Financial Correlation Modelling- Bottom Up Approaches

[QuantMan2318]

Dear David

I have attached a spreadsheet in which I have calculated copula correlations for a Bi Variate Normal distribution and have inserted a chart on the same. I would be extremely happy if you could have a look at the same and tell me if my understanding is correct. I have used the same example from Gunther Meissner's Ch. 4 as provided in GARP material, pg. 175.

Thanks and Regards
Manikandan V R

 

[QuantMan2318]

Dear David

Did you go through the above spreadsheet? Were you able to check the calculations and the graph? Its been a month since I posted that file and this thread is silent
I would be extremely grateful if you could confirm the accuracy of the spreadsheet as it would help me in understanding the concepts better

Thanks and Regards
Manikandan V R

 

[David Harper CFA FRM]

Hi @QuantMan2318 Sorry for long delay, but although I agree that your setup matches most of the input assumptions (except for the default correlation) of Meissner's Gaussian Copula 4.3.1., I can't see how you incorporate the default correlation (which in Meissner's examples in assumed to be 0.40) into bivariate normal distribution. I may be dense but, with respect to your matrix values at C44:L53, I don't understand this significance of these values; e.g., 169.98 is found at the intersection of default probabilities 6.01% (year 4 for first asset) and 5.13% (year 6 for second asset), but what does this value want to mean? If it means to be the density function, it appears to be too large. Related, your pearson rho value (0.738 ...) is computed as the correlation between the two assets' default probabilities. Okay, that looks to be the problem: the rho assumption should be an input variable. It is the default correlation between the two assets; ie, not the correlation between their respective sets of one-year default probabilities. I can see how this appears to be the same concept! But the one-year default probabilities which are also inputs (6.51% in year 1, 7.65% in year 2, ...) by definition are the unconditional default probabilities. They represent the probability of default during the year without respect to any dependence on the other credit, and they do not inform the correlation parameter). In this context, the default correlation can also be called the bivariate copula correlation coefficient; but it's an input.

Here is are two spreadsheets which should be helpful:

• Meissner's own: https://www.dropbox.com/s/314hczpaldrwnlj/2-asset_default_time_Copula-Meissner.xlsm?dl=0
  • I retrieved this at www.wiley.com/go/correlationriskmodeling or http://www.wiley.com/WileyCDA/Section/id-819643.html
• My own modification to Hull's: https://www.dropbox.com/s/59hu5pubnwjdmr6/gaussian_copula_v2.xlsm?dl=0
  • Notice that both of these spreadsheets return 3.44% in response to the question, What is the joint default probability Q of companies B and Caa in the next year assuming a one-year Gaussian default correlation of 0.4? I hope this helps!

 

[QuantMan2318]

Thank you so much @David Harper CFA FRM CIPM!
I had always wanted to create that copula correlation and that multivariate standard normal distribution ever since I encountered that in part I,some gaps were always making me unable to create a complete multivariate normal distribution, your excel sheets are just what the doctor ordered! thank you so much, do give me some time to go through those and I shall also see how to modify mine

Hi @QuantMan2318 but what does this value want to mean? If it means to be the density function, it appears to be too large. Related, your pearson rho value (0.738 ...) is computed as the correlation between the two assets' default probabilities. Okay, that looks to be the problem: the rho assumption should be an input variable. It is the default correlation between the two assets; ie, not the correlation between their respective sets of one-year default probabilities. I can see how this appears to be the same concept! But the one-year default probabilities which are also inputs (6.51% in year 1, 7.65% in year 2, ...) by definition are the unconditional default probabilities. They represent the probability of default during the year without respect to any dependence on the other credit, and they do not inform the correlation parameter). In this context, the default correlation can also be called the bivariate copula correlation coefficient; but it's an input.

The matrix I created is the density function for a multivariate normal distribution using the default probabilities ( the abscise values ) and the pearson rho (which is of course not the default correlation as you stated, however I had used it as an input to the density function of a bi-variate normal)

I shall check the excel sheets and get back to you

Thanks very much!

 

[QuantMan2318]

Dear @David Harper CFA FRM, I have now carefully looked at your files, I understood the first one but your John Hull's reworking file was not completely comprehensible to me and I will look at it a couple of more times; However, I have come up with a revised excel sheet for modelling the Gaussian Copula. I have taken the default correlation now as given at 0.4 as it seems to have been derived from a different formula than the Pearson one. Furthermore, my previous file only incorporated the Bivariate Normal density function; now based on that and a new formula that I came across in a Risk Metric material, I have reworked the Matrix. Of Course, its not correct yet, can you tell me where I am making the mistake? Sorry if I am too persistent and an ignoramus, I really do want to model a Copula on Excel

If the Gaussian Copula comes out well, I plan to make another Excel sheet for the Frank's Copula, I have gone through @jairamjana's wonderful material on Copulas. I wish him thanks for the same.

Here are the files that I found on the net pertaining to Gaussian Copulas, one is by David Li himself, though a little bit more advanced than what is required for FRM, it has gems here and there that a student might find useful. The other is a research paper by Christian Meyer that has the formulae that I found useful. Finally, my reworked Excel file.

I know that you are busy, just think of it as one cranky pastime of your student who has cleared the FRM, therefore, please look at it after you look at other member's queries and only when you have some time. This thing follows you and me like a dormant volcano eh?

 

[jairamjana]

It's really great to hear my contribution has helped the community.. Though all I did was compile existing material into flowing PDFs.. If I learn copulas I will definitely learn further from your worksheet.. ..