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# First to default put ( crouhy)

#### saurabhpal49

##### New Member
Hi David
Could you please explain the below mentioned sentence :
> First to default spread will lie between the spread of the worst individual credit and the sum of the spreads or all the credits- closer to the latter if correlation is low and closer to the former if correlation is high.

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @saurabhpal49 I'd explain with a simple illustration. Imagine the basket contains only two credits, each with (unconditional) default probability of, respectively say, 10.0% and 7.0%. Also, let's simplify and approximate spread by S = PD*LGD but assume LGD = 100%, so that spread approximated default probability. Now compare:
• If these credits are uncorrelated, what is the approximate spread on a first-to-default (1tD) CDS? It is the probability of either credit defaulting which is given by 1 - 90%*93% = 16.30%. Notice that's almost 10 + 7%, but these are uncorrelated and default by either triggers the basket! (as opposed to joint default which is probability = 10%*7% and would trigger a 2nd-to-default)
• Now push correlation up to 1.0. Now if either defaults, the other defaults also. But this is actually better for the writer of this 1TD basket because it goes the other way too (!): if one survives, the other survives also. Now the probability of either triggering is really just the probability of "the worst individual credit" which is 10%. So here as correlation increases, the spread decreases. I hope that's helpful!

#### GeniusGenie

##### New Member
@David Harper CFA FRM that is a neat explanation, thank you.

I think this illustration should be added to P1.T1_Chapter 04 notes. Appreciate your help, nonetheless.

Thank you.