IRB Approach¶

The Internal Ratings-Based (IRB) approach allows banks with regulatory approval to use their own risk estimates for calculating RWA. This provides greater risk sensitivity than the Standardised Approach.

Overview¶

Two IRB variants are available:

Approach	PD	LGD	EAD	CCF
Foundation IRB (F-IRB)	Bank	Supervisory	Supervisory	Supervisory
Advanced IRB (A-IRB)	Bank	Bank	Bank	Bank

Basel 3.1 Restrictions

Under Basel 3.1, A-IRB is no longer permitted for: - Large corporates (revenue > £500m) - Banks/Financial Institutions - Equity exposures

IRB Formula¶

The core IRB formula calculates the capital requirement (K). See irb/formulas.py for the pure Polars implementation.

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

Where: - N() = Standard normal cumulative distribution function - G() = Inverse standard normal distribution function - R = Asset correlation - PD = Probability of Default - LGD = Loss Given Default - MA = Maturity Adjustment (for non-retail)

Then:

RWA = K × 12.5 × EAD × Scaling Factor

Stats Backend¶

The IRB formulas require statistical functions (normal CDF and inverse CDF). The calculator uses polars-normal-stats for native Polars CDF/PPF expressions.

Installation

uv add polars-normal-stats

Capital K Formula Implementation (formulas.py)

See the _polars_capital_k_expr() function in src/rwa_calc/engine/irb/formulas.py for the full Polars expression implementation.

Risk Parameters¶

Probability of Default (PD)¶

PD is the likelihood of default within one year.

Floors:

Exposure Class	CRR	Basel 3.1
Corporate	0.03%	0.05%
Large Corporate	0.03%	0.05%
Bank/Institution	0.03%	0.05%
Retail Mortgage	0.03%	0.05%
Retail QRRE (Transactor)	0.03%	0.03%
Retail QRRE (Revolver)	0.03%	0.10%
Retail Other	0.03%	0.05%

PD_effective = max(PD_estimated, PD_floor)

Loss Given Default (LGD)¶

LGD is the percentage of exposure lost after recoveries.

F-IRB Supervisory LGD:

Exposure Type	CRR	Basel 3.1
Senior Unsecured	45%	40%
Subordinated	75%	75%
Secured by Financial Collateral	0%	0%
Secured by Receivables	35%	20%
Secured by CRE/RRE	35%	20%
Secured by Other Collateral	40%	25%

A-IRB LGD Floors (Basel 3.1 only, PRA PS1/26):

Collateral Type	LGD Floor
Unsecured Senior	25%
Unsecured Subordinated	50%
Financial Collateral	0%
Receivables	10%
Commercial Real Estate	10%
Residential Real Estate	5%
Other Physical	15%

Exposure at Default (EAD)¶

F-IRB: - On-Balance Sheet: Gross carrying amount - Off-Balance Sheet: Regulatory CCFs apply based on risk_type:

Risk Type	SA CCF	F-IRB CCF	Notes
FR (full_risk)	100%	100%	Guarantees, credit substitutes
MR (medium_risk)	50%	75%	Committed undrawn, NIFs, RUFs
MLR (medium_low_risk)	20%	75%	Documentary credits, trade finance
LR (low_risk)	0%	0%	Unconditionally cancellable

CRR Art. 166(8): Under F-IRB, MR and MLR categories both use 75% CCF.

CRR Art. 166(9) Exception: Short-term letters of credit arising from the movement of goods retain 20% CCF under F-IRB. Flag these exposures with is_short_term_trade_lc = True.

A-IRB: - Bank estimates EAD using internal models - Provide modelled CCF in ccf_modelled column (0.0-1.5, can exceed 100% for Retail IRB) - When ccf_modelled is provided, it takes precedence over risk_type lookup - Subject to CCF floors under Basel 3.1

Maturity (M)¶

Effective maturity affects capital through the maturity adjustment.

Range: 1 year (floor) to 5 years (cap)
Retail exemption: No maturity adjustment for retail

# Effective maturity calculation
M = max(1, min(5, weighted_average_life))

Asset Correlation¶

Asset correlation (R) determines how sensitive the exposure is to systematic risk. See _polars_correlation_expr() for the pure Polars implementation.

Corporate/Bank Correlation¶

R = 0.12 × (1 - exp(-50 × PD)) / (1 - exp(-50)) +
    0.24 × [1 - (1 - exp(-50 × PD)) / (1 - exp(-50))]

This produces: - R = 24% for very low PD - R = 12% for high PD

SME Size Adjustment¶

For SME corporates (turnover €5m-€50m). The turnover_m column is provided in GBP millions; the calculator converts to EUR using config.eur_gbp_rate:

# Convert GBP turnover to EUR
S_eur = turnover_gbp / eur_gbp_rate  # e.g. £25m / 0.8732 = €28.6m

# Clamp to range and apply adjustment
S = min(50, max(5, S_eur))
R_adjusted = R - 0.04 × (1 - (S - 5) / 45)

This reduces correlation by up to 4 percentage points for smaller firms.

