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Estimating and validating long run probability of default

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Multi-period Pluto and Tasche model allows us to fulfill Basel committee requirements regarding long-term LRDF calibration even for portfolios with no observable defaults.

The main drawback of that approach is a very strict requirement for the sample: only borrowers that are observable to the bank within each point on long-term horizon could be used as observations.

Both systematic and idiosyncratic factors are standard normally distributed, and idiosyncratic factors are i.i.d., while joint distribution of is completely determined by the correlation matrix RC (which could be expressed using time-distance function). Wagner (2003) [5] , conditional on particular systematic factor realization default probability in a given year could be expressed as the following link function: Solving inequality (6) in each rating class separately for PD values given Equation (12) leads us to PD estimates according to original multi-period Pluto & Tasche model.

Borrower i’s default occurs in year t if: with denoting the standard normal distribution function. Above described P&T model has obvious drawbacks in case of multi-period calibration.

Keywords: Credit Risk, Low Default Portfolio, PD Calibration1.

Introduction This paper is dedicated to ultra-low default portfolios.

For detail illustrations, please refer to OSFI document: “Risk Quantification of IRB Systems at IRB Banks” [6] and Miu P. Since there is no explicit analytical Equation to relate the mode of the distribution with the underlying LRPD, this approach involves changing the assumed value of LRPD until we obtain a mode (through simulations) of the distribution of the simulated average default rate, which is identical (or close enough) to the observed average default rate average.

Assuming unconditional independence of the default events, the probability of observing no defaults turns out to be, where denotes number of borrowers in the rating class i.

In order to find PD estimate for rating class A, we should solve the following inequality (given some confidence level for estimation):.

replacing the one-year number of borrowers in a grade with the sum of the borrower numbers of this grade over the years (analogously for the numbers of defaulted borrowers).

However, when turning to the case of cross-sectionally and intertemporally correlated default events, pooling does not allow for an adequate modelling—there is no way to incorporate into the model information about correlation between borrowers and autocorrelation of the systemic factor.