LLN-type approximations for large portfolio losses

We are concerned with the loss from defaults of a large portfolio of defaultable obligors. A static structural model is constructed, in which for each obligor i its default is driven by a latent variable Ui and its loss given default (LGD) is driven by another latent variable Vi through a general loss settlement function G. In this way, the default indicator 1Ui>a, with a denoting a default threshold, and the LGD G(Vi) are not necessarily comonotonic, hence essentially different from the ones used in some recent works. It is further assumed that the two latent variables Ui andVi are correlated in the way that they share a common systematic risk factor but each has its own idiosyncratic risk factor. We employ the law of large numbers (LLN) to derive the exact limit distribution of the portfolio loss as the portfolio size becomes large. As applications, we also derive exact approximations for the TVaR and moments of the portfolio loss.