Hi, hopefully you have had a chance to look at what we talked about in the recap, and the good news is the recap and what we did last week is very data-driven. Having said that, I think the onus on you, I think I've said this about these two weeks, the pressure on you is to do a lot of the assessments as we go along, more so for than the previous parts. And the reason is what I'm doing in these two weeks is giving you the essence of risk and return. And although I'm using data and giving the intuition, I'm not going to do the execution of how to estimate betas, how to do covariances, and so on. And I tell you, you may or may not need to do this, that's so awesome for finance. But I encourage you, I encourage you to do those assessments and homeworks, because what it will do is it will make you learn statistics while you're doing finance, so it's like a two-for-one deal, right? So please, I hope you have a good grasp of the measurement of things. And we'll today, this week again, we'll focus largely using numbers. As a take off, I use numbers, real numbers. And then, we'll also go to the website and do some analysis. So there'll be numbers in real world data everywhere, but I'm not going to spend a lot of time, if any, on calculating means and variances and covariances of data, okay? So let's get started. This is the second week of risk and return. Diversification is the sole basis of everything. And I said so, we like return, we dislike risk, we diversify, therefore, we hold portfolios. And we saw what that does is it reduces the risk of 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, a bond, a house, whatever. So 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, most everybody, is an exciting choice. But therein lies the Catch-22, right? 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. And what you can do is simply get returns over the last 20 years like we did and estimate Google's volatility. And the risk will be measured by, and we know this, right? Sigma squared, GOOG, by the way, that is the symbol that GOOG has on the stock exchange. I believe, Google is called GOOG. And if I screw that up, you can always Google. Sorry, unintended pun. Okay, but you'll see that we'll use this many times, too. So I'm going to interchangeably, if I talk about risk as variance with the standard deviation, don't get too thrown off by that, because one is just the square root of the other. And conceptually, where do we square it? Because otherwise, if we added deviations from the mean, they would all be zero, add up to zero. How would you measure the risk of your portfolio? So remember, your portfolio risk was called Sigma squared P, R standard deviation was Sigma P. So notice what I've done. I put all of my resources, money, in one investment, which is, by the way, physically impossible to do these days. And 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 Google. In other words, what's going on is 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 have only one investment, okay? So 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're equal. So we now are going to step to two assets. But before we go there, I mentioned to you, it's almost impossible that you'll have your investment in one asset. And 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. So you [INAUDIBLE] about, right? You may have invested in a house. Think broadly when you think investment, don't think just a stock. So in some senses, this artificial one asset example was created to make you think about risk and return. So 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? So take our Google example. The risk of Google was a risk of your portfolio when you had only Google. What kind of risk does everything have? Two generic risks. Market, Goggle goes up or down regardless of Google's own actions or specific things, because the market is going up and down. Why do the market go up and down? If I knew the answer to that, I don't know what to say. Some people think about how markets go up and down, and I'll tell you a little story in a second, but when the time is right for it. So markets go up and down, Google goes up and down. But this is the second reason that I'm going up and down with Google. And that is Google makes mistakes, of course it does, I think everybody does. Google makes good decisions and mistakes, and they'll go up and down due to things specific to Google. It had nothing to do with the marketplace. When you're holding only Google, your portfolio variance is the same as Google variance, so standard deviations are the same, and therefore, you're going to bounce up and down due to both reasons of risk. Now lets go to two. Suppose you have two securities in your portfolio, you have one is Google and the other is Yahoo. Just 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 caps a bit, too, and you'll see one is a gorilla and the other is relatively small. And which is which, right? So suppose I say, okay, I'll have a big guy and a small guy in a portfolio. How would you measure the risk of each of the two stocks, right? So what would you do? You would have sigma, sorry. You will have sigma square Goog or sigma Goog and sigma square Yahoo and sigma Yahoo. Those are two alternative ways, they're the same, one is the square root of the other. So do you understand what I am saying? Is that we saw it in our data, that you could measure the risk of each one separately and that's called the standard deviation of variance. Now let us look at what happens next. How would you measure the average risk of the two? We know how to do it. So 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%. 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 weights, whatever you like but with one caveat. You have to worry about how much money went into Google and Yahoo. So let's do a simple average if you put 50/50 in each, okay? Very simple, so let me just write it out here. You'll just average half sigma Goog plus sigma Yahoo. I'm just mimicking what we just saw in the data earlier on, right? Fair enough? Let's now ask you the following, how would you measure the risk of your portfolio? And if you are understand, I mean, I'm throwing out questions I'm going to now, next questions are deeper. So I can't just do it, so that's where we are 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. 