Jaume Belles-Sampera has a PhD in Business Studies at the University of Barcelona (UB). He has a Degree in Mathematics and a Master Degree in Research in Business, Finance and Insurance from the UB. He combines his daily job in the insurance industry with specific academical and research assistance to the research group Riskcenter - IREA. His main interests are capital allocation, risk measures and decision making behavioral, although his interests embraces performance attribution and several econometric issues, as well as the study of the role that aggregation functions play in risk management. He has a broad professional experience as an advisor for insurance companies and asset management firms, developed at an international audit firm, and nowadays he is professionally involved in the actuarial function of an international insurance group. He was awarded with the prize for the best academic record at Master level. He is a certified Financial Risk Manager (FRM) by the Global Association of Risk Professionals (GARP). He received the Ferran Armengol i Tubau prize (2014) to the best study about insurance, awarded by the Catalan Society of Economy.|Montserrat Guillén was received a Master of Science in Mathematics and Mathematical Statistics in 1987 and a PhD in Economics from UB in 1992. She received a MSc in Data Analysis from the University of Essex (United Kingdom). She was Visiting Research faculty at the University of Texas at Austin (USA) in 1994. She also holds a Visiting Professor position at the University of Paris II, where she teaches Insurance Econometrics. Since April, 2001 she has been chair professor of the Department of Econometrics at the University of Barcelona. She was awarded the ICREA Academia distinction. Her research focuses on actuarial statistics and quantitative risk management. She has published many scientific articles, contributions to book chapters and books on insurance and actuarial science. She is an Associate Editor for the Journal of Risk and Insurance - the official journal of the American Risk and Insurance Association, a senior editor of Astin Bulletin - the official journal of the International Actuarial Association and chief editor of SORT-Statistics and Operations Research Transactions She was awarded by the Casualty Actuarial Society and received the International Insurance Prize. She is a highly cited academic in the field of risk management and insurance. She was elected President of the European Group of Risk and Insurance Economists, the Geneva Association, in 2011. She has served in many scientific boards, international programs and steering committees and she has also conducted R&D joint programmes with many companies. She is member of the Royal Academy of Doctors.|Miguel Santolino has a PhD in Business Studies, MA Actuarial Science and MA Economics from the University of Barcelona and MSc in Financial and Actuarial Engineering from the Katholieke Universiteit Leuven (Belgium). His academic position is senior lecturer in the Department of Econometrics, Statistics and Spanish Economy in the University of Barcelona. His research focuses on risk measurement, the resolution of disputes, including ADR methods, and assessment of bodily injuries. His research is published in Risk Analysis, Accident Analysis and Prevention, Insurance: Mathematics and Economics, Journal of Risk Research, Group Decision and Negotiation, International Review of Law and Economics, European Journal of Law and Economics and national insurance journals. He received the Ferran Armengol i Tubau prize (2008) to the best study about insurance, awarded by the Catalan Society of Economy.

• 4 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

Jaume Belles-Sampera has a PhD in Business Studies at the University of Barcelona (UB). He has a Degree in Mathematics and a Master Degree in Research in Business, Finance and Insurance from the UB. He combines his daily job in the insurance industry with specific academical and research assistance to the research group Riskcenter - IREA. His main interests are capital allocation, risk measures and decision making behavioral, although his interests embraces performance attribution and several econometric issues, as well as the study of the role that aggregation functions play in risk management. He has a broad professional experience as an advisor for insurance companies and asset management firms, developed at an international audit firm, and nowadays he is professionally involved in the actuarial function of an international insurance group. He was awarded with the prize for the best academic record at Master level. He is a certified Financial Risk Manager (FRM) by the Global Association of Risk Professionals (GARP). He received the Ferran Armengol i Tubau prize (2014) to the best study about insurance, awarded by the Catalan Society of Economy.|Montserrat Guillén was received a Master of Science in Mathematics and Mathematical Statistics in 1987 and a PhD in Economics from UB in 1992. She received a MSc in Data Analysis from the University of Essex (United Kingdom). She was Visiting Research faculty at the University of Texas at Austin (USA) in 1994. She also holds a Visiting Professor position at the University of Paris II, where she teaches Insurance Econometrics. Since April, 2001 she has been chair professor of the Department of Econometrics at the University of Barcelona. She was awarded the ICREA Academia distinction. Her research focuses on actuarial statistics and quantitative risk management. She has published many scientific articles, contributions to book chapters and books on insurance and actuarial science. She is an Associate Editor for the Journal of Risk and Insurance - the official journal of the American Risk and Insurance Association, a senior editor of Astin Bulletin - the official journal of the International Actuarial Association and chief editor of SORT-Statistics and Operations Research Transactions She was awarded by the Casualty Actuarial Society and received the International Insurance Prize. She is a highly cited academic in the field of risk management and insurance. She was elected President of the European Group of Risk and Insurance Economists, the Geneva Association, in 2011. She has served in many scientific boards, international programs and steering committees and she has also conducted R&D joint programmes with many companies. She is member of the Royal Academy of Doctors.|Miguel Santolino has a PhD in Business Studies, MA Actuarial Science and MA Economics from the University of Barcelona and MSc in Financial and Actuarial Engineering from the Katholieke Universiteit Leuven (Belgium). His academic position is senior lecturer in the Department of Econometrics, Statistics and Spanish Economy in the University of Barcelona. His research focuses on risk measurement, the resolution of disputes, including ADR methods, and assessment of bodily injuries. His research is published in Risk Analysis, Accident Analysis and Prevention, Insurance: Mathematics and Economics, Journal of Risk Research, Group Decision and Negotiation, International Review of Law and Economics, European Journal of Law and Economics and national insurance journals. He received the Ferran Armengol i Tubau prize (2008) to the best study about insurance, awarded by the Catalan Society of Economy.

