Diversification is the sole basis of everything. I said so, we like return, we dislike risk, we diversify, therefore we hold portfolios. We saw what that does is it reduces the risk that we confront as measured by standard deviation. Now, please pay attention to this. Suppose I wanted to show you diversification and I have one asset over there. By that, I mean a stock or bond, a house, whatever. Suppose you hold one asset security and suppose that security's name is Google and it's your portfolio. What do I mean by that? That there is just one investment in your portfolio and you put all your money into Google. At least I'm picking something that everybody would agree is an exciting choice. But therein lies the catch-22, you get excited. How would you measure the risk of Google? Remember I showed you some numbers and I didn't show you Google purposely. Go for yourself, take the time before, now, or whenever, and find out the standard deviation of Google. What you can do is simply get returns over the last 20 years like we did and estimate Google's volatility. The risk will be measured by, and we know this, Sigma square, GOOG, by the way, that is the symbol that GOOG has on the stock exchange. I believe Google is called GOOG. If I screw that up, you can always Google. But you'll see that we'll use this many times too. Interchangeably, if I talk about risk as variance versus standard deviation, don't get too thrown off by that because one is just the square root of the other. Conceptually, why do we square? Because otherwise if we added deviations from the mean, they would all add up to zero. How would you measure the risk of your portfolio? Remember, your portfolio risk was called Sigma squared p, or standard deviation was Sigma p. Notice what I've done, I put all my resources, money, in one investment, which is by the way, physically impossible to do these days. I'll say that in a second. But just imagine you've done that. What will be this? This will be equal to, depending on your measure, GOOG or standard deviation of GOOG. In other words, I want you to recognize what's going on is that the portfolio's risk is identical to Google's risk. Does that make sense? Why? Because you've only one investment. Let me just make sure I have a clean slate to work on and go on to the next point. What is the relationship between the two we just saw? They are equal. We now are going to step 2 two assets. But before we go there, I mentioned to you it's almost impossible that you will have your investment in one asset. The major reason is diversification but the other reason is whether you like it or not, you're investing in yourself. Whether you like it or not, you have investments that you don't even know. You're all about, you may have invested in the house. You and I think broadly when you think investment, don't think just that stock. In some senses, this artificial one asset example was created to make you think about risk and return. If you have one asset investment, what are you confronting? You are confronting all the risk of that asset. In other words, all your eggs are in one basket. What are the two sources of risk you're confronting for each separate investment? Take our Google example. The risk of Google was a risk of your portfolio when you had only Google. What kind of risks does everything have? Two generic risks, market. Google goes up or down regardless of Google's own actions or specific things because the market is going up and now. Why does the market go up and down? If I knew the answer to that, do you know? I don't know what to say. Some people think about how markets go up and down. I'll tell you a little story in a second, but when the time is ripe for it. Markets go up and down, Google goes up and down. But there's a second reason that I'm going up and down with Google, and that is Google makes mistakes. Of course, it does, everybody does. Google makes good decisions and mistakes and will go up and down due to things specific to Google. Had nothing to do with the marketplace. When you're holding only Google, your portfolio variance is the same as Google variance or standard deviations are the same, and therefore, you're going to bounce up and down due to both reasons of risk. Now, let's go to two. Suppose you have two securities in your portfolio and one is Google, and the other is Yahoo. If we have time, I'll do this, but I will clearly remember to do Google and Yahoo as we go along, but look at market gaps of the two, and you'll see one is a gorilla and the other is relatively small. Which is, which? You know, right? Suppose I say, "Okay, I love a big guy and a small guy in a portfolio." How would you then measure the risk of each of the two stocks? What would you do? You will have Sigma square Goog or Sigma Goog, and Sigma square Yahoo, and Sigma Yahoo. Those are two alternative ways. They are the same. One is a square root of the other. Do you understand what I'm saying? Is that we saw it in our data, that you could measure the