Difference between the Basel IRB and the Vasicek formula

Question

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 1

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.