What is the maturity adjustment in the IRB formula and why does longer maturity need more capital?

Question

AcadiFi · Accepted Answer

The IRB maturity adjustment accounts for credit migration risk — the possibility that a borrower's credit quality deteriorates over a longer horizon, reducing asset value even without default. A 10-year loan carries more risk than a 1-year loan of the same PD because there's more time for the rating to drop.

**Basel formula:**
b(PD) = [0.11852 − 0.05478 × ln(PD)]²

MA(M, PD) = (1 + (M − 2.5) × b(PD)) / (1 − 1.5 × b(PD))

where M is effective maturity (capped at 5 years, floored at 1 year). Note the adjustment is normalized at M = 2.5 (gives MA = 1).

Example for a PD = 1.00% loan:
b = (0.11852 − 0.05478 × ln(0.01))² = (0.11852 + 0.2523)² = 0.1376

| Maturity M | MA |
|---|---|
| 1.0 | 0.815 |
| 2.5 | 1.000 |
| 5.0 | 1.518 |

A 5-year loan gets 51.8% more capital than a 2.5-year baseline. For the same PD = 1% applied to a $50M corporate loan with LGD 45%, R = 0.19:

- K at M = 2.5: 7.85%, RWA = $49.1M
- K at M = 5.0: 11.91%, RWA = $74.4M

```mermaid
flowchart LR
  A[Effective maturity M] --> B[b PD function]
  C[PD] --> B
  B --> D[MA formula]
  D --> E[Apply to K]
  E --> F[Higher RWA for long-dated]
```

Note: higher-PD borrowers have LOWER b(PD), meaning maturity adjustment is smaller for distressed credits. Intuition: a borrower already at PD = 20% has little room to deteriorate further, so time matters less. For investment-grade with PD = 0.05%, b is large and MA amplifies capital for long-dated exposures.

What is the maturity adjustment in the IRB formula and why does longer maturity need more capital?

Master Part II with our FRM Course

Related Questions

Practice Questions