R-squared Stuff ... a continuation of best Line Fit

Okay, here's what we read, again and again about the infamous notorious very useful R-squared (sometimes called the Coefficient of Determination).

A statistical measure that represents the percentage of a fund's or security's movements that are explained by movements in a benchmark index.
For fixed-income securities the benchmark is the T-bill, and for equities the benchmark is the S&P 500.
R-squared values range from 0 to 100. An R-squared of 100 means that all movements of a security are completely explained by movements in the index.

You've forgotten already?
Remember the Pearson Correlation r?
      r = COVAR[x,y] / SD[x]SD[y] = {M[xy] - M[x] M[y]}/{SD[x]SD[y]} = (1/n) Σ (x - M[x]) (y - M[y]) / {SD[x]SD[y]}
where M[x] stands for the mean of the set of returns denoted by x, SD[x] their Standard Deviation and COVAR[x,y] the Covariance between two sets of returns.

>And Σ means you add 'em all up, eh?
Yes ... and R-squared = r2.
Now here's where that statement, above, comes from:
  • We plot the returns for one asset, namely x1, x2, ... versus the returns for the second asset: y1, y2, ...
  • We plot the "best straight line fit" to the points (x1, y1), (x2, y2), ...
  • We calculate the deviations of the ys from that regression line and the root-mean-square of those deviations (which we'll call the Error):
    Then   Error2 = SD2[y] (1 - r2) = SD2[y] (1 - R-squared)
See that R-squared pop up?
It sort of measures how closely the points lie on a straight line.
If R-squared = 1, for example, then all the points lie on that line. When R-squared is close to 1, we assume that the behaviour of the y-values is, at least in part, determined by the x-values.
If the x-values are Market Returns, say the S&P500 as a benchmark, we're tempted to say (when R-squared is close to 1) that the y-values are following the market. Indeed, we're tempted to say things like ...
>The percentage of an asset's movements that are explained by movements in a benchmark index.
Yes, that's what we're tempted to say. In fact, for the situation in the picture, R-squared = 0.872 = 0.76
So we might conclude (with many others) that 76% of the asset's movement are due to market fluctuations and ...

>And the rest, 24%, is due to ... what?
If the x-values are the S&P returns and the y-values are my portfolio returns, then the 24% is due to my extraordinary trading skills.
If the y-values are the returns of, say, Ford Motors or Exxon, the remaining 26% may be due to the price of gasoline.
If the y-values are ...

>In other words, for large R-squared, much of the fluctuations in the y-asset is just due to market fluctuations, right?
If the x-values are Market Returns ... then that's what they say.
>Just because the Error is 0 when R-squared = 1, why would one say that R-squared gives the percentage of movement due to the benchmark?    
Remember when we talked about the two vectors that describe the two assets, here?
X has components (xk - M[x] ) / SD[x]√n   and   Y has components (yk - M[y] ) / SD[y]√n.
They each have length equal to "1". That is, ||X|| = ||Y|| = 1.
The component of Y in the direction of X is cos(θ) and that's the Pearson Correlation r.
So, staring at the diagram on the right, what fraction of Y is due to the influence of the benchmark vector X?
>Is it the vector A ... with length cos(θ)?
No, it'd be the vector B with length cos2(θ). Note that the lengths B + C add up to "1", the length of Y.
>I assume you got B = cos2(θ) with some trigonometry bumpf, eh?
Yeah ... so what've we got?
R-squared = r2 = cos2(θ)

>And you believe all this stuff, eh?
Of course!

Here are some more examples and I'll let you draw your own conclusions:

  • The R-squared between sunspot activity and the S&P 500 is ...

>C'mon! You're not telling me that sunspots are influenced by the S&P500?
Well, maybe S&P returns are influenced by sunspots ... ever think of that?
Anyway, continuing:

  • The R-squared between Microsoft returns and the length of women's skirts is ...


Okay, so here are some excerpts from other applications that I found, for investigating "causal" relationships using R-squared:

  1. This paper examines children who live with both biological parents and analyzes whether parental marriage confers educational advantages to children relative to cohabitation in Sweden.
  2. In this paper, a statistical model is developed to investigate the behaviour of supply of loans in Iranian banks in terms of the causal relationship between the main factors, which affect the supply of loans. The results indicate that government intervention has played a more important role than that of economic factors
  3. Specifically, the paper will look at whether or not the presence of market dominant minorities affects both the frequency and intensity of ethnic hatred and violence.
  4. When you do this with start-of-year p/e ratios and subsequent one-year stock returns, the R-squared is 0.03, meaning maybe 3% of stock price movement can be explained by p/e ratios.
  5. This paper empirically investigates the impact of international tourism receipts on the long-run economic growth of Turkey.
  6. This paper ...

>Okay, I get the idea.
Good for you.

Here's something neat.
Let i be an operation peformed on vectors which lie in the plane of X and Y.
The effect of applying the i-operation is to rotate a vector counterclockwise by 90 degrees.    
In the diagram on the right, we see some vector V as well as iV and i2V.

>I assume that i2V is the result of applying ...
Applying the operation twice: iiV = i2V

>Okay, I'm looking at the diagram and I'm seeing that i2V looks very much like ...
Very much like -V! In fact, it's exactly equal to -V. That is: i2V = -V so i2 = -1.

>Are we talking complex numbers here?
Why not? We just put i = √(-1).
Now look again at an earlier picture ... which I'll repeat here, for convenience  

>That ain't the same picture!
Pay attention !

Note that, since the length of vectors X and Y equal "1", then P = cos(θ)X.
That is, it's in the direction of X and had length cos(θ) ... since X has length = 1.

>Don't tell me! Q has length sin(θ), right?
Actually it's left ... or, to put it differently, it's in the direction of "X rotated 90 degrees left", so Q = sin(θ) iX.

Now, since Y is the vector sum of P and Q, that is: Y = P + Q, we get:
Y = ( cos(θ) + isin(θ) )X = exp(iθ) X

Isn't that neat!?



Since we're talking about correlations and stuff, while I was reading this article by Geoff Considine ...
(Geoff does a really neat analysis of Warren Buffett's BRK holdings.Why is ol' Buffett so successful, eh?)
... anyway, I was thinking "correlations".

>Yeah? So what else is new?
Here's collection of correlations for the top 20 holdings, as noted in the article:

What's interesting is the low correlations between these holdings.

>Is that good?
Well, sure. Diversification ya know. You should diversify across assets with low correlation in order to reduce portfolio volatility.

>And that Berkshire stuff, does it have low volatility?
Read the article yourself! You should learn to do things for yourself! Use your cerebral prowess! Why do I always have to ...
>Did you know that your chart is called Birkshire-Hathaway.gif? You can't even spell it right!