modeling dependence : correlation and copulas

[prasadhegde1]

Hi David

In Dowd (market risk ) reading (modeling dependence : correlation and copulas) he mentions that correlation is a good measure of dependence when random variables are normally distributed but it is not an adequate measure for multivariate normal distribution , a zero correlation implies that the variables are independent .

but doesnt correlation explain linear dependency and when its zero it means that there is no linear dependence but it may not mean non linear dependency is absent , could you please explain why returns having multivariate normal distribution assume zero correlation = independent

secondly , why do we need copulas ? is it for the same reason that correlation is inadequate for multivariate normal distribution ?

It would be very helpful if you could clarify

 

[David Harper CFA FRM]

Hi prasad,

I don't *think* he says "correlation is not an adequate measure for multivariate normal distribution"
(please correct me if I am wrong, but can you cite the page number?)

Rather, I think he says (e.g., A5.1.1 Correlation Good with Elliptical Distributions, A5.1.2 Correlation Bad with Non-elliptical Distributions)

1. "(linear) correlation is a good measure of dependence when our random variables are distributed as multivariate elliptical" As elliptical's most prominent member is normal, this is equivalent to: "(linear) correlation is a good measure of dependence when our random variables are distributed as multivariate normal."

2. Zero correlation does not imply independence, although independence --> rho = 0.

Point #2, IMO, is what we care about. It is part of the list of drawbacks/weaknesses of the correlation as a restricted, linear measure of dependence.

Please note he makes the following statement:
"If risks are independent, we will have a correlation of zero, but the reverse does not necessarily hold except in the special case where we have a multivariate normal distribution: so zero correlation does not imply that risks are independent unless we have multivariate normality"

...this is not inconsistent with point #2. Rather, he's saying that in the special case of multivariate normal, then rho = 0 implies independence, but more generally (i.e., without the multivariate normal), we cannot draw this conclusion
... the reason owes to the limiting definition of correlation as a measure that only captures "first moments." The variables could associate in higher moments (e.g., variance) but our correlation measure does not consider those moments. It is useful to recall that normality is special, in part, because it only uses mean & variance; measures that only use these first two moments implicitly assume normality!

Re copulas, I think that is a great way to look at copulas (I view copulas as a giant super-class of functions, with much flexibility, that includes in one small corner the person's product moment correlation that we are constantly working with): as a measure of dependence designed to overcome the long list of limitations of the linear correlation.

So, Dowd's key advantage is that copulas can be used for non-elliptical distributions (why is this vital in risk measurement? b/c most of our heavy-tailed distributions are non-elliptical). But there are several other flexibility advantages, including notably "Copulas enable us to extract the dependence structure from the joint distribution function, and so separate out the dependence structure from the marginal distribution functions" (Dowd p 147); i.e., the marginal distributions are separated as "inputs" ... i would say the correlation coefficient is a specific, highly limited linear metric while copulas is more like an brorad concept/approach with sub-classes that each use a different function to characterize dependence.

...I hope that helps...copulas are deep, please don't expect to digest them instantly (I have much to learn about them, too!)...in regard to FRM 2009 don't expect deep query on copulas, this is brand new FRM topic, so I would not spend a whole lot of time worrying copula details (the correlation coefficient and its limits are more important, currently)

David