Actuarial Theory for Dependent Risks

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 xiiiPreface xvPart I the Concept of Risk 11 Modelling Risks 31.1 Introduction 31.2 The Probabilistic Description of Risks 41.2.1 Probability space 41.2.2 Experiment and universe 41.2.3 Random events 41.2.4 Sigma-algebra 51.2.5 Probability measure 51.3 Independence for Events and Conditional Probabilities 61.3.1 Independent events 61.3.2 Conditional probability 71.4 Random Variables and Random Vectors 71.4.1 Random variables 71.4.2 Random vectors 81.4.3 Risks and losses 91.5 Distribution Functions 101.5.1 Univariate distribution functions 101.5.2 Multivariate distribution functions 121.5.3 Tail functions 131.5.4 Support 141.5.5 Discrete random variables 141.5.6 Continuous random variables 151.5.7 General random variables 161.5.8 Quantile functions 171.5.9 Independence for random variables 201.6 Mathematical Expectation 211.6.1 Construction 211.6.2 Riemann–Stieltjes integral 221.6.3 Law of large numbers 241.6.4 Alternative representations for the mathematical expectation in the continuous case 241.6.5 Alternative representations for the mathematical expectation in the discrete case 251.6.6 Stochastic Taylor expansion 251.6.7 Variance and covariance 271.7 Transforms 291.7.1 Stop-loss transform 291.7.2 Hazard rate 301.7.3 Mean-excess function 321.7.4 Stationary renewal distribution 341.7.5 Laplace transform 341.7.6 Moment generating function 361.8 Conditional Distributions 371.8.1 Conditional densities 371.8.2 Conditional independence 381.8.3 Conditional variance and covariance 381.8.4 The multivariate normal distribution 381.8.5 The family of the elliptical distributions 411.9 Comonotonicity 491.9.1 Definition 491.9.2 Comonotonicity and Fréchet upper bound 491.10 Mutual Exclusivity 511.10.1 Definition 511.10.2 Fréchet lower bound 511.10.3 Existence of Fréchet lower bounds in Fréchet spaces 531.10.4 Fréchet lower bounds and maxima 531.10.5 Mutual exclusivity and Fréchet lower bound 531.11 Exercises 552 Measuring Risk 592.1 Introduction 592.2 Risk Measures 602.2.1 Definition 602.2.2 Premium calculation principles 612.2.3 Desirable properties 622.2.4 Coherent risk measures 652.2.5 Coherent and scenario-based risk measures 652.2.6 Economic capital 662.2.7 Expected risk-adjusted capital 662.3 Value-at-Risk 672.3.1 Definition 672.3.2 Properties 672.3.3 VaR-based economic capital 702.3.4 VaR and the capital asset pricing model 712.4 Tail Value-at-Risk 722.4.1 Definition 722.4.2 Some related risk measures 722.4.3 Properties 742.4.4 TVaR-based economic capital 772.5 Risk Measures Based on Expected Utility Theory 772.5.1 Brief introduction to expected utility theory 772.5.2 Zero-Utility Premiums 812.5.3 Esscher risk measure 822.6 Risk Measures Based on Distorted Expectation Theory 842.6.1 Brief introduction to distorted expectation theory 842.6.2 Wang risk measures 882.6.3 Some particular cases of Wang risk measures 922.7 Exercises 952.8 Appendix: Convexity and Concavity 1002.8.1 Definition 1002.8.2 Equivalent conditions 1002.8.3 Properties 1012.8.4 Convex sequences 1022.8.5 Log-convex functions 1023 Comparing Risks 1033.1 Introduction 1033.2 Stochastic Order Relations 1053.2.1 Partial orders among distribution functions 1053.2.2 Desirable properties for stochastic orderings 1063.2.3 Integral stochastic orderings 1063.3 Stochastic Dominance 1083.3.1 Stochastic dominance and risk measures 1083.3.2 Stochastic dominance and choice under risk 1103.3.3 Comparing claim frequencies 1133.3.4 Some properties of stochastic dominance 1143.3.5 Stochastic dominance and notions of ageing 1183.3.6 Stochastic increasingness 1203.3.7 Ordering mixtures 1213.3.8 Ordering compound sums 1213.3.9 Sufficient conditions 1223.3.10 Conditional stochastic dominance I: Hazard rate order 1233.3.11 Conditional stochastic dominance II: Likelihood ratio order 1273.3.12 Comparing shortfalls with stochastic dominance: Dispersive order 1333.3.13 Mixed stochastic dominance: Laplace transform order 1373.3.14 Multivariate extensions 1423.4 Convex and Stop-Loss Orders 1493.4.1 Convex and stop-loss orders and stop-loss premiums 1493.4.2 Convex and stop-loss orders and choice under risk 1503.4.3 Comparing claim frequencies 1543.4.4 Some characterizations for convex and stop-loss orders 1553.4.5 Some properties of the convex and stop-loss orders 1623.4.6 Convex ordering and notions of ageing 