How do I calculate unilateral CVA for a derivative?

Question

AcadiFi · Accepted Answer

Unilateral CVA (UCVA) assumes only the counterparty can default (ignoring your own default). The formula is:

UCVA = LGD × Σ[EE(ti) × PD(ti-1, ti) × DF(ti)]

Where EE is expected exposure at time ti, PD is the marginal probability of default in interval (ti-1, ti), and DF is the discount factor. LGD is loss given default (typically 60%).

Example: Calder Financial enters a 3-year interest rate swap with Northbridge Corp (BBB counterparty). Simplify with annual time steps.

| Year | EE ($M) | Cumulative PD | Marginal PD | DF | Contribution |
|---|---|---|---|---|---|
| 1 | 2.0 | 1.2% | 1.2% | 0.97 | 0.6 × 2.0 × 0.012 × 0.97 = $0.014M |
| 2 | 3.5 | 2.5% | 1.3% | 0.94 | 0.6 × 3.5 × 0.013 × 0.94 = $0.026M |
| 3 | 2.5 | 3.9% | 1.4% | 0.91 | 0.6 × 2.5 × 0.014 × 0.91 = $0.019M |

UCVA = $0.014 + $0.026 + $0.019 = $0.059M = $59K

```mermaid
flowchart LR
  A[MtM simulation] --> B[EE path]
  C[CDS spread curve] --> D[PD path]
  B --> E[CVA integral]
  D --> E
  E --> F[UCVA]
```

Key FRM mechanics:

1. **EE from Monte Carlo** — simulate rate paths, reprice the swap at each time step, take the positive MtM average across paths.
2. **PD from CDS spreads** — bootstrap hazard rates from Northbridge's 1Y, 2Y, 3Y CDS quotes.
3. **Risk-neutral vs real-world PDs** — CVA always uses risk-neutral PDs (from CDS), not rating-based PDs.
4. **LGD assumption** — market standard is 60% (40% recovery) for senior unsecured.

The UCVA becomes a deduction from the swap's mark-to-market, effectively a 'credit charge' the bank holds against the counterparty.

How do I calculate unilateral CVA for a derivative?

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