Correlation vs Dependence or CoVariance

Hey FRM’ers

I was going through some old notes and found something that was really helpful in building intuition in my career and may be something helpful for both part I and part II candidates on the exam.

It took me a long time to understand the limitations of correlation ( whether in trading or in risk management) and how/why correlation breaks down. I think it is best explained in the form of a question:

If y = x^2 and dx is normally distributed, what is the coefficient of correlation, not the dependence, between x and y?

A. .5

B. 0

C. -.5

D. 1

Answer: B

It is critical to understand that correlation is a measure of linear dependence. When y =x^2 there is clearly a dependence between x and y. To calculate correlation, we need the expectation of the product of x and y - E(XY) - , substitute x^2 for y and we have E(x^3).

Now a standard normal expectation is always zero. Remember how I always say standard normal distributions add linearly. From this, we can see the correlation between these two variables – the actual, calculated correlation coefficient, is zero. ( 0 raised to the 3rd power is still zero)

This shows how we can know for certain there is dependence, but no correlation. Why? Correlation is a linear measure. If there is only a non linear relationship, then there is no linear relationship and by definition, zero correlation.

It helps if you don’t think of “correlation” as a “connection” between two things but a very, very specific type of a connection. Specifically, only a linear connection. The special name for that linear connection is correlation.

This is probably the defining principle of the entire FRM curriculum: Correlation does not capture non linear connections in a world that runs on non-linear relationships. I don’t mean correlation is a bad estimate of a non-linear connection, I mean correlation explicitly ignores relationships that are non-linear.

And this is why correlation breaks down in the copulas of Part II

Probably one of the greatest “aha” moments I’ve had.

Anyway, thought it might help.

This is actually perfect timing since I’m just starting the quantitative analysis sections.

Thanks for the post.

Hi Christian,

Thanks for the post. Could you please guide in terms of how to prepare for Quantitative analysis section.

how did you practice the questions.

Regards,

Quant was definitely not my strong suite early in my career so I had to think of new ways to put those pieces together. That’s one of the reasons I think what I do in the quant section and how I explain the material like building blocks helps build intuition and visual thinking around complex topics. This does three things: first, raises your confidence, builds intuition around math, and sets you up for what I am going to do in part II - the really tough sections on credit and copulas. There is a quant intro lecture I want you to see. I’ll see if we cant put it up on the website.

Hi Christian

I am following your lectures video, and I realized in Quantitative Analysis session, some videos are not in the same sequence as in GARP handbooks. Do you make this arrangement on purpose, for easier connecting the dots? Or it’s some mistakes in Wiley portal?

BTW, I love the way you explain the quantitative sessions. It helps a lot! Thanks!

HI there, thank you for the compliment. A ton of thought and effort goes into every lecture.

I make sure everything: both notes, lectures, and physical copy questions (The online learning can be out of order depending on what you study next) are in lock step with the GARP outlines. The only non-Garp lecture in quant is the introductory lecture but Ill double check to make sure everything else is in order.

So if something is out of order, it shouldn’t be and I’ll have a look.

I just wrapped up quant and didn’t see anything out of order.

Thanks, I didn’t either. Did you like the way I handled the quant section?

Hey Christian,

Sorry, just saw this. I did. I will definitely have to go through my notes once or twice more and drill your practice questions because my math background is not very strong, but I appreciate how you’ve taken the time to explain things in layman’s terms instead of just throwing up a bunch of greek letters and expecting us to understand it.

On Mac and windows computers, they are in order. But not on iPhone or iPad. Was going through financial products theses day, chapter 20 for rating agency is not even showing in the list if viewing on iPhone or iPad. This is something you should ask Wiley to check, I think.

ok thanks