I'm quite confused about the distribution of default rate.
I understand the default can be seen as from binomial distribution
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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.