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Statistics need to understand us and our attitude toward risk. So, very quickly

Â repeating something, we like more return but we dislike risk. We therefore, will

Â not put all our eggs in one basket. We will hold portfolios, we will diversify.

Â We'll keep coming back to this. This simple fact is underlying all the

Â statistics we'll do and, everything that follows. And, by the way, this whole

Â setup. This simple risk return relationship dictated by this phenomena,

Â our attitude towards risk, underlies all the profound work done by finance people

Â over the last 50 years. And, at least two Nobel prizes have to do with this, what we

Â are going to talk about for the rest of this week and the following week. Now, for

Â some statistics, and I'm going to use a pen and paper to write a lot. And by that,

Â I mean, an electronic pen and electronic paper. So, let me start off with the

Â following. I'm going to draw a graph. And I'm going to draw a distribution, okay?

Â And I'm going to call this point, something, it will be a central point of

Â central tendency. Then, I will try to characterize this behavior, departures

Â from here. And then, I will try to do something else called how do you measure

Â things and relationship between things? That's the goal. But starting off, there's

Â a distribution. Can somebody tell me, what does this distribution look like? I'm,

Â I've been pretty cool about the drawing. It's called a normal distribution. And we

Â are going to largely stick with normal distributions, because a lot of things

Â start of looking norm, normal, very strange behavior but when you, when you

Â look at distribution with a lot of numbers, lot of phenomena, they tend to

Â converge to normal. That's why this is called normal. But the most important

Â thing about this is this. On this axis, I've got probabilities. And on this axis,

Â I'll say, I have got the phenomena, and I'll call the phenomena, y. Now, very

Â quickly, what could this be about? And if it's okay with you, I do not want to teach

Â Statistics in a dry fashion. You have chapters from books in F inance that I'm

Â asking you to look at. And you also have books on Statistics that I'm sure you're

Â aware of, or can Google. What I wanted to tell you again, is the essence of it. And

Â today's a little bit dry. But I want you practice whatever we are doing, okay? I

Â keep repeating that. So what is y? Think of y as anything. And I'm going to call it

Â yi. So, think of y as a distribution of heights on all the people taking this

Â class. Do you agree that it will be distributed all over the place, right?

Â Hopefully, nobody has a negative height. So, we are not going in that direction.

Â So, I'm going to take height as an example. So, you have a distribution, not

Â everybody's exactly five feet tall, right? If it were, what would this height be?

Â Remember, this is probability. This height would be what? One. And there would be no

Â tails, no distribution. All of this would collapse into this one height. And

Â everybody's height, if it would be exactly five feet, we wouldn't need to worry about

Â statistics. It turns out, real world is not like that. Distributions are around

Â some normal behavior and look normal. We are going to assume that largely for

Â finance, okay? So, this is basically deflecting the fact that I do not know

Â something for sure. Going back to our example of a government bond giving me a

Â return of three%, then the probability is one, simply because I know that even

Â though real world could be bad or good, these possibilities have been knocked out,

Â 4:46

okay? So, that's the notion of a distribution. I'm now going to talk about

Â few characteristics of this distribution which may be very familiar for people who

Â are, have a statistical background but not familiar for all this, okay? So, let's

Â stick with our problem. And let's suppose, I know, the distribution of possible

Â heights. I want to calculate what is the normal. Imagine, if in my head, I had to

Â keep the heights of all people taking all classes in the university, it would be, it

Â would be mind boggling. So, what do we do? A distribution characterizes all

Â possibilities, but t hen I ask myself, what is the average chance? What is the

Â average height, sorry, right? And this, we call many times, expectation. And if the

Â distribution is normal, we can only worry about mean. It turns out the beauty of a

Â normal distribution is if I divide this, if I divide it over, if I carry this over,

Â it'll look like one perfect line, right? It's very symmetric. So, the mean is right

Â in the middle, and the min will also be equal to mode and median. I am getting a

