Hi David,
I have a small doubt about correlation in Nth default swap.In chapter 22 of Jorion (page 504) he mentions that type of protection depends upon correlation between credit events(is he referring credit events to default).Lower the correlation more expensive the swap(higher cost of protection) and vice versa. Does this imply basket swap premium is higher with low correlation of default? He then mentions in the next para that when N is large the higher default correlation would lead to higher cost of protection. It seems contradicting to me.
I have a doubt please correct me if Iam wrong CDS is more of an option as compared to it being a swap. The reason that the CDS seller (protection buyer) makes regular payment to CDS buyer (protection seller) until a credit event happens. The TROR is more of a swap as payments are made by both the parties simultaneously and also it covers economic risk (including fluctuation in market interest rates).

Regards,
Sunil

Hi Sunil,

On the apparent contradiction, Jorion refers to the difference (if the example is, say, a reference basket of 100 credits) between a 1st-to-default and a 99th- or 100th-to-default. On this issue (default correlation in the basket), we could refer to either a basket CDS or a CDO. You may find this post from last year helpful. So here we can equate:

high nth-to-default basket CDS ~ senior CDO tranches
medium nth-to-default ~ mezzanine CDO tranches
low nth-to-default (e.g., 1st-to-default) ~ junior/equity tranches

And now the “classic argument” is:
high correlation makes senior tranches (i.e., high n to default for basket CDS) more expensive (all other things equal) and junior tranches (e.g., 1st to default) less expensive;
low correlation makes senior tranches less expensive and junior tranches more expensive. You can see in my post that the binomial distribution can be used to illustrate this.

In a nutshell, if you have 100 credits with each 5% PD, consider a 100th-to-default (analogous to senior CDO tranche): if uncorrelated, the probability of triggering this basket CDS is virtually zero; = 5%^100 = ~ zero. Now increase correlation to 1.0 (perfect) and PD goes all the way up to 5%! Ergo, for the senior tranche, increase correlation increases the spread.

Now consider a 1st to default (junior tranche), what is the probability of triggering 1/100? Fully 99+% = 1-95%^100. Now increase correlation to perfect (1.0) and the probability DECREASES to only 5%! Ergo, for the junior tranche, increase correlation decreases the spread.

(The above is the classic line and the exam will not go further than this. If you are interested, smart folks at Kamakura have pushed this issue further).

“CDS is more of an option as compared to it being a swap.”
Yes, agreed, it is sometimes called an option rather than a swap. But on the other hand, notice Hull’s valuation consists of equating two swaps: the certain premium payments with the uncertain, contingent payoff. Aside from Culp, in my opinion, this point you make is why most authors use different terms to describe the counterparties. The TROR is truly a swap so you see payer/receiver, but the CDS is (arguably) a true credit derivative so you see protection buyer/seller.

“TROR is more of a swap as payments are made by both the parties simultaneously and also it covers economic risk (including fluctuation in market interest rates).”
Yes, true, so from a risk perspective, this is why Meissner shows that the TROR is a swap that covers more risk types (default, credit deterioration, and market risk). He would categorize a drop in interest rates as a MARKET RISK: The TROR hedges such a risk, the CDS does not. Further, the TROR hedges credit deterioration (downgrade) and a CDS may or may not (yes, if market to market, no if otherwise)

David

Hi David,

Credit default swaps written on multiple names, the first-of-basket to default swap give the protection buyer to deliver one and only one defaulted security out of a baset of selected securites, ie the trigger event.What is the pay off to the protection buyer ? Does it cover the first default asset only ie.par minus the recovery value)?. If it is 9th to default swap, the cover amount will be on the 9th protected asset only (those defaults before 9th default assets will not be cover? Thus, spreads for 2rd to default swaps will be less than 1st to default swaps?

Regards,
Daniel

Hi Daniel,

Yes, the nth-to-default is exactly as you say; e.g., in a 3rd-to-default, nothing happens upon the first two defaults in the basket (regardless of which credits default), then upon the 3rd default there is a credit event trigger and the buyer payoff is just like a single-name CDS: if physical, the protection buyer can deliver the defaulted credit and receive par/face, or if cash, the buyer receives LGD or “net loss” not gross loss (par/face of defaulted bond only minus recovery on defaulted bond only, not the basket). Either way, the protection buyer is receiving LGD (net loss) on the defaulted credit only, not the cumulative (in the vanilla design, anyway), just like a single name CDS.

So, ceteris paribus, the 3rd-to-default must earn the seller a lower premium than the 2nd-to-default which in turn must earn a lower premium than 1st-to-default (FTD).

The pricing of nth-to-default is very interesting as the correlation above enters strongly. For example, if the basket has 5 credits and each single name CDS has a premium of 20 bps, then if the default correlation is perfect (1.0), the FTD basket premium would also be 20 bps (default on 1 = default of all!); but if default correlation is zero, then FTD basket premium = ~ 100 bps (20 * 5) ... as above, lower correlation implies higer spread on the junior tranche (FTD CDS ~ equity tranche in CDO).

Now the 2nd-to-default, for the SAME GIVEN CORRELATION, must offer a lower premium. However, default/asset correlation is a very strong determinant; the 2nd-to-default premium could be higher than a FTD premium, if the basket is sufficiently large and the correlation is lower for the 2nd-to-D (or if it’s a small basket, and the 2nd-to-default is more mezzanine, correlation can act the other way!)...so in regard to any mezzanine tranche (e.g., 2nd-to-default, 3rd-default) we generally say the response to correlation is “complicated” and depends on the particulars

David

Hi David,

If I invested in the senior tranches asset, I am protected due to the subordination of credit enhancement for the senior claim. I will receive less coupon income compared with equity trench.  If the assets have higher default correlation, it means senior tranches become more risky. Therefore, I have received less coupon income but to incur higher risk (with high assets default correlation) for the senior trench of asset I hold. However, I should be compensated with high risk I take. I see I have problem in the logic. Could you help?

Regards,
Daniel

Hi Daniel,

I snapped a picture (with my new droid) from o’kanes
http://www.amazon.com/Modelling-Single-name-Multi-name-Derivatives-Finance/dp/0470519282/

What you say is exactly true of both dynamics (seniority and default correlation). They co-exist as inputs, so when talking about default correlation as a first pass, we typically really mean “impact of correlation for a given tranche (ceteris paribus)” and when talking about credit spread it’s “credit spread up/down for a given correlation”

David