Asymptotic Single Risk Factor Model (ASRF):

 Key assumptions: asymptoticity, a single risk factor, and normality.

PD assumptions and violations:

Probability of observing D defaults over N (total number of exposures in the credit portfolio) independent random draws follows a binomial distribution.
PD ASRF model Binomial Distribution assumptions:
  i.  Each asset in the rating grade has default probability P.
 ii.  Each pair of assets has default correlation ρ
iii.  The conditional correlation between any two assets is constant even if the number of defaults increases.
iv.   Normal distribution assumption for the systematic factor
 
ASRF model assumptions may get violated:

Assumptions (i) and (ii):
Let x1, ..,xn be random indicator variables representing the default behavior of the assets where xj =1 indicates the default of asset j. Define as the probability of default of asset j given that assets 1 to j-1 are known to have defaulted.
Assumptions (i) and (ii) imply that:
P1 = P and P2 = P + (1 - P )*ρ
When assets are independent ρ = 0 than these assumptions lead to the Binomial distribution with Pj = P.
However, If ρ > 0, then x1, ..,xn are not independent than the assumption of Binomial distribution is violated.
Assumption (iii):
If the conditional correlation between any two assets increase as the number of defaults increases will lead to increase in default probability.
The increasing default probability given other defaults results in fatter tails of the Correlated Binomial distribution. Contrast assumption (3) with the Binomial distribution where the independence assumption implies that ρj = ρ for all j=1,..,n assets
 
Assumption (iv):
Systematic factor may follows an autoregressive process.