Basel II IRB Capital Requirements as a Buffer for Unexpected Credit Loss
Summary
The document asks how unexpected loss is treated under Basel II and presents the advanced internal ratings-based framework’s expected-loss expression, correlation and maturity adjustments, capital requirement, and risk-weighted asset calculation. Its accepted response interprets the capital requirement K as covering unexpected credit loss, while expected loss is treated as a cost to be provisioned or reflected in pricing. The answer connects K and exposure at default to the capital framework rather than treating unexpected loss as a separate standalone limit.
The material is an informal explanation, not a complete regulatory derivation. The respondent explicitly says they do not know how the normal distribution terms in the formula arise and points to further explanatory material. The stated capital adequacy ratio denominator is also presented as total assets, whereas Basel regulatory ratios are generally defined using risk-weighted assets; readers should verify the applicable Basel text and terminology before using these formulas for compliance or calculation.
Key ideas
- Expected loss is expressed using probability of default, exposure at default, and loss given default.
- The IRB capital requirement formula includes default probability, asset correlation, loss severity, and a maturity adjustment.
- The accepted answer interprets capital K as covering unexpected loss, with expected loss handled separately.
- The explanation does not derive the formula’s distribution terms and is not a complete regulatory reference.
- The capital ratio description should be checked against the applicable Basel definition.
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Full text
# Is Unexpected Loss ever used in Basel II?
# Is Unexpected Loss ever used in Basel II?
In Basel II, EL is useful. It's calculated as
$$EL = PD \cdot EAD \cdot LGD $$
in advance IRB (internal rate-based approach),
Correlation $$R = 0.12 \frac{1 – e^{-50 \cdot PD}}{1 – e^{-50}} + 0.24 \cdot (1- \frac{ 1 – e^{-50 \cdot PD}} {1 – e^{-50}} )$$
Maturity adjustment
$$b = [0.11852 – 0.05478 \ln(PD)]^2$$
Capital requirement $$K = \{ LGD \cdot N(\sqrt{\frac{1}{1 – R}} \cdot G(PD) + \sqrt{\frac{R}{1 – R}} \cdot G(0.999)) – PD \cdot LGD\} \cdot \frac{1 + (M – 2.5) b}{1 – 1.5 b} $$
here Ln denotes the natural logarithm; N(x) denotes the cumulative distribution function for a standard normal random variable; G(z) denotes the inverse cumulative distribution function for a standard normal random variable (i.e. the value of x such that N(x) = z).
Afterwards,
Risk-weighted assets $$RWA = K \cdot 12.5 \cdot EAD$$
then
$$CAR = \frac{Tier 1 capital + Tier 2 capital}{Total Asset}$$
-- Basel II defines limits on CAR.
But, for unexpected loss, did Basel II make any restriction on it?
FRM has a set of formula calculating UL from LGD, EAD etc... Unexpected Loss $$UL = EAD \sqrt{PD\cdot \sigma_{LGD}^2 + LGD^2 \cdot \sigma_{PD}}$$
## Answer by athos (score 3, accepted)
https://quant.stackexchange.com/a/8053
let me try answer my own questions, partially, from below that are exerpted from FRM exam notes.
So actually the K above, is UL, though it derives only from PD and maturity, but the G, N and 0.999, actually are calculating the VaR and UL.
So, CAR is defined based on EAD and K, while K means UL. the essence is, CAR is to cover Unexpected Loss -- captical reserved is not for EL, EL shall be calculated in the cost already.
However, how the K formular using G, and N comes from PD, I don't know... maybe need dig some papers.
## Answer by Quartz (score 2)
https://quant.stackexchange.com/a/8112
For more explanations you can also try out "An explanatory note on the Basel II IRB Risk Weight Functions", or if you read german, "Die IRB Formel".Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.