Statistical Analysis of Operational Risk Data

By Giovanni De Luca, Danilo Carità and Francesco Martinelli

This concise book for practitioners presents the statistical analysis of operational risk, which is considered the most relevant source of bank risk, after market and credit risk. The book shows that a careful statistical analysis can improve the results of the popular loss distribution approach. The authors identify the risk classes by applying a pooling rule based on statistical tests of goodness-of-fit, use the theory of the mixture of distributions to analyze the loss severities, and apply copula functions for risk class aggregation. Lastly, they assess operational risk data in order to estimate the so-called capital-at-risk that represents the minimum capital requirement that a bank has to hold. The book is primarily intended for quantitative analysts and risk managers, but also appeals to graduate students and researchers interested in bank risks.

• Language: English
• Publisher: Springer
• Release date: Feb 24, 2020
• ISBN: 9783030425807

Book preview

SpringerBriefs in Statistics

More information about this series at http://​www.​springer.​com/​series/​8921

Giovanni De Luca, Danilo Carità and Francesco Martinelli

Statistical Analysis of Operational Risk Data

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Giovanni De Luca

Department of Management and Quantitative Sciences, Parthenope University of Naples, Naples, Italy

Danilo Carità

Department of Management and Quantitative Sciences, Parthenope University of Naples, Naples, Italy

