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Estimating PD, LGD and PDxLGD correlation

Basel II unexpected loss rate. Loss rate on a loan portfolio in a time horizon can be expressed as the default rate times the average loss given default Loss rate distribution depends on the joint distribution of the two variables DR and LGDBasel II simplifies the estimation of the unexpected loss

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Estimating PD, LGD and PDxLGD correlation

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    1. Estimating PD, LGD and PDxLGD correlation

    2. Basel II unexpected loss rate Loss rate on a loan portfolio in a time horizon can be expressed as the default rate times the average loss given default Loss rate distribution depends on the joint distribution of the two variables DR and LGD Basel II simplifies the estimation of the unexpected loss rate to 2

    3. Vasicek formula UDR is given by a very specific formula involving a BII default correlation parameter Default of a receivable is determined by a systematic and an idiosyncratic N(0,1) variable value The default rate on a large (asymptotic) portfolio then depends only on the systematic factor value and on the parameters 3

    4. Basle II unexpected default rate and the default correlation Basle II unexpected default rate depend heavily on the correlation parameter 4

    5. How to estimate the PD correlation? 5 Given a large retail portfolio monthly series of observed default rates (and LGDs) we may use MLE inverting the function

    6. Estimating PD correlation Given a and value calculate the latent systematic factors Take a possible AR(1) autocorrelation into account And express the likelihood function The resulting ?=3,63%, can be compared with Basle II 6

    7. LGD correlation Analogous approach: account level LGD determined by a normalized factor decomposed into a systematic and idiosyncratic one Given a distribution for the account level LGD variable we know and asymptotic portfolio average LGD 7

    8. Estimating LGD correlation Similarly to PD taking a possible autocorrelation of the systematic factors into account and using a smoothed observed account level LGD distribution we may evaluate the likelihood function The resulting LGD correlation estimate is 3,9% with a bootstrapping s.e. 0.8% 8

    9. PDxLGD correlation There is an empirical evidence (mostly from the bond market) of a positive correlation (see e.g. Altman) The notion of PDxLGD correlation depends on the model we choose The well-known models used in literature employ only one systematic factor Frye: one systematic, two idiosyncratic Tasche: one systematic, one idiosyncratic Pykhtin: one systematic, two correlated idiosyncratic 9

    10. One-factor models cannot capture PDxLGD correlation In all cases listed above and where is the systematic factor Hence PDxLGD correlation implied by the model is close to 100% For example in the case of the Frye model with the implied PDxLGD correlations is 98% In spite of this fact it is interesting to note that the one-factor model has been also used to capture the PDxLGD correlation (Frye) 10

    11. Two-factor model Hence to model the PDxLGD correlation we propose the following natural two-factor model LGD is influenced not only by the PD systematic parameter but also by an independent systematic parameter Hence and 11

    12. Maximum likelihood estimation Given the PD and LGD time series and certain correlation values calculate and decompose the systematic factors as above The PD likelihood function is not changed, the LGD likelihood function must incorporate the PDxLGD relationship 12

    13. Empirical Results 13 The estimation can be done in two steps (first maximize and then ) or just in one step (maximize ) Stability of the estimates can be tested with a bootstrapping

    14. Loss rate simulation The calibrated model can be used to obtain not only quantiles, i.e. unexpected values of PD and LGD, but also to generate the loss rate distribution 14

    15. Loss rate simulation - Examples Our model with the estimated parameters gives surprisingly a lower unexpected loss at the 99.9% probability level then the Basel II formula This is caused by a higher regulatory PD correlation 5.9%. If we used this value for and then 15

    16. Conclusion We have proposed a model and an estimation technique for LGD and PDxLGD correlations Estimated values on retail banking data confirmed positive correlations The two-factor model can be used to obtain a realistic loss rate distribution In practice the two-factor model can be generalized to capture different PD and LGD rating classes The PD,LGD time series should cover different economic cycles 16

    17. Thank you for your attention 17

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