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.
Autorentext
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.
Inhalt
Foreword xiii
Preface xv
Part I the Concept of Risk 1
1 Modelling Risks 3
1.1 Introduction 3
1.2 The Probabilistic Description of Risks 4
1.2.1 Probability space 4
1.2.2 Experiment and universe 4
1.2.3 Random events 4
1.2.4 Sigma-algebra 5
1.2.5 Probability measure 5
1.3 Independence for Events and Conditional Probabilities 6
1.3.1 Independent events 6
1.3.2 Conditional probability 7
1.4 Random Variables and Random Vectors 7
1.4.1 Random variables 7
1.4.2 Random vectors 8
1.4.3 Risks and losses 9
1.5 Distribution Functions 10
1.5.1 Univariate distribution functions 10
1.5.2 Multivariate distribution functions 12
1.5.3 Tail functions 13
1.5.4 Support 14
1.5.5 Discrete random variables 14
1.5.6 Continuous random variables 15
1.5.7 General random variables 16
1.5.8 Quantile functions 17
1.5.9 Independence for random variables 20
1.6 Mathematical Expectation 21
1.6.1 Construction 21
1.6.2 Riemann-Stieltjes integral 22
1.6.3 Law of large numbers 24
1.6.4 Alternative representations for the mathematical expectation in the continuous case 24
1.6.5 Alternative representations for the mathematical expectation in the discrete case 25
1.6.6 Stochastic Taylor expansion 25
1.6.7 Variance and covariance 27
1.7 Transforms 29
1.7.1 Stop-loss transform 29
1.7.2 Hazard rate 30
1.7.3 Mean-excess function 32
1.7.4 Stationary renewal distribution 34
1.7.5 Laplace transform 34
1.7.6 Moment generating function 36
1.8 Conditional Distributions 37
1.8.1 Conditional densities 37
1.8.2 Conditional independence 38
1.8.3 Conditional variance and covariance 38
1.8.4 The multivariate normal distribution 38
1.8.5 The family of the elliptical distributions 41
1.9 Comonotonicity 49
1.9.1 Definition 49
1.9.2 Comonotonicity and Fréchet upper bound 49
1.10 Mutual Exclusivity 51
1.10.1 Definition 51
1.10.2 Fréchet lower bound 51
1.10.3 Existence of Fréchet lower bounds in Fréchet spaces 53
1.10.4 Fréchet lower bounds and maxima 53
1.10.5 Mutual exclusivity and Fréchet lower bound 53
1.11 Exercises 55
2 Measuring Risk 59
2.1 Introduction 59
2.2 Risk Measures 60
2.2.1 Definition 60
2.2.2 Premium calculation principles 61
2.2.3 Desirable properties 62
2.2.4 Coherent risk measures 65
2.2.5 Coherent and scenario-based risk measures 65
2.2.6 Economic capital 66
2.2.7 Expected risk-adjusted capital 66
2.3 Value-at-Risk 67
2.3.1 Definition 67
2.3.2 Properties 67
2.3.3 VaR-based economic capital 70
2.3.4 VaR and the capital asset pricing model 71
2.4 Tail Value-at-Risk 72
2.4.1 Definition 72
2.4.2 Some related risk measures 72
2.4.3 Properties 74
2.4.4 TVaR-based economic capital 77
2.5 Risk Measures Based on Expected Utility Theory 77
2.5.1 Brief introduction to expected utility theory 77
2.5.2 Zero-Utility Premiums 81
2.5.3 Esscher risk measure 82
2.6 Risk Measures Based on Distorted Expectation Theory 84
2.6.1 Brief introduction to distorted expectation theory 84
2.6.2 Wang risk measures 88
2.6.3 Some particular cases of Wang risk measures 92
2.7 Exercises 95
2.8 Appendix: Convexity and Concavity 100
2.8.1 Definition 100
2.8.2 Equivalent conditions 100
2.8.3 Properties 101
2.8.4 Convex sequences 102
2.8.5 Log-convex functions 102
3 Comparing Risks 103
3.1 Introduction 103
3.2 Stochastic Order Relations 105
3.2.1 Partial orders among distribution functions 105
3.2.2 Desirable properties for stochastic orderings 106
3.2.3 Integral stochastic orderings 106
3.3 Stochastic Dominance 108
3.3.1 Stochastic dominance and risk measures 108
3.3.2 Stochastic dominance and choice under risk 110
3.3.3 Comparing claim frequencies 113
3.3.4 Some properties of stochastic dominance 114
3.3.5 Stochastic dominance and notions of ageing 118
3.3.6 Stochastic increasingness 120
3.3.7 Ordering mixtures 121
3.3.8 Ordering compound sums 121
3.3.9 Sufficient conditions 122
3.3.10 Conditional stochastic dominance I: Hazard rate order 123
3.3.11 Conditional stochastic dominance II: Likelihood ratio order 127
3.3.12 Comparing shortfalls with stochastic dominance: Dispersive order 133
3.3.13 Mixed stochastic dominance: Laplace transform order 137
3.3.14 Multivariate extensions 142
3.4 Convex and Stop-Loss Orders 149
3.4.1 Convex and stop-loss orders and stop-loss premiums 149
3.4.2 Convex and stop-loss orders and choice under risk 150
3.4.3 Comparing claim frequencies 154
3.4.4 Some characterizations for convex and stop-loss orders 155
3.4.5 Some properties of the convex and stop-loss orders 162
3.4.6 Convex ordering and notions of …