Reconciling Basel IRB and Vasicek Conditional Default Probabilities

Summary

The document compares the conditional default probability in the Basel IRB capital formula with the conditional probability expression in the Vasicek model. The apparent sign difference comes from the probability threshold used: the IRB expression uses the 0.999 quantile, while the explanation in the answer relates it to a default probability of 0.001.

Because the standard normal distribution is symmetric, the quantile for 0.999 is the negative of the quantile for 0.001. Rewriting one quantile in terms of the other accounts for the opposite sign in the formulas. This resolves the notation issue without introducing a different model. The document gives a concise algebraic explanation, but does not derive the full capital formula or discuss model assumptions and applications.

Key ideas

• The Basel IRB formula uses a high confidence quantile, whereas the Vasicek expression is framed using a low default probability.
• Normal quantile symmetry makes the 0.999 and 0.001 quantiles equal in magnitude and opposite in sign.
• The differing signs in the expressions can therefore reflect different probability conventions rather than a substantive model conflict.

Tags

# Difference between the Basel IRB and the Vasicek formula


# Difference between the Basel IRB and the Vasicek formula












The well known Basel IRB formula is as follows:

$${\displaystyle K=LGD*\left[N\left({\sqrt {\frac {1}{1-R}}}*G(PD)+{\sqrt {\frac {R}{1-R}}}*G(0.999)\right)-PD\right]}$$

where the term below is the conditional probability of default:

$$PD^c = N\left({\sqrt {\frac {1}{1-R}}}*G(PD)+{\sqrt {\frac {R}{1-R}}}*G(0.999)\right)$$

However, after going trough the referenced Vasicek(2002) paper there is the following formula for conditional PD on page 3, which has a minus instead of plus between the two terms:

$$ p(Y) = N\left( \frac{N^{-1}(p) - Y \sqrt{\rho}}{\sqrt{1 - \rho}} \right) $$

Am I missing something obvious?

Edit: The different notation is the following:

- probability of default: $PD = p $

- inverse normal distribution: $G = N^{-1}$

- correlation between assets: $R = \rho$

- $Y$ is a normally distributed random variable

## Answer by buckner (score 3, accepted)

https://quant.stackexchange.com/a/38259

The paper continues "The quantity p(Y) provides the loan default probability under the given scenario."

But the default probability is 0.001, not 0.999 as in the IRB version. So G(0.999) = -G(1 - 0.999) and that is where the minus comes in.

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.