• 4 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

Jaume Belles-Sampera has a PhD in Business Studies at the University of Barcelona (UB). He has a Degree in Mathematics and a Master Degree in Research in Business, Finance and Insurance from the UB. He combines his daily job in the insurance industry with specific academical and research assistance to the research group Riskcenter - IREA. His main interests are capital allocation, risk measures and decision making behavioral, although his interests embraces performance attribution and several econometric issues, as well as the study of the role that aggregation functions play in risk management. He has a broad professional experience as an advisor for insurance companies and asset management firms, developed at an international audit firm, and nowadays he is professionally involved in the actuarial function of an international insurance group. He was awarded with the prize for the best academic record at Master level. He is a certified Financial Risk Manager (FRM) by the Global Association of Risk Professionals (GARP). He received the Ferran Armengol i Tubau prize (2014) to the best study about insurance, awarded by the Catalan Society of Economy.|Montserrat Guillén was received a Master of Science in Mathematics and Mathematical Statistics in 1987 and a PhD in Economics from UB in 1992. She received a MSc in Data Analysis from the University of Essex (United Kingdom). She was Visiting Research faculty at the University of Texas at Austin (USA) in 1994. She also holds a Visiting Professor position at the University of Paris II, where she teaches Insurance Econometrics. Since April, 2001 she has been chair professor of the Department of Econometrics at the University of Barcelona. She was awarded the ICREA Academia distinction. Her research focuses on actuarial statistics and quantitative risk management. She has published many scientific articles, contributions to book chapters and books on insurance and actuarial science. She is an Associate Editor for the Journal of Risk and Insurance - the official journal of the American Risk and Insurance Association, a senior editor of Astin Bulletin - the official journal of the International Actuarial Association and chief editor of SORT-Statistics and Operations Research Transactions She was awarded by the Casualty Actuarial Society and received the International Insurance Prize. She is a highly cited academic in the field of risk management and insurance. She was elected President of the European Group of Risk and Insurance Economists, the Geneva Association, in 2011. She has served in many scientific boards, international programs and steering committees and she has also conducted R&D joint programmes with many companies. She is member of the Royal Academy of Doctors.|Miguel Santolino has a PhD in Business Studies, MA Actuarial Science and MA Economics from the University of Barcelona and MSc in Financial and Actuarial Engineering from the Katholieke Universiteit Leuven (Belgium). His academic position is senior lecturer in the Department of Econometrics, Statistics and Spanish Economy in the University of Barcelona. His research focuses on risk measurement, the resolution of disputes, including ADR methods, and assessment of bodily injuries. His research is published in Risk Analysis, Accident Analysis and Prevention, Insurance: Mathematics and Economics, Journal of Risk Research, Group Decision and Negotiation, International Review of Law and Economics, European Journal of Law and Economics and national insurance journals. He received the Ferran Armengol i Tubau prize (2008) to the best study about insurance, awarded by the Catalan Society of Economy.

• 4 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