Price vs Return Correlations

When one talks about the correlation between two stocks, it's usually the correlation between the weekly (monthly? yearly?) returns.
If the correlation is large, say 90%, one assumes the two stocks behave in a similar manner.

>Similar manner? What's that mean?
If a return for stock A is positive (negative), one would expect the return for stock B to be positive (negative).
In other words, you'd expect the stock prices to go up and down together.

>Exactly?
No, not exactly, but they should behave similarly ... if the correlation is large.
In fact, let's suppose we have any two sets of numbers (which may or may not be stock returns):
x1, x2, x3 ... xN   and   y1, y2, y3 ... yN.

Then the Pearson Correlation Coefficient is defined by:
where we're talking about the sum of the products of the deviations of the x's and y's from their mean (or average) value, divided by the product of the two standard deviations.

>Are we talking about stock returns?
Not necessarily. The numbers could be anything ... the mid-day temperatures in two cities (over a year) or maybe two sets of baseball scores or maybe ...

Okay, let's define the vector:
X = [ (x1-M(x))/ SD(x)√N, (x2-M(x))/ SD(x)√N ... (xN-M(x))/ SD(x)√N]
Each component is a deviation from the Mean M(x), divided by the Standard Deviation (of all N numbers) multiplied by √N, the square root of N.
Notice that the square of the magnitude of this vector is: |X|2 = (1/N){(x1-M(x))2 + (x2-M(x))2 + ... + (xN-M(x))2} / SD2(x).
But the Standard Deviation is defined such that SD2(x) = the average squared deviation.
That means that our vector X had magnitide "1".
It's a unit vector.

Similarly we can define Y, from the set of y-values.
Then we stare intently at the Correlation Coefficient and recognize that it's just the dot product: XY.
 >zzzZZZ Wait! We're almost there! The dot product is a number equal to: XY= |X| |Y| cos(θ)   where θ is the angle between the two vectors X and Y.       In our case these are unit vectors, so the correlation is just the cosine of the angle so ... >So it's between -1 and +1, eh? Glad to see you're awake again.
Anyway, when both vectors point in roughly the same direction, the correlation is high. If they point in directions which are quite different, then it's low.
>And if one points East and the other points West, then ...
The correlation is -1.
>And what's all this about return and price correlations?
Aah, yes. Glad you reminded me.

If we consider the two sets of returns, for two stocks, we'd get some correlation.
If we consider, instead, the two sets of stock prices, do we get similar correlations?
Okay, here's some pictures: the evolutionof two \$1.00 stocks and the correlation of returns and of prices:

Note the two vectors, depending upon whether we consider returns or prices. Sometimes (like in the left chart) they're similarly oriented vectors.
BUT sometimes, like in the right chart, they ain't. The angle between (determined by the correlation) is quite different.

Okay, so here's the Big Question:
 When trying to identify sister stocks, with prices which tend to move in the same direction, should one look at correlation between Returns or Prices?

>I vote for Prices.
Check these:

>Like I said ... I vote for Prices!
Yeah, it looks like a better indication of sister stocks and ...
>What's all this sister stocks stuff?
Patience. We're getting closer ...

 Note: If you'd like to compare PRICE and RETURN correlations, a spreadsheet is here. It looks like this. >Hey! Look at MSFT, a year ago. It dropped like a rock and ... Yes ... like a rock compared to the DOW. See?   Notice, too, that the drop in price correlation preceded the drop in MSFT prices ... compared to ^DJI. On the other hand the ... >But the return correlation predicted the drop as well, no? Sorta ... in April, 2006, but what about Nov/06. The return correlation gave a false signal and in Mar/07 there was no signal at all. In fact, it looks like we find a place where the price correlation goes negative. Then, when it again goes positive we buy. Since we have a pefect 1-sample demonstration of this stratgery, that's statistically significant, right? >You buy too early. If I were you I'd wait until the price correlation got a mite above zero. Then you'd buy nearer the bottom. Okay, if you say so ... >You really believe in this stuff, eh? Well, to tell the truth, I use this
 Here's some Canuck Stuff:   >Hmmm ... looks strange. Can I trust your stuff? Unlikely.

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