risk and you can measure the risk of each one separately. That's called standard deviation of variance. Let us look at what happens next. How would you measure the average risk of the two? We know how to do it. Assume you have put the same amount of money in both. This is going to be important down the road, so I just wanted to highlight it. As soon as you put all your money in one thing, you don't have to worry about weights or how much, what proportion you put, it's 100 percent. But as soon as you put in two things, you have to worry about the standard deviations of the two, and the average risk of the two will be the average standard deviation of average variance, whatever you like. But with one caveat, you have to worry about how much money went into Google and Yahoo. So you'll [inaudible] and simple average if you put 50-50 in each, okay? Very simple. Let me just write it out here. If you'll just average half Sigma Goog plus Sigma Yahoo. I'm just mimicking what we just saw in the data earlier on, fair enough? Let's now ask you the following, How would you measure the risk of your portfolio? If you all understand, I'm throwing out questions I'm going to, next questions are deeper so I can't just do it. That's where we're headed. How would you measure the risk of your portfolio? The intuitive answer is actually not right. The intuitive answer would be what? Let's just think about this for a second. The intuitive answer would be, Sigma p should be equal to this, and that's not the answer. Because if that were the case, what would we have seen in the 11 securities and the S&P 500? The average of the 11 securities was much higher than the Sigma p and the reason was diversification. We'll come to that. That's one of the major things we have to show. Is the portfolio risk the same as the average of the two? The answer we've just said if it were the same, there is nothing to talk about. We just saw last week and we revisited the whole data. The one amazing thing was their Sigma p was the lowest in that particular context of all. That notion is called diversification. Again, using data to motivate theory. Let's move on and see what the heck is going on. By the way, now comes a little bit of pain. Let me go back to last week when I said diversification is the fundamental driver of definition of risk and definition of the relationship between risk and return, and not just definition, actual measurement of it. What did I say? Is that, all of us will hold portfolios. Portfolios are what? A group of things. Imagine one person in a room, Google. How many personalities? One. Think of its variance as capturing that personality. Now you have two things in your portfolio, Yahoo and Google. How many personalities? Uniqueness, two standard deviations, yes? Two people, Mr. Google and Yahoo in a room. Their personalities are important because they are two of them. However, there are two more things will crop up. What are those? When Yahoo's moving, how's Google moving? When Google is moving, how is Yahoo moving with each other? Those are called relations. Please keep that at the back of your mind and that's where we are headed and we know how to measure relationships. How? Covariances? But covariances have the tragedy of being unit dependent, so correlations are more intuitive. Let's move on. Some definitions. Some definitions I have already done is that we are going to talk about Sigma P, we are going to talk about covariance which is also called Sigma. If it's a and b, we'll call it Sigma a, b, and if it's correlation, we'll call it correlation a, b. This is called correlation, and how do we measure it? Sigma a, b divided by Sigma a, Sigma b. Not comma, sorry. Multiplication. What happens as a result, we get it unit free, and this number is between minus 1, 0_a, b, 1. These are the definitions. One last definition which is very important over everything we'll do is we will say x_a is the amount of investment in a as a fraction of your total investments. Is this clear? We are going to focus largely on risk, but remember, when you're holding a portfolio, not only are you measuring risk, what will you also measure? Return, and something is very straightforward about that, and I want to just write out another set of notes for you. One more set of definitions and you'll be all set. Till now, everything more or less was about returns, so let me ask you about risk. Let me ask you about return on a portfolio, and this is so simple. Return on a portfolio would be say two assets, x_a, return on a plus x_b, return on b. What is x_a? Proportion in Google, x_b proportion in Yahoo in our particular example. What are these? Return on Google, return on Yahoo. Now the nice thing is if I have the last 50 months or 60 months of return on Google, and 50 or 60 months return on Yahoo, what can I do? I can measure the average return on Google. We did it: mean return. We can also measure the average return on Yahoo. Suppose this is 15 percent and suppose this is 12 percent, the