Foundation IRB Specification¶

Foundation IRB calculation with supervisory LGD, PD floors, and correlation formulas.

Regulatory Reference: CRR Articles 153-154, 161-163, 178

Test Group: CRR-B

Requirements Status¶

ID	Requirement	Priority	Status
FR-1.3	F-IRB capital requirement (K): PD, supervisory LGD, maturity adjustment	P0	Done
FR-1.8	Defaulted exposure treatment: F-IRB (K=0)	P0	Done

Default Definition — Art. 178

F-IRB defaulted exposures (PD = 100%, K = 0) are driven by the Art. 178 default trigger. The two-limb trigger (unlikeliness-to-pay and 90 DPD), UTP indicators, materiality threshold, suspension rules, and 3-month cure / 1-year distressed- restructuring probation are documented in the shared Default Definition (Art. 178) specification.

Supervisory LGD Values (CRR Art. 161)¶

Under F-IRB, LGD is prescribed by the regulator based on collateral type:

Art. 161(1) LGD Values¶

Category	Supervisory LGD	Reference
Senior unsecured	45%	Art. 161(1)(a)
Subordinated unsecured	75%	Art. 161(1)(b)
Covered bonds (Art. 129(4)/(5) eligible)	11.25%	Art. 161(1)(d)
Senior purchased corporate receivables	45%	Art. 161(1)(e)
Subordinated purchased corporate receivables	100%	Art. 161(1)(f)
Dilution risk of purchased corporate receivables	75%	Art. 161(1)(g)

Art. 161(1)(c) provides that institutions may recognise funded and unfunded credit protection in the LGD in accordance with Chapter 4.

Purchased Receivables (Art. 161(1)(e)–(g))

Art. 161(1)(e) and (f) apply where the institution cannot estimate PD for the purchased receivables pool (or estimates do not meet Section 6 requirements). When PD is estimable, the standard senior (45%) or subordinated (75%) LGD from (a)/(b) applies instead. Art. 161(1)(g) covers dilution risk — the risk that receivables amounts are reduced through credits or allowances to the obligor. It always applies to the dilution component regardless of PD estimation capability.

Not Yet Implemented — Purchased Receivables LGD

The code does not implement separate LGD paths for Art. 161(1)(e)/(f)/(g). Purchased receivables exposures currently receive the standard unsecured LGD (45% senior / 75% subordinated). The 100% subordinated purchased receivables LGD and the 75% dilution risk LGD are not applied. See D3.10.

Art. 230 Table 5 LGDS Values (Foundation Collateral Method)¶

When exposures are secured by eligible collateral, the LGD* formula (Art. 230) uses the following supervisory LGDS values for the secured portion:

Collateral Type	LGDS (Senior)	LGDS (Subordinated)	C*	C**	Reference
Financial collateral	0%	0%	0%	—	Art. 230 Table 5
Receivables	35%	65%	0%	125%	Art. 230 Table 5
Residential / commercial RE	35%	65%	30%	140%	Art. 230 Table 5
Other physical collateral	40%	70%	30%	140%	Art. 230 Table 5

Where C* is the minimum collateralisation threshold (below which the collateral is not recognised) and C** is the overcollateralisation level at which the full LGDS applies to the entire exposure.

Covered Bond LGD (Art. 161(1)(d))

CRR Art. 161(1)(d) provides a permissive ("may be assigned") 11.25% LGD for covered bonds eligible under Art. 129(4) or (5). Basel 3.1 restructures this into a separate paragraph Art. 161(1B) with the same 11.25% value. Covered bonds use the Art. 161 mechanism, not the Art. 230 Table 5 LGDS/overcollateralisation framework.

Art. 161 vs Art. 230 Distinction

Art. 161(1)(a)–(g) covers unsecured LGD, subordinated LGD, covered bonds, purchased receivables, and dilution risk. The per-collateral-type LGDS values (0%/35%/40%) for the secured portion of the LGD* formula come from Art. 230 Table 5 (Foundation Collateral Method), not Art. 161. The Art. 230 LGDS subordinated column (65%/70%) applies when the underlying exposure is a subordinated claim — these are distinct from the Art. 161(1)(b) subordinated unsecured LGD of 75%.

