Copulas

[RobWat]

Hi, I have a couple of questions on copulas...

1) It seems that we often have to discuss whether the Clayton, Gumbel or Frank copulas are appropriate based on the likely presence or not of upper/lower tail dependence. The Clayton has lower tail dependence and the Gumbel has upper tail dependence, while the Frank has neither. If you're considering a joint distribution that it likely to exhibit either upper or lower tail dependence, but not both, couldn't you use either the Clayton or Gumbel copulas by simply switching the definition of the upper tail in your distributions?

For example, say you're considering the joint behaviour in the tails of the return on two stocks that you think might exhibit lower tail dependence, but not upper tail dependence. Couldn't you use the Gumbel copula (with upper tail dependence) and simply define X (your random variable) to be the return multiplied by -1?


2) Do we need to know how to generate random variables from a multivariate distribution other than the multivariate normal distribution? For a multivariate normal distribution you can use Cholesky decomposition or principal component analysis. I can see how this could be extended to the case where you have a Gaussian copula, with non-normal marginals. I can also see how you would extend it to multivariate normal mixture distributions (which is discussed briefly in the notes). But what if, say you're using an Archimedean copula? How would that work? Could we be expected to know how it works in the exam?

Thanks.

 

[Edwin]

Hi, I have a couple of questions on copulas...

1) It seems that we often have to discuss whether the Clayton, Gumbel or Frank copulas are appropriate based on the likely presence or not of upper/lower tail dependence. The Clayton has lower tail dependence and the Gumbel has upper tail dependence, while the Frank has neither. If you're considering a joint distribution that it likely to exhibit either upper or lower tail dependence, but not both, couldn't you use either the Clayton or Gumbel copulas by simply switching the definition of the upper tail in your distributions?

For example, say you're considering the joint behaviour in the tails of the return on two stocks that you think might exhibit lower tail dependence, but not upper tail dependence. Couldn't you use the Gumbel copula (with upper tail dependence) and simply define X (your random variable) to be the return multiplied by -1?


2) Do we need to know how to generate random variables from a multivariate distribution other than the multivariate normal distribution? For a multivariate normal distribution you can use Cholesky decomposition or principal component analysis. I can see how this could be extended to the case where you have a Gaussian copula, with non-normal marginals. I can also see how you would extend it to multivariate normal mixture distributions (which is discussed briefly in the notes). But what if, say you're using an Archimedean copula? How would that work? Could we be expected to know how it works in the exam?

Thanks.

The answer to your first question is; what happens if you fit a Gumbel copula with upper tail dependence to two stocks that you think might exhibit lower tail dependence ....then STOP! And test goodness of fit, say using a visual ad-hoc test like QQ plot? (before multiplying by negative 1)

For part 2 of your question how would you extend cholesky to a simulate from a Gaussian copula?

 

[Edwin]

Assuming you are currently looking at the COSO framework & so don't have time to do difficult Maths or busy trying to remember the Core - reading, like me...

However, here is my take at the second part of your question...

...First I don't think this can be asked, but the idea is to realise that Cholesky relies on the inverse transform method of Monte Carlo i.e the simulation of the Z ~ N(0,1) Vector i.e X = u + C''*Z , hence you generate uniform random vectors with dependence and then apply the inverse of a distribution function to the uniform marginals (Inverse Transform method ) to get a multivariate vector with different marginals.

for the t coupla it is something like this.
let's say you want to generate a vector from the t copula with dependence rho=0.5 ,matirx =[1 rho;rho 1];with 4 -degrees of freedom. after simulating the the bivariate vector apply the T_CDF with 4-degrees of freedom. this will generate your uniform bivariate vector ,now apply the Inversion method to the marginals with your choice of marginal distribution taking into account the distribution parameters,you will get a bivariate vector with an imposed T copula and the choice of marginals you made.

This can be extended to Archimedian Copulas;-

http://economics.nd.edu/assets/1341...of_a_collateralized_debtsubmissioncopulas.pdf

 

[RobWat]

Thanks for your reply Edwin.

For part 2 of your question how would you extend cholesky to a simulate from a Gaussian copula?

I think you've answered your own question here:

the idea is to realise that Cholesky relies on the inverse transform method of Monte Carlo i.e the simulation of the Z ~ N(0,1) Vector i.e X = u + C''*Z , hence you generate uniform random vectors with dependence and then apply the inverse of a distribution function to the uniform marginals (Inverse Transform method ) to get a multivariate vector with different marginals

In regards to a t-copula:

for the t coupla it is something like this.
let's say you want to generate a vector from the t copula with dependence rho=0.5 ,matirx =[1 rho;rho 1];with 4 -degrees of freedom. after simulating the the bivariate vector apply the T_CDF with 4-degrees of freedom. this will generate your uniform bivariate vector ,now apply the Inversion method to the marginals with your choice of marginal distribution taking into account the distribution parameters,you will get a bivariate vector with an imposed T copula and the choice of marginals you made.

how would you simulate the bivariate vector? What joint distribution (and hence copula) would this be? I think you can generate a multivariate t-distribution from a multivariate gaussian as it can be defined as multivariate normal mixture distribution.

http://economics.nd.edu/assets/13410...ioncopulas.pdf

Taking a skim read at the paper you linked to, I think this says that you can use any "exchangeable bivariate copula" (whatever that is) as your starting block for generating other non-gaussian copulas. However, I can't pretend to totally follow their method. They refer to a book by Alexander J. McNeil called "Quantitative Risk Management: Concepts, Techniques, and Tools". I'd imagine it gives more detail in there.

Anyway, I'm only slightly curious. It just seems like something you'd obviously want to do in practice if you were using copulas for anything other than the most basic closed form application (i.e. if you're doing a Monte Carlo simulation). I'm surprised that the notes/textbooks don't at least say something along the lines of "this is something you'd probably want to do in practice, but is beyond the scope of these notes/textbook. Please refer to McNeil if you're interested."

Thanks.

 

[Edwin]

The book is not calibrated for practise, it is for exams, it's a story of Mathematics...

...for any insights you need to dig deeper. They can improve by bringing a CD with the book with example applications. But that's not their idea...

...there is a big gap between that book and implementation, you are going to get another headache when doing the things in real life, but just relax, download R and set google as your home page.

 

[RobWat]

The book is not calibrated for practise, it is for exams, it's a story of Mathematics...

...for any insights you need to dig deeper. They can improve by bringing a CD with the book with example applications. But that's not their idea...

...there is a big gap between that book and implementation, you are going to get another headache when doing the things in real life, but just relax, download R and set google as your home page.

Fair point, but IMHO the exams should be calibrated for practice, or a as close as reasonably possible.

 

[Edwin]

Fair point, but IMHO the exams should be calibrated for practice, or a as close as reasonably possible.

I agree fully, it's disappointing to find that the profession neglects practise, especially in the space of modelling...they just give us theory.

But maybe it's because part of the job description in Actuarial Science is

To take exams