Abstract
Financial product prices often depend on unknown parameters. Their estimation introduces the risk that a better informed counterparty may strategically pick mispriced products. Understanding estimation risk, and how to properly price it, is essential. We discuss how total estimation risk can be minimized by selecting a probability model of appropriate complexity. We show that conditional estimation risk can be measured only if the probability model predictions have little bias. We illustrate how a premium for conditional estimation risk may be determined when one counterparty is better informed than the other, but a market collapse is to be avoided, using a simple example from pricing regime credit scoring. We empirically examine the approach on a panel data set from a German credit bureau, where we also study dynamic dependencies such as prior rating migrations and defaults.
Acknowledgements
The authors thank Paul Glasserman for many valuable comments, and Ji Zhu for sharing his implementation of kernelized logistic regression in R.
References
[1] E. Altman, Financial ratios, discriminant analysis and the prediction of corporate bankruptcy, J. Finance 23 (1968), no. 4, 589–609. 10.1111/j.1540-6261.1968.tb00843.xSuche in Google Scholar
[2] B. Baesens, T. van Gestel, S. Viaene, M. Stepanova, J. Sukens and J. Vanthienen, Benchmarking state-of-the-art classification algorithms for credit scoring, J. Oper. Res. Soc. 54 (2003), no. 6, 627–635. 10.1057/palgrave.jors.2601545Suche in Google Scholar
[3] J. Bellovary, D. Giacomino and M. Akers, A review of bankruptcy prediction studies: 1930 to present, J. Financial Educ. 33 (2007), no. 4, 1–42. Suche in Google Scholar
[4] A. Blöchlinger and M. Leippold, Economic benefit of powerful credit scoring, J. Banking Finance 30 (2006), no. 3, 851–873. 10.1016/j.jbankfin.2005.07.014Suche in Google Scholar
[5] A. Blöchlinger, M. Leippold and B. Maire, Are ratings the worst form of credit assessment apart from all the others?, Research Paper, Swiss Finance Institute, 2012. 10.2139/ssrn.2012277Suche in Google Scholar
[6] A. Buja, W. Stuetzle and Y. Shen, Loss functions for binary class probability estimation and classification: Structure and applications, Working Paper (2005), http://www-stat.wharton.upenn.edu/~buja/PAPERS/paper-proper-scoring.pdf. Suche in Google Scholar
[7] C. Carter and J. Catlett, Assessing credit card applications using machine learning, IEEE Expert 2 (1987), no. 3, 71–79. 10.1109/MEX.1987.4307093Suche in Google Scholar
[8] A. Christmann and I. Steinwart, Consistency and robustness of kernel-based regression in convex risk minimization, Bernoulli 13 (2007), no. 3, 799–819. 10.3150/07-BEJ5102Suche in Google Scholar
[9] C. Cortes and V. Vapnik, Support-vector networks, Mach. Learn. 20 (1995), no. 3, 273–297. 10.1007/BF00994018Suche in Google Scholar
[10] D. Durand, Risk elements in consumer installment financing, Technical Report, National Bureau of Economic Research, 1941. Suche in Google Scholar
[11] B. Efron, Bootstrap methods: Another look at the Jackknife, Ann. Statist. 7 (1979), no. 1, 1–26. 10.1007/978-1-4612-4380-9_41Suche in Google Scholar
[12] T. Gneiting and A. Raftery, Strictly proper scoring rules, prediction, and estimation, J. Amer. Statist. Assoc. 102 (2007), no. 477, 359–378. 10.1198/016214506000001437Suche in Google Scholar
[13] D. Hand, Modelling consumer credit risk, IMA J. Manag. Math. 12 (2001), no. 2, 139–155. 10.1093/imaman/12.2.139Suche in Google Scholar
[14] D. Hand and W. Henley, Statistical classification methods in consumer credit scoring: A review, J. R. Stat. Soc. Ser. A Stat. Soc. 160 (1997), no. 3, 523–541. 10.1111/j.1467-985X.1997.00078.xSuche in Google Scholar
[15] W. Härdle, R. Moro and D. Schäfer, Predicting bankruptcy with support vector machines, Statistical Tools for Finance and Insurance, Springer, Berlin (2005), 225–248. 10.1007/3-540-27395-6_10Suche in Google Scholar
[16] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, New York, 2009. 10.1007/978-0-387-84858-7Suche in Google Scholar
[17] D. Hawley, J. Johnson and D. Raina, Artificial neural systems: A new tool for financial decision-making, Financial Anal. J. 46 (1990), no. 6, 63–72. 10.2469/faj.v46.n6.63Suche in Google Scholar
