Non linearity issues in PD modelling. Amrita Juhi Lucas Klinkers

Transcription

1 Non linearity issues in PD modelling Amrita Juhi Lucas Klinkers May 2017

2 Content Introduction Identifying non-linearity Causes of non-linearity Performance 2

3 Content Introduction Identifying non-linearity Causes of non-linearity Performance 3

4 Introduction Economic and / or regulatory capital held for credit risk is driven by three parameters: 1 Probability of Default The probability that a client will be unable to meet its debt obligations in the next 12 months PD (%) 2 Exposure at Default What is the expected exposure at the moment of default? EAD ( ) 3 Loss Given Default How much of the outstanding exposure should we expect to lose? LGD (%) 4

5 Introduction Economic and / or regulatory capital held for credit risk is driven by three parameters: 1 Probability of Default The probability that a client will be unable to meet its debt obligations in the next 12 months PD (%) PD modelling can be divided in two parts Clients are ranked on the basis of creditworthiness (scorecard model). Scores are converted in probability of defaults (PD model). 3 EAD ( ) LGD (%) A lot of attention in literature has been focused on the ranking performance of the model (Gini coefficient), while attention on the accuracy of PD prediction is less widespread. 5

6 Introduction In business operations accuracy of PD important for capital calculation purposes (AIRB & Vasicek). Increased accuracy of PD will decrease the Margin of Conservatism (MoC) and regulatory capital. PD s are overestimated which increases the MoC that has to be incorporated in the model *High log-odds mean low PD 6

7 Introduction Logistic regression maps PD on a [0,1] scale based on explanatory variables. Relationship between PD and explanatory variables is non-linear as shown in the example below. 7

8 Introduction Linearity assumption logistic regression Explanatory variables are linearly related to the log odds of the PD Risk driver A Risk driver B Risk driver C 8

9 Introduction Non-linearity can still be present Risk driver A Risk driver B Risk driver C PD Model (Risk driver A, B & C) Even if the individual risk drivers are linearly related to the log-odds, the PD model predictions deviate nonlinearly from observed logodds. 9

10 Content Introduction Identifying non-linearity Causes of non-linearity Performance 10

11 Identifying non-linearity 11

12 Identifying non-linearity 12

13 Identifying non-linearity Value of Gamma 2 parameter? Significance of Gamma 2 parameter? 13

14 Identifying non-linearity Run regression with squared factor. Expected values if linearity holds: Constant (γ 0 ) 0 Score (γ 1 ) 1 Score 2 (γ 2 ) 0 14

15 Identifying non-linearity Example dataset Run regression with squared factor: Constant: 0.15 (p < 0.01) Score: 0.85 (p < 0.01) Score 2 : (p < 0.01) Non-linear parameter significantly different from zero. Confirming non-linearity present in this data. 15

16 Identifying non-linearity Example dataset 16

17 Content Introduction Identifying non-linearity Causes of non-linearity Performance 17

18 Causes non-linearity Lloyds banking data from McDonald, Ross et al. (2012): McDonald, Ross et al. (2012) identified possible causes: Differences in distributions of default data and no default data. Correlation between risk drivers and their effect on maximum likelihood conversion. Missing values extrapolate the correlation issue. 18

19 Causes non-linearity Differences in distribution Difference between distribution of scores of default data and no-default data. 19

20 Causes non-linearity (1/2) Correlation Maximum likelihood converges well in the case of low correlation. X3 X1 20

21 Causes non-linearity (2/2) Correlation High correlation causes the parameters to be interchangeable, so hard to define the optimal set. X4 X2 21

22 Content Introduction Identifying non-linearity Causes of non-linearity Performance 22

23 Performance Example dataset Bucketing prediction error decreases with non-linear prediction: Bucket Prediction error Non-linear prediction error 1-1.7% 0.3% 2-12% -8% 3-3.9% -3.8% 4 2.9% 0.01% 5 14% 7.7% 6 35% 23% 7 49% 30% 8 36% 12% 9 33% 1.7% % -27% Mean error 16.3% 3.5% Mean absolute error 19.8% 11.4% 23

24 Conclusions Presence of non-linearity within PD modelling can be identified. Correcting for non-linearity outperforms in terms of prediction error and therefore decreases the MoC. Regulatory Capital PD accuracy 24

25 Next steps More research into causes of non-linearity and improved identification. Research into different methods of correcting for non-linearity. 25

26 Questions?

27 Appendix - Data statistics Correlation 27

28 Appendix - Bucket performance 28

29 Appendix - Skewness as indicator 29

30 Appendix Example data normality 30

31 Appendix Predicted vs Observed Squared transformation 31

32 Appendix AIC & SBIC 32

33 Appendix Math behind score^2 33