FINM3405 Revision - CDS

1. Basic definitions and concepts for credit default swaps.
2. More precise description of mechanics, including notions of:

• 2.1. Analogy with insurance contracts.
• 2.2. Parties: protection buyer and seller.
• 2.3. Reference entity, assets and events.
• 2.4. Payout, including physical vs cash delivery.
• 2.5. Recovery rate and loss given default.
• 2.6. CDS coupon or spread or premium.
• 2.7. Single name vs multi name, basket, CDS indices.
• 2.8. Idea that CDS spreads reflect credit risk perceptions, and relation between breakeven CDS spread and reference entity’s risk premium.
• 2.9. Survival and default probabilities.

Notation

So far we have the following CDS notation and terminology:

• $F$ is the notional principal or face value.

• $R$ is the recovery rate.

• $k$ is the CDS spread.

• $kFd$ is the CDS premium, where $d$ is the time between premium payments in years (say $d = 1/4$ for quarterly).

• $(1 − R)F$ is the payout upon occurrence of a reference event.

• $t_1 , . . . , t_N$ is the premium payment and payout dates.

• $r_1 , . . . , r_N$ is the risk-free rate yield curve for these dates.

• $q_i$ is the (risk-neutral) probability of default in period i.

  • Probability of a reference event occurring in period i.

• $s_i$ is the survival probability up to time $t_i$ .

  • Probability that no reference event occurred up to time $t_i$.

• $d$ is the time between premium payments in years

2. Mechanics

2.1 analogy

• The CDS buyer “takes out insurance” against a credit event (e.g. default) of a company or sovereign state and in return pays the CDS seller a regular, fixed “insurance” premium for the insurance.
• The CDS seller receives the CDS premium and agrees to pay the buyer a compensation amount upon occurrence of the credit event.
• If the credit event occurs, the compensation is paid, the CDS buyer pays the period’s accrued premium, and the CDS ceases to exist.
• If no credit event occurs, the CDS buyer keeps paying the periodic premium until the CDS maturity date.

2.2. Parties

• The CDS buyer is called the protection buyer.
• The CDS seller is called the protection seller or swap writer.

2.3. Reference entity, assets and events

Reference Entity

• The company or sovereign in the CDS is the reference entity

Reference Assets

• The particular debt securities (bonds, loans, floating rate notes, etc) of the reference entity over which the CDS is written are called reference assets or bonds or securities or obligations.

Reference Events

• The credit events relating to the reference entity’s reference assets as specified and defined in the CDS are called reference events.

Reference events specified in a CDS include, but are not limited to:

• Default: Failure of the reference entity to pay interest/loan/coupon payments or the principal or face value of the reference asset.
• Bankruptcy: Official or legal/judicial administration, liquidation or winding up of the reference entity rendering it unable to meet, either partially or wholly, its obligations of the reference asset.
• Restructuring: Corporate or debt restructuring that changes the nature of the reference asset in terms of its original structure and the original agreement between the reference entity and its creditors.
• Repudiation/moratorium: The reference entity disputing aspects of or failing to recognise the reference asset and its obligations, thereby disputing, delaying or refusing payment (common when the reference entity is a sovereign or sovereign state).
• Covenants: The reference entity’s breaching or noncompliance with debt covenants, including due to deterioration in financial position or the devaluation or writing off of assets used as security/collateral.
• Ratings downgrade: The downgrading of a reference entity by a credit ratings agency (such as S&P, Moody’s, Fitch, etc).

2.4. Payout, including physical vs cash delivery

The payout upon the occurrence of a reference event is defined as:

• A CDS is written over a notional principal or face value $F$.

CDS can be physically deliverable or cash settled upon the occurrence of a reference event (after which the CDS vanishes):

• Physical delivery: The protection seller agrees to buy the reference asset holding from the protection buyer at its total face value $F$.
• Cash settled: The protection seller agrees to pay the protection buyer the difference $F − V$ between the face value $F$ and the market value $V$ of the reference asset.

