What is the IRB formula for Expected Loss and how does it differ from Unexpected Loss?

Question

AcadiFi · Accepted Answer

Under IRB, **Expected Loss** is the statistical mean credit loss over a one-year horizon:

**EL = PD × LGD × EAD**

Example: Cavendish Regional Bank has a $20M commercial loan with PD = 2.5%, LGD = 35%, EAD = $20M.
EL = 0.025 × 0.35 × $20M = $175,000

This is the annual expected loss — covered by loan loss provisions and priced into credit spreads (the lender's expected cost of credit is embedded in the loan rate).

**Unexpected Loss** is the loss amount exceeding EL at a chosen confidence level (99.9% under Basel). UL represents extreme but plausible losses that provisions cannot absorb. Capital must cover UL.

Basel formula for IRB capital requirement K (per unit exposure):

K = LGD × [Φ((Φ⁻¹(PD) + √R × Φ⁻¹(0.999)) / √(1 − R)) − PD] × MA(M, PD)

where R is correlation, M is maturity, MA is maturity adjustment.

```mermaid
flowchart LR
  A[Loss distribution] --> B[EL = mean]
  A --> C[UL = 99.9 percentile - EL]
  B --> D[Covered by provisions]
  C --> E[Covered by capital]
  D --> F[Priced in spread]
  E --> G[RWA × 8%]
```

RWA = K × 12.5 × EAD. For the Cavendish loan with R = 0.15, M = 2.5, effective K might be 8.2%, giving RWA = $20.5M and minimum capital = $1.64M at 8% ratio.

Key insight: EL is budgeted cost (provisions), UL is true capital driver. Provision shortfall deducted from CET1 under Basel III final rules prevents double-counting.

What is the IRB formula for Expected Loss and how does it differ from Unexpected Loss?

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