First, we'll recall the magic formulas from Part I: M = { N Σxy - Σx Σy } / { N Σx2 - ( Σx )2 } K = { Σx2 Σy - Σx Σxy } / { N Σx2 - ( Σx )2 } where we are fitting a straight line, y = Mx + K, to a collection of points (xn, yn) and Σx = x1 + x2 + ... + xN and Σxy = x1y1 + x2y2 + ... + xNyN etc. etc.

Recall that M and K were chosen so as to minimize the Mean Squared Error:    E₁ = (1/N)Σ{y_n - (M x_n + K) }²

Note, too, that if we divide both numerator and denominator by N², in the expression for M, we get:
    M = { (1/N)Σxy - {(1/N)Σx} {(1/N)Σy} }/SD²(x)

where SD(x) is the Standard Deviation of the x_n.

Further, if the Standard Deviation for the y_n is SD(y), then consider the ratio:

r = M SD(x)/SD(y)	= M SD2(x) / {SD(x) SD(y)}
	= {(1/N)Σxy - {(1/N)Σx} {(1/N)Σy} } /{ SD(x)SD(y) }
	= { Mean(xy) - Mean(x) Mean(y) } /{ SD(x)SD(y) }

>Mamma mia!
Well, maybe it looks better if we write the Average of something with an overbar, like so:     = (1/N)Σxy
Note that = (1/N) ( x₁y₁+x₂y₂+...+x_Ny_N )

Forging ahead, we get:

r	= { - } /{ SD(x)SD(y) }
	= /{ SD(x)SD(y) }

>Want my opinion? It don't look any better!
Pay attention. Notice that, if the deviations from their means are identical, then the numerator is just SD² and so is the denominator
so the ratio r = +1. Also, if the deviations are the negative, one of the other, then ...

>One of the other?
If, when x is higher than its mean, then y is smaller than its mean ... by the same amount ... in this case the numerator is the negative of the denominator
and the ratio r = -1.

In fact, r is called the Pearson product-moment correlation coefficient, after Karl Pearson, the guy who first coined the phrase Standard Deviation.

It measures how correlated the y_n are, to the x_n. It varies from -1 to +1 and

r = +1 means perfect (linear) correlation r = 0 means no correlation r = -1 means perfect inverse correlation

>So?
So one normally uses the square of r and calls it ...
>R-squared, right?
Right!
>Is there some simple formula for ...?
Sure, we stare at the expression for r, fiddle ... then we come up with:

>That's simple?

We might also point out that the numerator is the square of the covariance of variables x and y.

>Which numerator?
This guy is the Covariance:    

If {x} and {y} are the same set of numbers, then the covariance is just the Variance of the set. That is, it's the Standard Deviation, squared.

One last thing. If the set {x} is the set of S&P 500 returns and the set {y} is a set of returns for some stock, we can compare the stock to "the market" ... sorta like R² ... but now it has a special name:

which, as we see, can be written as:

Covariance(x,y) / Variance(x)     or     {SD(y)/SD(x)} * r

Remember what r is?
>zzzZZZ
Well, R-squared = r², so we have that:

Beta²/R-squared = {SD(y)/SD(x)}²

In any case, because beta involves the ratio of Standard Deviations (or Volatilities), for a large beta one is tempted to say that the stock has a volatility larger than the S&P 500 ... but the r correlation may be small. For example, if this Pearson correlation coefficient is negative, it means the stock price, though perhaps volatile, tends to move in a direction opposite to the S&P 500. Interesting, eh? But, the most interesting point is this:

Beta is the slope of the regression line

>zzzZZZ
Here are some examples where, in each case, Beta is the slope of the "best-fit" regression line (shown in red):

>Finally! Some pictures!
In the middle chart, the y-values change at roughly twice the x-value rate (note that beta is roughly 2) and, in the rightmost chart ...
>The y-value changes are the negative of the x-value changes.
Roughly. Here's another, where the axes are labelled in percentages, so S&P500 returns run from -15% to +11%
and General Electric returns run from -13% to +19%:

>And beta is 1.15, right?
Right.

>But the points can be very widely scattered - far from the regression line - and still have a beta of 1.0, right?
Right. The points may be close to the regression line ... or far. Here, R-squared plays a role.

And y'all looky here In this case we're plotting something against itself! Of course, the match is perfect. >And the Covariance number? That's now the Variance of the data. Take its square root, about 0.047 (or 4.7%) and you've got the Standard Deviation (or Volatility) of the data (which could be stock returns). >Volatility is the same as risk, right? Wrong! Haven't you learned anything about risk?

for Part 3