What is the ASRF model and why is it the backbone of Basel IRB capital?

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AcadiFi · Accepted Answer

The Asymptotic Single Risk Factor (ASRF) model, based on Vasicek (2002) and Gordy (2003), underlies Basel IRB capital. Its core assumption: portfolios are infinitely granular (no concentration) and all systematic risk is captured by a single common factor.

Obligor i's asset return follows:
Xᵢ = √R · Y + √(1−R) · εᵢ

where Y is the common factor (macroeconomic), εᵢ is idiosyncratic, R is correlation to factor. Default occurs if Xᵢ falls below threshold Φ⁻¹(PDᵢ).

For an infinitely granular portfolio, the Law of Large Numbers eliminates idiosyncratic risk, leaving only systematic risk. The portfolio loss rate at confidence α equals the conditional PD given Y = −Φ⁻¹(α):

**Conditional PD = Φ((Φ⁻¹(PD) + √R · Φ⁻¹(α)) / √(1−R))**

With α = 99.9% (Basel's confidence level), Φ⁻¹(0.999) ≈ 3.09.

Example: Kentwell Commercial Bank's corporate exposure with PD = 1% and R = 0.20:
Conditional PD = Φ((Φ⁻¹(0.01) + √0.20 × 3.09) / √0.80)
= Φ((−2.326 + 1.382) / 0.894) = Φ(−1.056) = 14.54%

Capital charge K = LGD × [Conditional PD − PD] × Maturity Adjustment.

```mermaid
flowchart TD
  A[Single factor Y] --> B[Obligor X_i]
  B --> C[Default if X lt threshold]
  C --> D[Conditional PD given Y]
  D --> E[Capital per obligor]
  E --> F[Portfolio invariant]
```

Key property: **portfolio invariance** — capital for each loan depends only on its own parameters, not the rest of the portfolio. This enables additive capital charges. Limitations: real portfolios have concentrations (single-name, sector, geographic), addressed via Pillar 2 add-ons and the Concentration Risk Adjustment.

What is the ASRF model and why is it the backbone of Basel IRB capital?

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