Risk Quantification and Allocation Methods for Practitioners

Preface v
Contents ix
List of Figures xii
List of Tables xii

PART I RISK ASSESSMENT 1
1 Preliminary concepts on quantitative risk measurement 3
1.1 Risk measurement - Theory 3
1.1.1 First definitions 3
1.1.2 Properties for risk measures 14
1.2 Risk measurement - Practice 16
1.2.1 ‘Liability side’ versus ‘asset side’ perspectives 17
1.2.2 Some misunderstandings to be avoided in practice 20
1.3 Exercises 26

2 Data on losses for risk evaluation 29
2.1 An example on three dimensional data 29
2.2 Basic graphical analysis of the loss severity distributions 31
2.3 Quantile estimation 33
2.4 Examples 33

3 A family of distortion risk measures 35
3.1 Overview on risk measures 37
3.2 Distortion risk measures 38
3.3 A new family of risk measures: GlueVaR 40
3.4 Linear combination of risk measures 41
3.5 Subadditivity 43
3.6 Concavity of the distortion function 44
3.7 Example of risk measurement with GlueVaR 45
3.8 Exercises 48

4 GlueVaR and other new risk measures 51
4.1 Analytical closed-form expressions of GlueVaR 51
4.1.1 Illustration: GlueVaR expression for Student t distribution 51
4.1.2 Analytical expressions for other frequently used distributions 52
4.1.3 The Cornish-Fisher approximation of GlueVaR 54
4.2 On the relationship between GlueVaR and Tail Distortion risk measures 56
4.3 On the relationship between GlueVaR and RVaR risk measures 57
4.4 Example 60
4.5 Exercises 63

5 Risk measure choice 65
5.1 Aggregate attitude towards risk 66
5.1.1 Local risk attitude 70
5.2 Application of risk assessment in a scenario involving catastrophic losses 76
5.2.1 Calibration of GlueVaR parameters 77
5.2.2 Data and Results 78
5.3 GlueVaR to reflect risk attitudes 81
5.4 Exercises 82

PART II CAPITAL ALLOCATION PROBLEMS 83
6 An overview on capital allocation problems 85
6.1 Main concepts and notation 85
6.2 Properties of capital allocation principles 89
6.3 Review of some principles 91
6.3.1 The gradient allocation principle 91
6.3.2 Other capital allocation principles based on partial contributions 101
6.3.3 The excess based allocation principle 106
6.4 Further reading 110
6.5 Exercices 111

7 Capital allocation based on GlueVaR 113
7.1 A capital allocation framework 113
7.2 The Haircut capital allocation principle 115
7.3 Proportional risk capital allocation principles using GlueVaR 117
7.3.1 Stand-alone proportional allocation principles using GlueVaR 118
7.3.2 Proportional allocation principles based on partial contributions using GlueVaR 118
7.4 An example of risk capital allocation on claim costs 119
7.5 Exercices 122

8 Capital allocation principles as compositional data 123
8.1 The simplex and its vectorial and metric structure 123
8.1.1 From capital allocation principles to compositional data 128
8.2 Simplicial concepts applied to capital allocation 128
8.2.1 The inverse of a capital allocation 129
8.2.2 Ranking capital allocation principles 130
8.2.3 Averaging capital allocation principles 131
8.2.4 An illustration 131
8.3 Exercises 135

Appendix 137
A.1 Equivalent expression for the GlueVaR distortion function 137
A.2 Bijective relationship between heights and weights as parameters for GlueVaR risk measures 138
A.3 Relationship between GlueVaR and Tail Distortion risk measures 138

Bibliography 141
Biographies of the authors 149
Index 151