Credit Default Swaps

Variant: Index CDS Default Payoff and Remaining Notional

Party X is the protection buyer in a five-year $100 million notional CDS index containing 125 equally weighted entities. One constituent, Company A, defaults. After default, its bonds trade at 30% of par. 1. What payoff does Party X receive? 2. What is the CDS index notional after the default?

What is happening?

A CDS index is like buying default insurance on a basket of companies rather than on one company.

$100m CDS Index
      |
      +-- Company 1
      +-- Company 2
      +-- ...
      +-- Company A  <-- defaults
      +-- ...
      +-- Company 125

Because the 125 companies are equally weighted, each company represents the same fraction of the $100 million index notional.

1. Find Company A's notional

We currently have:

• Total index notional = \(100\) million
• Number of entities = \(125\)

Therefore:

So Party X has effectively bought protection on $800,000 of Company A debt.

2. Understand the 30% bond price

After default, Company A's bonds trade at 30% of par.

This is the recovery rate:

If $1 of debt defaults but is still worth $0.30, the investor has actually lost:

So the loss given default (LGD) is:

3. Calculate the CDS payoff

Intuition: Party X had $800,000 of Company A exposure. After default, $240,000 is recoverable:

The protection seller therefore replaces the missing $560,000:

4. Calculate the remaining index notional

Once Company A defaults, its $0.8 million notional is removed from the continuing index:

The remaining 124 companies continue to be covered.

Reusable pattern

Variant: Calculating Survival Probability from Annual Hazard Rates

Consider a five-year senior CDS on Xeon Corp. Its hazard rate is 2% in Year 1 and increases by 1 percentage point each year. Calculate the probability that Xeon survives all five years without defaulting.

What is the question asking?

We want the survival probability: the probability that Xeon makes it through every year without defaulting.

The annual hazard rates are:

What is a hazard rate?

A hazard rate is the probability of default during that year, conditional on the company having survived up to the start of that year.

So if Year 1 hazard is 2%:

Therefore:

1. Convert every hazard rate into a survival probability

2. Surviving five years means surviving Year 1 AND Year 2 AND ... AND Year 5

Therefore multiply the conditional survival probabilities:

Intuition: Imagine starting with 100 companies. Roughly 98 survive Year 1. Of those survivors, 97% survive Year 2, then 96% of those survive Year 3, and so on. Each year's surviving population becomes the starting population for the next year.

Start
  |
100%
  | × 0.98
  v
98%
  | × 0.97
  v
95.06%
  | × 0.96
  v
91.26%
  | × 0.95
  v
86.69%
  | × 0.94
  v
81.49%

Reusable rule

For annual hazard rates \(h_1,h_2,\ldots,h_T\):

Recognition pattern: hazard rates across several periods + probability of surviving to maturity → convert each hazard into \(1-h\) and multiply.

Variant: Cheapest-to-Deliver Bond in a CDS

Party X is the protection buyer in a $10 million notional senior CDS on Alpha, Inc. Alpha defaults. After default:

• Bond P: subordinated unsecured, trading at 15% of par
• Bond Q: senior unsecured, trading at 25% of par
• Bond R: senior unsecured, trading at 30% of par

What is the payoff on the CDS?

What is the question really asking?

After Alpha defaults, the CDS protection buyer is compensated for the loss on an eligible bond of the reference company.

The key phrase is senior CDS.

That means we cannot automatically choose the bond with the lowest market price. We must first ask:

Which bonds have the required seniority?

• Bond P = subordinated → not eligible
• Bond Q = senior unsecured → eligible
• Bond R = senior unsecured → eligible

1. Find the cheapest eligible bond

Among the eligible senior unsecured bonds:

Therefore:

Why? A $100 face-value Bond Q can be obtained in the market for only $25, versus $30 for Bond R.

The fact that Bond P trades at only $15 does not matter because its subordinated status makes it unsuitable for this senior CDS.

2. Translate 25% of par into recovery

Bond Q trades at 25% of par.

Think of a $1 claim:

• Face value promised = $1.00
• Value recovered after default = $0.25
• Economic loss = $0.75

Therefore:

3. Calculate the CDS payoff

The CDS protects $10 million of notional.

Substitute:

Intuition: The protected claim was worth $10 million before default, but the cheapest eligible defaulted obligation is now worth only:

The CDS fills the $7.5 million shortfall:

Reusable recognition pattern

Default occurs
      |
      v
Which bonds are eligible?
      |
      v
Keep only correct seniority
      |
      v
Choose lowest-priced eligible bond
      |
      v
CTD price = recovery rate
      |
      v
Payoff = Notional × (1 - Recovery)

Rule: Eligibility first → cheapest bond second → recovery third → CDS payoff last.

Variant: Upfront Premium and Price of a CDS

A 10-year CDS on Alpha, Inc. has a fixed coupon of 5.0%, while the current market CDS spread is 3.5%. The CDS duration is 7. Calculate the approximate upfront premium and the price of the CDS.

What is the question asking?

A CDS has two different rates here:

• CDS coupon = 5.0% → the contractual annual premium the protection buyer must pay.
• CDS spread = 3.5% → the premium the market currently thinks is fair for Alpha's credit risk.

But the buyer is contractually being asked to pay 5% when fair compensation is only 3.5%.

So the CDS coupon is too high.

The protection buyer must therefore receive money upfront to compensate for these excessive future coupon payments.

1. Measure how far the coupon is from the fair market spread

The negative sign means the buyer's contractual coupon is 1.5 percentage points too high each year.

2. Why multiply by duration?

The 1.5% difference occurs over many future premium payments.

Duration = 7 approximately converts this annual spread difference into the present value of all those future differences.

Think of it as:

Therefore:

Substitute:

For every $100 of CDS notional, the buyer therefore receives approximately:

Intuition: You agree to pay $5 per year for protection that the market says should cost only $3.50. Because you are overpaying in the future, the seller gives you compensation today.

3. Convert the upfront premium into a CDS price

The pricing convention is:

Here the upfront premium is negative:

So the CDS is quoted at:

Reusable rule

Spread vs Coupon
      |
      +-- Spread > Coupon --> buyer pays upfront --> Price < 100
      |
      +-- Spread = Coupon --> no upfront       --> Price = 100
      |
      +-- Spread < Coupon --> buyer receives upfront --> Price > 100