Basket CDS Value & Correlation

[RomanS]

Hi,
another question that left me puzzled due imprecise question format from Schweser's Practice Exam 2010 L2 E1 Q78.

Question
A particular basket credit default swap (CDS) has 100 reference entitites and makes a payoff when the 25th reference entity defaults. Which of the following statements regarding this basket CDS is correct?

Answer Options
a) If the default correlation is equal to 1, the value of the basket CDS will be greater than the value of a 1st-to-default CDS.
b) If the default correlation is negative, the value of the basket CDS will be greater than the value of a 1st-to-default CDS.
c) If the default correlation is zero, the value of the basket CDS will be less than the value of a 1st-to-default CDS.
d) As default correlation increases, the value of the basket CDS will decrease.

Correct Answer
c)

My problem here is not so much what happens when correlations change but the question doesn't say from which perspective it wants the candidate to look at the swaps, i.e. from the protection buyer or protection seller. From the answer I take that they want us to take the protection buyer perspective. Would you agreed?

Perspective as a Protection Buyer
Perspective as a Protection Seller

a) If the default correlation is equal to 1, the value of the basket CDS will be greater than the value of a firts-to-default CDS.
>>> incorrect from either perspective b/c it should hold roughly that V(1st) = V(25th) since p=1.

b) If the default correlation is negative, the value of the basket CDS will be greater than the value of a firts-to-default CDS.
>>> incorrect: redcucing the default correlations implies a relatively larger likelihood that the 1st-to-default basket CDS will make a payment, i.e. as a protection buyer V(1st) > V(25th).
>>> correct: redcucing the default correlations implies a relatively larger likelihood that the 1st-to-default basket CDS will make a payment, i.e. as a protection seller V(1st) < V(25th).

c) If the default correlation is zero, the value of the basket CDS will be less than the value of a firts-to-default CDS.
>>> correct: redcucing the default correlations implies a relatively larger likelihood that the 1st-to-default basket CDS will make a payment, i.e. as a protection buyer V(1st) > V(25th).
>>> incorrect: redcucing the default correlations implies a relatively larger likelihood that the 1st-to-default basket CDS will make a payment, i.e. as a protection seller V(1st) < V(25th).

d) As default correlation increases, the value of the basket CDS will decrease.
>>> incorrect: increasing the default correlations implies a relatively larger likelihood that the 25th-to-default basket CDS will make a payment, i.e. as a protection buyer V(25th) will increase.
>>> correct: increasing the default correlations implies a relatively larger likelihood that the 25th-to-default basket CDS will make a payment, i.e. as a protection seller V(25th) will decrease.

 

[David Harper CFA FRM]

Hi RomanS,

You raise an important point that often leads to confusion. There is a difference between (i) the probability of event/default and (ii) the price or value of CDS. The question refers to the VALUE (a.k.a., price) of the swap and this is the correct way to avoid confronting the difference between the two buyer/seller perspective; i.e., the (mark to market) fair value/price will be the same to both, even as we can AGREE about the perspective differences with respect to default likelihoods, so it does not matter if we view from perspective of buyer/seller. Our assumption is that, upon a change in correlation or underlying PD, the mark-to-market price clears to the same price for both (the new PV of the premium leg = new PV of the contingent payoff leg).

Consider option (d): as default correlation increases:
* the senior tranche (high nth-to default) is more likely to trigger
* the cost of protection is therefore higher: the CDS spread must increase (protection seller must earn more). Or, equivalently, the present value of the spread payments increases
* for both buyer/seller, the market-to-market swap is more expensive; i.e., higher value, higher price
* so, we say: for senior tranches, increase in default correlation implies an increase in price/value

(fwiw, my only slight quibble with an otherwise STRONG question is that mezzanine tranches have a complex, ambivalent reaction, and without the PDs, I would forgive (IMO) the assumption that 25th is acting like mezzanine. I guess with any realistic PD, the 25th is distributionally in the "senior tail" so it's a weak quibble....)

Thanks, David

 

[RomanS]

Hi David,
Thanks for opening my eyes. I kind of have the feeling that it is starting to make sense. Could you check my revised reasoning below and help me b) if correaltion is negative, please:

Question
A particular basket credit default swap (CDS) has 100 reference entitites and makes a payoff when the 25th reference entity defaults. Which of the following statements regarding this basket CDS is correct?

