How does Basel set the correlation parameter R in the IRB formula?

Question

AcadiFi · Accepted Answer

Basel prescribes the correlation parameter R based on empirical studies showing that higher-credit-quality obligors have HIGHER systematic correlation (they're more sensitive to macro shocks), while lower-credit-quality obligors have HIGHER idiosyncratic risk.

**Corporate/Sovereign/Bank formula:**
R = 0.12 × (1 − e⁻⁵⁰·PD) / (1 − e⁻⁵⁰) + 0.24 × [1 − (1 − e⁻⁵⁰·PD) / (1 − e⁻⁵⁰)]

R smoothly interpolates between 24% (low PD) and 12% (high PD).

**Small-Medium Enterprise adjustment:** For borrowers with sales < €50M, subtract up to 4% from R (SME carve-out).

**Residential mortgage:** R = 0.15 (fixed)

**Qualifying revolving retail (QRRE):** R = 0.04 (lowest, reflects diversification)

**Other retail:** R follows formula similar to corporate but with bounds 3%–16%.

Example for Thornbury Corporate Bank:
- Obligor A (investment grade, PD = 0.10%): R ≈ 23.66%
- Obligor B (crossover, PD = 1.50%): R ≈ 15.71%
- Obligor C (distressed, PD = 10.00%): R ≈ 12.03%

```mermaid
flowchart LR
  A[PD low 0.1%] --> B[R high ~24%]
  C[PD med 1%] --> D[R ~18%]
  E[PD high 10%] --> F[R low ~12%]
```

Intuition: Prime corporate borrowers (Apple, Microsoft) all struggle together when the economy tanks — high systematic co-movement. Subprime borrowers default for idiosyncratic reasons (job loss, health) that don't all coincide — lower systematic correlation.

Basel III finalization kept these formulas but introduced the output floor (72.5% of standardized RWA) to cap IRB benefits.

How does Basel set the correlation parameter R in the IRB formula?

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