Statistics needs to understand us and our attitude towards risk. Very quickly, repeating something, we like more return but we dislike risk. We therefore will not put all our eggs in one basket. We'll hold portfolios, we'll diversify, we'll keep coming back to this. This simple fact is underlying all the statistics field and everything that follows. By the way, this whole setup, the simple risk return relationship dictated by this phenomena, our attitude towards risk underlies all the profound work done by finance people over the last 50 years. At least two Nobel prizes have to do with this, what we are going to talk about for the rest of this week and the following week. Now for some statistics and I'm going to use a pen and paper to write a lot and by that I mean an electronic pen and electronic paper. Let me start off with the following. I'm going to draw a graph and I'm going to draw the distribution. I'm going to call this point something, it'll be a central point of central tendency. Then I'll try to characterize this behavior, departures from here and then I'll try to do something else called, how do you measure things and relationships between things. That's the goal. But starting off, there's a distribution. Can somebody tell me what does this distribution look like? I've been pretty cool about the drawing. It's called a normal distribution and we are going to largely stick with normal distributions because a lot of things start off looking non-normal, very strange behavior. But when you look at distribution with a lot of numbers, a lot of phenomena, they tend to converge to normal. That's why this is called normal. But the most important thing about this is this. On this axis, I've got probabilities and on this axis I'll say I have got the phenomena and I'll call the phenomena y. Now, very quickly, what could this be about, and if it's okay with you, I do not want to teach statistics in a dry fashion. You have chapters from books in finance that I'm asking you to look at. You also have books on statistics that I'm sure you're aware of or can Google. What I want to tell you again is the essence of it and today's a little bit dry but I want you to practice whatever we are doing. I keep repeating that. What is y? Think of y as anything and I'm going to call it yi. Think of y as a distribution of heights on all the people taking this class. Do you agree that it will be distributed all over the place. Hopefully nobody has a negative height, so we're not going in that direction. I'm going to take height as an example. You have a distribution, not everybody is exactly five feet tall. If it were, what would this height be? Remember this is probability. This height would be what? One, and there would be no tails, no distribution. All of this would collapse into this one height and everybody's height if it were exactly five feet, we wouldn't need to worry about statistics. Turns out, real-world is not like that. Distributions are around some normal behavior and look normal. We're going to assume that largely for finance. This is basically reflecting the fact that I do not know something for sure. Going back to our example of a government bond giving me a return of 3 percent, then the probability is one simply because I know that even though real world could be bad or good, these possibilities have been knocked out. That's the notion of a distribution. I'm now going to talk about a few characteristics of this distribution which may be very familiar for people who have a statistical background but not familiar for others. Let's stick with our problem and let's suppose I know the distribution of possible heights. I want to calculate what is the normal. Imagine if in my head I had to keep the heights of all people taking all classes in the university. It will be mind boggling. So what do we do? A distribution characterizes all possibilities. But then I ask myself, what is the average height? This we call many times expectation and if the distribution is normal, we can only worry about mean. Turns out the beauty of a normal distribution is if I divide it over, if I carry this over, it'll look like one perfect line. It's very symmetric. The mean is right in the middle, and the mean will also be equal to mode and median. I'm getting a little geeky now. These are two other ways of measuring what is called central tendency. Why am I interested in what's happening on average? Because that's what most people think about the future. Hey, we're on average, what will be my cash flow next year? Hundred. But will it be exactly 100? No, it could be 90, it will be 110. I hate the 90, but I like the 110. That's where the hypocrisy comes in. So mean, median, mode, and sticking with heights, let's figure out how do you calculate that mean. I've given you examples with returns and so on because we're doing finance, but I'm just getting a little excited here. The way you'll figure out y mean, and you'll call it y-bar, and theoretically it's also called expectation of y, will be equal to this. What will you do? You will take the values of y, all values of y_i, all the values, and you multiply them by p_i, which is what? The probability. So you will take each y, multiply it by its probability and sum. Over how many? All n possible. So if i goes from 1 through n, n is the sample size. What is it saying? It's saying, multiply the probability by the height, and if the probability of being 5 feet 7 is 1 percent, that's how you get the first data point and so on. I want to just emphasize this way of doing things because I think people forget that the usual way of saying it, and I'm going to write it up here, is this, summation y_i divided by n. That's the usual way you'll see it done, even in Excel. When you do mean or average in Excel, you tell them what the y's are, they are already in your spreadsheet going from A1 through A100, if there are 100 observations, you just sum them and divide by n. You're making an assumption when you do that. The assumption is, what is each p_i? 1 over n? That means that the chance of each height entered in your Excel spreadsheet, and by the way, there's a note that tells you how to do that in Excel. It's so simple, Excel says, do average, and when we have the time towards the end, we may do that. But I'm not inclined to do that right now. I just want you to understand, it's very straightforward. Now the assumption built-in on a normal average that you calculate, so what is the average rainfall this year? What