In order to simulate Vasicek random sample, we invert (1) resolving it for s:

(2)

Simulating random value w that is uniformly [0,1] distributed, we generate Vasicek sample.

Vasicek [1991, 2002] has described the main property of this distribution. He characterized it as “highly skewed,” which is correct unless p = 0.5. The mean is equal to p (a trivial fact), whereas the variance is
N₂ [N ^-1(p), N ^-1(p), r] – p ² where N₂ denotes the standard two-dimensional normal cumulative distribution function.

The Vasicek model is underpinned by several key assumptions. One of them is the identity of default rates among all the loans. One can easily give a counter-example (see the next section) in which, with this assumption omitted, the model leads to an incorrect result. Another assumption – infinite pool size – leads to a full diversification of individual defaults, except driven by a common factor. If such a factor does not exist, i.e r is 0, then s = p, the entire distribution of possible default rates collapses into a single value.

An APD extension of the Vasicek model
Assume now that the pool consists of two populations, active and passive; each of them separately follows the Vasicek model having own parameters p and r. Let y denotes the active portion of the pool. Assume further that the key factor affecting default rate (such as global home price index) is identical for both groups. This assumption does not assume that asset values of active and passive borrowers are perfectly correlated. However, each value of this common factor points to a single default rate in the infinitely diversified active sub-pool and passive sub-pool.

Hence, we can formally generate default rates for active and passive populations using formula (2):

(2a)

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