0:00

Okay. We are in the middle of about half hour,

Â 45 minute of Statistics. Actually, it's more, because we'll keep

Â going back to it, after we introduce the concepts.

Â But I'm also assuming, by the way, if you have never done Statistics before, you

Â will, would have before watching the next step, made yourself very comfortably at

Â calculating means and standard deviations which includes calculating variants.

Â Before each two relationships, because the reason I'm saying that is because I've

Â given you a note on the side which goes through numbers.

Â I shouldn't spend time in the video going through numbers doing mean and variance.

Â I believe that is taking up more valuable time, doing and talking about things that

Â are, I think, more important. Means and variances are more difficult.

Â Having said that, please do examples and the easier number crunching in your

Â assessment. Okay.

Â Now, I think I tried to convince you a little bit about why relationships matter.

Â If you are not fully convinced, believe me after you are done about portfolio theory.

Â In fact, by the end of next week, you will see that that's the only thing that

Â matters. Not just in life, relationships not just

Â matter in life. They matter in Finance.

Â No wonder I love it. Okay, so let's start with relationships.

Â There are two things you listen, there are multiple ways you listen, but two

Â fundamental things about relationships is this, how do you measure them, right?

Â So, suppose I have y over here and I have x over here.

Â As soon as I have relationships, I have two, right?

Â At least two. So, I'm going to call this y and you'll

Â see in a second why I'm calling this y and x.

Â Because, in Finance, in Statistics there is a fundamental structure and symbols

Â which are very similarly used in almost every textbook.

Â And y and x is very common. So, let's make y now.

Â 2:30

What is bushels of corn? The amount, it's a, it's a measurement

Â unit of corn. And let's call this x inches of rainfall.

Â Again, I'm doing an example which you can relate to whether really it's important to

Â Finance or not, right? Because that's the beauty of Statistics.

Â I sometimes feel like Statistics is may go slightly ahead of Finance in the love

Â hierarchy of things. I can't believe I said it.

Â But anyways, so, that's the way, okay. So now, why are you interested in this

Â relationship? You are interested in this relationship

Â simply to figure out whether rainfall has impact on the production of corn, one, and

Â what is the nature of the impact, okay? So, the reason why we are interested in

Â the relationships in a portfolio in the Finance context will become more apparent

Â once we have started doing the financial application.

Â First thing you do is calculate the average value, y bar.

Â Second thing you do here is calculate the average value, x bar.

Â Another way of saying the x bar and y bar are the values you expect to happen.

Â So, on average, how many bushels of corn are you expected to produce?

Â And on average, how many inches of rainfall are you expected to get?

Â And we can throw in numbers there if you want, you know, 120, 60, whatever.

Â I'm, I'm, I'm sure I'm just 60 inches of rainfall, maybe a lot, a little, I don't

Â have a clue, right? You're not interested in the averages.

Â Stand alone you are, but now what you are asking is the following question.

Â How are these two related? So, the very simple way of thinking about

Â is rain falling. Ask yourself, when rainfall is below the

Â average? And remember, here are the probabilities.

Â When rainfall is below the average, what the heck is corn doing?

Â Let's assume that for this particular data point, corn, it happens to be above its

Â average, right? For this data point, how would you measure

Â that deviation, this deviation? It will be y, sorry, x.

Â Xi - x bar is the amount, measurement of this.

Â 5:20

Then, you multiply that by yi - y bar. In this case, what is xi - x bar?

Â It happens to be positive, and but accidentally or maybe somehow, the yi

Â minus y bar is negative. But when you do most of these, what will

Â you tend to see? That this tends to be positive on this

Â side, right? So, you, what will you do?

Â You'll then sum all these up and you'll get what is called covariance.

Â And what it's measuring is, multiplied by the probabilities, of course.

Â What it's measuring? What is the tendency of corn to be when

Â rainfall is off its normal behavior? Corn, off it's normal behavior, too.

Â So, I took a perverse example here, when rainfall was positive, corn became little

Â negative. That could happen, right?

Â If there's excess rainfall. However, on average, what do you expect

Â this to be? This number to be greater than zero.