Basel 3.1 F-IRB LGD Changes (PRA PS1/26 Art. 161(1))¶

Under Basel 3.1, senior unsecured LGD is differentiated by whether the counterparty is a financial sector entity (FSE):

Art. 161 LGD Comparison¶

Category	CRR	Basel 3.1	Reference
Senior unsecured (non-FSE)	45%	40%	Art. 161(1)(a) → Art. 161(1)(aa)
Senior unsecured (FSE)	45%	45%	Art. 161(1)(a)
Subordinated unsecured	75%	75%	Art. 161(1)(b)
Covered bonds	11.25%	11.25%	Art. 161(1)(d) → Art. 161(1B)
Senior purchased receivables	45%	40%	Art. 161(1)(e)
Subordinated purchased receivables	100%	100%	Art. 161(1)(f)
Dilution risk	75%	100%	Art. 161(1)(g)

Art. 230 LGDS Comparison (Secured Portions)¶

Collateral Type	CRR LGDS	Basel 3.1 LGDS	Reference
Financial collateral	0%	0%	Art. 230 Table 5 / Art. 230(2)
Receivables	35%	20%	Art. 230 Table 5 / CRE32.9
Residential RE	35%	20%	Art. 230 Table 5 / CRE32.10
Commercial RE	35%	20%	Art. 230 Table 5 / CRE32.11
Other physical	40%	25%	Art. 230 Table 5 / CRE32.12

FSE Definition

Financial sector entity includes banks, building societies, investment firms, insurance companies, and any entity primarily engaged in financial intermediation (general FSE definition: CRR Art. 4(1)(27)). The "large FSE" total-assets threshold driving the 1.25x correlation multiplier (Art. 153(2)) is EUR 70 billion under CRR Art. 142(1)(4) and GBP 79 billion under Basel 3.1 (PS1/26 Glossary p. 78, with Note "corresponds to Article 142(1)(4) of CRR").

Key B31 Changes to Purchased Receivables / Dilution

Basel 3.1 aligns the senior purchased receivables LGD with the new non-FSE rate (45% → 40%, Art. 161(1)(e)). The dilution risk LGD increases from 75% to 100% (Art. 161(1)(g)), reflecting the PRA's view that dilution losses are not mitigated by collateral recovery. The subordinated purchased receivables LGD remains at 100% (Art. 161(1)(f)).

B31 Art. 230 — Subordinated LGDS Distinction Removed

CRR Art. 230 Table 5 has separate "senior" and "subordinated" LGDS columns (e.g., receivables 35%/65%). PRA PS1/26 Art. 230(2) replaces this with a single LGDS per collateral type (20%/20%/25%) with no subordinated distinction. The subordination effect is captured solely through the LGDU term (75% per Art. 161(1)(b)).

PD Floor¶

CRR: Single floor of 0.03% (3 basis points) for all non-defaulted exposure classes (Art. 160(1) for corporate/sovereign/institution; Art. 163(1) for retail).

Basel 3.1 PD Floors by Exposure Class (PRA PS1/26 Art. 160/163)¶

Under Basel 3.1, PD floors are differentiated by exposure class:

Exposure Class	CRR PD Floor	Basel 3.1 PD Floor	Reference
Corporate / SME	0.03%	0.05%	Art. 160(1)
Sovereign	0.03%	0.05%	Art. 160(1)
Institution	0.03%	0.05%	Art. 160(1)
Retail — mortgage	0.03%	0.10%	Art. 163(1)(b)
Retail — QRRE (transactor)	0.03%	0.05%	Art. 163(1)(c)
Retail — QRRE (revolver)	0.03%	0.10%	Art. 163(1)(a)
Retail — other	0.03%	0.05%	Art. 163(1)(c)

Sovereign Row is Regulatory Dead Letter under Basel 3.1 (Art. 147A(1)(a))

Under Basel 3.1, sovereign exposures (Art. 147(2)(a)) are restricted to the Standardised Approach by Art. 147A(1)(a); PS1/26 provides no grandfathering or transitional carve-out for pre-existing sovereign IRB models. The 0.05% sovereign row in the Basel 3.1 column is retained for completeness only — under Basel 3.1 it cannot bind on any live exposure.

Institutions (Art. 147(2)(b)) are capped at F-IRB by Art. 147A(1)(b) (A-IRB unavailable; SA applies only where permission has been granted under Art. 148 or Art. 150). The 0.05% institution PD floor applies normally to F-IRB institution exposures.

