Bayesian Inference for Predicting the Default Rate Using the Power Prior

Abstract

Commercial banks and other related areas have developed internal models to better quantify their financial risks. Since an appropriate credit risk model plays a very important role in the risk management at financial institutions, it needs more accurate model which forecasts the credit losses, and statistical inference on that model is required. In this paper, we propose a new method for estimating a default rate. It is a Bayesian approach using the power prior which allows for incorporating of historical data to estimate the default rate. Inference on current data could be more reliable if there exist similar data based on previous studies. Ibrahim and Chen (2000) utilize these data to characterize the power prior. It allows for incorporating of historical data to estimate the parameters in the models. We demonstrate our methodologies with a real data set regarding SOHO data and also perform a simulation study.

Keywords

• Default rate;
• Bayesian approach;
• power prior;
• AR(1) model;
• historical data;
• Gibbs sampling

File

Download PDF

References

1. Berger, J.O. and Pericchi, L.R. (1996). The Intrinsic Bayes Factor for Model Selection and Prediction. Joumal of American Statistical Association, Vol. 91, 109-122 https://doi.org/10.2307/2291387
2. Caouette, J.B., Altman, E.I., and Narayanan, P. (1998). Managing Credit Risk, John Wiley & Sons, Inc
3. Carey, M. (1998). Credit Hisk in Private Debt Portfolios, J. Finance, LIII, 1363-1387
4. Chen, M.-H., Ibrahim, J.G., and Shao, Q.-M. (1999). Prior Elicitation, Variable Selection and Bayesian Computation for Logistic Regression Models. Journal of Royal Statistical Society B, Vol. 61, Part 1, 223-242 https://doi.org/10.1111/1467-9868.00173
5. Dianconis, P. and Ylvisaker, D. (1979). Conjugate Priors for Exponential Families. Annals of Statistics, Vol. 7, 269-281 https://doi.org/10.1214/aos/1176344611
6. Emmer, S. and Tasche, D. (2003). Calculating Credit Risk Capital Charges with the One-factor Model, The European Investment Review Third Annual Conference Geneva, September 25
7. Fons, J.S. (1991). An Approach to Forecasting Default Rates, A Moody's Special Report
8. Gelfand, A.E. and Smith, A.F.M. (1990). Sampling Based Approaches to Calculating Marginal Densities. Journal of the American Statistical Association, Vol. 87, 523-532 https://doi.org/10.2307/2290286
9. Gilks, W.R. and Wild, P. (1992). Adaptive Rejection Sampling for Gibbs Sampling. Applied Statistics, Vol. 41, No.2, 337-348 https://doi.org/10.2307/2347565
10. Gordy, M. (2000). A Comparative Anatomy of Credit Risk Models. Journal of Banking & Finance, Vol. 24, 119-149 https://doi.org/10.1016/S0378-4266(99)00054-0
11. Ibrahim, J.G. and Chen, M.-H. (2000). Power Prior Distributions for Regression Models. Statistical Science, Vol. 15, 46-60 https://doi.org/10.1214/ss/1009212673
12. Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems, Proceedings of the Royal Society London. Ser. A, Vol. 186, 453-461
13. Kim, S.W. and Ibrahim, J.G. (2000). On Bayesian Inference for Proportional Hazards Models Using Noninformative Priors. Life Time Data Analysis, Vol. 6, 331- 341 https://doi.org/10.1023/A:1026505331236
14. Morris, C.N. (1983). Natural Exponential Families with Quadratic Variance Functions: statistical theory. Annals of Statistics, Vol. 11, 515-529 https://doi.org/10.1214/aos/1176346158
15. Nickell, P., Perraudin, W., and Varotto, S. (200l). Ratings versus Equity-based Credit Risk Modelling: an empirical analysis, Working paper, Bank of England, Birkbeck College
16. Thomas, L.C., Edelman, D.B., and Crook, J.N. (2002). Credit Scoring and Its Applications, SIAM, Philadelphia