【正文】 s number squared is a big number. The more x moves, the bigger the variance is. There39。s also another variance measure, which we use in the sampleor also Var is used sometimesand this is ∑amp。sup2。. There39。s also another variance measure, which is for the sample. When we have n observations it39。s just the summation i = 1 to n of (x x bar)amp。sup2。/n. That is the sample variance. Some people will divide by n–1. I suppose I would accept either answer. I39。m just keeping it simple here. They divide by n1 to make it an unbiased estimator of the population variance。 but I39。m just going to show it in a simple way here. So you see what it isit39。s a measure of how much x deviates from the mean。 but it39。s squared. It weights big deviations a lot because the square of a big number is really big. So, that39。s the variance. So, that pletes central tendency and dispersion. We39。re going to be talking about these in finance in regards to returns becausegenerally the idea here is that we want high returns. We want a high expected value of returns, but we don39。t like variance. Expected value is good and variance is bad because that39。s risk。 that39。s uncertainty. That39。s what this whole theory is about: how to get a lot of expected return without getting a lot of risk. Another concept that39。s very basic here is covariance. Covariance is a measure of how much two variables move together. Covariance iswe39。ll call itnow we have two random variables, so I39。ll just talk about it in a sample term. It39。s the summation i = 1 to n of [(x – xbar) times (y – ybar)]/n. So x is the deviation for the isubscript, meaning we have a separate xi and yi for each observation. So we39。re talking about an experiment when you generateEach experiment generates both an x and a y observation and we know when x is high, y also tends to be high, or whether it39。s the other way around. If they tend to move together, when x is high and y is high together at the same time, then the covariance will tend to be a positive number. If when x is low, y also tends to be low, then this will be negative number and so will this, so their product is positive. A positive covariance means that the two move together. A negative covariance means that they tend to move opposite each other. If x is high relative to xbarthis is positivethen y tends to be low relative to its mean ybar and this is negative. So the product would be negative. If you get a lot of negative products, that makes the covariance negative. Then I want to move to correlation. So this is a measureit39。s a scaled covariance. We tend to use the Greek letter rho. If you were to use Excel, it would be correl or sometimes I say corr. That39。s the correlation. This number always lies between 1 and +1. It is defined as rho= [cov(xiyi)/SxSy] That39。s the correlation coefficient. That has kind of almost entered the English language in the sense that you39。ll see it quoted occasionally in newspapers. I don39。t know how much you39。re used to itWhere would you see that? They would say there is a low correlation between SAT scores and grade point averages in college, or maybe it39。s a high correlation. Does anyone know what it is? But you could estimate the corrit39。s probably positive. I bet it39。s way below one, but it has some correlation, so maybe it39。s .3. That would mean that people who have high SAT scores tend to get higher grades. If it were negativeit39。s very unlikely that it39。s negativeit couldn39。t be negative. It couldn39。t be that people who have high SAT scores tend to do poorly in college. If you quantify how much they relate, then you could look at the correlation. I want to move to regression. This is another concept that is very basic to statistics, but it has particular use in finance, so I39。ll give you a financial example. The concept of regression goes back to the mathematician Gauss, who talked about fitting a line through a scatter of points. Let39。s draw a line through a scatter of points here. I want to put down on this axis the return on the stock market and on this axis I want to put the return on one pany, let39。s say Microsoft. I39。m going to have each observation as a year. I shouldn39。t put down a name of a pany because I can39。t reproduce this diagram for Microsoft. Let39。s not say Microsoft, let39。s say Shiller, Inc. There39。s no such pany, so I can be pletely hypothetical. Let39。s put zero here because these are not gross returns these are returns, so they39。re often negative. Suppose that in a given yearand say this is minus five and this is plus five, this is minus five and this is plus fiveSuppose that in the first year in our sample, the pany Shiller, Inc. and the market both did 5%. That puts a point right there at five and five. In another year, however, the stock market lost 5% and Shiller, Inc. lost 7%. We would have a point, say, down here at five and seven. This could be 1979, this could be 1980, and we keep adding points so we have a whole scatter of points. It39。s probably upward sloping, right? Probably when the overall stock market does well so does Shiller, Inc. What Gauss did was said, let39。s fit a line through the pointthe scatter of pointsand that39。s called the regression line. He chose the line so thatthis is Gausshe chose the line to minimize the sum of squared distances of the points from the lines. So these distances are the lengths of these line segments. To get the best fitting line, you find the line that minimizes the sum of squared distances. That39。s called the regression line and the intercept is called alphathere39。s alpha. And the slope is called beta. That may be a familiar enough concept to you, but in the context of finance this is a major concept. The way I39。ve written it, the beta of Shiller, Inc. is the slope of this line. The alpha is just the intercept of this curve. We can also do this with excess returns. I will get to this later, where I have the return minus the interest rate on this axis and the market r