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.