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Show how Vasicek's model supports the estimation of capital requirements in the internal-ratings-based (IRB) approach of Basel II.

FRM® Part II · Credit Risk Measurement and Management · Credit Value at Risk · FRM2.CR.10.d

The idea

Vasicek's model assumes each borrower's asset value is driven by one systematic factor and an idiosyncratic factor, with asset correlation ρ, in a large granular portfolio. Conditional on a bad factor outcome, the default rate becomes the worst-case default rate. WCDR = N[(N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ)]. Here N is the standard normal cumulative distribution function and N⁻¹ is its inverse. The Basel II IRB approach applies it at 99.9 percent over one year. Expected loss, PD × LGD, is handled by provisions, which is why PD is deducted from the worst-case rate. Capital per unit of exposure, before any maturity adjustment, is then calculated as K = LGD × (WCDR − PD). A maturity adjustment multiplies K where the approach requires it. Risk-weighted assets are then given by RWA = K × 12.5 × EAD. Common errors are using ρ instead of its square root, and omitting the deduction of PD.

Formulas

Vasicek worst-case default rate at 99.9%

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

N
the standard normal cumulative distribution function, with N⁻¹ its inverse
PD
the borrower's one-year probability of default
ρ
the asset correlation with the systematic factor

A large, granular portfolio driven by one factor, at the IRB 99.9% one-year level. The correlation enters as √ρ on top and √(1 − ρ) below; using ρ itself understates the WCDR.

IRB capital per unit of exposure

K = LGD × (WCDR − PD)

LGD
loss given default
WCDR
the 99.9% worst-case default rate
PD
the probability of default

Before the maturity adjustment, which multiplies K where the approach requires it. PD is deducted because expected loss is covered by provisions.

Risk-weighted assets from the IRB capital requirement

RWA = K × 12.5 × EAD

K
capital per unit of exposure, after any maturity adjustment
EAD
the exposure at default

Converts a capital charge into risk-weighted assets. Leaving out the 12.5 reports the capital itself, and skipping the maturity adjustment understates K.

A practice question

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

In Vasicek's model as used in the Basel II IRB approach, the worst-case default rate (WCDR) is:

  1. the long-run average default rate, equal to the PD.Why not: Confuses the worst case with the unconditional average; the WCDR is much larger than PD.
  2. the one-year default rate that is not exceeded with 99.9% probability given the systematic factor. - correct
  3. the default rate in the worst year in the bank's own history.Why not: Uses a historical extreme; the WCDR is model-based at a stated confidence level.
  4. PD multiplied by LGD.Why not: This is expected loss as a fraction of exposure, not a default rate.

Conditional on a bad draw of the systematic factor (a 0.1% tail), the large portfolio's default rate is the WCDR; capital covers WCDR minus PD, scaled by LGD.

A harder one

Using Vasicek's model with PD of 2%, asset correlation of 0.15 and a 99.9% confidence level, the worst-case default rate is closest to (N^-1(0.02) = -2.054, N^-1(0.999) = 3.090):

  1. 4.2%Why not: Uses rho (0.15) in place of its square root in the numerator.
  2. 19.6%Why not: Omits the division by sqrt(1 - rho).
  3. 17.6% - correct
  4. 10.6%Why not: Uses the 99% quantile (2.326) instead of 99.9%.

WCDR = N[(-2.054 + sqrt(0.15) x 3.090) / sqrt(0.85)] = N[(-2.054 + 1.197) / 0.922] = N(-0.929) = 17.6%. The correlation enters as sqrt(rho) and the denominator is sqrt(1 - rho).

The hardest tier

A bank has EAD of USD 50 million, PD 1%, LGD 45% and a Vasicek WCDR of 14.55%. The maturity adjustment is (1 + (M - 2.5) x b) / (1 - 1.5 x b) with M = 4 and b = 0.1375. Risk-weighted assets are K x 12.5 x EAD. RWA is closest to:

  1. USD 57.9 million - correct
  2. USD 38.1 millionWhy not: Leaves out the maturity adjustment of about 1.52.
  3. USD 4.6 millionWhy not: Reports the capital requirement and omits the 12.5 multiplier.
  4. USD 46.0 millionWhy not: Applies only the numerator of the maturity adjustment, 1 + 1.5 x 0.1375 = 1.206, and omits the denominator.

K before adjustment = 0.45 x (14.55% - 1%) = 6.10%. MA = 1.20625 / 0.79375 = 1.5197. K adjusted = 9.27%. RWA = 9.27% x 12.5 x 50 = 57.9 million.

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