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.".
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.
That is why I mentioned CoIntegration which is a better estimate of co-movement of securities than correlation. The problem is that you need lots of data. nitro
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.
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.