Answer Options
a) If the default correlation is equal to 1, the value of the basket CDS will be greater than the value of a 1st-to-default CDS.
b) If the default correlation is negative, the value of the basket CDS will be greater than the value of a 1st-to-default CDS.
c) If the default correlation is zero, the value of the basket CDS will be less than the value of a 1st-to-default CDS.
d) As default correlation increases, the value of the basket CDS will decrease.

Reasoning
a) p=1, incorrect b/c this is when the a 1st-to default basket CDS is worth the most
b) p<0, ???
c) p=0, correct since a 25th-to-default basket CDS is very unlikely to be triggered and is therefore worth very little whereas a 1-st-to-default basket CDS is veryl likely to be triggered and consequently the later is worth more
d) p increases, incorrect since a 25th-to-default basket CSD becomes more likely to be triggered, i.e. its value will increase

Best Roman

 

[David Harper CFA FRM]

Hi Roman,

Sure thing.

First idea is that the price [1st to default] must be at least greater than or equal to (>=) price [25th to default] regardless of correlation. If you and I each buy, respectively, a 1st-to-default and 25th-to-default on the same reference basket, the 1st-to-default must trigger first and necessarily before the 25th-to-default.

a) if rho = 1, then values are equal as default of the 1st credit implies default of all credits and therefore the 25th-to-default is triggered; this is the only (unrealistic) condition where values are equal

b) the value of the 25th- is always <= value of 1st, lower correlation to negative correlation just makes the more true. As correlation tends to --> -1.0, value of 25th becomes extremely low

c) agreed

d) agreed

Thanks, David

 

[RomanS]

Hello David,
That finally helped, I guess! I entirely focussed on the correaltion and forgot about the basic property that you now mentioned ( V(1-st) >= V(25th) regardless of correlation ) ! MANY THANKS !

Reasoning
a) cannot be true, see property above
b) cannot be true, see property above
c) can and most likley is true
d) incorrect as higher correlation implies higher trigger-likelihood and hence greater value

Best Roman

 

[Plirts]

Hi!
Thank you for this explanation, was very helpful, indeed. But I still have some Qn on this topic, hope you can clarify. Lower correlation --> senior tranches less risky, lower CDS premium, fine. But is this CDS premium yield+ (in this case according to price/yield relationship, price should be lower)? Also I can not intuitively understand "price[1st to default] >= price[25th to default]". 1st to default is more risky, therefore I would pay LESS for it (and demand higher return/yield).
Many thanks if you could come to this topic once again.
Best,
Kaari

 

[David Harper CFA FRM]

Hi Kaari,

I think the concept of "yield" is unnecessary here in the CDS, and may confuse due to the inverse price/yield relationship in the bond context.

If I could ask you to imagine that you are the CDS buyer such that your are buying credit protection from me, for example, on a SENIOR tranche. Under the current regime, you may pay me up-front X bps + 100 bps per year (quarterly); let's say for a senior tranche, you will pay under current assumptions up-front of 200 bps + 100 bps per year for five years. We might say the value or price of this CDS protection is the present value:
PV (CDS) = 200 + 100/(1+r)^1 + 100/(1+r)^2 + ... + 100/(1+r)^5; r is maybe LIBOR or something near risk-free
i.e., much like we price a bond by discounting to PV (and this is how Hull values the CDS leg in his CDS valuation, only there is no up-front)

Now let's assume that default correlation drops and the senior tranche becomes less risky:
With less likelihood of a payout, I cannot charge you as much premium, so maybe the price of the protection goes down to:
PV (CDS) = 90 + 100/(1+r)^1 + 100/(1+r)^2 + ... + 100/(1+r)^5; same (r) to discount
i.e., lower value/price, but still discounting at a similar ~ riskfree rate

Re price[1st] > price[25th]:
If you are the protection buyer and I am the seller, we must agree that you will pay me higher premiums for the 1st-to-default, as there is a much higher likelihood that I will need to make the contingent payoff to you. The PV[your CDS premiums] should roughly ~= the PV[my probability-weighted expected continent payout]. As the probability of my payout increases, you must be more willing to make large "insurance" payments, which increases the value of the CDS (from both of our perspectives!)

I hope that helps, thanks!

 

[Plirts]

David,
many thanks indeed! It helped a lot! "Spread" was the most confusing part/word for me, "premium" makes more sence.
Best,
Kaari