Computing value of a CDS, Hull chapter 24

[Hend Abuenein]

Hi David,

I hope you're doing well

First: With regards to solving for CDS spread, I don't understand the element of the accrual payments, step 3.

If in step one we assumed that payments will be made in full end of every year , since reference did not default and Swap is still alive, then why are any payments accrued if default occurs mid year?

I'm probably not making any sense, but I'm trying to sound my confusion about this.

If end of year 1 we (buyer of Swap) must pay a spread weighted by probability of survival, then in year 2 reference defaults, an amount will accrue to seller of swap. But the approach says to add up both accruals weighted by PDt, and payments weighted by survival probabilities, both in PV FOR ALL YEARS . How could we consider and sum up scenarios of paying (survival) and a counter scenario of default (accruals) through the life of the same swap?

Shouldn't there be scenarios equal to T years, life of swap, were accrual would be added only after/if default occurs in previous year?...then what do we do with scenarios

I got things pretty scrambled here. Please help clarify.

Second: the point of equating PV of payments from buyer to seller, with payoff to buyer from seller in step 4, is that a Swap should not provide an arbitrage opprtunity to either side.
But is that realistic? What would be in it for all the CDSs issuers then?

Thank you

 

[David Harper CFA FRM]

Hi Hend,

Thanks, I hope you are doing well too!

These Hull tables are difficult (you are probably aware that my reconstruct of the same is located in XLS at http://www.bionicturtle.com/how-to/spreadsheet/2011.t6.c.4.-cds-valuation)

Here is how i cope with it: exactly as you say, the spread solves for PV of both legs equal to the same value. The calculations, on both side, are performing a discounted expected value; it is very much like the discounted expected value that occurs in the binomial, in the sense that we are weighting all the future outcomes (which collectively sum to 1.0 or 100%, but in the case, over time so each year has its own probability) and discounting that average value.

So, in the case of the protection buyer, let's say that is you, and I am the seller, you will be making annual CDS premium payments, unless and until there is a default. If default, under the MODEL's assumption, a default occurs at mid-year. So, if there is a default, you must pay me the final 0.5*CDS premium, then I will settle up with you by making the PAYMENT.

So, with respect to discounted expected value, we can omit the "discounting" temporarily and just focus on the expected premium. What is your expected premium? It is just a weighted average; and like any probability, we do it correctly if the probabilities sum to 1.0. So, you are looking at the following future:

As of valuation (today), unconditional probabilities are, if annual PD = 2%:
Year 1: 98% survive or 2% default; ... if default, we are done,
... but if survive, the 98% survive contains two possibilities:
Year 2: 98%^2 = 96.06% survive or 98%*2% = 1.96% default conditional on survival; ... if default, we are done,
... but if survive, the 96.06% (unconditional) survival probability contains two possibilities in year 3:
Year 3: 98%^3 = 94.1% survive or 2%*96.06% default conditional on survival

(It's helpful to see that the survival probabilities are unconditional and the PDs are, defaults conditional on survival)

But, all this does is "create a map" of the future possibilities (not totally unlike a binomial tree!):

Year 1: survive or default (sum to 100%); expected payment is weighted average
Year 2: survive or default (sum to probability of Year 1 survival); expected payment is weighted average
Year 3: survive or default (sum to probability of Year 2 survival); expected payment is weighted average
.... it actually could be a full binomial tree, imo, if we imagine that all default nodes contains (implicitly) further default nodes with zero values, so it's like the default section of the tree is empty because you won't pay premiums in that space.

That's how i cope: I see it as a big weighted average, over time.
(if we included the zeros for triangle of years subsequent to default, then we would have 5 years * 100% for each year = 500% probability, such that all payments are weighted by their probability, including a triangle of zeros )

Your second point, imo, is excellent and CORRECT, and consistent with other FRM readings (nice!): we have no reason to expect the CDS seller to be risk neutral such that he/she (you) will sell me the CDS for the no-arbitrage price. Any more than we expect a future contract to trade at the expected future spot. We should expect, just as you imply, the seller to demand compensation for assuming the risk. The only thing is, you've now opened the door to other factors not in the no-arbitrage model, yes? If you are going to do that, well, me as the buyer, I get to charge for my counterparty risk (which is not in Hull's model! We should deduct a CVA charge) ... we can keep going with additional fundamental factors (your risk aversion charge, my counterparty risk charge would be fundamental) and then we still have technical factors not in the model (supply/demand). For the reason, the so-called CDS basis has many factors.

I hope that helps, thanks!

 

[David Harper CFA FRM]

I just want to add to that final point, only because it only came into focus in the FRM after the crisis with the addition of the (excellent, imo) Canabarro reading on CVA. Previously, we frankly didn't even think about counterparty risk in the CDS value model (it is still nowhere in Hull!).

But we could CVA-adjust this model; i.e., the protection seller has de minimus counterparty risk such that we might round down to zero. Then, the CDS buyer would deduct a CVA charge = Exposure * EL.

So, Hend, this would offset your risk compensation; e.g., if we agreed to a PV no-arbitrage price of X for the CDS, you would be right to want to increase it for compensation (i.e., risk aversion as opposed to risk neutrality) but I would offset with a deduction for my CVA (counterparty risk that you won't make the payment). Both concepts are ripe in FRM P2 but neither are in Hull's model.

 

[Hend Abuenein]

Thanks David,

it actually could be a full binomial tree, imo, if we imagine that all default nodes contains (implicitly) further default nodes with zero values, so it's like the default section of the tree is empty because you won't pay premiums in that space.

This is where I lose it...if the default section of the tree is empty when we're making payments due because of survival, then why does the model add default weighted payments (accrual) to survival weighted payments for the same years?
It's like saying : year one has expected payment for default AND another for survival! But default and survival are two mutually exhaustive probabilities. meaning that payments for a single year should be either because debt survived OR because debt defaulted, not both payments for the same year, every year.

Do you see my point?

Thank you for your reply on second, understood.

 

[David Harper CFA FRM]

Hi Hend,

Yes, I think i see your point because it is what gives me trouble ... they ARE INDEED mutually exclusive WITHIN a year, but note they don't sum, within a year, to more than 100%. Just like your expected payment for a coin toss is = [50% * (whatever you pay me if heads)] + 50% (whatever you pay me if tails)], your first year's expected payment = (CDS premium)*98% + (accrual)*2%.
Then, next year, the two probabilities are similarly mutually exclusive: Your expected payment in Year 2 = (CDS premium)*98%^2 + (accrual)*(98%*2%); or, we could fairly say that EACH YEAR has a 100% probability such that year 2 is given by:
Expected value of your payment = (CDS premium)*98%^2 + (accrual)*(98%*2%) + (zero)*2%; i.e., as of today (i.e., unconditionally), you have three possible outcomes for your year 2 payment:
1. either you will be paying the premium with p = 98%^2, or mutually exclusively,
2. you will default in year 2, such that you pay the accrual with p = 98%*2%, or mutually exclusively
3. you will have earlier defaulted, such that you will be paying 0 (in year 2) with p = 2%.
Sum of those p = 100%. But each of five years does have it's own p = 100%

Thanks,

 

[Hend Abuenein]

Thank you