Bernoulli Mixture Models: One-Factor Model and the Vasicek Distribution

The Gaussian threshold model of the previous lesson has an equivalent mixture-model representation that is far more convenient for simulation, statistics, and large-portfolio analysis. In a Bernoulli mixture model, defaults are independent conditional on the realisation of a common economic factor $\Psi$. Unconditional default dependence arises entirely through the shared factor: a deep recession drives every firm's conditional PD upward simultaneously. The mixing distribution of the conditional PD $Q = p(\Psi)$ determines the unconditional portfolio loss distribution.

The key equivalence: the one-factor Gaussian threshold model is identical to the probit-normal Bernoulli mixture model. Both are parameterised by the same quantities (marginal PD $p_i$ and asset correlation $\beta_i$), and both produce the same joint default distribution. In the probit-normal case, the conditional PD is $Q = \Phi(\mu + \sigma\Psi)$ for $\Psi \sim \mathcal{N}(0,1)$. For a homogeneous portfolio, the loss distribution converges as portfolio size $m\to\infty$ to the Vasicek distribution — the key building block of the Basel IRB formula.

Learning objectives. By the end of this note you should be able to: