How are risk-weighted assets calculated under the Internal Ratings-Based approach?

The IRB (Internal Ratings-Based) approach allows banks to use their own risk parameters to compute capital requirements, subject to regulatory formulas. The key formula converts PD, LGD, EAD, and maturity into a capital charge, which then maps to risk-weighted assets.

The IRB Capital Formula (Corporate Exposures)

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

Where:

• N() = Standard normal CDF
• G() = Inverse standard normal
• R = Asset correlation
• MA = Maturity adjustment

Step-by-Step Calculation

Suppose Alderton International Bank rates a corporate borrower with:

• PD = 1.5%
• LGD = 40%
• EAD = $50 million
• Maturity = 3 years

Step 1: Asset Correlation (R)

R = 0.12 x [(1 − e^(−50xPD))/(1 − e^(−50))] + 0.24 x [1 − (1 − e^(−50xPD))/(1 − e^(−50))]

R = 0.12 x [(1 − e^(−0.75))/(1 − e^(−50))] + 0.24 x [1 − ...]

With PD = 0.015:

(1 − e^(−0.75)) = 0.5276

R = 0.12 x 0.5276 + 0.24 x (1 − 0.5276) = 0.0633 + 0.1135 = 0.1768

Step 2: Conditional PD at 99.9% Confidence

Conditional PD = N((1/(1−0.1768))^0.5 x G(0.015) + (0.1768/(1−0.1768))^0.5 x G(0.999))

= N(1.1027 x (−2.1701) + 0.4635 x 3.0902)

= N(−2.3930 + 1.4333)

= N(−0.9597) = 0.1687 or 16.87%

Step 3: Base Capital Charge

K_base = LGD x (Conditional PD − PD) = 0.40 x (0.1687 − 0.015) = 0.40 x 0.1537 = 6.15%

Step 4: Maturity Adjustment

b = [0.11852 − 0.05478 x ln(PD)]^2 = [0.11852 − 0.05478 x (−4.1997)]^2 = [0.3485]^2 = 0.1215

MA = (1 + (M − 2.5) x b) / (1 − 1.5 x b) = (1 + 0.5 x 0.1215) / (1 − 0.1822) = 1.0607 / 0.8178 = 1.297

Step 5: Final Capital Requirement

K = 6.15% x 1.297 = 7.98%

Step 6: Risk-Weighted Assets

RWA = K x 12.5 x EAD = 0.0798 x 12.5 x $50M = $49.9 million

Capital required = 8% x RWA = 8% x $49.9M = $3.99 million

Key Insights:

• Higher PD leads to higher conditional PD and more capital
• Longer maturity increases capital through the maturity adjustment
• The 0.999 quantile reflects Basel's 99.9% confidence level — capital covers losses in a 1-in-1000 year event
• Asset correlation R decreases with PD: riskier borrowers are assumed to be more idiosyncratic

Exam Tip: You won't need to memorize the full formula, but understand the intuition: the IRB approach models portfolio credit risk as a single systematic factor (the Vasicek model), and the correlation parameter R determines how much each borrower's default is driven by the common factor vs. idiosyncratic risk.

For more Basel capital problems, try our FRM Part II question bank.