A Generalized Single Common Factor Model of Portfolio Credit Risk

Abstract

The Gaussian copula model, with all of its flaws, has become the most common approach for estimating the loss distribution of a portfolio of risky credits. In general, extensive calculations are still required, although simplification is often obtained by using an asymptotic single factor model. The model assumes idiosyncratic risks are fully diversified and dependence across names comes entirely from their joint dependence on a single common factor. However, credit risk exposure in a portfolio, such as one held by a bank, does not depend only on the probability of default, but also on uncertainty over loss given default (LGD) and even uncertainty over face amounts, or the exposure at default (EAD), in the case of revolving credits. This article extends the asymptotic model by formally introducing creditors’ draw rates and recoveries in case of default, both of which are tied to the same common factor. A closed-form asymptotic solution incorporating all three components of risk is obtained when distributions are continuous. The author also derives an approximate solution for the discrete distribution case and uses it to show how the overall loss distribution is affected by the properties of the LGD, in particular, its skewness.