See Framework Differences and IRB Approach Restrictions for the full Art. 147A(1) class mapping.

Asset Correlation Formula (CRR Art. 153)¶

Corporate, Institution, Sovereign¶

PD-dependent correlation with exponential decay factor of 50:

f(PD) = (1 - exp(-50 x PD)) / (1 - exp(-50))
R = 0.12 x f(PD) + 0.24 x (1 - f(PD))

SME Firm-Size Adjustment¶

For corporates with turnover < EUR 50m, correlation is reduced:

CRR (Art. 153(4)):

s = max(5, min(turnover_EUR, 50))
adjustment = 0.04 x (1 - (s - 5) / 45)
R_adjusted = R - adjustment

Turnover is stored in GBP and converted to EUR via the configured FX rate (default: 0.8732).

Basel 3.1 (PRA PS1/26): Thresholds converted to GBP:

Parameter	CRR (EUR)	Basel 3.1 (GBP)
SME threshold	EUR 50m	GBP 44m
Floor turnover	EUR 5m	GBP 4.4m
Adjustment range	45	39.6

s = max(4.4, min(turnover_GBP, 44))
adjustment = 0.04 x (1 - (s - 4.4) / 39.6)
R_adjusted = R - adjustment

Retail Mortgage¶

Fixed correlation: R = 0.15

Qualifying Revolving Retail (QRRE)¶

Fixed correlation: R = 0.04

Other Retail¶

PD-dependent correlation with exponential decay factor of 35:

f(PD) = (1 - exp(-35 x PD)) / (1 - exp(-35))
R = 0.03 x f(PD) + 0.16 x (1 - f(PD))

FI Scalar (CRR Art. 153(2))¶

A 1.25x multiplier applied to the asset correlation coefficient (R) for large financial sector entities (LFSEs) (total assets ≥ EUR 70 billion per CRR Art. 142(1)(4)) and unregulated financial sector entities (per CRR Art. 153(2)).

Two distinct thresholds — do not conflate

LFSE total-assets threshold — EUR 70 billion under CRR Art. 142(1)(4); GBP 79 billion under Basel 3.1 (PS1/26 Glossary p. 78, which cites CRR Art. 142(1)(4) as the corresponding pre-revocation provision). Triggers the 1.25x correlation multiplier (Art. 153(2)) for large FSEs and all unregulated FSEs under both frameworks.
GBP 440m annual revenue → F-IRB only approach restriction (Art. 147A(1)(e), Basel 3.1 only). Does not affect correlation.
The Art. 147A(1)(e) F-IRB restriction applies to all FSEs regardless of size — it is separate from the correlation uplift which only applies to large or unregulated FSEs.

Capital Requirement Formula¶

K = LGD x N[(1-R)^(-0.5) x G(PD) + (R/(1-R))^(0.5) x G(0.999)] - PD x LGD

Where:

N(x) = cumulative normal distribution function
G(x) = inverse normal CDF
G(0.999) = 3.0902323061678132
K is floored at 0

Effective Maturity (CRR Art. 162)¶

Applied to non-retail exposures only (retail exposures use MA = 1.0).

Art. 162(1) — F-IRB Fixed Supervisory Maturities¶

Institutions that have not received permission to use own LGDs and own conversion factors (i.e. F-IRB firms) shall assign:

Exposure Type	Supervisory Maturity	Reference
Repo-style transactions (repos, securities/commodities lending or borrowing)	0.5 years	Art. 162(1)
All other exposures	2.5 years	Art. 162(1)

0.5-Year Repo Maturity — Implemented

F-IRB exposures flagged with is_sft = True on the Facility/Loan/Contingent input row receive M = 0.5 years under CRR, overriding any maturity_date-derived value (engine/irb/transforms.py prepare_columns). The override is gated on the CRR framework only — Basel 3.1 deleted Art. 162(1), so B31 F-IRB calculates M per Art. 162(2A). Exposures without is_sft (or with is_sft = False) retain the existing 2.5-year default. Regression: tests/unit/irb/test_firb_sft_maturity.py.

Alternatively, the competent authority may require the institution to calculate M for each exposure using the A-IRB methods in Art. 162(2).

