Therefore we lose significant amount of information for stable portfolios and decreasing stability of PD estimates.Moreover, in case of shifts in the portfolio (like significant growth of the portfolio or high prepayment rate) the model is totally inapplicable.In case of available (number of defaults should be less or equal to the number of borrowers in each rating class:) historical defaults we simply replace probability estimate by the CDF (cumulative distribution function) of binomial distribution: As the result, solving (6) for each separately leads us to original P&T one-period PD calibration approach that obviously don’t comply with Basel committee requirement for long-term LRPD estimation period  .In order to comply with the requirements, one can use pooling, e.g.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.
The cross-sectional dependence of the default events stems from the presence of the systematic factor.
Equation (15) provides us with the values of that correspond to claimed Mode LRDF calibration approach.
In order to prove the efficiency of the proposed estimator we could use Monte-Carlo simulations.3.
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.