Modeling Portfolio Default Rates with Binomial and Vasicek Distributions

Summary

A default indicator for one borrower can be modeled as a Bernoulli trial. For a portfolio of loans, the default count is binomial when defaults are treated as independent trials with a common probability. This gives a straightforward way to describe portfolio defaults, but it does not capture dependence among borrowers on its own.

The Vasicek framework adds a normally distributed systemic factor that influences borrowers’ asset values and their chances of crossing default thresholds. Conditional on that factor, the portfolio default count follows a binomial distribution; across possible factor outcomes, the default rate varies with the shared economic condition. The document identifies this framework as a basis for the Basel credit risk capital formula and analytical CDO pricing approximations. It sketches the model rather than deriving it, and notes that normality applies to the systemic factor, not directly to the bounded default rate. The usefulness of the resulting distribution depends on assumptions about asset values and default dependence.

Key ideas

• A single borrower’s default can be represented as a Bernoulli outcome.
• A portfolio default count is binomial when defaults are independent with a common probability.
• A Vasicek model introduces a shared systemic factor that affects default likelihoods.
• Conditional on the systemic factor, portfolio defaults follow a binomial distribution.
• The model is used in credit capital calculations and analytical CDO pricing approximations.

Tags

# Does the default rate follow normal distribution or binomial distribution?


# Does the default rate follow normal distribution or binomial distribution?












I'm quite confused about the distribution of default rate.

I understand the default can be seen as from binomial distribution

## Answer by Magic is in the chain (score 2)

https://quant.stackexchange.com/a/54934

Normal rv can take negative values so won’t work for default rate (which is positive) without some form of transformation- classic approach is Vasicek, which by making assumptions about the default process ( default occurs when asset value falls below some thresholds) and then the dynamics of the asset process (e.g. geometric brownian), produces the standard single systemic factor model for the default rates, and eventually portfolio loss distribution.. The systemic factor is in a way assumed to be normal.

This is the model behind the Basel capital formula for Credit Risk, and is also used in the analytical approximation of CDOs prices.

Re binomial, default/No default is a Bernoulli trial, and if you have a portfolio of loans, then you are dealing with sum of Bernoulli variables, which is binomial. Linking this to the Vasicek, you can say that the number of defaults in the portfolio conditional on the realisation of the systemic factor will follow Binomial.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.