Retail Correlations¶

Retail Type	Correlation
Residential Mortgage	15%
QRRE	4%
Other Retail	3-16% (PD-dependent)

Other Retail:

R = 0.03 × (1 - exp(-35 × PD)) / (1 - exp(-35)) +
    0.16 × [1 - (1 - exp(-35 × PD)) / (1 - exp(-35))]

Actual Correlation Implementation (formulas.py)

See _polars_correlation_expr() in src/rwa_calc/engine/irb/formulas.py for the full Polars expression implementation including SME adjustment.

Maturity Adjustment¶

For non-retail exposures. See _polars_maturity_adjustment_expr() for the pure Polars implementation.

# Maturity factor b
b = (0.11852 - 0.05478 × ln(PD))^2

# Maturity adjustment
MA = (1 + (M - 2.5) × b) / (1 - 1.5 × b)

Where M is effective maturity in years.

Actual Maturity Adjustment (formulas.py)

See _polars_maturity_adjustment_expr() in src/rwa_calc/engine/irb/formulas.py for the full Polars expression implementation.

Example values:

PD	M=1yr	M=2.5yr	M=5yr
0.03%	0.853	1.000	1.221
0.10%	0.880	1.000	1.177
1.00%	0.934	1.000	1.099
5.00%	0.966	1.000	1.050

Scaling Factor¶

Framework	Scaling Factor
CRR	1.06
Basel 3.1	1.00 (none)

# CRR
RWA = K × 12.5 × EAD × MA × 1.06

# Basel 3.1
RWA = K × 12.5 × EAD × MA

Expected Loss (EL)¶

IRB requires calculation of Expected Loss:

EL = PD × LGD × EAD

EL is compared to provisions: - EL > Provisions: Shortfall deducted from capital - EL < Provisions: Excess added to Tier 2 (with limits)

Detailed Calculation Example¶

Exposure: - Corporate loan, £50m - Bank-estimated PD: 0.50% - F-IRB (LGD = 45%) - Maturity: 3 years - Counterparty turnover: £25m (SME)

Step 1: Apply PD Floor

PD = max(0.0050, 0.0003) = 0.0050  # 0.50%

Step 2: Calculate Asset Correlation

# Base correlation
R_base = 0.12 × (1 - exp(-50 × 0.005)) / (1 - exp(-50)) +
         0.24 × (1 - (1 - exp(-50 × 0.005)) / (1 - exp(-50)))
R_base = 0.12 × 0.221 + 0.24 × 0.779 = 0.214

# SME adjustment (turnover £25m, converted to EUR)
S_eur = 25 / 0.8732 = 28.63  # GBP → EUR conversion
S = min(50, max(5, 28.63)) = 28.63
adjustment = 0.04 × (1 - (28.63 - 5) / 45) = 0.04 × 0.475 = 0.019
R = 0.214 - 0.019 = 0.195

Step 3: Calculate Capital Requirement (K)

# Intermediate calculations
G_PD = norm.ppf(0.005) = -2.576
G_999 = norm.ppf(0.999) = 3.090

term1 = (1 - R)^(-0.5) × G_PD = 1.115 × (-2.576) = -2.871
term2 = (R / (1-R))^0.5 × G_999 = 0.492 × 3.090 = 1.521

K_pre = LGD × N(term1 + term2) - LGD × PD
K_pre = 0.45 × N(-1.350) - 0.45 × 0.005
K_pre = 0.45 × 0.0885 - 0.00225
K_pre = 0.0398 - 0.00225 = 0.0376

Step 4: Calculate Maturity Adjustment

b = (0.11852 - 0.05478 × ln(0.005))^2 = (0.11852 + 0.290)^2 = 0.167
MA = (1 + (3 - 2.5) × 0.167) / (1 - 1.5 × 0.167)
MA = 1.0835 / 0.7495 = 1.446

Step 5: Calculate RWA

# CRR
RWA_CRR = K × 12.5 × EAD × MA × 1.06
RWA_CRR = 0.0376 × 12.5 × 50,000,000 × 1.446 × 1.06
RWA_CRR = £36,019,860

RW_CRR = RWA / EAD = 72.0%

# Basel 3.1 (no scaling)
RWA_B31 = 0.0376 × 12.5 × 50,000,000 × 1.446 × 1.00
RWA_B31 = £33,981,000

RW_B31 = 68.0%

Step 6: Check Output Floor (Basel 3.1)