1663.4.7 Stochastic (increasing) convexity 1673.4.8 Ordering mixtures 1693.4.9 Ordering compound sums 1693.4.10 Risk-reshaping contracts and Lorenz order 1693.4.11 Majorization 1713.4.12 Conditional stop-loss order: Mean-excess order 1733.4.13 Comparing shortfall with the stop-loss order: Right-spread order 1753.4.14 Multivariate extensions 1783.5 Exercises 182Part II Dependence Between Risks 1894 Modelling Dependence 1914.1 Introduction 1914.2 Sklar’s Representation Theorem 1944.2.1 Copulas 1944.2.2 Sklar’s theorem for continuous marginals 1944.2.3 Conditional distributions derived from copulas 1984.2.4 Probability density functions associated with copulas 2014.2.5 Copulas with singular components 2014.2.6 Sklar’s representation in the general case 2034.3 Families of Bivariate Copulas 2044.3.1 Clayton’s copula 2054.3.2 Frank’s copula 2054.3.3 The normal copula 2074.3.4 The Student copula 2084.3.5 Building multivariate distributions with given marginals from copulas 2104.4 Properties of Copulas 2134.4.1 Survival copulas 2134.4.2 Dual and co-copulas 2154.4.3 Functional invariance 2164.4.4 Tail dependence 2174.5 The Archimedean Family of Copulas 2184.5.1 Definition 2184.5.2 Frailty models 2194.5.3 Probability density function associated with Archimedean copulas 2204.5.4 Properties of Archimedean copulas 2214.6 Simulation from Given Marginals and Copula 2234.6.1 General method 2234.6.2 Exploiting Sklar’s decomposition 2244.6.3 Simulation from Archimedean copulas 2244.7 Multivariate Copulas 2254.7.1 Definition 2254.7.2 Sklar’s representation theorem 2254.7.3 Functional invariance 2264.7.4 Examples of multivariate copulas 2264.7.5 Multivariate Archimedean copulas 2294.8 Loss–Alae Modelling with Archimedean Copulas: A Case Study 2314.8.1 Losses and their associated ALAEs 2314.8.2 Presentation of the ISO data set 2314.8.3 Fitting parametric copula models to data 2324.8.4 Selecting the generator for Archimedean copula models 2344.8.5 Application to loss–ALAE modelling 2384.9 Exercises 2425 Measuring Dependence 2455.1 Introduction 2455.2 Concordance Measures 2465.2.1 Definition 2465.2.2 Pearson’s correlation coefficient 2475.2.3 Kendall’s rank correlation coefficient 2535.2.4 Spearman’s rank correlation coefficient 2575.2.5 Relationships between Kendall’s and Spearman’s rank correlation coefficients 2595.2.6 Other dependence measures 2605.2.7 Constraints on concordance measures in bivariate discrete data 2625.3 Dependence Structures 2645.3.1 Positive dependence notions 2645.3.2 Positive quadrant dependence 2655.3.3 Conditional increasingness in sequence 2745.3.4 Multivariate total positivity of order 2 2765.4 Exercises 2796 Comparing Dependence 2856.1 Introduction 2856.2 Comparing Dependence in the Bivariate Case Using the Correlation Order 2876.2.1 Definition 2876.2.2 Relationship with orthant orders 2886.2.3 Relationship with positive quadrant dependence 2896.2.4 Characterizations in terms of supermodular functions 2896.2.5 Extremal elements 2906.2.6 Relationship with convex and stop-loss orders 2906.2.7 Correlation order and copulas 2926.2.8 Correlation order and correlation coefficients 2926.2.9 Ordering Archimedean copulas 2926.2.10 Ordering compound sums 2936.2.11 Correlation order and diversification benefit 2946.3 Comparing Dependence in the Multivariate Case Using the Supermodular Order 2956.3.1 Definition 2956.3.2 Smooth supermodular functions 2966.3.3 Restriction to distributions with identical marginals 2966.3.4 A companion order: The symmetric supermodular order 2976.3.5 Relationships between supermodular-type orders 2976.3.6 Supermodular order and dependence measures 2976.3.7 Extremal dependence structures in the supermodular sense 2986.3.8 Supermodular, stop-loss and convex orders 2986.3.9 Ordering compound sums 2996.3.10 Ordering random vectors with common values 3006.3.11 Stochastic analysis of duplicates in life insurance portfolios 3026.4 Positive Orthant Dependence Order 3046.4.1 Definition 3046.4.2 Positive orthant dependence order and correlation coefficients 3046.5 Exercises 305Part III Applications to Insurance Mathematics 3097 Dependence in Credibility Models Based on Generalized Linear Models 3117.1 Introduction 3117.2 Poisson Credibility Models for Claim Frequencies 3127.2.1 Poisson static credibility model 3127.2.2 Poisson dynamic credibility models 3157.2.3 Association 3167.2.4 Dependence by mixture and