Bilateral CVA (BCVA) is the valuation framework that accounts for the default risk of both parties to a derivative contract, consisting of the sum of CVA (the cost of counterparty default) and DVA (the benefit from the party’s own default). Unlike unilateral CVA, which assumes only the counterparty can default, BCVA incorporates the “first-to-default” effect: each party’s adjustment is conditioned on the survival of the other party up to the default time.
The BCVA formula in continuous form is BCVA = CVA + DVA, where CVA involves an integral of the counterparty’s default intensity times EPE times a risky discount factor (discounting at r + lambda_C + lambda_P for the survival of both parties), and DVA involves the analogous integral over the party’s own default intensity times ENE. In discrete form, the CVA term multiplies EPE by the counterparty’s marginal default probability and the party’s survival probability, while DVA multiplies ENE by the party’s marginal default probability and the counterparty’s survival probability. The inclusion of survival probabilities in the discount factor is sometimes referred to as “contingent” BCVA.
BCVA creates theoretical price symmetry: since one party’s CVA equals the negative of the other’s DVA, the bilateral valuations can agree. A simple approximation is BCVA ~ -EPE x (Spread_C - Spread_P), so that a party charges its counterparty for the difference in credit spreads. However, BCVA raises practical concerns. Three interconnected subtleties affect the formulas: (i) whether to include survival probabilities (contingent vs non-contingent), (ii) default dependency between the two parties, and (iii) close-out assumptions (risk-free vs risky close-out). Market practice is mixed on these points. Furthermore, the market has largely moved to replace the DVA component with FBA in a symmetric FVA framework for pricing, while retaining BCVA for accounting purposes.
Key Details
- BCVA = CVA + DVA, where CVA is negative (cost) and DVA is positive (benefit)
- First-to-default effect: the risky discount factor includes both parties’ default intensities
- Contingent vs non-contingent: including survival probabilities changes the CVA/DVA magnitudes but the net BCVA is less affected
- For equal credit spreads, BCVA is independent of the ordering and netting of transactions
- Risky close-out (including CVA/DVA in the close-out amount) creates a recursive problem; risk-free close-out avoids this at the cost of theoretical inconsistency
- Price symmetry from BCVA: BCVA ~ -EPE x (Spread_C - Spread_P) — weaker counterparties pay stronger ones
- In practice, most banks use CVA + symmetric FVA for pricing rather than BCVA
Textbook References
The xVA Challenge (Gregory, 2020)
- Section 17.3.1 (pp. 498—499): Accounting background. BCVA driven by FAS 157 (2006) and IFRS 13 (2013). US and Canadian banks adopted DVA reporting from 2006; most other large banks followed from 2013.
- Section 17.3.3 (pp. 500—502): Bilateral CVA formulas. Continuous form (Eq. 17.7a—c) and discrete form (Eq. 17.8a—b). Table 17.3 shows numerical examples for interest rate swaps: the OTM receive-fixed swap has positive BCVA overall due to large DVA.
- Section 17.3.3 (p. 502): Simple BCVA approximation (Eq. 17.9): BCVA ~ -EPE x (Spread_C - Spread_P). If the party’s own spread exceeds the counterparty’s, the party should pay for the privilege of trading.
- Section 17.3.4 (pp. 502—503): Close-out and default correlation. Three interconnected subtleties: survival adjustment, default dependency, and close-out conventions. Gregory and German (2013) find that the formulas without survival probabilities are the best approximation under risky close-out. Market participants are split between contingent and non-contingent calculations.