If you were to find a perfect correlation between two investments, Google and Yahoo, what would be the main source of risk in both? Would it be the market? Or would it be specific? MARK is standing for market. I hope you recognize that given a broad two definitions of risk, one which affects everything and one which is specific. Which one will it be? It has to be this. Because market is common to both, and relationships happen because of common things. Whereas if you were totally different personalities, i.e., Google is in a totally different industry, has no market effect, and so is Yahoo, it would be a different story. But that's unlikely to happen. Doesn't this make sense? It makes a lot of sense, and that's why I think I keep saying this, it's the most fascinating subject. You don't need data, almost to convince yourself of what's going on, right. Okay, so let me show you some graphs similar to the regressions. Remember I showed you some dots in regression? And look what's on the various axes. If we look at any specific graph, and I'm going to purposely look at the middle one, because it's the most transparent, and please go back and forth. What do I have? Ra, rb. Think of ra and rb as the two securities in your portfolio, or the relationship between two. And it's good to visually show two relationships, right? Because if you have three things going on it becomes a little bit difficult. So that's why I'm spending a lot of time on the relationship between two things. And this you will see repeatedly happening. And as I make the formula more complex. Okay, so let's stare at this and my question to you is, I'm going to have some fun with you. So my question to you is the following. What is on the top left graph? What am I showing on the top left graph? All the dots are lying on the straight line. Yes, this is called a perfect positive correlation. And within our context when will it happen? When the things common to both are the only thing driving, and that we call the market risk. Okay? So this is perfect positive. What is on the right bottom? Same thing, but now the relationship is what? Negative. And how can you tell that? First of all I've said it's negative, perfect. Perfect is an art if dots are on the straight line, but now the straight line is shaped like this instead of shaped like that. So now you've seen the two extremes. What's in the middle? What's going on here? This has dots all over the place, and I cannot see any relationship between a and b. It is measured by a relationship of zero. Another way to think about it is I can draw any line through this, it'll seem to make sense. Okay, good news is you can estimate this, you can go into excel as we have put up the note, and say what? Correlation equals correlation. And then show, you can also do this, equals correlation and then show array 1, array 2. That means, show me the data on a, show me the data on b. Where is it? And a will be in the a column, b will be in the b column, or whatever. And depending on the data if you're 60 you'll say row 1 through 60. If you're 50 you'll say row 1 to 50. The only thing to worry about is they should be matched with each other. So you can't have Google's return in 1975 match with Yahoo's return in 1985, that's a silly thing to do. Okay? Because if you do that, you're likely to show up here. Okay, now, what's happening here? A negative relationship. And what is happening here? A positive relationship. But not perfect. So let me ask you. In reality, which is the most likely one of these graphs? On average, if you picked two stocks, which is the most likely graph you'll see? Chances are this is almost impossible, and that's the beauty of diversification. Why? Because two things cannot be identical. In every respect. They have to have some unique reasons for moving. Similarly, probably the right low part is unlikely to happen. I would actually say that this is also unlikely to happen. Simply because almost everything that we see Is effected by the common market in a positive way, perhaps. But you could see scenarios in that this is possible. So what I'm saying is perfect relationships I'm ruling out. I'm just saying probabilistically finding this is much lower than right top. And the reason for the right top is things are not perfectly related but they would have a common thing, and that common thing is called? The market. And they have a positive relationship with the market typically, most companies. So this gives you a sense of what's going on in the real world. I'm going to ask you one question which has nothing to do with finance. Where do you think love is? Think about the person you love the most, and you are A, they are B, he or she. Which graph signifies love? And it is also, I believe, the graph on the right top. Because if you’re looking for a perfect relationship, there’s no such thing. And in fact it’s the wrong thing to look for. You are looking for yourself in the other person. You know, that’s probably not the right thing to approach love. It’s noisy, there's tension, but hopefully the relationship is positively inclined. Okay? So bless you, let's take a break. I hope you enjoyed these graphs. We'll come back and move on to three assets. With the following, this is a short piece. I wanted you to look at data visually and get a sense of where we are headed. And we'll come back. While you are taking a break, think about, where is Yahoo and Google likely to be? Where are they likely to be on this graph and closer to which of these graphs? You have some hints and you can intuitively think. That's the awesomeness of this. So see you. Quickly, take a quick break, unless you're doing some exercises and so on, which is fine. But I expect you to think about this as we go along.