Correlation vs dependence

[wrongsaidfred]

Hi David,

I am a little confused about the discussion relating correlation to covariance in the notes for the quantitative section. You say that zero covariance means zero correlation, but that the reverse is not necessarily true. Mathematically, from cov=corr * std dev (x) * std dev (y), it would seem that if one is zero, the other must also be zero.

I know you are trying to tell us something about independence, but I am getting a little lost between the pure mathematics and the general concept. Any explanation would be greatly appreciated.

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

You are correct (about your math and that it clearly implies "if one is zero, the other must also be zero").
My apologies, and I don't know how it survived, but that statement on page 49 of quant notes is incorrect. It really should read something like:

independence --> zero covariance (--> zero correlation);

but the converse is not true:
zero correlation (or zero covariance, interchangeable) --> does NOT imply independence

...where the idea (and this is FRM thematic) is that "dependence" is a broader (and more useful) measure than correlation because correlation is merely linear; so correlation can be viewed as a special case of dependence.

You are absolutely right about the math. The key formula (really important in the FRM) is covariance = correlation * vol * vol, and they do indeed imply zeros on each other. The idea is: we can calculate a zero covariance or correlation, but we cannot conclude independence because the relationship can be non-linear

Thanks,
(I will correct the mistake, yikes....)

David