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Work out Basel II credit risk capital with the internal ratings-based (IRB) approach.

FRM® Part II · Operational Risk and Resilience · Capital Regulation Before the Global Financial Crisis · FRM2.ORR.21.f

The idea

The IRB formula measures unexpected loss at a 99.9 percent one-year confidence level. The worst-case default rate, written WCDR, uses N for the standard normal distribution and ρ for the asset correlation, which for corporates falls from 0.24 at very low PD to 0.12 at high PD. Altogether, the worst-case default rate is WCDR = N((N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ)). EAD is exposure at default, LGD loss given default and MA the maturity adjustment. Capital per exposure is K = EAD × LGD × (WCDR − PD) × MA. The maturity adjustment depends on the default probability through a factor b = (0.11852 − 0.05478 × ln PD)². With M the effective maturity, the maturity adjustment is MA = (1 + (M − 2.5) × b) ÷ (1 − 1.5 × b). PD is subtracted because expected loss is covered by provisions and pricing, so capital covers only unexpected loss. Risk-weighted assets follow by multiplying the capital requirement K by 12.5, so RWA = 12.5 × K. Common errors are forgetting to subtract PD, dropping MA, or omitting the 12.5 factor when asked for RWA.

Formulas

IRB worst-case default rate

WCDR = N((N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ))

N
the cumulative standard normal distribution, with N⁻¹ its inverse
PD
the average default probability
ρ
the asset correlation

The 99.9 percent one-year default rate. For corporates ρ falls from 0.24 at very low PD to 0.12 at high PD.

IRB capital for an exposure

K = EAD × LGD × (WCDR − PD) × MA

EAD
the exposure at default
LGD
the loss given default
WCDR
the worst-case default rate at 99.9 percent
PD
the probability of default
MA
the maturity adjustment

PD is subtracted because expected loss is covered by provisions and pricing. RWA = 12.5 × K; dropping MA or the 12.5 factor are common errors.

Maturity adjustment factor b

b = (0.11852 − 0.05478 × ln PD)²

PD
the probability of default

ln PD is negative, so b rises as PD falls. Feeds the maturity adjustment.

IRB maturity adjustment

MA = (1 + (M − 2.5) × b) ÷ (1 − 1.5 × b)

M
the effective maturity in years
b
the maturity factor from PD

Above 1 when M exceeds 2.5 years. At M of 2.5 the numerator is 1.

A practice question

From the question bank, with the reason each wrong option is wrong.

The Basel II IRB worst-case default rate is calibrated to which confidence level and horizon?

  1. 99 percent over 10 daysWhy not: This is the market risk VaR standard of the 1996 amendment.
  2. 99.9 percent over one year - correct
  3. 99.5 percent over one yearWhy not: This is the Solvency II SCR calibration, not the Basel II IRB standard.
  4. 95 percent over one yearWhy not: The IRB formula uses a much higher confidence level, 99.9 percent.

The IRB formula estimates the default rate that will not be exceeded with 99.9 percent confidence over one year, a one-in-1000-year stress.

A harder one

For a corporate loan the IRB inputs are EAD of USD 80 million, LGD of 45 percent, PD of 0.8 percent, WCDR of 9.5 percent and maturity adjustment of 1.10. The risk-weighted assets are closest to:

  1. USD 43.1 million - correct
  2. USD 3.4 millionWhy not: This is the capital requirement K; it is not multiplied by 12.5 to give RWA.
  3. USD 47.0 millionWhy not: Uses WCDR without subtracting PD, so charges capital on expected loss as well.
  4. USD 39.1 millionWhy not: Leaves out the maturity adjustment of 1.10.

K = 80 x 0.45 x (0.095 - 0.008) x 1.10 = 3.445. RWA = 12.5 x K = 43.1.

The hardest tier

A corporate exposure has EAD of USD 50 million, PD of 0.5 percent, LGD of 45 percent and effective maturity of 4 years. The asset correlation from the Basel formula is 0.2135, giving a worst-case default rate of 9.77 percent. With b = (0.11852 - 0.05478 x ln PD)² and MA = [1 + (M - 2.5) x b] / (1 - 1.5 x b), the IRB risk-weighted assets are closest to:

  1. USD 26.1 millionWhy not: Leaves out the maturity adjustment, which is well above 1 for a four-year exposure.
  2. USD 43.5 million - correct
  3. USD 50.0 millionWhy not: Uses the maximum correlation of 0.24 instead of the value stated for this PD, giving a higher WCDR.
  4. USD 45.9 millionWhy not: Does not subtract PD from WCDR, charging capital for expected loss as well.

b = (0.11852 + 0.05478 x 5.2983)² = 0.1671; MA = (1 + 1.5b)/(1 - 1.5b) = 1.6689. K = 50 x 0.45 x (0.0977 - 0.005) x 1.6689 = 3.482. RWA = 12.5K = 43.5.

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