motivated by old age
Until recently I accepted the notion of correlation as described here, namely:
Perfect positive correlation (a correlation coefficient 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 ( x_{k}  M(x)) ( y_{k}  M(y))
/ {
SD(x) SD(y)
}
where x and y refer to vectors of numbers, where x = [x_{1}, x_{2}, ... 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  SD^{2}(x)
} = +/1 ... depending upon whether a is positive or negative.
>Huh?
Remember the definition of Standard Deviation?
SD^{2}(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 = [x_{1}, x_{2}, ... x_{N}]
and y = [y_{1}, y_{2}, ... y_{N}].
 Introduce two new vectors:
u = [x_{1}M(x), x_{2}M(x), ... x_{N}M(x)] / N^{1/2}SD(x)
v = [y_{1}M(y), y_{2}M(y), ... y_{N}M(y)] / N^{1/2}SD(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}
/ SD^{2}(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,
[x_{1}M(x), x_{2}M(x), ... x_{N}M(x)] / n^{1/2}SD(x) =
[y_{1}M(y), y_{2}M(y), ... y_{N}M(y)] / n^{1/2}SD(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 x_{1}, x_{2}, ... x_{N} and y_{1}, y_{2}, ... y_{N}
have correlation +1 or 1 if and only if y_{k} = a x_{k} + b for some constants "a" and "b".

>Huh? So what if your portfolio has two assets with perfect ... uh, anticorrelation? Can you still make money?
Perfect anticorrelation? 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 SD^{2}(portfolio) = SD^{2}(px) + SD^{2}(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:
Xreturns 
34.5%  10.6%  18.6%  15.8%  51.3%  33.3%  1.3%  22.7%  12.8%  6.5% 
Yreturns 
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
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."
"neutral"
"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 setup."
>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.
