Perfect Correlation
motivated by old age

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.
>And you blame that on old age?
Uh ... that's right. However, I now know better (after playing with copulas).
In fact, for two sets of number (for example), the (Pearson) correlation actually measures how close they are to being linearly related.

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

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

where x and y refer to vectors of numbers, where x = [x1, x2, ... xN] 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 yk = a xk + 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   yk - M(y) = a ( xk - M(x))
[2b]       SD(y) = | a | SD(x)

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

Remember the definition of Standard Deviation?   SD2(x) = (1/N) Σ ( xk - 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, ... xN]   and   y = [y1, y2, ... yN].
  • Introduce two new vectors:
    u = [x1-M(x), x2-M(x), ... xN-M(x)] / N1/2SD(x)
    v = [y1-M(y), y2-M(y), ... yN-M(y)] / N1/2SD(y)
  • Note that the dot or inner product is:
    uv = (1/N) Σ (xk - M(x)) (yk - M(y)) / { SD(x) SD(y) } = r
  • Note, too, that:
    uu = (1/N) Σ (xk - 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), ... xN-M(x)] / n1/2SD(x) = [y1-M(y), y2-M(y), ... yN-M(y)] / n1/2SD(y)
    meaning that (xk-M(x)) / SD(x) = (yk-M(y)) / SD(y) and that means that yk = a xk + b for some constants "a" and "b".

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?
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?
Answers included:
"- about 90% positively correlated over the time period shown."
"I would say the correlation of returns is close to zero to slightly negative -"
"I would guess: r = .922"
"I say slight positive."
"Look up the definition of correlation and calculate it."
"a low correlation"
"... would indicate to me a positive correlation."
"I'd guess the correlation is negative"
"I would say negative..."

and the best response was:
"I feel like this is a set-up."

>And the answer is ... what?
The correlation is -100% !!

>So now you don't feel so schtupid, right?
Right! I got some company

>But the question asks to guess the correlation between returns, but the chart asks to guess the correlation between stocks.
Uh ... yeah. I guess that's confusing, eh? Some might think they were guessing the correlation between stock prices.

>Now you feel schtupid again, right?
Uh ... right.