In credit risk models, the loss given default (LGD)1 is either incorporated deterministically (as in Credit Risk+) or stochastically (as in CreditMetrics). In the latter case, the LGD may be drawn from a beta distribution.2 In both cases, no correlation between default and LGD is considered.
In economic downturns, not only do probabilities of default (PDs) increase, but recovery rates also decrease. This pattern can be seen in historical data (see, for example, Frye, 2000, or Altman, et al., 2003). To incorporate this relationship into credit risk models, several basic approaches have been proposed. Frye (2000) and Pykthin (2003) give first insights into a single-factor model for PD and recovery rate containing systematic and idiosyncratic risk. Dullmann & Trapp (2004) compare various transformations for the recovery rate. All of these authors use a single-factor model for PD as well as for recovery rate. They assume that the systematic risk that drives PDs influences recovery rates in the same way. Frye (2000) and Dullmann & Trapp (2004) estimate the PD model and use the realisations of the systematic risk factor as an input factor to the recovery rate model. Then the intercept, the parameter for the systematic and the idiosyncratic risk are estimated using a maximum-likelihood approach. Chava, Stefanescu & Turnbull (2006) use issuer-specific, time-dependent data to model default and recovery rates simultaneously. They also use the same systematic risk factor for defaults and recovery rates.