Â little geeky now. These are two other ways of measuring what is called central

Â tendency. So, why am I interested in what's happening on average? Because

Â that's what most people think about the future. Hey, well, on average what will be

Â my cash flow next year? 100. But will it be exactly 100? No. It could be 90, it

Â could be 110. I hate the 90 but I like the 110. That's where the hypocrisy comes in,

Â okay? So, min, median and mode. I am sticking with heights. Let's figure out

Â how do we calculate that min, okay? And I've given you examples with returns and

Â so on, so, because we are doing finance, but I'm just getting a little excited

Â here. So, the way you'll figure out why, min, and you'll call it y bar. And

Â theoretically, it's also called expectation of y, will be equal to this.

Â What will you do? You'll take the values of y, all values of yi, all the values.

Â And you'll multiply them by Pi, which is what? The probability. So , you'll take

Â each Y, multiply it by its probability and sum, over how many? All n possible. So, if

Â i goes from one through n. N is the sample size okay? So, so, the, so what, what is

Â it saying? It's saying, multiply the probability by the chance, by the height.

Â And if the probability of being five feet seven, is one%, that's how you get the

Â first data point and so on. I want to just emphasize this way of doing things,

Â because I think people forget, that the usual way of, saying it, and I'm going to

Â write it up here, is this, summation yi / n. That's the usual way. You'll see it

Â done even in Excel. So, when you do min or average, i t's called in Excel, you tell

Â them what the ys are. They're already in a spreadsheet going from A1 through A100, if

Â there are 100 observations. You just sum them and divide by n. You're making an

Â assumption when you do that. And the assumption is, what is HPi? One . That

Â means that the chance of each height entered in your Excel spreadsheet, and by

Â the way, there's a note that tells you how to do that in Excel. It's so simple. You

Â just say, Excel says, do average. And then, we have the time towards the end, we

Â may do that. But I'm not inclined to do that right now. I just want you to

Â understand. It's very straightforward. Now, the assumption will tend on a normal

Â average that you calculate, right? So, what is the average rainfall this year?

Â What will they do on the website, on a weather website? They'll add up all the

Â rainfall for each day and divide by 365. They're assuming that the likelihood of

Â each thing is equal. And that's an important assumption. If you have a large

Â data set, it usually is an okay assumption, right? Because, it doesn't

Â matter what value of one / n is that much. I want to emphasize this so that you

Â understand. So, you've calculated. Okay, what do I expect will happen? However,

Â that's not the only story. I also have to worry about uncertainty or variance,

Â right? In this case, worrying about the variance of height is, doesn't seem that

Â traumatic but let's just stick with it, just as an example. Worrying about

Â variance of returns is very traumatic, right? Especially, if they're going in a

Â negative direction. So, this is what you have. So, what have I calculated? Let's

Â assume, I've already calculated y-bar. Remember, probabilities are here, Ps. Now,

Â what am I, I look at this and I say, ask you the following. Okay, are you sure? And

Â suppose the average height, in all the classes I've ever taught is five foot,

Â eight inches. And I, somebody asked me, but Gautam, are you sure that's the

Â height? I said, obviously I am not sure, right? The only way I'd be sure is this

Â height was how much? Exactly one. Then you wouldn't have a distribution, right? So,

Â you, are you sure? And the answer is, obviously, you're not. Some people are

Â here, some people are here. So, what do you do? You do this. You take a yi, each

Â yi. And suppose that's this one. And you subtract y-bar from it. Why? Because y-bar

Â is the normal, the center of gravity of this behavior, the normal behavior. So,

Â you got a deviation from it. In this, case it's positive, in this case, it's

Â negative, right? Now, you have to multiply that by the chance of this happening,

Â right? This data point happening, okay? But you have to do another thing, you have

Â to square it. And then, you sum across all possibilities. And that's called variance.

Â And the symbol used is sigma i^2. Quick question, think about it for a second. Why

Â do I not sum these? Why do I square them? And the reason is, I just gave you a hint.

