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# credit Swap Pricing

#### jyothi1965

##### New Member
David

I picked this one up from the PRM set of questions:

A firm has two outstanding bond issues: a 6 percent coupon bond with one year to maturity
trading at a spread of 88 bps over Treasuries and a 9 percent coupon bond with ten years to
maturity trading at a spread of 340 bps over Treasuries. The one-year and ten-year Treasuries
are trading at 4% and 5% respectively. Both the bonds rank pari passu and have an estimated
recovery rate of 30%. What is the minimum upfront premium that a dealer will charge to sell a
one year credit swap to an owner of the 10-year bond?

A. 85 basis points.
B. 88 basis points.
C. 327 basis points.
D. 340 basis points.

The answer is simplistic: (considering that reading the question takes a couple of minutes!!!)

18. Correct answer: A
Since all the bonds rank equally, they would default at the same time. Therefore, the dealer
could hedge a one-year credit swap on the firm (for whichever bond) by selling the one-year
bond and purchasing one-year Treasuries. This results in a cost of 88 bps at the end of the
year, or 85 bps upfront {= 88 / (1 + 4%)}.

This seems an intelligent question, and and I am curious to know how to interpret the answer.

(How can equal ranking mean that they would default at the same time?)- unless it is a credit event.

Thanks as always

J

#### David Harper CFA FRM

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

Hmmm...I would get this wrong, I must not get something. (The setup leads to wonder if i am supposed to compute par yields?) But the n-year spread should be near the spread over n-year treasury (or LIBOR - XX bps more or less). So, I like the concept of the answer, which I interpret simply based on no-arbitrage idea: owning the risky bond + long the CDS should = owning the default-free bond. That is, the market spread should approximate the CDS premium.

But I don't understand why the cross hedge applies just because they are pari passu. If the CDS @ 85 bps, then don't i have an arbitrage: if i buy the +340 bps bond plus protection @ +85/55 bps, I profit more than risk-free rate? So i don't get that!? What makes more sense to me is 340 bps/(1+ 4 or 5%). Unless the 9-6 = 300 bps implies some conversion but the terms don't quite suggest that

I admit I've not seen the conceptual link from PP to equivalent PD...I may misunderstand...is not that contradicted by the market implied (marginal) probability of default? Sorry..the question leads to questions that seem to outpace the simplicity of the answer. (that's why bond questions, more than any other, need to be very specific in their terms)

David

#### jyothi1965

##### New Member
David

The solution (which struck me when I was casually browsing through Messiner) is as follows:

The no arbitrage condition for a CDS pricing is:

Return on Risk free bond = Return on Risky bond - Default swap premium or

default swap premium = Return on Risky bond - Return on Risk free bond

By this eqn, the swap premium for asset 1 is 88 bps, which when paid upfront is discounted by the risk free rate and hence the answer is 85 bps.

The more important issue is that the owner is buying a CDS on the one year bond rather than on the ten year bond.

Why?

If they are ranked paripassu, the default on any one of them will be considered a credit event and the payout will happen.

Therefore the owner of the ten year bond would rather buy a CDS on the 1 year bond (because it is cheaper) even though he owns the ten year bond - because default on the one year will automatically lead to default on the ten year bond.

The question is the tenure of the CDS is for one year. So will the investor keep buying a one year CDS each year for the ten years (assuming that he holds the ten year bond to maturity)- strip hedging?

Jyothi

#### David Harper CFA FRM

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

The no arbitrage part makes sense

But, the "more important issue" still confuses me: a single name CDS protects the bond - it's not a 1st-to-default basket that covers both the 1- and the 10-year bond. So, this would still be an (imperfect) cross-hedge. I'm not clear on why pari passu earns them equivalence (PP = unsecured, contractually equivalent. So, one could still default before the other...)

same conundrum, the hedger earns +340 over treasury and pays 88 bps (in arrears), so the hedger is + 252 bps over treasury with a cross-hedge....free lunch or...not quite a free lunch due to (i) the cross hedge and (ii) the roll-over from year to year. That's a great point on the strip hedge...

your question does say "minimum" so perhaps this is the LOWER BOUND...

(but this part i like: "I was casually browsing through Messiner"...some people flip thru magazines )

UPDATE: you know, unless default of a pari passu bond does qualify as a credit event under ISDA criteria; e.g., cross default can be a trigger...UPDATE #2: I woke up this AM i realized the cross default doesn't make sense: you buy the single-name CDS on the 1-year bond, if the 10-year defaults, although it may trigger credit event, delivery would still be based on net recovery of the 1-year, so I am back to not believing this question...