# SA equivalent RWA (assume 100% RW corporate)
RWA_SA = 50,000,000 × 100% = £50,000,000

# Output floor (72.5%)
Floor = 50,000,000 × 0.725 = £36,250,000

# Final RWA
RWA_final = max(33,981,000, 36,250,000) = £36,250,000

Implementation¶

Using the IRB Namespace (Recommended)¶

The IRB module provides a Polars namespace extension for fluent, chainable calculations:

import polars as pl
from datetime import date
from rwa_calc.contracts.config import CalculationConfig
import rwa_calc.engine.irb.namespace  # Registers .irb namespace

config = CalculationConfig.crr(reporting_date=date(2026, 12, 31))

# Create sample exposure data
exposures = pl.LazyFrame({
    "exposure_reference": ["EXP001"],
    "pd": [0.005],
    "lgd": [0.45],
    "ead_final": [50_000_000.0],
    "maturity": [3.0],
    "exposure_class": ["CORPORATE"],
    "turnover_m": [25.0],  # Annual turnover in millions
})

# Fluent IRB calculation pipeline
result = (
    exposures
    .irb.classify_approach(config)   # Determine F-IRB vs A-IRB
    .irb.apply_firb_lgd(config)      # Apply supervisory LGD for F-IRB
    .irb.prepare_columns(config)     # Ensure required columns exist
    .irb.apply_all_formulas(config)  # Run full IRB calculation
    .collect()
)

# Access results
print(result.select([
    "pd_floored", "correlation", "k", "maturity_adjustment", "rwa", "expected_loss"
]))

Individual formula steps can also be chained:

result = (
    exposures
    .irb.apply_pd_floor(config)
    .irb.apply_lgd_floor(config)
    .irb.calculate_correlation(config)
    .irb.calculate_k(config)
    .irb.calculate_maturity_adjustment(config)
    .irb.calculate_rwa(config)
    .irb.calculate_expected_loss(config)
)

Using the IRB Calculator¶

from rwa_calc.engine.irb.calculator import IRBCalculator
from rwa_calc.contracts.config import CalculationConfig

# Create calculator
calculator = IRBCalculator()

# Calculate — takes a CRMAdjustedBundle, returns a LazyFrameResult
result = calculator.calculate(
    data=crm_adjusted_bundle,
    config=CalculationConfig.crr(reporting_date=date(2026, 12, 31))
)

# Result is a LazyFrameResult containing the LazyFrame and any errors
irb_rwa_df = result.data.collect()
print(irb_rwa_df.select("rwa", "expected_loss"))

Using IRB Formulas Directly¶

The IRB formulas are implemented in irb/formulas.py. Both scalar functions and pure Polars expression functions are available.

Implementation Architecture

Vectorized expressions: Pure Polars expressions for bulk processing (used by namespace and calculator)
Scalar wrappers: Thin wrappers around vectorized expressions for single-value calculations

from rwa_calc.engine.irb.formulas import (
    calculate_k,
    calculate_correlation,
    calculate_maturity_adjustment,
    calculate_irb_rwa,  # Convenience function for single exposures
)

# Calculate components
R = calculate_correlation(
    pd=0.005,
    exposure_class="CORPORATE",
    turnover_m=25.0,  # Turnover in GBP millions (converted to EUR internally)
)

MA = calculate_maturity_adjustment(pd=0.005, maturity=3)

K = calculate_k(pd=0.005, lgd=0.45, correlation=R)

# Calculate RWA
rwa = K * 12.5 * ead * MA * scaling_factor

# Or use the convenience function for complete calculation
result = calculate_irb_rwa(
    ead=50_000_000,
    pd=0.005,
    lgd=0.45,
    correlation=R,
    maturity=3.0,
    apply_scaling_factor=True,  # CRR 1.06 factor
)
print(f"RWA: {result['rwa']:,.0f}")

Scalar K Calculation (formulas.py)

See calculate_k() in src/rwa_calc/engine/irb/formulas.py for the scalar wrapper around the Polars expression.

Expected Loss Calculation¶

from rwa_calc.engine.irb.formulas import calculate_expected_loss

el = calculate_expected_loss(
    pd=0.005,
    lgd=0.45,
    ead=50_000_000
)
# el = 0.005 × 0.45 × 50,000,000 = £112,500

F-IRB vs A-IRB Comparison¶

Parameter	F-IRB	A-IRB
PD	Bank estimate	Bank estimate
LGD	Supervisory (45%/75%)	Bank estimate (floored)
EAD	Regulatory	Bank estimate
CCF	Regulatory	Bank estimate
Typical RW	Higher	Lower
Complexity	Lower	Higher
Approval	Easier	Harder

Regulatory References¶

Topic	CRR Article	BCBS CRE
IRB approach overview	Art. 142-150	CRE30
K formula	Art. 153	CRE31
PD estimation	Art. 178-180	CRE32
LGD estimation	Art. 181	CRE32
Correlation	Art. 153	CRE31
Maturity adjustment	Art. 162	CRE31
Supervisory LGD	Art. 161	CRE32

Next Steps¶

Standardised Approach - Compare with SA
Credit Risk Mitigation - CRM under IRB
Supporting Factors - SME correlation adjustment