Art. 162(2) — A-IRB Effective Maturity Calculation¶

Institutions permitted to use own LGDs and CCFs (A-IRB) must calculate M per exposure. M shall not exceed 5 years (except under Art. 384(1) for CVA). Key methods:

Method	Applies To	Formula / Rule	Minimum M
(a) Cash-flow schedule	Instruments with known cash flows	`M = max(1, min(Σ(t × CF_t) / Σ(CF_t), 5))`	1 year
(b) Derivatives (MNA)	Derivatives under master netting agreement	Notional-weighted average remaining maturity	1 year
(c) Fully collateralised derivatives + margin lending (MNA)	Daily remargined and revalued (Annex II)	Weighted average remaining maturity	10 days
(d) Repo-style transactions (MNA)	Daily remargined and revalued repos/SFTs	Notional-weighted average remaining maturity	5 days
(e) Purchased corporate receivables	Drawn amounts (own PD permitted)	Exposure-weighted average maturity	90 days
(f) Other instruments	When (a) cannot be calculated	Max remaining time to discharge obligations	1 year
(g) IMM netting sets	Longest-dated contract > 1 year	IMM formula with effective EE	1 year
(j) Double-default protection	Art. 153(3) credit protection	Effective maturity of protection	1 year

Art. 162(3) — One-Day Maturity Floor Exceptions¶

Where documentation requires daily re-margining and daily revaluation with provisions for prompt liquidation or set-off of collateral, M shall be at least one day (overriding the longer minimums in paragraph 2) for:

(a) Fully/nearly-fully collateralised derivatives (Annex II)
(b) Fully/nearly-fully collateralised margin lending
(c) Repurchase transactions, securities or commodities lending or borrowing

The same one-day floor applies to qualifying short-term exposures not part of ongoing financing, including:

(a) FX settlement exposures to institutions
(b) Self-liquidating short-term trade finance (residual maturity ≤ 1 year, Art. 4(1)(80))
(c) Securities settlement within usual delivery period or 2 business days
(d) Cash settlement/electronic payment exposures, including failed-transaction overdrafts

Implementation Note — has_one_day_maturity_floor Flag + the ccr_effective_maturity carrier

The code implements Art. 162(3) via a boolean flag has_one_day_maturity_floor. When the flag is set, the IRB maturity chain (engine/irb/transforms.py::_build_maturity_exprs) does drive the effective-maturity column down to one day (1/365 ≈ 0.00274 years), and the maturity adjustment then suppresses its own 1-year floor so the sub-1-year M survives into MA. The flag has a second, independent consumer — CRM maturity-mismatch ineligibility (Art. 237(2): any mismatch zeroes protection value for one-day-floor exposures) — but it is not only a CRM-side flag.

For synthetic CCR / SFT rows the one-day signal is not a firm input. The FCCM SFT producer and the SA-CCR derivative producer compute the Art. 162 effective maturity at netting-set grain and surface it on a dedicated ccr_effective_maturity Float64 carrier (declared on CCR_EXIT_EDGE, propagated through the classifier / CRM / RE-split CCR edges). A new rung in _build_maturity_exprs coalesces that carrier into the IRB M column (AIRB-gated, so it never displaces the F-IRB fixed M), and sets has_one_day_maturity_floor from the winning rung — so when the carrier resolves to the Art. 162(3) one-day value, the maturity adjustment uses the actual sub-1-year M rather than re-flooring to 1 year. Because the flag is derived inside the IRB chain (downstream of CRM), CRM never sees a CCR one-day signal and no collateral is silently zeroed by Art. 237(2).

The F-IRB 0.5-year repo carve-out (Art. 162(1)) reaches CCR / SFT rows via a widened gate: _apply_firb_sft_supervisory_maturity now fires on (is_sft OR risk_type == CCR_SFT) — so a CCR_SFT row routed to F-IRB receives M = 0.5 years without ever setting the lending-side is_sft flag (which stays a CRM-only input). CCR_DERIVATIVE rows are deliberately excluded from this gate.

Art. 162(4) — SME Maturity Simplification¶

For exposures to corporates situated in the UK with consolidated sales and consolidated assets < EUR 500 million, institutions may consistently apply the F-IRB fixed maturities from Art. 162(1) instead of calculating per Art. 162(2). The EUR 500m threshold rises to EUR 1,000m for corporates that primarily own and let non-speculative residential property.

