this topic is cropping up a lot.is it correct to say correlations can be observed to be less than minus one / greater than one when combining two skewed distributions?
or do I just say correlations may not be meaningful… (i.e. the famous MCS is the answer)
Correlation coefficients _ cannot _ be less than −1.0 nor greater than +1.0.
Period.
fine. but how do i describe the interactions?
e.g. if i combine an instrument with a skewed distribution, with another which nets out at the risk free rate.
Apart from the point made by S2000magician, I would say that’s out of topic for the exam. We learn how to compute and analyze and use correlations in the case of normal distributions. We learn why the correlations given or computed assuming normality may not be correct because normality finally does not apply. But we don’t work any further on the computation or analysis of correlations in the context of non-normality.
I have no background in statistics, so I can’t help you further.
Correlation doesn’t assume normality.
The correlation coefficient we calculate in the CFA curriculum is known as the Pearson moment correlation; it measures the strength of the linear relationship between two paired samples or two paired populations. There’s no reason that the samples need to come from normal distributions (nor, even, from populations with similar probability distributions); the relationship may be close to linear (|ρ| is close to 1.0) or not (|ρ| is close to 0.0).
Ok I need to review this!!
I mean, I know it does not in general (that was not the point in my first post), but I thought that using the specific computation approach that we apply in the course to derive the correlation, we needed the normality assumption. I need to work on this again.
I wanted to add to this because it seemed to be a point of confusion in a prior thread regarding normality in the sharpe ratio but not the pearson correlation coefficient.
You’re right in that it does not assume normality (only in doing a test of hypothesis might we need to assume normality). However, the pearson correlation coefficient will only guarantee an accurate description of the relationship between the two variables if they are bivariate normal in their distribution (simplest case of multivariate normality). In other words, if you don’t have bivariate normality, you can still use the pearson correlation coefficient, but it might not be the best representation of the relationship between the variables-- as you said, it is a measure of linear association. By even one of the variables being non-normal, this allows for the possibility of a non-linear relationship between them.
So, you can use the pearson correlation without normality, because it doesn’t assume normality. Maybe this is where some of the confusion came from?
Could you be more specific in your use of “interactions”? There is a statistical concept of variable interactions, but I’m not sure if that’s what you’re intending.
Also, what about a skewed distribution seems to indicate |correlations| exceeding 1?
I think that you meant “_ . . . |correlations| exceeding 1? _”
As you well know, |1| = 1. Correlations can be negative; 1 rarely is.
Indeed, I did mean what you wrote. I made the correction. Thanks for the catch!
guys… your all going too deep, i just want a line to write if I need to formulate an answer on this topic.
Lets say the question is something like…
“portfolio manager holds a long futures position and wants to immunise the position for one week due to an expected market shock. the collar has a std deviation matching the futures position, but he calculates the correlation as minus 0.79”
choose & justify the appropriate immunisation strategy.
A) the collar is appropriate.
The correlation is not minus one because it is calculated asssuming a linear relationship ( pearson linear moment).Due to the non-linear returns of the option pair, and possibly implied volatility differences in the put & call option.
3 points?
i would say 0 points - because you did not answer the question. You went off on a lecture about correlation + assumed normality - while they asked you if the collar was appropriate or not.
That isn’t why it’s not -1. It’s simply because each time one variable deviates above (below) its mean, the other doesn’t always deviate below (above) its respective mean. (Think about the fact that linearly related variables aren’t all -1 or +1 in terms of a pearson correlation.)
Using the pearson correlation under non-normality doesn’t guarantee that the relationship is non-linear, it just means it’s possible for it to be non-linear (in which case the coefficient wouldn’t accurately capture the relationship).
In any case, I agree, my point may have been beyond the scope of the exam, but you’d be better off knowing you can’t write things like “the skewed distribution implies a correlation greater than +1”, for example (or like what you wrote above). I also mentioned it because it (normality as an assumption for pearson correlation) seemed to be a point of confusion in another thread.