Michel Denuit & Jan Dhaene 
Actuarial Theory for Dependent Risks 
Measures, Orders and Models

The increasing complexity of insurance and reinsurance products has
seen a growing interest amongst actuaries in the modelling of
dependent risks. For efficient risk management, actuaries need to
be able to answer fundamental questions such as: Is the correlation
structure dangerous? And, if yes, to what extent? Therefore tools
to quantify, compare, and model the strength of dependence between
different risks are vital. Combining coverage of stochastic order
and risk measure theories with the basics of risk management and
stochastic dependence, this book provides an essential guide to
managing modern financial risk.

* Describes how to model risks in incomplete markets, emphasising
insurance risks.

* Explains how to measure and compare the danger of risks, model
their interactions, and measure the strength of their
association.

* Examines the type of dependence induced by GLM-based credibility
models, the bounds on functions of dependent risks, and
probabilistic distances between actuarial models.

* Detailed presentation of risk measures, stochastic orderings,
copula models, dependence concepts and dependence orderings.

* Includes numerous exercises allowing a cementing of the concepts
by all levels of readers.

* Solutions to tasks as well as further examples and exercises can
be found on a supporting website.

An invaluable reference for both academics and practitioners alike,
Actuarial Theory for Dependent Risks will appeal to all those eager
to master the up-to-date modelling tools for dependent risks. The
inclusion of exercises and practical examples makes the book
suitable for advanced courses on risk management in incomplete
markets. Traders looking for practical advice on insurance markets
will also find much of interest.

Cuprins

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 Case Study.

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 Correlation Order.

6.3 Comparing Dependence in the Multivariate Case Using the Supermodular Order.

6.4 Positive Orthant Depenedence Order.

6.5 Exercises.

PART III: APPLICATIONS TO INSURANCE MATHEMATICS.

7. Depenedence in Credibility Models Based on Generalized Linear Models.

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 in Risk Theory.

9.10 Compound Poisson Approximation for a Portfolio of Dependent Risks.

9.11 Exercises.

References.

Index.

Despre autor

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.