will they do on the website, on a weather website? They'll add up all the rainfall for each day and divide by 365. They're assuming that the likelihood of each thing is equal. That's an important assumption. If you have a large dataset, it usually is an okay assumption, because it doesn't matter what value of 1 over n is that much. I want to emphasize this so that you understand. So you calculated, okay, what do I expect will happen? However, that's not the only story. I also have to worry about uncertainty or variance. In this case, worrying about the variance of height doesn't seem that traumatic, but let's just stick with it just as an example. Worrying about variance of returns is very traumatic, especially if they're going in the negative direction. This is what you have, so what have I calculated? Let's assume I've already calculated y-bar. Remember probabilities are here, p's. I look at this and I ask you the following, okay, are you sure? Suppose the average height in all the classes I've ever taught is 5 foot, 8 inches. Somebody asked me, but Gotham are you sure that's the height? Answer is obviously I'm not sure. The only way I'd be sure is if this height was how much? Exactly 1, then you wouldn't have a distribution. So are you sure? The answer is obviously not. Some people are here, some people are here, so what do you do? You do this, you take each y_i, and suppose that's this one, and you subtract y-bar from it. Why? Because y-bar is the normal, the center of gravity of this behavior, the normal behavior. So you've got a deviation from it. In this case it's positive, in this case it's negative. Now, you have to multiply that by the chance of this data-point happening. But you have to do another thing. You have to square it, and then you sum across all possibilities, and that's called variance. The symbol used is sigma i squared. Quick question. Think about it for a second. Why do I not sum these? Why do I square them? The reason is I just gave you a hint, the mean is the center of gravity. So what will happen? The positives and the negatives will cancel each other out. What will you get every time? Zero. There's no point to saying zero variance because zero variance is only true for what? Something you're 100 percent sure about everybody's five feet eight inches tall, or I'm going to get my money for sure. So the variance is a measure of uncertainty. However, look at its units. The units of average are what? Inches. The unit of variance or uncertainty about your estimate or average height is square. What do we do? To make it the same unit, we do square root of Sigma square i, which we call Sigma i, which we call standard deviation. By the way, one thing very important to note about normal distributions is just like the mean is the average is also the median, is also the mode. Similarly, the only measure of uncertainty is standard deviation. If you do more strange distributions, you'll get things like skewness, kurtosis. I don't want to get into those because that's not the purpose of this class. High level in possibilities of including skewness and all we do in finance, it depends on your assumption about the distribution. But for now, let's stick with standard deviation. I'm going to keep going and I'm going to first emphasize now, why will we not stop here. Think about it. Normal distribution, you know the expected value, you know the uncertainty, why? We're done with it. We know the measure of risk. Because we know variance will be zero in the cash-flows of each instrument if you're holding a treasury bill. But if you're holding a corporate bond, what will the variance be? Positive. Why worry about anything more? Why not just simply stick with not knowing the world and characterizing expectation by mean or average and uncertainty by variance? Well, there's a reason for it and I'm going to just give you a flavor of the reason before I do the statistics because we are going to get into the details of this concept, big time next week when we talk about measurement of risk. Why variance of security versus portfolio turns out because we are risk-averse, we are averse to risk, we hold portfolios. In fact, I don't know anybody in the world who has money to invest, who doesn't hold portfolios. It would be silly to put all your eggs in one basket assuming you're risk-averse and human behavior is risk-averse. There's enough data to show it, and I'll show you more as we go along, including today. Because we hold portfolios, portfolios are a collection of things. They are not single things. Imagine a world in which each one of us was holding just one thing, either Apple, Google, GE, and so on, and that was our behavior. That's not what the world is like. Then variances and means would be enough. Turns out, I know ahead of time. In fact, we knew it in the cave when she was ready to leave hunting outside for the first time, guess what the guy said. "Hey, don't put all your eggs in one basket." That means diversify. Try to do different things so that you have different ways of collecting food so that you survive. Risk aversion implies holding portfolios. Portfolios mean a collection of things, not single things, and that means relations. We have to figure out relationships and how to measure them. Let me ask you this simple example. I know I use a very bizarre examples. Again, not to do with finance. Suppose a human being could survive by themselves, just by themselves. Each single person, nothing to do with anybody else. Well, that's one world. But what happens? We believe that especially in business schools, we teach group work. Imagine if you were the only person in a group, you're only one thing. Now let me ask you if you have a collection of things is called team. Think about it. I could look at your behavior alone if you were the only thing determining everything. You operate individually. But if you operate in groups and let's take a group of two, what have I done? How many personalities? Two personalities. But what else have I introduced? I've introduced relationships. How many? Me and Ryan are a team doing this. What is important now? Not just his personality and my personality, what's important is my relationship with him and his relationship with me. As soon as collective things in a portfolio or collections matter. We got to be able to measure Relationships. That's what after a break, we try to do using statistics. Please take a break and we'll come back to how do you measure relationships.