Estimating Probabilities of Default for Low Default Portfolios Katja Pluto and Dirk Tasche∗† April 4, 2005
Abstract For credit risk management purposes in general, and for allocation of regulatory capital by banks in particular (Basel II), numerical assessments of the credit-worthiness of borrowers are indispensable. These assessments are expressed in terms of probabilities of default (PD) that should incorporate a certain degree of conservatism in order to reﬂect the prudential risk management style banks are required to apply. In case of credit portfolios that did not at all suﬀer defaults, or very few defaults only over years, the resulting naive zero or close to zero estimates would clearly not involve such a suﬃcient conservatism. As an attempt to overcome this issue, we suggest the most prudent estimation principle. This means to estimate the PDs by upper conﬁdence bounds while guaranteeing at the same time a PD ordering that respects the diﬀerences in credit quality indicated by the rating grades. The methodology is most easily applied under an assumption of independent default events but can be adapted to the case of correlated defaults.
A core input to modern credit risk modeling and managing techniques are probabilities of default (PD) per borrower. As such, the accuracy of the PD estimations determines the quality of the results of credit risk models. One of the obstacles connected with PD estimations can be the low number of defaults, especially in the better rating grades. Good rating grades might experience many years without any defaults. And even if some defaults occur in a given year, the observed default rates might exhibit a high degree of volatility, due to the relatively low number of borrowers in that grade. But even entire portfolios with low or no defaults are not uncommon in reality. Examples include Deutsche Bundesbank, Postfach 10 06 02, 60006 Frankfurt am Main, Germany E-mail: firstname.lastname@example.org, email@example.com † The opinions expressed in this note are those of the authors and do not necessarily reﬂect views of the Deutsche Bundesbank. ∗
portfolios with an overall good quality of borrowers (e.g. sovereign or bank portfolios) as well as high-volume-low-number portfolios (e.g. specialized lending). Usual bank practices for deriving PD values for such exposures often focus on qualitative mapping mechanisms to bank-wide master scales or external ratings. These practices, while widespread in the industry, do not entirely satisfy the desire for a statistical foundation of the assumed PD values. One may “believe” that the PDs per rating grade appear correct, as well as believe that the ordinal ranking and the relative spread between the PDs of two grades is right, but information about the absolute PD ﬁgures is lacking. Lastly, it could be questioned whether these rather qualitative methods of PD calibration fulﬁll the minimum requirements set out in BCBS (2004a). The issue has, amongst others, recently been raised in BBA (2004). In that paper, applications of causal default models and of exogenous distribution assumptions on the PDs across the grades have been proposed. In a recent paper, Schuermann and Hanson (2004) present a methodology to estimate PDs by means of migration matrices (“duration method”, cf. also Jafry and Schuermann, 2004). This way, non-zero PDs for high-quality rating grades can be estimated more precisely by counting the borrower migrations through the lower grades to eventual default and using Markov chain properties. This paper focuses on a diﬀerent issue of PD estimations in low default portfolios. We present a methodology to estimate PDs for portfolios without any defaults, or a very low number of defaults in the overall portfolio. The proposal by Schuermann and Hanson does not provide a solution for such cases, because the duration method requires a certain number of defaults in at...