Actuarial Theory for Dependent Risks (eBook)

Michel Denuit - Michel Denuit is Professor of Statistics and Actuarial Science at the Université catholique de Louvain, Belgium. His major fields of research are risk theory and stochastic inequalities. He (co-)authored numerous articles appeared in applied and theoretical journals and served as member of the editorial board for several journals (including Insurance: Mathematics and Economics). He is a section editor on Wiley's Encyclopedia of Actuarial Science. Jan Dhaene, Faculty of Economics and Applied Economics KULeuven, Belgium. Marc Goovaerts, Professor of Actuarial Science (Non-life Insurance) at University of Amsterdam (The Netherlands) and Catholique University of Leuven (Belgium) Rob Kaas, Professor of Actuarial Science (Actuarial Statistics), U. Amsterdam, The Netherlands.

Foreword.

Preface.

PART I: THE CONCEPT OF RISKS.

1. Modelling Risks.

1.1 Introduction.

1.2 The Probabilitsic Description of Risks.

1.3 Indepenance for Events and Conditional Probabilities.

1.4 Random Variables and Vectors.

1.5 Distribution Functions.

1.6 Mathematical Expectation.

1.7 Transforms.

1.8 Conditional Ditsributions.

1.9 Comonotonicity.

1.10 Mutual Exclusivity.

1.11 Exercises.

2. Measuring Risk.

2.1 Introduction.

2.2 Risk Measures.

2.3 Value-at-Risk.

2.4 Tail Value-at-Risk.

2.5 Risk MEasures Based on Expected Utility Theory.

2.6 Risk Measures Based on Distorted Expectation Theory.

2.7 Exercises.

2.8 Appendix: Convexity and Concavity.

3. Comparing Risks.

3.1 Introduction.

3.2 Stochastic Order Relations.

3.3 Stochastic Dominance.

3.4 Convex and Stop-Loss Orders.

3.5 Exercises.

PART II: DEPENDANCE BETWEEN RISKS.

4. Modelling Dependence.

4.1 Introduction.

4.2 Sklar's Representation Theorem.

4.3 Families of Bivariate Copulas.

4.4 Properties of Copulas.

4.5 The Archimedean Family of Cpoulas.

4.6 Simulation from Given Marginals and Copula.

4.7 Multivariate Copulas.

4.8 Loss-Alae Modelling with Archimedean Copulas: A CaseStudy.

4.9 Exercises.

5. Measuring Depenence.

5.1 Introduction.

5.2 Concordance Measures.

5.3 Dependence Structures.

5.4 Exercises.

6. Comparing Depe6.1 Introduction.

6.2 Comparing in the Bivariate Case Using the CorrelationOrder.

6.3 Comparing Dependence in the Multivariate Case Using theSupermodular Order.

6.4 Positive Orthant Depenedence Order.

6.5 Exercises.

PART III: APPLICATIONS TO INSURANCE MATHEMATICS.

7. Depenedence in Credibility Models Based on Generalized LinearModels.

7.1 Introduction.

7.2 Poisson Static Credibility for Claim Frequencies.

7.3 More Results for the Static Credibility Model.

7.4 More Results for the Dynamic Credibility Models.

7.5 On the Depenedence Induced By Bonus-Malus Scales.

7.6 Credibility Theory and Time Series for Non-Normal Data.

7.7 Exercises.

8. Stochastic Bounds on Functions of Dependent Risks.

8.1 Introduction.

8.2 Comparing Risks with Fixed Depoenedence Structure.

8.3 Stop-Loss Bounds on Functions of Dependent Risks.

8.4 Stochastic Bounds on Functions of Dependent Risks.

8.5 Some Financial Applications.

8.6 Exercises.

9. Integral Orderings and Probability Metrics.

9.1 Introduction.

9.2 Integral Stochastic Oredrings.

9.3 Integral Probability Metrics.

9.4 Total-Variation Distance.

9.5 Kolmogorov Distance.

9.6 Wasserstein Distance.

9.7 Stop-Loss Distance.

9.8 Integrated Stop-Loss Distance.

9.9 Distance Between the Individual and Collective Models inRisk Theory.

9.10 Compound Poisson Approximation for a Portfolio of DependentRisks.

9.11 Exercises.

References.

Index.