Monotonic Estimation for the Survival Probability over a Risk-Rated Portfolio by Discrete-Time Hazard Rate Models

Yang, Bill Huajian (2019): Monotonic Estimation for the Survival Probability over a Risk-Rated Portfolio by Discrete-Time Hazard Rate Models. Forthcoming in: International Journal of Machine Learning and Computing

Abstract

Monotonic estimation for the survival probability of a loan in a risk-rated portfolio is based on the observation arising, for example, from loan pricing that a loan with a lower credit risk rating is more likely to survive than a loan with a higher credit risk rating, given the same additional risk covariates. Two probit-type discrete-time hazard rate models that generate monotonic survival probabilities are proposed in this paper. The first model calculates the discrete-time hazard rate conditional on systematic risk factors. As for the Cox proportion hazard rate model, the model formulates the discrete-time hazard rate by including a baseline component. This baseline component can be estimated outside the model in the absence of model covariates using the long-run average discrete-time hazard rate. This results in a significant reduction in the number of parameters to be otherwise estimated inside the model. The second model is a general form model where loan level factors can be included. Parameter estimation algorithms are also proposed. The models and algorithms proposed in this paper can be used for loan pricing, stress testing, expected credit loss estimation, and modeling of the probability of default term structure.

• Item Type: MPRA Paper
• Language: English
• Keywords: loan pricing, survival probability, Cox proportion hazard rate model, baseline hazard rate, forward probability of default, probability of default term structure
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C40 - General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C44 - Operations Research ; Statistical Decision Theory; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C55 - Large Data Sets: Modeling and Analysis; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling

References

[1] Barlow, R. E.; Bartholomew, D. J.; Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions; The Theory and Application of Isotonic Regression. New York: Wiley. ISBN 0-471-04970-0.

[2] Friedman, J. and Tibshirani, R. (1984). The Monotone Smoothing of Scatterplots. Technometrics, Vol. 26 ( 3), pp. 243-250. DOI: 10.2307/1267550

[3] Potharst, R. and Feelders, A. J. (2002). Classification Trees for Problems with Monotonicity Constraints, SIGKDD Explorations, Vol. 14 (1), 1-10, 2002

[4] Kotlowski, W. and Slowinski, R. (2009). Rule Learning with Monotonicity Constraints, Proceedings of the 26th Annual International Conference on Machine Learning, pp. 537-544, 2009

[5] Eichenberg, T. (2018). Supervised Weight of Evidence Binning of Numeric Variables and Factors, R-Package Woebinning.

[6] You, S.; Ding, D.; Canini, K.; Pfeifer, J. and Gupta, M. (2017). Deep Lattice Networks and Partial Monotonic Functions, 31st Conference on Neural Information Processing System (NIPS), 2017

[7] Allison, P. D. (1992). Discrete-Time Methods for the Analysis of Event Histories. Sociological Methodology, Vol. 13, pp. 61-98.

[8] Yang, B. H. (2017). Point-in-Time PD-Term Structure Models for Multi-Period Scenario Loss Projections. Journal of Risk Model Validation, Vol. 11 (1), pp. 73-94.

[9] Cox, D. R. (1972). Regression Models and Life Tables. Journal of the Royal Statistical Society, Series B, 20, pp.187–220.

[10] SAS Institute Inc. (2015). SAS/STAT(R) 14.1 User’s Guide.

[11] Breslow, N. E. (1972). Discussion of Professor Cox’s Paper. J. Royal Stat. Soc. B, 34, pp.216–217.

[12] Merton, R. (1974). On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance, Vol. 29 (2), pp. 449-470.

[13] Vasicek, O. (2002). Loan Portfolio Value. RISK, December 2002, pp. 160 - 162.

[14] Rosen, D. S. (2009). Analytical Methods for Hedging Systematic Credit Risk with Linear Factor Portfolios. Journal of Economic Dynamics & Control, 33, pp. 37-52.

[15] Yang, B. H. (2018). Smoothing Algorithms by Constrained Maximum Likelihood. Journal of Risk Model Validation, Vol. 12 (2), pp. 89-102.

[16] Wolfinger, R. (2008). Fitting Nonlinear Mixed Models with the New NLMIXED Procedure. SAS Institute Inc.