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Work out the DVA and the BCVA, then restate BCVA as a spread.

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

The idea

To compute the DVA, use the expected negative exposure (absolute value) and the institution's own marginal default probabilities and loss given default. CVA uses expected positive exposure and the counterparty's PD, so the DVA is its mirror image. ENE is the discounted expected negative exposure, PD the own marginal default probability. DVA = LGDown × Σ[ENE(ti) × PD]. The net charge is written BCVA, and it is negative when own credit risk dominates. BCVA = CVA − DVA. A negative BCVA means the adjustment adds value, so it raises the value of the position to the institution. To restate BCVA as a running spread, divide by the notional and the risky annuity. Spread = BCVA ÷ (Notional × Annuity). Where survival conditioning is included, each leg is multiplied by the survival probability of the other party. Typical mistakes are adding DVA to CVA, using the counterparty's PD in the DVA leg, using cumulative PDs, and dividing by the risk-free annuity.

Formulas

Debit value adjustment

DVA = LGDown × Σ ENE(tᵢ) × PD(tᵢ)

LGDown
the institution's own loss given default, 1 − own recovery
ENE(tᵢ)
the discounted expected negative exposure at tᵢ, as an absolute value
PD(tᵢ)
the institution's own marginal default probability for the interval ending at tᵢ
tᵢ
each exposure date

The mirror image of CVA. Using the counterparty's PD or the positive exposure in this leg, or cumulative PDs, are the traps.

Bilateral credit valuation adjustment

BCVA = CVA − DVA

CVA
the cost of the counterparty's possible default
DVA
the benefit of the dealer's own possible default

A negative BCVA means own credit risk dominates and the adjustment adds value. Adding DVA to CVA is the usual mistake.

BCVA as a running spread

BCVA spread = BCVA ÷ (Notional × Annuity)

BCVA
CVA − DVA, as an upfront amount
Notional
the trade's notional
Annuity
the risky annuity per unit of notional

Net DVA first, then convert. Dividing by the risk-free annuity understates the spread.

A practice question

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

Which formula gives the DVA in the simple setting with independent exposure and default?

  1. DVA = (1 - counterparty recovery) x sum of [discounted expected negative exposure x counterparty marginal default probability]Why not: Uses the counterparty's loss and default inputs rather than the institution's own.
  2. DVA = own recovery x sum of [discounted expected negative exposure x own marginal default probability]Why not: Uses recovery in place of loss given default.
  3. DVA = (1 - own recovery) x sum of [discounted expected positive exposure x own marginal default probability]Why not: Uses the positive exposure, which drives CVA; DVA needs the negative exposure.
  4. DVA = (1 - own recovery) x sum of [discounted expected negative exposure x own marginal default probability] - correct

DVA mirrors CVA: own LGD x discounted expected negative exposure x own marginal default probability, summed over dates.

A harder one

On a 5-year swap with notional USD 30 million, a bank computes a CVA of USD 410,000 and a DVA of USD 265,000. The risky annuity per unit of notional is 4.2. The bilateral adjustment expressed as a running spread is closest to:

  1. 48.3 bp per yearWhy not: Divides by notional only, so the result is an upfront amount rather than a running spread.
  2. 32.5 bp per yearWhy not: Converts the CVA alone and ignores the DVA offset.
  3. 11.5 bp per year - correct
  4. 53.6 bp per yearWhy not: Adds DVA to CVA instead of netting it.

BCVA = 410,000 - 265,000 = 145,000. Spread = 145,000 / (30,000,000 x 4.2) = 11.5 bp.

The hardest tier

A bank has a 3-year swap with notional USD 40 million. Expected exposures (undiscounted) at years 1, 2, 3 are: positive USD 2.0, 3.0, 2.5 million; negative USD 1.0, 1.8, 2.2 million (absolute values). The discount rate is 2% continuously compounded. The counterparty's CDS spread is 250 bp and the bank's own is 120 bp, both flat, recovery for both 40%, with default probabilities 1 - exp(-s t / (1 - R)). Ignore joint defaults. The risky annuity is the sum of discount factor times counterparty survival probability. The BCVA expressed as a running spread (positive means a cost to the bank) is closest to:

  1. 10.7 bp per year - correct
  2. 5.4 bp per yearWhy not: Uses the counterparty's default probabilities in the DVA leg instead of the bank's own.
  3. 9.8 bp per yearWhy not: Divides by the risk-free annuity instead of the risky annuity.
  4. 21.1 bp per yearWhy not: Adds DVA to CVA instead of subtracting it.

CVA = USD 168,743, DVA = USD 55,436, BCVA = USD 113,307. Risky annuity = 2.6553. Spread = 113,307 / (40,000,000 x 2.6553) = 10.7 bp.

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