common mixture models 3207.2.5 Dependence in the Poisson static credibility model 3237.2.6 Dependence in the Poisson dynamic credibility models 3257.3 More Results for the Static Credibility Model 3297.3.1 Generalized linear models and generalized additive models 3297.3.2 Some examples of interest to actuaries 3307.3.3 Credibility theory and generalized linear mixed models 3317.3.4 Exhaustive summary of past claims 3327.3.5 A posteriori distribution of the random effects 3337.3.6 Predictive distributions 3347.3.7 Linear credibility premium 3347.4 More Results for the Dynamic Credibility Models 3397.4.1 Dynamic credibility models and generalized linear mixed models 3397.4.2 Dependence in GLMM-based credibility models 3407.4.3 A posteriori distribution of the random effects 3417.4.4 Supermodular comparisons 3427.4.5 Predictive distributions 3437.5 On the Dependence Induced by Bonus–Malus Scales 3447.5.1 Experience rating in motor insurance 3447.5.2 Markov models for bonus–malus system scales 3447.5.3 Positive dependence in bonus–malus scales 3457.6 Credibility Theory and Time Series for Non-Normal Data 3467.6.1 The classical actuarial point of view 3467.6.2 Time series models built from copulas 3467.6.3 Markov models for random effects 3487.6.4 Dependence induced by autoregressive copula models in dynamic frequency credibility models 3497.7 Exercises 3508 Stochastic Bounds on Functions of Dependent Risks 3558.1 Introduction 3558.2 Comparing Risks With Fixed Dependence Structure 3578.2.1 The problem 3578.2.2 Ordering random vectors with fixed dependence structure with stochastic dominance 3588.2.3 Ordering random vectors with fixed dependence structure with convex order 3588.3 Stop-Loss Bounds on Functions of Dependent Risks 3608.3.1 Known marginals 3608.3.2 Unknown marginals 3608.4 Stochastic Bounds on Functions of Dependent Risks 3638.4.1 Stochastic bounds on the sum of two risks 3638.4.2 Stochastic bounds on the sum of several risks 3658.4.3 Improvement of the bounds on sums of risks under positive dependence 3678.4.4 Stochastic bounds on functions of two risks 3688.4.5 Improvements of the bounds on functions of risks under positive quadrant dependence 3708.4.6 Stochastic bounds on functions of several risks 3708.4.7 Improvement of the bounds on functions of risks under positive orthant dependence 3718.4.8 The case of partially specified marginals 3728.5 Some Financial Applications 3758.5.1 Stochastic bounds on present values 3758.5.2 Stochastic annuities 3768.5.3 Life insurance 3798.6 Exercises 3829 Integral Orderings and Probability Metrics 3859.1 Introduction 3859.2 Integral Stochastic Orderings 3869.2.1 Definition 3869.2.2 Properties 3869.3 Integral Probability Metrics 3889.3.1 Probability metrics 3889.3.2 Simple probability metrics 3899.3.3 Integral probability metrics 3899.3.4 Ideal metrics 3909.3.5 Minimal metric 3929.3.6 Integral orders and metrics 3929.4 Total-Variation Distance 3939.4.1 Definition 3939.4.2 Total-variation distance and integral metrics 3949.4.3 Comonotonicity and total-variation distance 3959.4.4 Maximal coupling and total-variation distance 3969.5 Kolmogorov Distance 3969.5.1 Definition 3969.5.2 Stochastic dominance, Kolmogorov and total-variation distances 3979.5.3 Kolmogorov distance under single crossing condition for probability density functions 3979.6 Wasserstein Distance 3989.6.1 Definition 3989.6.2 Properties 3999.6.3 Comonotonicity and Wasserstein distance 4009.7 Stop-Loss Distance 4019.7.1 Definition 4019.7.2 Stop-loss order, stop-loss and Wasserstein distances 4019.7.3 Computation of the stop-loss distance under stochastic dominance or dangerousness order 4019.8 Integrated Stop-Loss Distance 4039.8.1 Definition 4039.8.2 Properties 4039.8.3 Integrated stop-loss distance and positive quadrant dependence 4059.8.4 Integrated stop-loss distance and cumulative dependence 4059.9 Distance Between the Individual and Collective Models in Risk Theory 4079.9.1 Individual model 4079.9.2 Collective model 4079.9.3 Distance between compound sums 4089.9.4 Distance between the individual and collective models 4109.9.5 Quasi-homogeneous portfolios 4129.9.6 Correlated risks in the individual model 4149.10 Compound Poisson Approximation for a Portfolio of Dependent Risks 4149.10.1 Poisson approximation 4149.10.2 Dependence in the quasi-homogeneous individual model 4189.11 Exercises 421References 423Index 439