2.5. Recovery rate and loss given default

We define the important concept of the recovery rate $R$:

• It is the market value $V$ of the reference asset expressed as a percent of the notional principal $F$ after a reference event:

$$V = RF$$

• From another perspective, R tells us “how much we would recover” if we held a portfolio of the reference asset of total face value F and then sold it after a reference event occurred: We receive $V = RF$. It also follows then that the payout $F − V$ for a cash settled CDS is

$$\text{CDS payout} = (1 − R)F$$

Loss Given Default

This payout $(1 − R)F$ is sometimes called the loss given default since it’s how much we’d lose on a holding of the reference asset.

2.6. CDS coupon or spread or premium.

The regular, fixed “insurance premium” paid by the protection buyer to the protection seller is called the CDS coupon or spread or premium.

• The premium $C$ is given as a percent $k$ of the notional principal:

$$\text{premium} = C = kFd$$

with $d = 1/2$ if semiannual premiums, $d = 1/4$ if quarterly premiums, etc. The market convention is premiums are usually paid quarterly.

• Here, $k$ is also called the CDS spread.

2.7. Single name vs multi name, basket, CDS indices.

The CDS can be single-name or multi-name:

• Single-name CDS are written over the reference assets of one single reference entity.
• Multi-name CDS, sometimes called basket CDS, are written over the reference assets of multiple reference entities, and the nature of their payouts can differ (add-up basket, kth -to-default basket, etc).

There’s also CDS indices (see wiki), which are effectively basket CDS:

• The CDX range (also see Investopedia and the CDS Indices Primer).
• The iTraxx range (also see Investopedia and wiki)

2.8. Idea that CDS spreads reflect credit risk perceptions, and relation between breakeven CDS spread and reference entity’s risk premium

Why is k called the CDS spread?

Let $r$ be the risk-free rate, or a reference rate such as Term SOFR or Euribor which are effectively risk-free rates. Also let $y$ be the yield on the reference entity’s reference asset.

• Then $y − r$ is the reference asset’s risk premium or spread.
• k rises (falls) when this perceived credit risk rises (falls).

2.9. Pricing

Buyer

From the perspective of the protection buyer:

• The regular premium payments $kFd$ are cash outflows.
• The payout $(1 − R)F$ is a cash inflow.

So the value of a CDS to the protection buyer is

$$\text{CDS value = PV( E [payouts]) − PV( E [premiums])}$$

Payouts

PV of expected payouts: At time $t_i$ there is a payout of:

• $(1 − R)F$ with probability $q_i$.
• $0$ otherwise.

Hence, the expected payout at time $t_i$ is

$$\mathbb{E}[\text{payout}_i]=q_i(1-R)F$$

Present value of all payouts is

$$\text{PV(}\mathbb{E}\text{[payouts])} = \sum ^N _{i=1} e^{-r_i t_i} q_i (1-R)F$$

Premiums

At time $t_i$ there is a premium of:

• $kFd$ with probability $s_i$.
• 0 otherwise (note that we’re ignoring any accrued premium).

Hence, the expected premium at time $t_i$ is

$$\mathbb{E}[\text{premium}_i] = s_i kFd$$

The present value of all premiums are:

$$\text{PV(}\mathbb{E}\text{[premiums])} = \sum ^N _{i=1} e^{-r_i t_i} s_i kFd$$

Pricing

The value of a CDS is:

$$V_{\text{CDS}} = \text{PV(}\mathbb{E}\text{[payouts])} - \text{PV(}\mathbb{E}\text{[premiums])}$$

The breakeven CDS spread k that which gives the CDS 0 initial value:

$$\text{PV(}\mathbb{E}\text{[payouts])} = \text{PV(}\mathbb{E}\text{[premiums])}$$

which we rearrange to get

$$k= \frac{\text{PV(}\mathbb{E}\text{[payouts])}}{\sum ^N _{i=1} s_iFd}$$

$$k= \frac{\sum ^N _{i=1} e^{-r_i t_i} q_i (1-R)F}{\sum ^N _{i=1} s_iFd}$$