Not Yet Implemented

The code does not implement the Art. 162(4) SME maturity simplification. All A-IRB exposures calculate maturity from maturity_date or default to 2.5 years.

Maturity Adjustment Formula¶

b = (0.11852 - 0.05478 x ln(PD))^2
MA = (1 + (M - 2.5) x b) / (1 - 1.5 x b)

Where M is clamped to the range [1.0, 5.0] years (per Art. 162(1)/(2) floor and cap).

Basel 3.1 Changes to Art. 162¶

PRA PS1/26 makes significant changes to Art. 162:

Aspect	CRR	Basel 3.1	Reference
F-IRB fixed maturities (§1)	0.5yr repo / 2.5yr other	Deleted — all IRB firms must calculate M	Art. 162(1)
Scope	A-IRB only (Art. 143 permission)	F-IRB and A-IRB (Art. 147A)	Art. 162(2)
Revolving exposures	Repayment date of current drawing	Max contractual termination date	Art. 162(2A)(k)
Mixed MNA (derivatives + repos)	Not addressed	10-day floor	Art. 162(2A)(da)
Purchased receivables minimum M	90 days	1 year	Art. 162(2A)(e)
Collateral daily condition	Re-margining and revaluation	Re-margining or revaluation	Art. 162(2A)(c)/(d)
SME simplification (§4)	Available (EUR 500m threshold)	Deleted	Art. 162(4)

See the Basel 3.1 F-IRB specification for full details.

Previous Description Was Wrong

This section previously stated "unconditionally cancellable revolving facilities are assigned a maturity of 1 year". Art. 162(2A)(k) actually requires the maximum contractual termination date — not a 1-year default. Using 1 year instead of the facility termination date would systematically understate maturity and therefore RWA for revolving corporate exposures.

RWA Calculation¶

CRR Corporate/Institution (Art. 153(1)(iii)): RWA = K x 12.5 x 1.06 x EAD x MA

CRR Retail (Art. 154(1)(ii)): RWA = K x 12.5 x 1.06 x EAD (MA = 1.0 for retail)

Under CRR the 1.06 scaling factor applies uniformly to corporate, institution, sovereign and retail IRB RWA. The factor is embedded in the risk weight formulae in both Art. 153(1)(iii) (corporate/institution/sovereign) and Art. 154(1)(ii) (retail) — each ends ... × 12,5 × 1,06.

Previous Spec Error Corrected

An earlier version of this section claimed that retail CRR RWA does not include the 1.06 factor. That was wrong — the factor is written explicitly into the Art. 154(1)(ii) retail formula in the onshored CRR. The 1.06 scaling is removed only under Basel 3.1 (PRA PS1/26 sets it to 1.0 for all IRB exposure classes); under CRR it applies to both Art. 153 and Art. 154 outputs.

Expected Loss¶

EL = PD x LGD x EAD

Used for comparison against provisions (see Provisions).

Key Scenarios¶

Scenario ID	Description	Key Parameters
CRR-B1	Corporate F-IRB, senior unsecured, low PD	PD=0.10%, LGD=45%, M=2.5y
CRR-B2	Corporate F-IRB, senior unsecured, high PD	PD=5.00%, LGD=45%, M=2.5y
CRR-B3	Subordinated debt — supervisory LGD 75%	PD=0.50%, LGD=75% (Art. 161(1)(b))
CRR-B4	Financial collateral — blended LGD	Collateral reduces effective LGD (Art. 161(1)(d))
CRR-B5	SME corporate with firm-size adjustment + supporting factor	Turnover < EUR 50m, correlation reduced (Art. 153(4))
CRR-B6	PD floor binding — input PD below 0.03% floor	Input PD=0.01% → floored to 0.03% (Art. 160(1))
CRR-B7	Long maturity — contractual 7Y capped to 5Y	M clamped to [1, 5] range (Art. 162(2))

FI Scalar Coverage

The 1.25x correlation multiplier for large/unregulated FSEs (Art. 153(2)) is validated within CRR-B5 and through the B31-B group (B31-B7 specifically tests FSE LGD differentiation). See FI Scalar for details.

Acceptance Tests¶

Group	Scenarios	Tests	Pass Rate
CRR-B: Foundation IRB	B1–B7	13	100% (13/13)