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State what incremental CVA and marginal CVA are, compute them and show how CVA is turned into a running spread.

FRM® Part II · Credit Risk Measurement and Management · CVA · FRM2.CR.20.i

The idea

Incremental CVA of a trade is the CVA of the netting set with the trade minus the CVA without it. It is what the new trade adds, and it is the natural price to charge a client. It depends on what is already in the set and on the order of trades, and incremental amounts do not generally sum to the total. Marginal CVA allocates the total CVA across the trades so the parts add up. The allocation is typically in proportion to each trade's contribution to expected exposure conditional on the net exposure being positive, and it can be negative for a trade that offsets others. To express a CVA as a running spread, divide by the notional and the risky annuity, called Annuity below. Written out, the running spread per year charged on the trade is Spread = CVA ÷ (Notional × Annuity). Do not use standalone CVA for a trade in a netting set.

Formulas

CVA as a running spread

CVA spread = CVA ÷ (Notional × Annuity)

CVA
the upfront CVA to convert (for a new trade in a netting set, its incremental CVA)
Notional
the trade's notional
Annuity
the risky annuity: the sum of discount factor × counterparty survival probability, per unit of notional

For a client price, convert the incremental CVA, not the standalone or the whole netting set's CVA, and divide by the risky annuity.

Incremental CVA of a new trade

Incremental CVA = CVA(set + trade) − CVA(set)

CVA(set + trade)
the netting set's CVA with the new trade
CVA(set)
the netting set's CVA without it

The client price for a trade in an existing netting set. It depends on the order trades are added, so incremental amounts need not sum to the total.

Marginal CVA of a trade

Marginal CVAᵢ = CVA × EEᵢ ÷ EE

CVA
the netting set's total CVA
EEᵢ
trade i's contribution to expected exposure, given the net exposure is positive
EE
the netting set's expected exposure, the sum of the contributions

The parts add up to the total. A trade that offsets others has a negative contribution and a negative marginal CVA; do not floor it at zero.

A practice question

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

The incremental CVA of a new trade added to an existing netting set is:

  1. the CVA of the new trade calculated on a standalone basis, as if it stood alone.Why not: Ignores the interaction with existing trades through netting.
  2. the total CVA of the netting set after the trade is added, taken as the charge.Why not: Reports the level of CVA, not the change caused by the trade.
  3. the netting-set CVA with the new trade minus the CVA without it. - correct
  4. the CVA of the existing netting set minus that of the new trade alone, as a net figure.Why not: Subtracts in the wrong order and mixes a set with a standalone trade.

Incremental CVA = CVA(set with trade) - CVA(set without trade). It measures what the trade adds to the counterparty's total charge, which is the relevant price for the client.

A harder one

A bank's netting set with a counterparty has a CVA of USD 450,000. When a new swap is added the netting-set CVA becomes USD 520,000. The swap's CVA on a standalone basis would be USD 200,000. What is the incremental CVA of the swap?

  1. USD 70,000 - correct
  2. USD 330,000Why not: Adds the netting benefit to the standalone CVA instead of deducting it, so the charge is overstated.
  3. USD 520,000Why not: Takes the new total as the charge instead of the change.
  4. USD 200,000Why not: Uses the standalone CVA, ignoring the netting benefit with existing trades.

Incremental CVA = 520,000 - 450,000 = USD 70,000. The difference from the standalone figure (200,000) is the netting benefit of 130,000.

The hardest tier

A bank's netting set has a CVA of USD 548,000 without a new trade and USD 612,000 with it. The new trade's standalone CVA is USD 150,000. The trade has a notional of USD 25 million; the risky annuity is 4.2 and the risk-free annuity is 4.6, both per unit of notional. The running spread the bank should charge the client for this trade's CVA is closest to:

  1. 6.1 bp per year - correct
  2. 14.3 bp per yearWhy not: Converts the standalone CVA and ignores the netting benefit.
  3. 5.6 bp per yearWhy not: Uses the risk-free annuity; the CVA spread is paid only while the counterparty survives.
  4. 58.3 bp per yearWhy not: Converts the total netting-set CVA, which includes the pre-existing trades.

Incremental CVA = 612,000 - 548,000 = 64,000. Running spread = 64,000 / (25,000,000 x 4.2) = 6.1 bp per year.

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