I Risk Assessment 1 Preliminary concepts on quantitative risk measurement 1.1 Risk measurement - Theory 1.1.1 First de_nitions 1.1.2 Properties for risk measures 1.2 Risk measurement - Practice 1.2.1 `Liability side' versus `asset side' perspectives 1.2.2 Some misunderstandings to be avoided in practice 1.3 Exercises 2 Data on losses for risk evaluation 2.1 An example on three dimensional data 2.2 Basic graphical analysis of the loss severity distributions 2.3 Quantile estimation 2.4 Examples 3 A family of distortion risk measures 3.1 Overview on risk measures 3.2 Distortion risk measures 3.3 A new family of risk measures: GlueVaR 3.4 Linear combination of risk measures 3.5 Subadditivity 3.6 Concavity of the distortion function 3.7 Example of risk measurement with GlueVaR 3.8 Exercises 4 GlueVaR and other new risk measures 4.1 Analytical closed-form expressions of GlueVaR 4.1.1 Analytical expressions for other frequently used distributions 4.1.2 The Cornish-Fisher approximation of GlueVaR 4.2 On the relationship between GlueVaR and Tail Distortion risk measures 4.3 On the relationship between GlueVaR and RVaR risk measures . 4.4 Example 4.5 Exercises 5 Risk measure choice 51 5.1 Aggregate attitude towards risk 5.1.1 Local risk attitude 5.2 Application of risk assessment in a scenario involving catastrophic losses 5.2.1 Calibration of GlueVaR parameters 5.2.2 Data and Results 5.3 GlueVaR to reect risk attitudes 5.4 Exercises II Capital Allocation Problems 6 An overview on capital allocation problems 6.1 Main concepts and notation 6.2 Properties of capital allocation principles 6.3 Review of some principles 6.3.1 The gradient allocation principle 6.3.2 Other capital allocation principles base on partial contributions 6.3.3 The excess based allocation principle 6.4 Further reading 6.5 Exercices 7 Capital allocation based on GlueVaR 7.1 A capital allocation framework 7.2 The Haircut capital allocation principle 7.3 Proportional risk capital allocation principles using GlueVaR 7.3.1 Stand-alone proportional allocation principles using Glue- VaR 7.3.2 Proportional allocation principles based on partial contributions using GlueVaR 7.4 An example of risk capital allocation on claim costs 7.5 Exercices 8 Capital allocation principles as compositional data 99 8.1 The simplex and its vectorial and metric structure 8.1.1 From capital allocation principles to compositional data 8.2 Simplicial concepts applied to capital allocation 8.2.1 The inverse of a capital allocation 8.2.2 Ranking capital allocation principles 8.2.3 Averaging capital allocation principles 8.2.4 An illustration 8.3 Exercises Appendix A.1 Equivalent expression for the GlueVaR distortion function A.2 Bijective relationship between heights and weights as parameters for GlueVaR risk measures A.3 Relationship between GlueVaR and Tail Distortion risk measures Bilbiography Biographies of the authors Index