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
1
FENS 2014 Lublin 14-16.05.2014
Plan of the talk

Introduction

Motivation

Description of models and methods

Data

Results

Conclusions
2
Introduction
According 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.
3
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.
4
Low default portfolio - definition
Low default portfolio (LDP) is a portfolio with only few
defaults or a portfolio free from any defaults
5
Probability of a Low Default Portfolio
Basel Committee on Banking Supervision proposed
several 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);
6
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 creditworthiness of the grades, in the sense of the following
inequality:
pA ≤ pB ≤ pC
7
Confidence regions for PDs
The 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:
𝑝𝐴 ≤ 1 − 1 − 𝛼
𝑝𝐵 ≤ 1 − 1 − 𝛼
𝑝𝐶 ≤ 1 − 1 − 𝛼
1 𝑛𝐴 +𝑛𝐵 +𝑛𝐶
1 𝑛𝐵 +𝑛𝐶
1 𝑛𝐶
The only key assumption is a correct ordinal rating of the
borrowers.
8
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.
9
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.
10
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.
11
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.
12
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
13
DEA Model
Assume we have n DMU
𝑗 = 1,2, … , 𝑛
xij denote inputs i=1,2,.., 𝑚
yrj denote outputs r=1,2,…,r
inputs
DMU
outputs
14
Basic CCR model in its dual form – Farrel
Model (1978)
𝜃 ∗ =min 𝜃
subject to
𝑛
𝑗=1 𝑥𝑖𝑗 𝜆𝑗
𝑛
≤ 𝜃𝑥𝑖0 𝑖 = 1,2, … , 𝑚
𝑦𝑟𝑗 𝜆𝑗 ≥ 𝑦𝑟0 𝑟 = 1,2, … , 𝑠
𝑗=1
𝜆𝑗 ≥ 0 𝑗 = 1,2, … , 𝑛
15
The BCC-0 model
𝜃 ∗ =min 𝜃
subject to
𝑛
𝑥𝑖𝑗 𝜆𝑗 ≤ 𝜃𝑥𝑖0 𝑖 = 1,2, … , 𝑚
𝑗=1
𝒏
𝒚𝒓𝒋 𝝀𝒋 ≥ 𝒚𝒓𝟎 𝑟 = 1,2, … , 𝑠
𝒋=𝟏
𝑛
𝜆𝑗 = 1
𝑗=1
𝜆𝑗 ≥ 0 𝑗 = 1,2, … , 𝑛
16
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)

17
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)
18
Applied methods

We performed DEA, PCM and cluster analysis to distinguish
groups of homogeneous elements - rating classes.
19
Results of DEA BCC-0. Example 1
Group
Number of
elements
in a group
Company
7
BUDIMEX. BUDOPOL, INSTAL_K, MOST_EXP,
MOST_PK, POLNORD, ULMA
2
8
AWBUD, CENNOWTE, ELBUDOWA, ELKOP.
ENERGOPL, ENMONTPD
MOST_ZAB, RESBUD
3
2
KOPEX, PEMUG
1
20
Company
MOST_EXP
ULMA
BUDIMEX
MOST_ZAB
ELKOP
CENNOWTE
ENERGOPL
ENMONTPD
KOPEX
PEMUG
MOST_PK
ELBUDOWA
INSTAL_K
POLNORD
RESBUD
AWBUD
BUDOPOL
PCM
score
206,00
190,23
180,39
149,84
134,62
134,49
134,49
122,89
117,13
108,20
97,47
87,72
87,17
83,60
71,20
70,86
62,245
DEA
Group
1
1
1
2
2
2
2
2
3
3
1
2
1
1
2
2
1
Efficiency
rating
1
1
1
0,68
0,74
0,997
0,998
0,74
0,68
0,756
1
0,994
1
1
0,933
0,99
1
21
22
Results of DEA BCC-0. Example 2

76 Production companies traded on WSE
23
Group
Number of
elements
1
9
2
13
3
15
4
12
5
10
6
17
Companies
ZELMER, PGE, EKO_EXP, HYDROTOR, PANITERE, BERLING,
WINDMOB, AC, MENNICA
SONEL, BSCDRUK, APATOR, CIGAMES, CITYINTE,
STALPROD, ESSYSTEM, PULAWY, MEGAR, WAWEL,
ZYWIEC, POLICE, IZOL_JAR
KETY, POLNA, PEPEES, BIOMAXIM, NOVITA
ZUK, RELPOL, IZOSTAL, BUDVAR, STOMIL_S
ALKAL, HUTMEN, INTERCAR, KPPD, DUDA
INTEGER, TAURON, MOJ, POZBUD, BORYSZEW,
PROJPRZM, PATENTUS, INVICO, FORTE, ZPUE, DEBICA,
LOTOS
GROCLIN, LENTEX, RAFAMET, PLASTBOX, FASING, FERRO,
RAFAKO, SYNEKTIK, ERG
AMICA
GRAJEWO, MUZA, KOELNER, RAWLPLUG, VISTULA,
ARMATURA, SUWARY, GRAAL, WOJAS, ENERGOIN,
MIESZKO, PAMAPOL, ZPC_OTM, FERRUM, ZUE, SNIEZKA,
WIELTON
24
LP
Company
PCM score Ranking PCM
DEA group
71
ZELMER
1029,41
1
1
48
28
65
24
42
41
25
34
57
66
36
56
5
52
PGE
INTEGER
TAURON
GRAJEWO
MUZA
MOJ
GROCLIN
KOELNER
RAWLPLUG
VISTULA
LENTEX
RAFAMET
ARMATURA
POZBUD
566,264
521,15
433,20
391,11
379,89
379,86
358,82
356,03
355,99
333,50
326,93
315,23
303,81
301,09
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
4
4
6
6
4
5
6
6
6
5
5
6
4
Ward
group
[GR=1]
[GR=2]
[GR=1]
[GR=1]
[GR=1]
[GR=2]
[GR=2]
[GR=1]
[GR=1]
[GR=1]
[GR=1]
[GR=2]
[GR=2]
[GR=1]
[GR=2]
Eliptical- ElipticalSeriation Seriation
(1) score (4) score
75
76
49
73
69
74
40
34
67
71
72
65
37
52
70
57
49
73
69
75
40
35
67
71
72
65
37
52
70
57
25
26
27
28
29
30
31
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.
32