[18] C.-L. Huang, M.-C. Chen and C.-J. Wang, Credit scoring with a data mining approach based on support vector machines, Exp. Sys. Appl. 33 (2007), no. 4, 847–856. 10.1016/j.eswa.2006.07.007Suche in Google Scholar
[19] T. Jaakkola and D. Haussler, Probabilistic kernel regression models, Proceedings of the 1999 Conference on Artificial Intelligence and Statistics, Morgan Kaufmann, Burlington (1999). Suche in Google Scholar
[20] N. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, 2nd ed., Wiley & Sons, New York, 1995. Suche in Google Scholar
[21] S. Keerthi, K. Duan, S. Shevade and A. Poo, A fast dual algorithm for kernel logistic regression, Mach. Learn. 61 (2005), no. 1, 151–165. 10.1007/s10994-005-0768-5Suche in Google Scholar
[22] A. Khandani, A. Kim and A. Lo, Consumer credit-risk models via machine-learning algorithms, J. Banking Finance 34 (2010), no. 11, 2767–2787. 10.1016/j.jbankfin.2010.06.001Suche in Google Scholar
[23] G. King and L. Zeng, Logistic regression in rare events data, Political Anal. 9 (2001), no. 2, 137–163. 10.1093/oxfordjournals.pan.a004868Suche in Google Scholar
[24] M. Maalouf and T. Trafalis, Robust weighted kernel logistic regression in imbalanced and rare events data, Comput. Statist. Data Anal. 55 (2011), no. 1, 168–183. 10.1016/j.csda.2010.06.014Suche in Google Scholar
[25] J. Min and Y.-C. Lee, Bankruptcy prediction using support vector machine with optimal choice of kernel function parameters, Exp. Sys. Appl. 28 (2005), no. 4, 603–614. 10.1016/j.eswa.2004.12.008Suche in Google Scholar
[26] J. Myers and E. Forgy, The development of numerical credit evaluation systems, J. Amer. Statist. Assoc. 58 (1963), no. 303, 799–806. 10.1080/01621459.1963.10500889Suche in Google Scholar
[27] J. Ohlson, Financial ratio and the probabilistic prediction of bankruptcy, J. Accounting Res. 18 (1980), no. 1, 109–131. 10.2307/2490395Suche in Google Scholar
[28] A. Oliver, V. Fumas and J. Saurina, Risk premium and market power in credit markets, Econom. Lett. 93 (2006), no. 3, 450–456. 10.1016/j.econlet.2006.06.021Suche in Google Scholar
[29] J. Platt, Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods, Adv. Large Margin Classifiers 10 (1999), no. 3, 61–74. Suche in Google Scholar
[30] C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning, MIT Press, Cambridge, 2006. 10.7551/mitpress/3206.001.0001Suche in Google Scholar
[31] V. Roth, Probabilistic discriminative kernel classifiers for multi-class problems, Joint Pattern Recognition Symposium, Springer, Berlin (2001), 246–253. 10.1007/3-540-45404-7_33Suche in Google Scholar
[32] C. Shannon, A mathematical theory of communication, Bell Sys. Technical J. 27 (1948), no. 7, 379–423. 10.1002/j.1538-7305.1948.tb01338.xSuche in Google Scholar
[33] M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), no. 1, 171–176. 10.2140/pjm.1958.8.171Suche in Google Scholar
[34] R. Stein, The relationship between default prediction and lending profits: Integrating ROC analysis and loan pricing, J. Banking Finance 29 (2005), no. 5, 1213–1236. 10.1016/j.jbankfin.2004.04.008Suche in Google Scholar
[35] V. Vapnik, Statistical Learning Theory. Vol. 2, Wiley, New York, 1998. Suche in Google Scholar
[36] C. Williams and D. Barber, Bayesian classification with Gaussian processes, IEEE Trans. Pattern Anal. Mach. Intelligence 20 (1998), no. 12, 1342–1351. 10.1109/34.735807Suche in Google Scholar
[37] J. Zhu and T. Hastie, Kernel logistic regression and the import vector machine, J. Comput. Graph. Statist. 14 (2005), no. 1, 185–205. 10.1198/106186005X25619Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A double clustering algorithm for financial time series based on extreme events
- Improved algorithms for computing worst Value-at-Risk
- Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price
- Company rating with support vector machines
- Loan pricing under estimation risk
Artikel in diesem Heft
- Frontmatter
- A double clustering algorithm for financial time series based on extreme events
- Improved algorithms for computing worst Value-at-Risk
- Testing for asymmetry in betas of cumulative returns: Impact of the financial crisis and crude oil price
- Company rating with support vector machines
- Loan pricing under estimation risk