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Show how exposure to credit risk is taken into account when valuing a CDS.

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

The idea

A protection buyer in a CDS faces two risks. The reference entity may default, which the CDS hedges, and the protection seller may default, which it does not. Counterparty exposure exists only when the CDS has positive value to the buyer, because then a seller default costs the replacement value. Valuation therefore subtracts a credit valuation adjustment, summed over the payment periods, where LGD is the seller's loss given default. Written out, the adjustment is CVA = LGD × Σ (probability the seller defaults in the period × expected positive exposure × discount factor). Wrong-way risk matters most. If the seller's default is positively correlated with the reference entity's, exposure is highest exactly when the seller is weakest, so the CDS is worth less and the buyer should pay a lower spread. Exposure is positive market value, not notional.

Formulas

CVA on a CDS bought from a dealer

CVA = LGD × Σ qᵢ × EPEᵢ × DFᵢ

LGD
the loss given default on the protection seller
Σ
the sum over the periods of the contract
qᵢ
the seller's marginal probability of defaulting in period i
EPEᵢ
the expected positive exposure to the seller in period i
DFᵢ
the discount factor for period i

Uses marginal, not cumulative, default probabilities. With wrong-way risk the independence behind this product understates the CVA.

Mark-to-market value of a CDS to the protection buyer

value = (current spread − contract spread) × risky annuity × notional

current spread
the market CDS spread now
contract spread
the spread fixed at inception
risky annuity
the present value of one unit of annual premium, allowing for default
notional
the CDS notional

Positive when spreads have widened, which is when the buyer is exposed to the seller. Leaving out the annuity gives only one year's spread difference.

A practice question

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

A bank buys five-year protection on a corporate bond from a dealer through a bilateral CDS. When does the bank face credit exposure to the dealer?

  1. Only at the moment the reference entity defaults, because before that moment the CDS has no value to either party.Why not: A CDS has a mark-to-market value before default, since spread changes make it worth something to one side.
  2. When the CDS has positive market value to the bank, since the dealer's default would then cost the replacement value. - correct
  3. Always, in the full notional amount, because the dealer might fail to pay the whole protected face value.Why not: Treats the notional as the exposure; exposure is the positive replacement value, not the face amount.
  4. Never, because the reference entity itself guarantees every CDS written on its own bonds and loans.Why not: Reference entities do not guarantee CDSs on their debt; a bilateral CDS carries the seller's counterparty risk.

Counterparty exposure equals the positive mark-to-market of the contract: if the CDS is worth nothing or is a liability to the bank, the dealer's default costs it nothing; if it is an asset, the loss is the replacement cost.

A harder one

A 20 million notional CDS was entered at a spread of 100 bp. The market spread is now 160 bp and the risky annuity (PV01 per unit of spread) of the remaining contract is 3.5. Which party now has credit exposure to the other, and how large is it, closest to?

  1. The protection seller, about USD 420,000Why not: Gets the direction wrong: spread widening makes the contract an asset to the buyer, so the seller owes value and the buyer is exposed.
  2. The protection buyer, about USD 120,000Why not: Omits the risky annuity: value is the spread change times the annuity times notional, not the spread change times notional.
  3. The protection buyer, about USD 420,000 - correct
  4. The protection buyer, about USD 1,120,000Why not: Uses the full current spread instead of the 60 bp change since inception.

Value to buyer = (160 - 100) bp x 3.5 x 20 million = 0.006 x 3.5 x 20,000,000 = 420,000. A positive value means the buyer would lose that replacement cost if the seller defaulted.

The hardest tier

A bank is valuing the counterparty adjustment on a three-year CDS bought from a dealer. At the end of years 1, 2 and 3 the expected positive exposure to the dealer is USD 0.40, 0.90 and 1.30 million. The dealer's marginal default probabilities in those years are 1.0, 1.5 and 2.0 percent, discount factors are 0.97, 0.94 and 0.91, and the loss given default is 55 percent. The CVA is closest to:

  1. USD 40,230Why not: Leaves out the loss given default; CVA is LGD x the sum of marginal PD x exposure x discount factor.
  2. USD 43,046Why not: Uses cumulative default probabilities for each year instead of the marginal probability of defaulting within that year.
  3. USD 23,925Why not: Skips discounting; future losses must be discounted at the year in which they are expected.
  4. USD 22,127 - correct

CVA = 0.55 x (0.01x0.40x0.97 + 0.015x0.90x0.94 + 0.02x1.30x0.91) = 0.55 x 0.04023 million = 22,127. Use marginal default probabilities, exposure, discount factors and LGD.

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