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

IntroductionAccording to the Capital Requirements Directive (2006, 2009, 2010,2013) banks applying the internal-rating based approach have to estimate probabilities of default (PDs) for their obligors. PDs are a core input to modern credit risk models. 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 obstacles connected with PD estimation is the low number of defaults, especially in high rating grades.

High rating categories might experience many years without any default.

A substantial part of bank assets consists of portfolios with low default rate, especially high rated portfolios are LDP.

5

Low default portfolio - definition

Low default portfolio (LDP) is a portfolio with only few defaults or a portfolio free from any defaults

6

Probability of a Low Default Portfolio

A. Forrest’s (2005); K. Pluto and D. Tasche’s (2005); N. M. Kiefer (2006);

M. Burgt’s (2007); D. Tasche’s (2009);

Basel Committee on Banking Supervision proposed several methods to estimate PD for LDP:

7

The model of K. Pluto and D. Tasche’s Assume that three rating classes are given: A, B and C. We assume that no default occurred.

Let pA be the unknown probability of default for grade A,

pB - probability of default of grade B, and pC of grade.

The probabilities should reflect the decreasing credit-worthiness of the grades, in the sense of the following inequality:

pA ≤ pB ≤ pC

8

Confidence regions for PDsThe confidence region for pA can be described as the set of all admissible values of pA with the property that the probability of not observing any default during the observation period is not less than 1 −α (for instance for α= 90%). Let nA, 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 get confidence intervals for desired probabilities:

The only key assumption is a correct ordinal rating 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 will compare the DEA driven results with results

obtained by PCM and a clustering method.

10

DEA- its origin and applications

Data Envelopment Analysis (DEA) is an OR approach for

evaluating the performance of a set of peer entities called

Decision Making Units (DMU). The first article on DEA

application by Cooper, Charnes and Rhodes was published in

1978. The work on the subject originated in the eraly 1970s in

response to the thesis effort of Rhodes. The aim of the thesis

was to evaluate the educational programs for disadvantaged

students.

11

DEA as a Benchmarking Tool

Benchmarking can 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 wide variety of activities. It can be used to evaluate the performance of: Governental agencies; Hospitals; Universities; Non-profit organizations; Banks; Firms.

13

Basic DEA Benchmarking InformationDEA 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

xij denote inputs i=1,2,..,

yrj denote outputs r=1,2,…,r

inputs outputs

DMU

15

Basic CCR model in its dual form – Farrel Model (1978)

=min

subject to

16

The BCC-0 model=min

subject to

17

Variable selection

Inputs: Assets turnover Total Liabilities/Total Assets (Debt Ratio)

Outputs: Return on assets (ROA) Return on equity (ROE) Current ratio (CR) Operating profit margin (OPM)

18

Data

17 Building companies traded on Warsaw Stock Exchange

(the the financial reports covered two years: 2001 and

2002)

76 Production companies traded on WSE (the financial

reports covered two years: 2011 and 2012)

19

Applied methods

We performed DEA, PCM and cluster analysis to distinguish groups of homogeneous elements - rating classes.

20

Group

Number of

elements in a group

Company

1 7BUDIMEX. BUDOPOL, INSTAL_K, MOST_EXP, MOST_PK, POLNORD, ULMA

2 8AWBUD, CENNOWTE, ELBUDOWA, ELKOP. ENERGOPL, ENMONTPDMOST_ZAB, RESBUD

3 2 KOPEX, PEMUG

Results of DEA BCC-0. Example 1

21

CompanyPCM score

DEA Group

Efficiency rating

MOST_EXP 206,00 1 1ULMA 190,23 1 1

BUDIMEX 180,39 1 1MOST_ZAB 149,84 2 0,68

ELKOP 134,62 2 0,74CENNOWTE 134,49 2 0,997ENERGOPL 134,49 2 0,998ENMONTPD 122,89 2 0,74

KOPEX 117,13 3 0,68PEMUG 108,20 3 0,756

MOST_PK 97,47 1 1ELBUDOWA 87,72 2 0,994INSTAL_K 87,17 1 1POLNORD 83,60 1 1RESBUD 71,20 2 0,933AWBUD 70,86 2 0,99

BUDOPOL 62,245 1 1

22

23

Results of DEA BCC-0. Example 2

76 Production companies traded on WSE

24

GroupNumber of elements

Companies

1 9ZELMER, PGE, EKO_EXP, HYDROTOR, PANITERE, BERLING, WINDMOB, AC, MENNICA

2 13SONEL, BSCDRUK, APATOR, CIGAMES, CITYINTE, STALPROD, ESSYSTEM, PULAWY, MEGAR, WAWEL, ZYWIEC, POLICE, IZOL_JAR

3 15KETY, POLNA, PEPEES, BIOMAXIM, NOVITAZUK, RELPOL, IZOSTAL, BUDVAR, STOMIL_SALKAL, HUTMEN, INTERCAR, KPPD, DUDA

4 12INTEGER, TAURON, MOJ, POZBUD, BORYSZEW, PROJPRZM, PATENTUS, INVICO, FORTE, ZPUE, DEBICA, LOTOS

5 10GROCLIN, LENTEX, RAFAMET, PLASTBOX, FASING, FERRO, RAFAKO, SYNEKTIK, ERGAMICA

6 17

GRAJEWO, MUZA, KOELNER, RAWLPLUG, VISTULA, ARMATURA, SUWARY, GRAAL, WOJAS, ENERGOIN, MIESZKO, PAMAPOL, ZPC_OTM, FERRUM, ZUE, SNIEZKA, WIELTON

25

LP Company PCM score Ranking PCM DEA group Ward

group

Eliptical-Seriatio

n (1) score

Eliptical-Seriation (4) score

71 ZELMER 1029,41 1 1 [GR=1] 75 76

48 PGE 566,264 2 1 [GR=2] 49 4928 INTEGER 521,15 3 4 [GR=1] 73 7365 TAURON 433,20 4 4 [GR=1] 69 6924 GRAJEWO 391,11 5 6 [GR=1] 74 7542 MUZA 379,89 6 6 [GR=2] 40 4041 MOJ 379,86 7 4 [GR=2] 34 3525 GROCLIN 358,82 8 5 [GR=1] 67 6734 KOELNER 356,03 9 6 [GR=1] 71 7157 RAWLPLUG 355,99 10 6 [GR=1] 72 7266 VISTULA 333,50 11 6 [GR=1] 65 6536 LENTEX 326,93 12 5 [GR=2] 37 3756 RAFAMET 315,23 13 5 [GR=2] 52 525 ARMATURA 303,81 14 6 [GR=1] 70 70

52 POZBUD 301,09 15 4 [GR=2] 57 57

26

27

28

29

30

31

32

Conclusions

DEA seems to be a promising tool, alternative to traditional scoring models.

It enables ranking of agents.

It can be used for distinguishing classes of homogeneous object , e.g., rating classes.

The rsults obtained with help of DEA differ from results obtained with clustering methods.