Quote from gummy:
Aha! Afraid to guess, eh?
My guess was way off!
Thanks to all who participated.
For the answer, check out:
http://www.gummy-stuff.org/perfect-correlation.htm#QUESTION
a Correlation Question
- Thread starter gummy
- Start date
The correlation is negative because a<0; it is perfect because ("almost surely"
) you chose the conditional variance of y to be zero. Your choice of b and mean(x) ensures a probable positive return for both stocks over the full time period.Edit: Nice website!
Actually, I invented ten random x- returns (a la NORMINV(), in Excel).
Then I generated the y-returns via:
y = ax + b (with your choice of "a" and "b").
The spreadsheet is available, to play with
Many think (me included, until recently!) that (for example) Investopedia's explanation is okay, 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.".
Quote from gummy:
Actually, I invented ten random x- returns (a la NORMINV(), in Excel).
Then I generated the y-returns via:
y = ax + b (with your choice of "a" and "b").
The spreadsheet is available, to play with
Many think (me included, until recently!) that (for example) Investopedia's explanation is okay, 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.".
Meant conditional on x: if you use y=ax+b+e, where e is independent of x (and zero mean), the correlation would be greater than -1. Your choice of stock "Y" is more like "short some of X" plus/minus "cash."
There are a jillion ways to generate y-returns, given the x-returns.
My purpose was to illustrate that it's possible for securities to go up and down together ... yet a correlation of -1.
Indeed, it's possible to go in opposite directions yet have a correlation = +1.
I've done that.
It's irrelevant how the y-returns were generated. They're inventions, to illustrate the possibility.
I guess the moral is:
<B>Don't place too much trust in the Pearson Correlation Coefficient!</B>
I suggest that Spearman correlation is sexier.
My purpose was to illustrate that it's possible for securities to go up and down together ... yet a correlation of -1.
Indeed, it's possible to go in opposite directions yet have a correlation = +1.
I've done that.
It's irrelevant how the y-returns were generated. They're inventions, to illustrate the possibility.
I guess the moral is:
<B>Don't place too much trust in the Pearson Correlation Coefficient!</B>
I suggest that Spearman correlation is sexier.
Agreed, wasn't trying to undo your basic point, which is very well taken.
Gummy,
Thanks. I find your posts refreshing. Please continue with these statistical puzzles.
IMO, looking for any type of correlations between 2 stocks or two instruments is meaningless- be it spearman or pearson.
The individual companies/indices have possibly different products/components and different weights to components, different business models within which they operate, different expectations of future cash flows, different market externalities that impact their valuations.
Just because a correlation number is calculated and is increasing or diverging from the past, I would not venture to assign any meaning to it, let alone trade on the info.
Thanks. I find your posts refreshing. Please continue with these statistical puzzles.
IMO, looking for any type of correlations between 2 stocks or two instruments is meaningless- be it spearman or pearson.
The individual companies/indices have possibly different products/components and different weights to components, different business models within which they operate, different expectations of future cash flows, different market externalities that impact their valuations.
Just because a correlation number is calculated and is increasing or diverging from the past, I would not venture to assign any meaning to it, let alone trade on the info.
I tend to agree.
However, if the returns of stock Y are uniquely determined by the returns of stock X as, for example, when
(y-returns) = (x-returns)^3
it's difficult to understand a statement that says they have a low correlation.
It's even more confusing (to me, at least) to see two stocks whose prices move up together
... and then find that their correlation is -100%.
Mamma mia!
(That's why I'd prefer Spearman to Pearson
)Thank you for sharing your time. I can sort of explain the anomaly (pardon my broadcasting my naivete). When you have an equation relating the returns of X and Y, you are making some sort of linear regression fit between the respective returns. Theoretically, they can have a perfect negative correlation. When X increases, Y can decrease, and vice versa. As long as the returns bounce up and down around the regression line, they would have satisfied both conditions- i.e have a negative correlation and a positive regression fit of their respective returns. In any case, your website is wonderful. am eternally grateful to that.
Best wishes.
Best wishes.
Quote from gummy:
I tend to agree.
However, if the returns of stock Y are uniquely determined by the returns of stock X as, for example, when
(y-returns) = (x-returns)^3
it's difficult to understand a statement that says they have a low correlation.
It's even more confusing (to me, at least) to see two stocks whose prices move up together
... and then find that their correlation is -100%.
Mamma mia!
(That's why I'd prefer Spearman to Pearson