T-Vasicek Credit Portfolio Loss Distribution

Abstract

We derive and discuss a new analytical credit loss distribution. This new model, T-Vasicek, is a variant of the well-known and highly useful Vasicek distribution. We inject a t-distribution extension into Vasicek that preserves the analytical convenience of Vasicek while providing a richer credit loss framework with fat tails and an additional user-specified parameter. This additional parameter is directly tied to the t-distribution and represents the uncertainty in the default probability estimate. The classical Vasicek limit, then, is the case in which the analyst knows the pool default probability with certainty. We show how one may impose desired correlation among all debt instruments in the t-distribution framework. We derive closed-form, numerical, and analytical forms for T-Vasicek and check the numerical results with Monte Carlo simulation. We also determine suitable maximum likelihood estimators for the T-Vasicek parameters of default probability (PD), correlation (ρ), and PD-uncertainty factor (ν).