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Account for how unilateral CVA (UCVA) differs from BCVA, and how unilateral DVA (UDVA) differs from BCVA.

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

The idea

Unilateral CVA (UCVA) assumes only the counterparty can default and treats the institution as default free. Unilateral DVA (UDVA) assumes only the institution can default and the counterparty is default free. Bilateral CVA (BCVA) accounts for both, so each leg is conditioned on the other party's survival, which is the first-to-default effect. The CVA term is the loss from the counterparty defaulting while the institution has survived. The DVA term is the gain from the institution defaulting while the counterparty has survived. Hence the CVA term in BCVA is smaller than UCVA, and the DVA term is smaller than UDVA. Because the DVA term is not negative, the net charge is below UCVA, and it is written BCVA = CVA − DVA. With independent defaults in one period, the bilateral CVA term is UCVA × (1 − own PD). The bilateral DVA term is UDVA × (1 − counterparty PD).

Formulas

CVA term of a bilateral valuation, one period

Bilateral CVA = UCVA × (1 − PDI)

UCVA
unilateral CVA, with the institution treated as default-free
PDI
the institution's own default probability

Independent defaults in one period. The loss counts only if the institution survives, so the term is below UCVA.

DVA term of a bilateral valuation, one period

Bilateral DVA = UDVA × (1 − PDC)

UDVA
unilateral DVA, with the counterparty treated as default-free
PDC
the counterparty's default probability

Independent defaults in one period. Weight by the counterparty's survival, not the institution's own.

A practice question

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

Unilateral CVA (UCVA) differs from bilateral CVA in that it:

  1. applies to collateralized trades only, because bilateral CVA requires an uncollateralized setting by definition.Why not: Invents a collateral condition; collateral status does not define unilateral versus bilateral.
  2. ignores the recovery rate and treats the full exposure as lost on default of the counterparty.Why not: Confuses unilateral with a zero-recovery assumption; recovery is still used.
  3. assumes only the counterparty can default. - correct
  4. assumes only the institution can default and the counterparty is default-free, mirroring the DVA case.Why not: Describes the assumption of unilateral DVA, not unilateral CVA.

Unilateral CVA is calculated as if the institution cannot default, so it ignores the possibility that the institution defaults first and so would never suffer the counterparty's default.

A harder one

In a bilateral calculation, the CVA component is compared with UCVA for the same exposure profile and the same counterparty default probabilities. Which statement is correct?

  1. The two are identical, because the institution's own default affects only the DVA component of the bilateral value.Why not: Own default does condition the CVA term through survival to the counterparty's default.
  2. The bilateral CVA component is smaller only if the counterparty is rated lower than the institution.Why not: Adds a rating condition; the first-to-default effect reduces the CVA term whenever own default probability is positive.
  3. The bilateral CVA component is larger, because the institution's own default risk raises the cost of the trade.Why not: Own default risk reduces, not raises, the CVA term through the first-to-default effect.
  4. The bilateral CVA component is smaller, because the loss arises only if the institution has not defaulted first. - correct

In the bilateral formula the counterparty's default loss is weighted by the probability that the institution survives; with positive own PD this is below one, so the CVA term is smaller than UCVA.

The hardest tier

In a single-period model, the counterparty's default probability is 6%, the institution's is 4% and the two defaults are independent. The expected positive exposure is USD 10 million and loss given default on the counterparty is 60%. The CVA component of the bilateral valuation (multiplying by the probability that the institution survives) is closest to:

  1. USD 14,400Why not: Multiplies by the joint probability of both defaulting instead of the probability of counterparty default with institution survival.
  2. USD 345,600 - correct
  3. USD 120,000Why not: Uses the difference between the two default probabilities, a net-spread shortcut that is not the bilateral formula.
  4. USD 360,000Why not: This is the unilateral CVA, which ignores the probability that the institution defaults first.

UCVA = 0.6 x 10,000,000 x 0.06 = USD 360,000. Bilateral CVA = UCVA x (1 - 0.04) = USD 345,600.

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