Overview

Two complementary modelling frameworks handle this:

• Threshold (latent variable) models: each obligor $i$ defaults if a latent "credit quality" variable $X_i$ falls below a threshold $d_i$. The joint default distribution is determined by the copula of $(X_1,\ldots,X_m)$. The Gaussian threshold model — a multivariate extension of Merton's structural model — is the industry standard and the basis of the Basel IRB capital formula.
• Bernoulli mixture models: obligors share common macro factors $\Psi$; conditional on $\Psi$, defaults are independent. Unconditional dependence arises through the shared factor — a recession raises every firm's conditional PD simultaneously. The probit-normal and logit-normal mixing distributions encompass the Gaussian threshold model as a special case.

For large portfolios, the Gordy-Vasicek asymptotics show that the $\alpha$-quantile of the relative loss converges to $\bar\ell(q_\alpha(\Psi))$ — the loss at the $\alpha$-quantile of the macro factor. This single-factor ASRF result is the derivation behind the Basel IRB formula $\text{RC}_i = c\delta_i e_i\,\Phi\!\left(\frac{\Phi^{-1}(p_i)+\sqrt{\beta_i}\,\Phi^{-1}(0.999)}{\sqrt{1-\beta_i}}\right)$.

By the end of this topic you should be able to:

• define a threshold model and state Lemma 11.2 (equivalence criterion: same marginals + same copula);
• set up the Gaussian threshold model with factor structure and compute default correlations;
• define a Bernoulli mixture model and verify that the Gaussian threshold model is a probit-normal mixture;
• derive the Vasicek large-portfolio loss distribution and its VaR limit;
• state the Basel IRB formula and trace each term to the ASRF derivation;
• explain exponential tilting (importance sampling) and how it shifts the factor distribution to oversample bad states;
• describe a CDO tranche, compute the tranche loss, and state why the Gaussian copula fails for super-senior tranches.

1. Contents
2. The big picture