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Judge how credit indices and fixed coupons are employed when CDS trades are priced.

FRM® Part II · Credit Risk Measurement and Management · Credit Derivatives · FRM2.CR.13.d

The idea

Credit indices such as CDX North America Investment Grade and iTraxx Europe are equally weighted portfolios of 125 single-name CDSs. They have standard maturities, and a new series is rolled roughly every six months. The index pays a standardised fixed coupon, and CDS trades generally use fixed coupons such as 100 bp or 500 bp, so contracts are interchangeable. Because the coupon rarely equals the market spread, the difference is settled by an upfront payment. To a good approximation, the upfront payment is given by Upfront = (market spread − coupon) × risky annuity × notional. If the market spread exceeds the coupon, the protection buyer pays the upfront. If it is below, the buyer receives it. When a name defaults, the index pays (1 − R) × that name's share of the notional and the notional falls. Do not treat the quoted spread as the running payment.

Formulas

Upfront payment on a fixed-coupon CDS

upfront = (par spread − fixed coupon) × risky annuity × notional

par spread
the spread that would make the contract worth zero
fixed coupon
the standard coupon the contract pays
risky annuity
the present value of one unit of annual premium, allowing for default
notional
the notional amount

Positive means the protection buyer pays. Using the whole spread rather than the excess over the coupon, or leaving out the annuity, are the slips.

Par spread from a quoted upfront

par spread = fixed coupon + upfront ÷ risky annuity

fixed coupon
the standard coupon the contract pays
upfront
the upfront as a share of notional, positive when the buyer pays
risky annuity
the present value of one unit of annual premium, allowing for default

When the buyer receives the upfront it is negative and the par spread is below the coupon: 500 bp with 2.4% received and an annuity of 3.8 gives 437 bp.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which feature describes a standard credit index such as CDX North America Investment Grade or iTraxx Europe?

  1. A portfolio of 125 names weighted by debt outstanding, so large issuers have a proportionally larger effect on the index spread.Why not: Index constituents are equally weighted, not weighted by outstanding debt.
  2. A first-to-default basket on 125 names that pays out once and then terminates after the first credit event.Why not: Index protection pays on each default and the notional falls; it does not terminate at the first default.
  3. An equally weighted portfolio of 125 single-name CDSs with a fixed coupon and standard maturities, rolled periodically. - correct
  4. A floating-coupon contract reset each quarter to the market spread, so that no upfront payment is ever required.Why not: Index contracts carry a fixed coupon; the gap to the market spread is settled by an upfront payment.

Credit indices are standardized, equally weighted baskets of 125 CDSs with a fixed coupon; a defaulted name is settled and removed, and a new series is introduced roughly every six months.

A harder one

An investor buys USD 250 million of protection on a 125-name equally weighted credit index with a 100 bp coupon. One name defaults and its bonds are valued at 25 percent of par. What does the investor receive on the defaulted name, and what annual coupon is paid on the remaining index?

  1. USD 1.5 million; annual coupon of USD 2.48 million on a reduced notional of USD 248 million - correct
  2. USD 0.5 million; an annual coupon of USD 2.48 million on the reduced notional of USD 248 million after the defaultWhy not: Pays the recovery (25 percent) of the defaulted name's 2 million instead of the loss (75 percent).
  3. USD 2.0 million; an annual coupon of USD 2.48 million on the reduced notional of USD 248 million after the defaultWhy not: Ignores recovery and pays the full 2 million share of notional for the defaulted name.
  4. USD 1.5 million; an annual coupon of USD 2.50 million on the original USD 250 millionWhy not: Forgets that the index notional drops by the defaulted name's share, so the coupon base falls to 248 million.

Each name carries 250/125 = USD 2 million. Payoff = 2 x (1 - 0.25) = 1.5 million. Notional falls to 248 million, so the coupon is 1% x 248 = 2.48 million.

The hardest tier

A CDS with a standard coupon of 500 bp is quoted by a dealer at an upfront payment of 2.4 percent of notional received by the protection buyer, with a risky annuity of 3.8. The equivalent par (running) spread is closest to:

  1. 563 bpWhy not: Adds the upfront conversion; because the buyer receives the upfront, the par spread is below the coupon, not above.
  2. 437 bp - correct
  3. 260 bpWhy not: Subtracts the upfront percentage directly from the coupon without dividing by the annuity.
  4. 63 bpWhy not: Reports only the annualised upfront and omits the 500 bp coupon it must be netted against.

A buyer who receives upfront is paying an above-market coupon. Par spread = coupon - upfront / annuity = 500 - 240/3.8 = 437 bp.

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