average return over the last 60 months. The good news is suppose this is half and suppose this is half, guess what the average return on your portfolio is? It is so easy. It's linear. It's half of 15 which is 7.5, half of 12 is 6 percent, so this is 13.5 percent. The average portfolio return is just linear. Suppose I have 500, well how will you figure out the portfolio's average return over the last 60 months as a predictor of what you think you'll get? Just the average of all the averages in the portfolio. The good news about returns is because there are no squares going on, it's linear. That's another way to think about why risk becomes bizarre, is because it's squared. Anyway. That's a silly way of thinking about it, but useful nevertheless. We're done with symbols, let's get on to what is the famous formula, and I apologize, there are a lot of symbols, but let's try to think about this. What's going on? Sigma square P is what? The variance of your portfolio. What is x_a square? The proportion of wealth in Google? What is Sigma a square? The variance of Google? What is x_b square? The weight or the proportion of wealth in Yahoo in our example, and Sigma b square is the variance of Yahoo. So we are okay till here. Remember we are okay. We know this, we know this, we know this, we know this intuitively. Now things start happening which are a little bit strange, and that is this shows up. This is the relationship or the covariance between Google and Yahoo. Why x_a x_b? Because they are like married to what's going on. Let me write out what the portfolio return is again. This is what R_p is. It's a combined return of the two, where x_a, x_b could be 1/2 and 1/2, or they could be 2/3, 1/3 depending on your investments and R_a, R_b are the unknown returns, the change over time. Everybody okay? Why am I squaring x_a squared, why is this squared because variance squares, because if you don't square, what happens? You add up all the deviations from the average, you get 0, right? The key here is this is there, this is there. What would Sigma a square b? Let's pause there for a second. Then would Sigma a square b equal to Sigma square b? When all your wealth is in a, let's assume that's Google. But you will never do that. Why? Because you're risk averse. Let's assume you pick up Yahoo. So what is Sigma b square Yahoo? The risk of Yahoo if you were holding Yahoo by itself. Those two unique characteristics, those variances have to be there because you have two securities now. But now look at what else you've added. You've added a relationship between the two. Why is it two times? Simply because, unlike human beings where my relationship with Ryan and Ryan's, relationship with me, are not identical. In data, they are, for Google and Yahoo. Sigma ab is equal to Sigma ba when we measure it. That's why the two comes. Whereas I really like Ryan, he may not like me. The relationship doesn't have to be identical for human beings and typically is not, but in the data it is. That's the beauty of dealing with inanimate nonhuman things like Google and Yahoo. You have two times that, but the unique relationships are the same, in life I'll separate the two out. How many total are two? How many unique? In this case is one because a and b are the same as b and a. Everybody, okay? What's the tragedy of Sigma ab, the covariance. The tragedy of covariance was it's unit dependent and its extent is not known and all that. We define something called Rho ab, as Sigma ab divided by Sigma a times Sigma b. What can I do? I can take this to the other side and replace Sigma ab by this. That's all I've done. Everything else is identical and please, I let you stare at this for a little while and we'll take a break after this simply because this is the one equation where it is very important to understand and then will only build on it. Why? Because we don't hold two security. What do we do? We do mutual fund investments, and it makes sense and I'll talk about that in a second. So, 2xa Rho a. Look at this relationship. I want you to pause. Given standard deviations are fixed of Google and Yahoo. Given those two numbers, what's driving this relationship up and down? Do this for yourself during the break. When will Sigma square R_p the risk of your portfolio, when will this be such that you will not diversify at all? Think of it. What is both intuitive and mathematical condition under which there is no benefit at all of putting Yahoo and Google together. Having said that, let's just one more time, see how many terms are there in my portfolio. Two personalities weighted, of course, by the Sigma a square, Sigma b square. How many relationships? Two, and they are exactly the same. We have four things going on in the portfolio, two personalities and two relationships. Same thing as what's happening in a group of two people, two personalities, and two relationships. Pause, think about this and then tell me when would you not benefit from diversification in this example? See you in a little.
