In a background note by the Basel Committee on Banking Supervision (2004) on loss given default (LGD), the Committee seeks input from the financial industry on defining and quantifying 'downturn' LGD. The main reason for this requirement is that the Vasicek model (Vasicek (2002)) used in the Basel Accord does not have systematic correlation between probability of default (PD) and LGD and, to compensate for this deficiency, downturn LGD estimates are required to be used as an input to the model. The idea here is that a credit risk model with systematic correlation between PD and LGD using long-run LGD inputs should give comparable capital to a credit risk model without correlated PD and LGD using downturn LGD inputs. One suggestion by the Basel Committee to help quantify downturn LGD is to establish a functional relationship between long-run and downturn LGD. Recently, Miu & Ozdemir (2006) used Monte Carlo to specifically tabulate such a relationship in terms of the 'LGD mark-up' required to achieve downturn from long-run LGD.
In this article, we extend the work by Miu & Ozdemir to develop an analytical relationship between long-run and downturn LGD so that credit risk is not overestimated or underestimated in the Vasicek model. We do this by introducing another fully granular credit risk model that contains systematic dependence between PD and LGD. This model is calibrated to historical default and recovery rate data using the Merton model for firm asset return, and recoveries are modelled with a three-parameter lognormal distribution for the value of the assets of the creditor, which may include secured or unsecured assets of any priority or seniority. The choice of a lognormal distribution is first introduced in Pykhtin (2003), and is a more natural choice for quantities that remain positive compared with the Gaussian distribution used by Frye (2000). To determine downturn LGD, we then solve for the downturn LGD input in the Vasicek model so that it gives identical credit risk capital to the model with systematically correlated PD and LGD. We also show that, to correctly compensate for the lack of systematic correlation between PD and LGD, the Vasicek model requires two.