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Modeling portfolio loss by interval distributions

1 Royal Bank of Canada, 155 Wellington St W, ON M5V 3H6 Toronto, Canada
2 Scotiabank, 4 King St W, ON M5H 1A1 Toronto, Canada
3 Department of statistics, Shenzhen University, 518000 Shenzhen, China

Models for a continuous risk outcome has a wide application in portfolio risk management and capital allocation. We introduce a family of interval distributions based on variable transformations. Densities for these distributions are provided. Models with a random effect, targeting a continuous risk outcome, can then be fitted by maximum likelihood approaches assuming an interval distribution. Given fixed effects, regression function can be estimated and derived accordingly when required. This provides an alternative regression tool to the fraction response model and Beta regression model.
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1. Basel Committee on Banking Supervision, An Explanatory Note on the Basel II IRB Risk Weight Functions, 2005. Available from: https://www.bis.org/bcbs/irbriskweight.htm.

2. Basel Committee on Banking Supervision, Guidance on credit risk and accounting for expected credit loss, 2015. Available from: https://www.bis.org/bcbs/publ/d350.htm.

3. Basel Committee on Banking Supervision, Minimum capital requirements for market risk, 2019. Available from: https://www.bis.org/bcbs/publ/d457.htm.

4. Cribari-Neto F and Zeileis A, (2010) Beta Regression in R. J Stat Software 34: 1-24.

5. Friedman J, Hastie T and Tibshirani R, (2001) The Elements of Statistical Learning, 2 Eds., New York: Springer series in statistics.

6. Gourieroux C, Monfort A and Trognon A, (1984) Pseudo maximum likelihood methods: Theory. Econometrica 52: 681-700.    

7. Huang X, Oosterlee CW and Mesters M, (2007) Computation of VaR and VaR contribution in the Vasicek portfolio credit loss model: a comparative study. J Credit Risk 3: 75-96.

8. Mullahy J, (1990) Regression models and transformations for Beta-distributed outcomes.

9. Murphy K, (2012) Machine learning: a probabilistic perspective, MIT press.

10. Papke LE and Wooldrige JM, (1996) Econometric methods for fractional response variables with an application to 401 (k) plan participation rates. J Appl Econometrics 11: 619-632.    

11. Ramponi FA and Campi MC, (2018) Expected shortfall: Heuristics and certificates. Eur J Operational Res 267: 1003-1013.    

12. Rosen D and Saunders D, (2009) Analytical methods for hedging systematic credit risk with linear factor portfolios. J Econ Dyn Control 33: 37-52.    

13. Vasicek O, (1991) Limiting loan loss probability distribution. KMV Corporation.

14. Vasicek O, (1991) The distribution of loan portfolio value. Risk 15: 160-162.

15. Wolfinger RD, Fitting nonlinear mixed models with the new NLMIXED procedure. Proceedings of the 24th Annual SAS Users Group International Conference (SUGI 24), 1999. Available from: https://pdfs.semanticscholar.org/.

16. Wu B, (2019) The Probability of Default Distribution of Heterogeneous Loan Portfolio. Curr Anal Econ Finance 1: 88-95.

17. Yamai Y and Yoshiba T, (2002) Comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition, and optimization. Monetary Econ Stud 20: 87-121.

18. Yang BH, (2013) Estimating Long-Run PD, Asset Correlation, and Portfolio Level PD by Vasicek Models. J Risk Model Validation 7: 3-19.

19. Yang BH, Wu B, Cui K et al. (2020) IFRS9 Expected Credit Loss Estimation: Advanced Models for Estimating Portfolio Loss and Weighting Scenario Losses. J Risk Model Validation 14: 19-34.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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