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#### jyothi1965

##### New Member
Hi David

While evaulating portfolio credit risk (in the sense of all the credit models), the concept is that higher positive correlations leads to higher portfolio risk, and negative corrleations lower portfolio risk.

Now when you consider a basket CDS where the reference asset is a basket of debt securities, the logic is inverted. Higher correlations actually lower the swap premium and vice versa.

The question is: Swap premiums (for either the 1st to default or N to default) are based on the number of defaults taken individually or taken as group? If correlations are high, then if one
asset defaults (in a nth to default) the nth default will be reached quickly..and then swap premiums have to be higher not lower...

Hope my question is clear......

Jyothi

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Jyothi,

The swap premium formula (the model-priced spread value of the nth-to-default CDS) must be aware of the group of reference assets, if i understand the question. Simplifying assumptions can be made (e.g., distribution is same for all assets) to get an easier formula, but the model boils down to something like "if it is a 2nd-to-default, what is the probability that cumulatively over the time period at least two of the assets will default." So, it is very much like asking of a (discrete) binomial distribution, what are the odds of 2 OR MORE "successes/failures" in ten trials. As such, to try and answer the question, it answers a cumulative probability based on finite individual events, but must include an "awareness" of the total reference basket, I mean, of all individual reference assets (e.g., if you add a even single speculative--high PD--asset to an otherwise high quality basket, that must have an impact on your nth-to-default probability)

But the first part of this, I struggle with too, frankly: for either a CDO or a basket CDS, higher correlations will lower the odds of at least one or two defaults (high correlation --> lower value of 1st or 2nd to default). But keep in mind we are talking just about the tail of the distribution, a non-normal distribution. The tail behaves differently than the center. So, for example if we have a 10 assets with 1% PD each, and they are perfectly correlated, the nth-to-default probability is 1%. (1st-, 2nd-, 9th-...). Now start to lower the correlation. At independence (correlation = 0), the probability of at least one default (1st to default) is about 9.5% (1 - 99%^10). Lower correlation increases the odds of at least one default, from 1% to 9.5%! But we are just at one end of the tail. What happened to the (ridiculous) 10th-to-default? Lower correlation decreased its chances from 1% to infinitesimal (1%^10).

Similarly, measures like unexpected portfolio loss won't behave directionally like the 1st or 2nd-to default. Higher correlation will tend to increase the credit portfolio's unexpected loss even as the 1st-to-default odds decrease...

hope that helps! as usual, you spot the most interesting issues....

David