Â Why? Because hopefully, rainfall helps the

Â production of corn. Maybe if I took rice here and I am just

Â out of my league, I am talking about agriculture, maybe the relationship is

Â more strong, positive, right, because it needs more water.

Â But you understand whatever I measure, I measured.

Â When I deviate from the average in this, how does this deviate exactly for that

Â situation? So, point by point, point by point, that's

Â where the probabilities are about. This is the fundamental measure of

Â relationships, okay? And its called covariance.

Â How do you co-vary with something, right? So, let's ask, lets assume, every time I

Â turn, Come on, on life and I have a smile on my face, you also automatically have

Â smile on your face. So, you tend to smile.

Â We have a positive covariance going on. On the other hand, if you dislike my guts

Â for some reason and every time I smile, you kind of frown.

Â Then maybe we don't have something positive going on.

Â So, what I am saying is you can use this same phenomena and the normal is no smile

Â at all, right? So, covariance captures that relationship.

Â But there's a tragedy with covariance, and that is, two things are wrong, wrong with

Â it, and I'll just emphasize that in a second.

Â Let me just start off with a [unknown]. So, covariance has one issue.

Â One issue is, magnitude is not communicated.

Â 8:16

Let me take a corn example. So, it's called sigma y, x.

Â Covariance of y with x is this. Summation Pi (yi - y bar) (xi - x bar),

Â right? It's very important to take deviations

Â from normal behavior. That's why I said, when I smile, it's,

Â what is my normal behavior? No smile, right?

Â Okay that's, that's [unknown]. And the probabilities are in there, right?

Â Look, what will, what will happen to this? Right now, I had this in bushels, if I

Â believe, yep. And I had this rainfall in inches.

Â Could I change this number, and suppose this number was 55, whatever, can I make

Â this number larger Just, without changing anything dramatically fundamental?

Â Answer is yes. Start measuring your rainfall instead of

Â inches, in millimeters. What will happen?

Â This number will become big. Because there are a lot more millimeters

Â in an inch, right? So, it does, it doesn't have magnitude.

Â It doesn't tell me anything. The only thing that's good is, it's

Â telling sign is okay. But magnitude is not reflective of

Â strength or weakness. The second tragedy with it is this, it is

Â unit-dependent. So, what are the units of covariance?

Â The units of the covariance are the units of both the things being measured,

Â alright? [laugh] In, when we do return analysis,

Â both the returns are measured in percentage.

Â So, it's not a big deal, but I wanted that's why to show you, why Statistics is

Â so awesome and why you have to deal with things which are more difficult than in

Â finance? So, bushel inches, what it is suppose to

Â mean? I mean, it shouldn't be.

Â An ideal measure of relationship shouldn't have units.

Â So, how can I compare bushel inches with, say, the productivity of people and

Â whether they have had a school education or not.

Â That units are totally different, correlation, it, the measurement, the

Â relationship should be comparable across different phenomena.

Â So, here's the tragedy of covariance. Though it's trying to measure the right

Â thing, its magnitude doesn't mean much and it's unit-dependent.

Â So, what did we do? What does the, the statisticians do?

Â A very little about this to talk about that's important to us, but this one is

Â extremely important. So, we took covariance of y and x, which

Â units was what? Bushels, four inches.

Â And magnitude was what? Didn't affect much, but sign was okay.

Â In other words, the, that it's positive or negative relationship is being reflected

Â because you're taking deviations from the mean and then multiplying.

Â So, what did we do? We came up with a measure called

Â correlation. And what is correlation?

Â Correlation simply takes sigma yx. Remember, what are the units?

Â 12:34

And a second thing happens, which is little bit, I am not going to go into a

Â correlation which is written like this, yx is between -one and +one.

Â So, you see the awesomeness of correlation.

Â It's taken covariance, retained its value of positive negative sign being important,

Â got rid of its issues of what, the units by dividing by the standard deviations of

Â the two, and creating a number that can be compared across different phenomena, okay?

Â So, let's take a break again and I'll come back with one last statistical and an

Â extremely important statistical concept called regression.

Â Which is needed in Finance and almost in any other discipline.

Â Again, a way of capturing relationships, but an important one for us.

Â See you.

Â