Until recently I accepted the notion of correlation as described here, namely:

Perfect positive correlation (a correlation co-efficient of +1) implies that as one security moves, either up or down, the other security will move in lockstep,in the same direction.

>What's that Pearson correlation again?
It's a number r (between -1 and +1) defined by:

[1]       r = (1/N) Σ ( x_k - M(x)) ( y_k - M(y)) / { SD(x) SD(y) }

where x and y refer to vectors of numbers, where x = [x₁, x₂, ... x_N] might be N returns for stock X and the y the set of N returns for stock Y
and M(x), M(y) and SD(x), SD(y) are the Means and Standard Deviations for the two sets of numbers.

If y_k = a x_k + b   (where "a" and "b" are constants; that is, the two sets are linearly related) then the Mean and Standard Deviation of the y's are related to those of the x's like so:

[2a]       M(y) = a M(x) + b   ... so   y_k - M(y) = a ( x_k - M(x))
[2b]       SD(y) = | a | SD(x)

so [1] then reads: r = (a/N) Σ ( x_k - M(x))² / { | a | SD²(x) } = +/-1 ... depending upon whether a is positive or negative.

>Huh?
Remember the definition of Standard Deviation?   SD²(x) = (1/N) Σ ( x_k - M(x))².

>So we've got "perfect correlation, eh?
Yes, if the two sets are linearly related. (Note that a / | a | = +1 if a > 0 and -1 if a < 0.)

>So if the correlation is +1 or -1, then we know that they're linearly related, right?
Well, I didn't realize that until recently because ...

>Because of old age, eh?
Yes, but now I understand that it's true. Look at it this way:

• Let x = [x₁, x₂, ... x_N]   and   y = [y₁, y₂, ... y_N].
   
• Introduce two new vectors:
  u = [x₁-M(x), x₂-M(x), ... x_N-M(x)] / N^1/2SD(x)
  v = [y₁-M(y), y₂-M(y), ... y_N-M(y)] / N^1/2SD(y)
   
• Note that the dot or inner product is:
  uv = (1/N) Σ (x_k - M(x)) (y_k - M(y)) / { SD(x) SD(y) } = r
   
• Note, too, that:
  uu = (1/N) Σ (x_k - M(x))² / SD²(x) = 1   so u and v are unit vectors
   
• Note, too, that uv = u v cos(θ) where u and v are the lengths of the vectors u and v and θ is the angle between them.
   
• If r = uv = +/-1, then
  u v cos(θ) = 1 and since u = v = 1, then cos(θ) = +/-1 so θ = 0 or π so the vectors are parallel unit vectors so v = +/-u
   
• Then, for r = +/-1, [x₁-M(x), x₂-M(x), ... x_N-M(x)] / n^1/2SD(x) = [y₁-M(y), y₂-M(y), ... y_N-M(y)] / n^1/2SD(y)
  meaning that (x_k-M(x)) / SD(x) = (y_k-M(y)) / SD(y) and that means that y_k = a x_k + b for some constants "a" and "b".

>zzzZZZ
Wait for the grand conclusion!

Sets of numbers x1, x2, ... xN   and   y1, y2, ... yN have correlation +1 or -1 if and only if yk = a xk + b for some constants "a" and "b".

>Huh? So what if your portfolio has two assets with perfect ... uh, anti-correlation? Can you still make money? Perfect anti-correlation? I assume that you mean correlation = -100%. Yes, you can make money. Indeed, both assets could have a positive compound annual growth rate. It's just that the returns are linearly related: y = ax + b with a < 0. See the chart? If a fraction p of your portfolio is devoted to Stock X and q = 1- p to Stock Y, and the correlation is -100%, then the Volatility (or Standard Deviation) of your portfolio would be the magnitude of: p SD(x) - q SD(y) . >And for correlation = +100%? Your portfolio would have SD(portfolio) = p SD(x) + q SD(y) .

Figure 1

>And for correlation = r?
It can always be calculated from SD²(portfolio) = SD²(px) + SD²(qy) + r SD(px) SD(qy)

>That chart ... it portrays 10 years, with p = 0.6 and q = 0.4?
Yes, starting with $1K.
>With annual rebalancing?
Uh ... yes. Did I forget to mention that?

>But the last three years they both go UP together? Is that possible with ...? With -100% correlation? Yes indeedy For example, if the two sets of returns are: X-returns 34.5% 10.6% 18.6% 15.8% 51.3% 33.3% 1.3% 22.7% 12.8% 6.5% Y-returns 16.5% 18.9% 18.1% 18.4% 14.9% 16.7% 19.9% 17.7% 18.7% 19.4% then their correlation is (surprise!) -100% ... yet look at Figure 2.

Figure 2

>What about +100% correlation?
There's a spreadsheet. You get to choose the allocations and the Mean & SD of Stock X and "a" and "b" (as in y = ax + b) and you get a different set of returns each time you press the F9 key.

Play with it.

>I assume I click on the picture to play?
Indeed.
You may be surprised to find that Stock X can go UP and Stock B go down, every single year, yet you can have +100% correlation.
Well ... I was surprised

the Question

Since I was somewhat surprised to find that stocks that go up together can have -100% correlation, I decided ...

>Surprised? Don't you mean shocked?
Well ... yes. So I decided to see if others thought differently.
To that end I posed the following question of several discussion forums:

Sam invests $1K in Stock X.
Sally invests $1K in Stock Y.
The performance of each is as shown here:

Would you say the correlation between returns is positive or negative?