Â The min is the center of gravity. So, what will happen? The positives and the

Â negatives will cancel each other out. And what will you get every time? Zero. So,

Â you, there's no point saying zero variance, because zero variance is only

Â true for what? Something is, you're a 100% sure about. Everybody's five feet, eight

Â inches tall or I'm going to get my money for sure. So, the variance is a measure of

Â uncertainty. However, look at its units. The units of average are what? Inches. The

Â unit of variance or uncertainty about your estimate or average height is squared. So,

Â what do we do? To make it the same unit, we do square root of sigma square i, which

Â we call sigma i, which we call standard deviation. By the way, one thing very

Â important to note about normal distributions is, just like the min is the

Â average, is also the median, is also the mode. Similarly, the only measure of

Â uncertainty is standard deviation. If you do more strange distributions, you'll get

Â things like skewness, kurtosis. I don't want to get into those, because that's not

Â the purpose of this class. High level possibilities of including skewness and

Â I'll be doing Finance. It depends o n your a ssumption about the distribution. But

Â for now, let's stick with standard deviation, okay? I'm going to keep going

Â and I'm going to first emphasize now why will we not stop here. Think about it.

Â Normal distribution, you know, the expected value, you know the uncertainty,

Â why? We are done with it. We know the measure of risk, right? Because we know

Â variance will be zero in the cash flows of each instrument if you're holding a

Â Treasury bill. But if you're holding a corporate bond, what will the variance be?

Â Positive, right? So, why worry about anything more? Why not just simply state

Â with not knowing the world and characterizing expectation by min, or

Â average, and uncertainty by variance? Well, there's a reason for it. And I'm

Â going to just give you a flavor of the reason before I do the Statistics because

Â we're going to get into the details of this concept, big time, next week when we

Â talk about measurement of risk. Why variance of a security versus portfolio?

Â Turns out, because we are risk averse, we are averse to risk. We hold portfolios. In

Â fact, I don't know anybody in the world who has money to invest who doesn't hold

Â portfolios. It will be silly to put all your eggs in one basket assuming you're

Â risk averse, and human behavior is risk averse. If there's enough data to show it,

Â and I'll show you more as we go along, including today. Because we hold port,

Â portfolios, portfolios are a collection of things. They are not single things. So,

Â not, so, imagine a world in which each one of us was holding just one thing. Either

Â Apple, Google, GE, and so on, and that was our behavior. That's not what the world is

Â like. Then, variances and mins would be enough. Turns out, I know ahead of time,

Â in fact, we knew it in the cave, when she was raised to live hunting outside for the

Â first time, guess what the guy said? Hey, don't put all your eggs in one basket.

Â That means, diversify, try to do different things so that you have different ways of

Â collecting food, so that you survive, right? So, risk aversion implies hol ding

Â po rtfolios. Portfolios means a collection of things, not single things. And that

Â means. Relations. We have to figure out relationships and how to measure them. Let

Â me ask you this simple example. I know I use very bizarre examples. Again, not to

Â do with Finance. Suppose, a human being could survive in, by themselves, just by

Â themselves, each single person, nothing to do with anybody else. Well, that's one

Â world. But what happens? We believe that, especially in business schools, we teach

Â group work. So, imagine, if you will, the only person in a, in a group, you're only

Â one thing, right? Now, let me ask you. If you have a collection of things, it's

Â called team. So, think about it. I could look at your behavior alone if you were

Â the only thing determining everything, right? You operate individually. But if

Â you operate in groups, and lets take a group of two, what have I done? How many

Â personalities? Two personalities. But what else have I introduced? I have introduced

Â relationships. How many? Me and Ryan, Ryan and me are a team doing this. What is

Â important now? Not just his personality and my personality. What's important is my

Â relationship with him, and his relationship with me. So, as soon as

Â collective, things in a portfolio or collections matter, we've got to be able

Â to measure relationships. And that's what, after a break, we'll try to do, using

Â Statistics. So please take a break, and we'll come back to how do you measure

Â