Basic CDS Pricing Concepts
Stock options are derivatives based on an underlying stock, and futures contracts are designed around delivery of some underlying, such as a commodity.
What do you think is the underlying in a credit default swap (CDS)?
No. While a bond can potentially be delivered in a CDS arrangement, consider what causes delivery in the first place.
Yes!
A derivative's value is based on the underlying, and the CDS value is based on credit of the reference entity. It's all about credit. Now a stock or a commodity trades and has a clear market value. But credit? Not really, no.
No. The entity exists, but it's not the value of the entity that makes the difference in a CDS.
Pricing a CDS is a bit of a challenge, and of course, a whole lot of fun. It all starts with a __probability of default (POD)__. Getting this concept in your mind is easier if you think about a tightrope walker who starts at one end and continues to the other, surviving three specific sections along the way. Perhaps there's a 1% chance of falling in the first section, a 2% chance of falling in the second section, and a 3% chance of falling in the third section. What do you know for sure if the tightrope walker is in the third section?
No. A 3% probability of falling here is just what it is. But you know that the first two sections were successful.
Absolutely.
This relates to the __hazard rate__, which is the probability that an event will occur given that it hasn't occurred already. The tightrope walker has a 3% hazard rate at this point. Looking at the whole rope, you still don't have a default probability, but you have three hazard rates.
With this idea in mind, look at a bond issue.
Maybe you're focused on a simple three-year bond with EUR 100 coupons and a EUR 1,000 face value. There are three payments of EUR 100, EUR 100, and EUR 1,100. Same hazard rates as the tightrope walker: 1%, 2%, and 3%. For each payment, these are the hazard rates that estimate conditional default probability. The issuer could "fall off the rope" at any of these cash flow obligations. Surviving each of these would have probabilities of 99%, 98%, and 97%, respectively, so the __probability of survival (POS)__ for the issuer is
$$\displaystyle 0.99 \times 0.98 \times 0.97 = 0.9411 $$.
What does that tell you about the default probability?
No. That's just the sum of the hazard rates here, but consider the probability of survival. The issuer either survives or it doesn't.
That's right.
Specifically, it's 5.89% since the issuer either defaults or doesn't default. Survive or fall. It's a binary. Here's a little picture to illustrate.
No. There's no reason here to suspect a greater than 6% probability of default.
Now with one more assumption, you can really do some calculations. Suppose that the recovery rate (RR) in default was 25%. So if default happened in Year 1, then instead of the regular payments, the bondholder would receive 25% of them: EUR 25 in Year 1, EUR 25 in Year 2, and EUR 275 in Year 3. But there's a 99% chance of surviving Year 1 and getting the first coupon. If default is in Year 2, then the cash flows would be EUR 100 in Year 1 (survived!), EUR 25 in Year 2 (default), and EUR 275 in Year 3.
Suppose default happened in Year 3. What would be the __loss given default__ in this case?
No. That would be the amount recovered in Year 3 but not the loss given default.
Exactly!
The issuer walked the tightrope for two years, but fell in the third. So a 25% recovery rate on that last cash flow of EUR 1,100 means that the loss given default here is
$$\displaystyle (1 - 0.25)1{,}100 = 825 $$.
Here is everything spelled out in the same tightrope style:
No. There's still a 25% recovery rate.
The expected loss is also obtainable here. It's a simple calculation.
$$\displaystyle \text{Expected Loss } = \text{ Loss Given Default } \times \text{ Probability of Default} $$
The first part is easy: a 1% chance of losing EUR 975 (that's 75% of all cash flows, ignoring the time value of money), or EUR 9.75. What is the probability of default in Year 2?
No. It can't be more than 2%. Walk the tightrope carefully.
Right.
There's a 99% chance of even reaching that point, then a 2% chance of "falling." So the probability of this default is
$$\displaystyle 0.99 \times 0.02 = 0.0198 $$.
And the loss given default here adds up to EUR 900. So this portion of the expected loss is
$$\displaystyle 0.0198 \times 900 = 17.82$$.
Then for Year 3.
$$\displaystyle 0.99 \times 0.98 \times 0.03 \times 825 = 24.0125 $$
The expected loss is then the sum of these pieces.
$$\displaystyle 9.75 + 17.82 + 24.01 = 51.58 $$
No. Walk the tightrope from the start to that point.
The same sort of simple probabilities can be applied to long-term bonds as well. The graphics and calculations would be more involved for most values, but not for default probability. For example, a 20-year annual coupon bond may be risky, with estimated probabilities of just 1% each year. That doesn't sound bad, but the math shows you how quickly these things add up.
$$\displaystyle \text{Probability of Default} = 1 - (1 - 1\%)^{20} = 1 - 0.8179 = 18.21 \% $$
Ultimately, the CDS spread quoted as a fair price of protection given both the recovery rate (RR) and probability of default (POD) can be expressed as:
$$\displaystyle \text{CDS Spread } = (1 - \text{RR}) \times \text{POD} $$
To summarize:
[[summary]]
The __credit valuation adjustment (CVA)__ is essentially the present value of credit risk for a loan. So what does that suggest as far as the value of a CDS hedge position?
That's not true; that would suggest that it's a risk-free loan.
Right. The hedge position should be a good estimate for the CVA.
There's really no need for a greater value, since it kind of represents the same thing.
Credit
A bond
The reference entity
The first two sections were successful
The probability of survival is now more than 97%
It's 6%
It's less than 6%
It's more than 6%
EUR 275
EUR 825
EUR 1,100
2%
A little less than 2%
A little more than 2%
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It should be zero
It should be the CVA or so
It must be greater than the CVA