Francesco Martinelli

Research Department, UBI Banca, Milan, Italy

ISSN 2191-544Xe-ISSN 2191-5458

SpringerBriefs in Statistics

ISBN 978-3-030-42579-1e-ISBN 978-3-030-42580-7

https://doi.org/10.1007/978-3-030-42580-7

Mathematics Subject Classification (2010): 91B3046F1062F1062H0562H30

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

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Contents

1 The Operational Risk 1

1.​1 Introduction 1

1.​2 Models for Operational Risk 2

1.​2.​1 Basic Indicator Approach 4

1.​2.​2 Standardized Approach 4

1.​2.​3 Advanced Measurement Approach 5

1.​3 Loss Distribution Approach 7

1.​4 DIPO Consortium 8

References 10

2 Identification of the Risk Classes 11

2.​1 Introduction 11

2.​2 Distributional Tests 11

2.​3 Application to DIPO Data 15

References 17

3 Severity Analysis 19

3.​1 Introduction 19

3.​2 Mixture of Three-Parameter Log-Normal Distributions 20

3.​3 Extreme Value Theory 21

3.​4 Application to DIPO Data 23

3.4.1 Mixture of $$ k $$ Log-Normal Distributions 23

3.​4.​2 Log-Normal–GPD Distribution 37

3.​4.​3 Comparison 47

References 50

4 Frequency Analysis 51

4.​1 Introduction 51

4.​2 Mixture of Poisson Distributions 51

4.​2.​1 The Poisson Distribution 51

4.​2.​2 Finite Poisson Mixture 52

4.​3 Mixture of Negative Binomial Distributions 53

4.​3.​1 The Negative Binomial Distribution 53

4.​3.​2 Relationship with Poisson Distribution 54

4.​3.​3 Maximum Likelihood Estimation 56

4.​3.​4 Finite Negative Binomial Mixture 57

4.​4 Application to DIPO Data 57

References 69

5 Convolution and Risk Class Aggregation 71

5.​1 Introduction 71

5.​2 Overall Loss Distribution 71

5.​3 Risk Class Aggregation and Copula Functions 73

5.​3.​1 Tail Dependence 74

5.​3.​2 Elliptical Copulae 75

5.​3.​3 Archimedean Copulae 77

5.4 Value-at-Risk Estimates Considering $$ t $$ -Copula 78

References 82

6 Conclusions 83

List of Figures

Fig. 2.1 Size of the eight business lines12

Fig. 3.1 BL1—Severity distribution26

Fig. 3.2 BL2—Severity distribution27

Fig. 3.3 BL3/ET1—Severity distribution27

Fig. 3.4 BL3/ET2—Severity distribution28

Fig. 3.5 BL3/ET3—Severity distribution28

Fig. 3.6 BL3/ET4—Severity distribution29

Fig. 3.7 BL3/ET5—Severity distribution29

Fig. 3.8 BL3/ET6—Severity distribution30

Fig. 3.9 BL3/ET7—Severity distribution30

Fig. 3.10 BL4/ET1—Severity distribution31

Fig. 3.11 BL4/ET2—Severity distribution31

Fig. 3.12 BL4/ET367—Severity distribution32

Fig. 3.13 BL4/ET4—Severity distribution32

Fig. 3.14 BL4/ET5—Severity distribution33

Fig. 3.15 BL5—Severity distribution33

Fig. 3.16 BL6—Severity distribution34

Fig. 3.17 BL7—Severity distribution34

Fig. 3.18 BL8/ET1—Severity distribution35

Fig. 3.19 BL8/ET27—Severity distribution35

Fig. 3.20 BL8/ET35—Severity distribution36

Fig. 3.21 BL8/ET4—Severity distribution36

Fig. 3.22 BL8/ET6—Severity distribution37

Fig. 3.23 BL1—Log-normal versus GPD fit38

Fig. 3.24 BL2—Log-normal versus GPD fit39

Fig. 3.25 BL3/ET1—Log-normal versus GPD fit39

Fig. 3.26 BL3/ET2—Log-normal versus GPD fit40

Fig. 3.27 BL3/ET3—Log-normal versus GPD fit40

Fig. 3.28 BL3/ET4—Log-normal versus GPD fit41

Fig. 3.29 BL3/ET5—Log-normal versus GPD fit41

Fig. 3.30 BL3/ET6—Log-normal versus GPD fit42

Fig. 3.31 BL3/ET7—Log-normal versus GPD fit42

Fig. 3.32 BL4/ET1—Log-normal versus GPD fit43

Fig. 3.33 BL4/ET2—Log-normal versus GPD fit43

Fig. 3.34 BL4/ET367—Log-normal versus GPD fit44

Fig. 3.35 BL4/ET4—Log-normal versus GPD fit44

Fig. 3.36 BL4/ET5—Log-normal versus GPD fit45

Fig. 3.37 BL5—Log-normal versus GPD fit45

Fig. 3.38 BL6—Log-normal versus GPD fit46

Fig. 3.39 BL7—Log-normal versus GPD fit46

Fig. 3.40 BL8/ET1—Log-normal versus GPD fit47

Fig. 3.41 BL8/ET27—Log-normal versus GPD fit47

Fig. 3.42 BL8/ET35—Log-normal versus GPD fit48

Fig. 3.43 BL8/ET4—Log-normal versus GPD fit48

Fig. 3.44 BL8/ET6—Log-normal versus GPD fit49

Fig. 4.1 BL1/ET—Frequency distribution62

Fig. 4.2 BL2/ET—Frequency distribution62

Fig. 4.3 BL3/ET1—Frequency distribution62

Fig. 4.4 BL3/ET2—Frequency distribution63

Fig. 4.5 BL3/ET3—Frequency distribution63

Fig. 4.6 BL3/ET4—Frequency distribution63

Fig. 4.7 BL3/ET5—Frequency distribution64

Fig. 4.8 BL3/ET6—Frequency distribution64

Fig. 4.9 BL3/ET7—Frequency distribution64

Fig. 4.10 BL4/ET1—Frequency distribution65

Fig. 4.11 BL4/ET2—Frequency distribution65

Fig. 4.12 BL4/ET367—Frequency distribution65

Fig. 4.13 BL4/ET4—Frequency distribution66

Fig. 4.14 BL4/ET5—Frequency distribution66

Fig. 4.15 BL5/ET—Frequency distribution66

Fig. 4.16 BL6/ET—Frequency distribution67

Fig. 4.17 BL7/ET—Frequency distribution67

Fig. 4.18 BL8/ET1—Frequency distribution67

Fig. 4.19 BL8/ET27—Frequency distribution68

Fig. 4.20 BL8/ET35—Frequency distribution68

Fig. 4.21 BL8/ET4—Frequency distribution68

Fig. 4.22 BL8/ET6—Frequency distribution69

List of Tables

Table 2.1 Pooling results using KS and AD tests for business lines 3, 4, 8 and percentages within each of the three business lines15

Table 2.2 Operational risk classes and percentages16

Table 3.1 Estimated mixtures according to the following criteria: BIC, argmin ( $$ p $$ -value > 0.05) 24

Table 3.2 Estimated mixtures according to the proposed criterion26

Table 3.3 GPD estimation on DIPO data38

Table 3.4 AD goodness-of-fit comparison between a mixture of $$ k $$ Log-normal distributions and a Log-normal and GPD combination 49

Table 4.1 Number of components and KS $$ p $$ -values for Poisson and Negative Binomial distributions 59

Table 4.2 Adjusted KS $$ p $$ -values for Poisson and Negative Binomial mixtures 60

Table 4.3 Selected distribution and number of components for the risk classes61

Table 5.1 Correlation between Severities and Frequencies72

Table 5.2 Pearson’s linear correlation matrix79

Table 5.3 Kendall’s rank correlation matrix80

Table 5.4 Years with maximum aggregated losses, maximum frequencies, and maximum single loss