Tail behavior of credit loss distributions

We derive the exact loss distribution for portfolios of bonds or cor-porate loans when the number of risks grows indefinitely. We show that in many cases this distribution lies in the maximal domain of attraction of the Weibull (Type III) limit law. Knowledge of the dis-tribution and its tail behavior is important for risk management in order not to over- or underestimate the likelihood of extreme credit losses for the portfolio as a whole. Conform to the credit risk literature, we assume that bond (or loan) defaults are triggered by a latent variable model involving two stochastic variables: systematic and idiosyncratic risk of the bond. It is shown that the tail behavior of these two variables translates into the tail behavior of the whole credit loss distribution. Surprisingly, even if both variables are thin-tailed, the credit loss distribution can have a finite tail index. Moreover, if idiosyncratic risk exhibits heavier tails than the systematic risk factor the tail index of the credit loss distribution can become extremely high, giving rise to a non-conventional shape of the credit loss distribution.