Is Unexpected Loss ever used in Basel II?

Question

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 1

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 2

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".