Copula Tail Dependence

[Hardlearner]

Hi David,

I have the following question concerning tail dependence:

Assume two random variables which are normally distributed with mean = 0 and deviation=1.
What is the conditional probability

P(X>=1 | Y>=1) given high tail dependence?

Can you show in detail the calculation?

Thank you in advance
Best regards

 

[David Harper CFA FRM]

Hi @Hardlearner

For the details you can search bivariate normal distribution; e.g., http://mathworld.wolfram.com/BivariateNormalDistribution.html

For conditional P(X>=1 | Y>=1), I might use conditional P(X>=1 | Y>=1) = joint P(X>=1 , Y>=1)/marginal P(Y>=1)
Here is my implementation of Hull's bivariate normal approximation https://www.dropbox.com/s/59hu5pubnwjdmr6/gaussian_copula_v2.xlsm
i.e., same mechanics as his cumulative bivariate normal distribution @ http://www-2.rotman.utoronto.ca/~hull/software/RMFIsoftware.htm

for example, at correlation (rho) = 0.5, the joint Pr(X < -1, Y <-1) = 6.25%
... compare to same joint under independence: =NORM.S.DIST(-1,TRUE)^2 = 2.52%

So under rho = 0.5, P(X>=1 | Y>=1) = 6.25%/NORM.S.DIST(-1,T) = 6.25%/39.40% = 39.40%. Thanks,

 

[Hardlearner]

Hi David,

thank you very much for your detailed answer and especially for the links to the files. These are very helpful for me in order to do calculations for other parameter sets. Additionally this reinforces my knowledge concerning the calculation of conditional probabilities, which might be a topic in the exam.

Again thank you very much for your help!

 

[David Harper CFA FRM]

Sure @Hardlearner , yes agreed about the conditional probability. When I first read your question, thought "I have sheet for the JOINT bivariate normal, but I don't have anything for the conditional bivariate normal" ... I almost forgot we can retrieve the conditional from the joint ... it's good not to forget a basic relationship: Pr(ab) = Pr(a|b)Pr(b)