4.9.4 Credit & Operational Risk

Basic Prerequisites: Credit Risk & the Merton Model (PD, LGD, distance-to-default, the Vasicek one-factor model) and 02 · Capital & RWA.

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1. Intuition & Practical Objective

Credit risk is the largest RWA category for almost every bank; operational risk is the smallest but the one where the regulator gave up on internal models. This page covers both, and the common theme is SA vs internal model: the standardised approach applies regulator-set weights, while the internal approach lets the bank supply its own parameters - and the difference in capital can be enormous.

• Credit risk. SA weights a loan by asset class and (where available) external rating: sovereign, bank, corporate, retail, mortgage. IRB instead plugs the bank’s own PD, LGD, EAD, and M into the ASRF (asymptotic single-risk-factor) closed form - the same Vasicek one-factor formula behind portfolio credit-VaR. Foundation IRB: bank supplies PD, supervisor supplies LGD/EAD. Advanced IRB: bank supplies all four.
• Operational risk. Basel II offered SA (Basic Indicator / Standardised) and the AMA (Advanced Measurement Approach), a full internal loss-distribution model. The AMA was abused (op-risk RWA fell, then losses appeared), and the 2017 finalisation removed it, replacing everything with one Standardised Measurement Approach (SMA): capital $=$ Business Indicator Component $\times$ Internal Loss Multiplier. Op risk is now the one Pillar-1 risk with no internal-model route.

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2. Mathematical Ground Truth & Derivations

2.1 SA credit risk weights (BCBS 2006)

A sample of the standardised grid (exposure × weight → RWA):

Exposure	Weight
Sovereign, AAA–AA	0%
Bank, AAA–AA (or good, unrated)	20%
Residential mortgage	35%
Retail	75%
Corporate, unrated	100%
Past-due / high-risk	150%+

2.2 IRB credit risk - the ASRF capital formula

For a corporate/retail exposure with probability of default $\mathrm{PD}$, loss-given-default $\mathrm{LGD}$, and maturity $M$:

$$\boxed{K=\underset{\text{conditional expected loss above PD}\cdot \text{LGD}}{\underbrace{[\mathrm{LGD}\cdot N(\frac{N^{-1}(\mathrm{PD})+\sqrt{R}{N}^{-1}(0.999)}{\sqrt{1-R}})-\mathrm{PD}\cdot \mathrm{LGD}]}}\times \mathrm{MA}}$$

with the asset correlation

$$R=0.12\frac{1-e^{-50\mathrm{PD}}}{1-e^{-50}}+0.24(1-\frac{1-e^{-50\mathrm{PD}}}{1-e^{-50}}),$$

and the maturity adjustment $b=(0.11852-0.05478\ln \mathrm{PD}{)}^2$, $\mathrm{MA}=\frac{1+(M-2.5)b}{1-1.5b}$. The numerator $N^{-1}(\mathrm{PD})+\sqrt{R}{N}^{-1}(0.999)$ is the Vasicek one-factor conditional-default-probability argument at the 99.9% confidence level (Hull eq. 24.10): the $N^{-1}(0.999)$ shock is the systematic factor, $\sqrt{R}$ its loading. Then $\mathrm{RWA}=K\times 12.5\times \mathrm{EAD}$ (and historically $\times 1.06$ for the Basel II scaling factor).

2.3 Operational risk - the SMA (BCBS 2017)

$$\mathrm{ORC}=\mathrm{BIC}\times \mathrm{ILM},{\mathrm{RWA}}_{\text{op}}=12.5\times \mathrm{ORC}.$$

The Business Indicator Component scales the Business Indicator $BI$ by marginal coefficients ${\alpha }_i$:

$$\mathrm{BIC}=\{\begin{array}{ll}0.12BI,& BI\le \text{EUR }1\text{bn}\\ 0.12\cdot 1+0.15(BI-1),& 1<BI\le \text{EUR }30\text{bn}\\ 0.12\cdot 1+0.15\cdot 29+0.18(BI-30),& BI>\text{EUR }30\text{bn}\end{array}$$

The Internal Loss Multiplier folds in the bank’s own loss history through the Loss Component $LC=15\times$ average annual losses:

$$\mathrm{ILM}=\ln (e-1+(\frac{LC}{\mathrm{BIC}}{)}^{0.8}),$$

so $\mathrm{ILM}=1$ when $LC=\mathrm{BIC}$, $\mathrm{ILM}>1$ for a lossy bank, $\mathrm{ILM}<1$ for a clean one. Capital literally rises and falls with a bank’s own realised losses.

