default correlation and 1st to default basket

[ajsa]

Hi David,

I wonder why high default correlation can make 1st to default basket less risky? will the 1st default's possibility decrease with high correlation?

thanks.

 

[David Harper CFA FRM]

Hi asja,

the classic example is: assume PD = 1% and 10 credits

if 0.0 correlation, PD (1st to default) = 1-(1-1%)^10 = about 10%
but if 1.0 correlation, PD (1st) goes all the way down to 1%
similarly, -1.0 correlation ensures a 1st to default (!)

binomial distribution can show similarly for senior tranche...
more here with links: http://forum.bionicturtle.com/viewthread/1812/

David

 

[David Harper CFA FRM]

....post with numerical example here: http://www.bionicturtle.com/learn/article/correlations_in_cdos_and_cds_baskets/

David

 

[ajsa]

Hi David,

Could you explain how this is derived? "but if 1.0 correlation, PD (1st) goes all the way down to 1%"

Sorry I am a little slow. Thanks.

 

[David Harper CFA FRM]

Hi asja,

if there is 1.0 correlation, there are only two outcomes:

* all default (together) with p = 99%, or
* none default with p = 1%

David

 

[ajsa]

Hi David,

If A and B's correlation is 1.
P(AB)=P(A|B)*P(B)=P(1)*P(B)=P(B)
P(AB)=P(B|A)*P(A)=P(1)*P(A)=P(A)=P(B)

So does it mean if A and B's correlation is 1, P(A) must equal to P(B)?

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

This just plain stumps me because your logic (syllogism) sure looks fine yet the conclusion *appears* onerous
...e.g., my perfect correlation copula returns a Minimum(p(A), p(B) function which implies, for example, PD(A) = 5% and PD(B) = 8% such that PD(AB) = min() = 5%
...so I feel the conclusion that P(A) must = P(B) is too strict but i cannot justify by finding a fallacy??
uggg...apparently simple problems can be so hard

David

 

[ajsa]

Hi David,

Practically thinking, I feel it makes senese. If A and B's correlation is 1, that means they must happen at the same time so have the same prob..

Just my 2 cents.. Thanks..

 

[David Harper CFA FRM]

Hi ajsa - I was thinking the exact same thing...although i did go to my copula XLS (http://www.bionicturtle.com/premium/spreadsheet/5.d.2._gaussian_copula/) and i input PD(5%), PD(8%) and default correlation of 1.0 (0.9999999) and it correctly returned joint PD of 5% (per the copula minimum function) .... ergo i cannot seem to reconcile ??! ...David