# Difference between revisions of "Vasicek Distribution"

## Definition

The Vasicek Distribution is a special probability distribution that emerges in the context of Threshold Models used in credit portfolio modelling. It was first introduced in[1]. There are two versions, the finite portfolio case and its limiting case wher the number of exposures in a portfolio assumed infinitely many / infinitely small.

## Finite Portfolio Case

In the finite case the probability mass of D=k defaults out of a pool of N credits with equal probability of default p is

$P[D=k] = \int_{0}^{\infty} \binom{N}{k} p^k(x) (1 - p(x))^{N-k} G(dx)$

where G(z) denotes the inverse cumulative distribution function

## Limit Case

The limit case arrises as a limit distribution of the sum of conditionally independent Bernulli variables

## Usage

Used extensively in as a simple model of portfolio loss (also in Basel II as the ASRF model)

## Seel Also

• Vasicek O., Probability of loss on loan portfolio, 1987