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3. Computational Implementation - IRB vs SA, and the op-risk SMA

Standard library (normal CDF from math.erf, inverse from a rational approximation).

Experiment 1 - IRB risk weights vs the flat SA weight.

Experiment 2 - operational risk via the SMA.

(The €35bn row reproduces the regulator’s own worked example - $\mathrm{BIC}=0.12+29\times 0.15+5\times 0.18=\text{EUR }5.37\text{bn}$ - a direct check that the bucket arithmetic is right.)

What the numbers say. (i) IRB is not uniformly cheaper: at PD 0.03% it yields a 14.4% RW but at PD 5% a 149.9% RW, so it is more risk-sensitive than SA in both directions. (ii) For a typical 1%-PD corporate the IRB RW (92.3%) is close to the flat SA 100%, which is why the arbitrage was never about plain corporates but about the tails and about low-PD assets. (iii) The SMA makes op-risk capital increase with the bank’s own losses - an elegant self-punishing design that removed the AMA’s incentive to under-model.

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4. Failure Modes & First-Principles Breakdowns

1. IRB parameters are the bank’s own. Foundation IRB isolates PD, but advanced IRB lets the bank set LGD, EAD, and M - and a low-balled LGD directly cuts capital. Supervisory validation is the only check; weak validation is a first-order failure (05).
2. Procyclicality is built into IRB. PD and LGD rise in downturns, so IRB RWA rises exactly when capital is scarce - the opposite of what a stabiliser should do. This is the canonical procyclicality failure and a core motivation for the output floor and buffers.
3. The AMA was gamed out of existence. Banks’ op-risk models predicted low capital; realised losses (conduct, litigation) exceeded them badly. The regulator’s response - abolishing the internal route - is a case study in why model-based capital needs a backstop.
4. SA is blunt but hard to game. Flat asset-class weights ignore real differences in borrower quality; a bank can hold high-quality and low-quality exposure at the same weight. Comparability rose; risk sensitivity fell. The post-crisis reform trades between the two.
5. The 1.06 scaling factor and floor mechanics. Small multiplicative factors and floors sound cosmetic but move RWA by billions; never ignore a “scaling factor” in the rules.

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5. References

• BCBS - Basel II: International Convergence of Capital Measurement and Capital Standards (2006). SA risk-weight grid, the IRB formula, the three pillars. Read from the corpus PDF.
• BCBS - Basel III: Finalising Post-Crisis Reforms (2017, d424). Revised SA credit risk weights, the removal of the AMA, and the SMA for operational risk (Business Indicator, BIC, ILM, ORC$=\mathrm{BIC}\times \mathrm{ILM}$). Read from the corpus PDF; the €35bn BIC example is checked numerically above.
• Hull, John C. - Options, Futures, and Other Derivatives (11th ed.)
• Vašíček, Oldřich - Probability of Loss on Loan Portfolio (1987, KMV). The one-factor Gaussian model underpinning the IRB capital formula.
• Gupton, Finger & Bhatia (J.P. Morgan) - CreditMetrics™ Technical Document (1
• Bluhm, Overbeck & Wagner - Introduction to Credit Risk Modeling (2nd ed., 2010). Accessible derivation of the ASRF formula and portfolio loss distributions.
• Panjer, Harry H. - Operational Risk: Modeling Analytics (2006, Wiley). The loss-distribution approach that the AMA was built on

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6. Connected Graph Bridges

• Back: 02 · Capital & RWA · Index Hub
• Prerequisite / sibling: Credit Risk & the Merton Model · Counterparty Risk & xVA (the CVA charge)
• Forward: 05 · Failure Modes · 06 · Advanced Extensions