Urszula Grzybowska i Marek Karwański Katedra Informatyki

1. Application of data envelopment analysis to calculating probability of default for high rated portfolio Urszula Grzybowska i Marek Karwański Katedra Informatyki Wydział Zastosowań Informatyki i Matematyki SGGW w Warszawie FENS 2014 Lublin 14-16.05.2014

2. Plan of the talk • Introduction • Motivation • Description of models and methods • Data • Results • Conclusions

3. Introduction According to the Capital Requirements Directive (2006, 2009, 2010,2013) banksapplying the internal-rating basedapproachhave to estimateprobabilities of default (PDs) for theirobligors. PDsare a coreinput to modern creditriskmodels. In credit risk estimation an obligor is assigned to one of several rating classes. The obligors with the same credit quality are assigned to the same risk group. There are from 8 to 18 rating categories that describe credit quality of agents. Following S&P the highest and the best rating category is AAA. An obligation rated AAA is judged to be the best quality, with the smallest degree of investment risk. On the other edge of the scale is category D, which is assigned to an obligation where a default has already occurred.

4. Introduction One of the obstaclesconnected with PD estimationis the lownumber of defaults, especially in high rating grades. High rating categoriesmightexperiencemanyyearswithoutanydefault. A substantialpart of bank assetsconsists of portfolios with lowdefaultrate, especially high rated portfolios are LDP.

5. Lowdefault portfolio - definition Lowdefault portfolio (LDP) is a portfolio with onlyfewdefaultsor a portfolio free from anydefaults

6. Probability of a LowDefault Portfolio Basel Committee on Banking Supervision proposedseveral methods to estimate PD for LDP: • A. Forrest’s (2005); • K. Pluto and D. Tasche’s (2005); • N. M. Kiefer (2006); • M. Burgt’s (2007); • D. Tasche’s (2009);

7. The model of K. Pluto and D. Tasche’s Assumethatthree rating classesaregiven: A, B and C. We assumethat no defaultoccurred. Let pA be the unknownprobability of default for grade A, pB - probabilityof defaultof grade B,and pCof grade. The probabilitiesshouldreflectthe decreasing credit-worthiness of the grades, in the sense of the following inequality: pA≤ pB≤ pC

8. Confidence regions for PDs The confidenceregion for pAcan be described as the set of all admissible values of pAwith the property that theprobability of not observing any default during the observation period is not less than 1 −α (forinstance for α= 90%).LetnA, nB, nC be the size of groups A, B and C respectively. Then, using the formula for probability of no success in Bernoulli trials we getconfidenceintervals for desiredprobabilities: The onlykeyassumptionis a correctordinalrating of the borrowers.

9. Motivation The aim of our research is to propose a method of rating which is based on efficiency measure given by DEA. We willcompare the DEA drivenresults with resultsobtained by PCM and a clusteringmethod.

10. DEA- itsorigin and applications Data Envelopment Analysis (DEA) isan OR approach for evaluating the performance of a set of peerentitiescalledDecisionMakingUnits (DMU). The firstarticle on DEA application by Cooper, Charnes and Rhodes was published in 1978. The work on the subjectoriginated in the eraly 1970s in response to the thesiseffort of Rhodes. The aim of the thesis was to evaluate the educationalprograms for disadvantagedstudents.

11. DEA as a BenchmarkingTool Benchmarkingcan be described as a process of defining valid measures of performance comparison among peer units, using them to determine the relative positions of the peer units and, ultimately, establishing a standard of excellence.

12. Applications of DEA DEA can be applied to a widevariety of activities. It can be used to evaluate the performance of: • Governentalagencies; • Hospitals; • Universities; • Non-profit organizations; • Banks; • Firms.

13. Basic DEA Benchmarking Information DEA gives • Efficiency rating, or score, for each DMU: Θ • Efficiency reference set: peer group • Target for the inefficient DMU • Information on how much inputs can be decreased or outputs increased to make the unit efficient – improving productivity and performance

14. DEA Model Assume we have n DMU xijdenoteinputs i=1,2,.., yrjdenoteoutputs r=1,2,…,r inputs DMU outputs

15. Basic CCR model in its dual form – Farrel Model (1978) =min subject to

16. The BCC-0 model =min subject to

17. Variableselection • Inputs: • Assetsturnover • Total Liabilities/Total Assets (Debt Ratio) • Outputs: • Returnon assets (ROA) • Return on equity(ROE) • Current ratio (CR) • Operating profit margin (OPM)

18. Data • 17 Buildingcompaniestraded on Warsaw Stock Exchange (the thefinancialreportscoveredtwoyears: 2001 and 2002) • 76 Productioncompaniestraded on WSE (the financialreportscoveredtwoyears: 2011 and 2012)

19. Applied methods • We performed DEA, PCM and clusteranalysis to distinguishgroups of homogeneouselements - rating classes.

20. Results of DEA BCC-0. Example 1

23. Results of DEA BCC-0. Example 2 • 76 Productioncompaniestraded on WSE

32. Conclusions DEA seems to be a promisingtool, alternative to traditionalscoringmodels. It enables ranking of agents. It can be used for distinguishingclasses of homogeneousobject , e.g., rating classes. The rsultsobtained with help of DEA differ from resultsobtained with clusteringmethods.