# The large homogeneous portfolio approximation with a two-factor Gaussian copula and random recovery rate

## Geon Ho Choe and Soon Won Kwon

#### Abstract

ABSTRACT

In this paper we consider the large homogeneous portfolio (LHP) approximation with a two-factor Gaussian copula and random recovery rate. In addition, we assume that the earlier the default occurs, the less the asset recovers; in other words, random recovery rate and individual default times have a positive rank correlation. Under the LHP assumption, the conditional cumulative loss of the reference portfolio is approximated by the product of loss given default and conditional default probability. In order to derive semianalytic formulas for the loss distribution and the expected tranche loss, we use a Gaussian two-factor model and assume that the recovery rate depends on one systematic factor. In addition, we consider stochastic correlation for a better fit to credit default swap index tranches. The derived semianalytic formula only involves integration with respect to the standard normal density and can be computed by Gauss-Hermite quadrature. Numerical tests show that the two-factor model with stochastic correlation and random recovery fits iTraxx tranche premiums better than other correlation or recovery assumptions under the Gaussian LHP framework.We also apply our model to credit risk assessment such as value-at-risk of the loss distribution.