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. So 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? And the answer, we had just said. If it were the same, there's nothing to talk about. We just saw last week, and we revisited the whole data. The one amazing thing was that sigma p was the lowest in that particular context of all, okay? And that motion's called diversification, again using data to motivate theory, right? Okay, so let's move on and see what the heck is going on. And by the way, now comes a little bit of pain. And 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? And portfolios are what? A group of things. So imagine one person in the room, Google. How many personalities? One, think of its variance as capturing that personality, all right? 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 the room. Their personalities are important because there are two of them. However, there are two more things that will crop up, what are those? When Yahoo's moving, how is Google moving? When Google is moving how is Yahoo moving with each other? Those are called relations. So please keep that in 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. So let's move on some definitions. So some definitions I have already done is that we are going talk about sigma p. We are going to talk about covariance or 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. And what happens as a result, we get a unit rate. And this number is between minus one, zero, one. I mean, sorry, zero a b. Okay, so these are the definitions. One last definition which is very important over everything we'll do. Is we will say xa is the amount of investment in a as a fraction of your total investments. Okay, is this clear? And 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 straight forward 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. So till now everything more or less was about returns. So let me ask you about risk, let me ask you about return on the portfolio. And this is so simple. So return on a portfolio would be set two assets, Xa, return on a, plus Xb, return on b. What is Xa? Proportion in Google. Xb, 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 of return on Yahoo, what can I do? I can measure the average return on Google, right? We did it. Mean return. We can also measure the average return on. Yahoo, yes. So suppose this is 15% and suppose this is 12%. 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 1/2 of 15 which is 7 and 1/2, 1/2 of 12 is 6% so this is 13 1/2%. So the average portfolio return is just too linear. So suppose I have 500, how you will 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. So the good news about returns is that because they are no squares going one, it's linear, right? That's another way to think about why risk becomes bizarre, it's because it's square. Anyway, so that's a kind of silly way of thinking about it but useful nevertheless. So we are done with symbols, let's get on to what is the famous formula and I'm apologize there are a lot of symbols, but let's try to think about this. So what's going on? Sigma square p is what? The variance of your portfolio, right? What is Xa squared? The proportion of wealth in global. What is sigma a squared? The variance of global. What is xb squared? The weight on proportion of wealth n, Yahoo in our example. And sigma b squared is the Is the variance of Yahoo. So we are okay til 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 what, relationship or the covariance between Google and Yahoo. Why XaXb? Because they are like married to what's going on. So let me write out what the portfolio return is again so that you are. This is what RB is. It's a combined return of the two. Where XX, XB can be half and half, or they could be two thirds one third, depending on your investments. And RA RBR, the unknown returns, the change over time, right. Everybody okay? So why am I squaring excess square? Why is this squared, why is this squared, because radiance squares. Because if you don't square, what happens. You add up all the deviations from the average you get zero, right? So the key here is this is there, this is there. What would sigma a square be? Let's pause for a second. When would sigma a square b equal to sigma square p? 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. So 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. So 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. Google and Yahoo. So sigma ab = sigma b a when we measure it. So that's why the 2 comes, right? Whereas I really like Ryan, he may not like me, right? So the relationship doesn't have to be identical for human beings, and typically it's not, but in the data it is, right? That's the beauty of dealing with inanimate, non-human things like Google and Yahoo. So you have two times that, but the unique relationships are the same. In life they could, 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, yeah? What's the tragedy of sigma AB, the covariance? The tragedy of covariance was it's unit dependent and it's extent is not known and all that. So we define something called rho ab as sigma ab divided by sigma times sigma b. So 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's very important to understand and then we only build on it. Why? Because we don't owe 2 security. What do we do? We do mutual fund investments. And it makes sense, and I'll talk about that in a second. Okay so 2 X a Row A. So look at this relationship. I want you to pause and given standard divisions 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 squared Rp, the risk of your portfolio, when will this, Be such that you will not diversify at all? Think of it. So what is the condition under which more 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 S square, sigma B square. How many relationships? Two. And they are exactly the same. Okay, so 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. Okay, pause, think about this, and then tell me when would you not benefit from diversification in this example? See you in a little.