<p>Actuarial Theory for Dependent Risks</p><p>PART I THE CONCEPT OF RISK 1<br/>1 Modelling Risks 3<br/>1.1 Introduction 3<br/>1.2 The Probabilistic Description of Risks 4<br/>1.2.1 Probability space 4<br/>1.2.2 Experiment and universe 4<br/>1.2.3 Random events 4<br/>1.2.4 Sigma-algebra 5<br/>1.2.5 Probability measure 5<br/>1.3 Independence for Events and Conditional Probabilities 6<br/>1.3.1 Independent events 6<br/>1.3.2 Conditional probability 7<br/>1.4 Random Variables and Random Vectors 7<br/>1.4.1 Random variables 7<br/>1.4.2 Random vectors 8<br/>1.4.3 Risks and losses 9<br/>1.5 Distribution Functions 10<br/>1.5.1 Univariate distribution functions 10<br/>1.5.2 Multivariate distribution functions 12<br/>1.5.3 Tail functions 13<br/>1.5.4 Support 14<br/>1.5.5 Discrete random variables 14<br/>1.5.6 Continuous random variables 15<br/>1.5.7 General random variables 16<br/>1.5.8 Quantile functions 17<br/>1.5.9 Independence for random variables 20<br/>1.6 Mathematical Expectation 21<br/>1.6.1 Construction 21<br/>1.6.2 Riemann–Stieltjes integral 22</p><p>1.6.3 Law of large numbers 24<br/>1.6.4 Alternative representations for the mathematical expectation<br/>in the continuous case 24<br/>1.6.5 Alternative representations for the mathematical expectation<br/>in the discrete case 25<br/>1.6.6 Stochastic Taylor expansion 25<br/>1.6.7 Variance and covariance 27<br/>1.7 Transforms 29<br/>1.7.1 Stop-loss transform 29<br/>1.7.2 Hazard rate 30<br/>1.7.3 Mean-excess function 32<br/>1.7.4 Stationary renewal distribution 34<br/>1.7.5 Laplace transform 34<br/>1.7.6 Moment generating function 36<br/>1.8 Conditional Distributions 37<br/>1.8.1 Conditional densities 37<br/>1.8.2 Conditional independence 38<br/>1.8.3 Conditional variance and covariance 38<br/>1.8.4 The multivariate normal distribution 38<br/>1.8.5 The family of the elliptical distributions 41<br/>1.9 Comonotonicity 49<br/>1.9.1 Definition 49<br/>1.9.2 Comonotonicity and Fréchet upper bound 49<br/>1.10 Mutual Exclusivity 51<br/>1.10.1 Definition 51<br/>1.10.2 Fréchet lower bound 51<br/>1.10.3 Existence of Fréchet lower bounds in Fréchet spaces 53<br/>1.10.4 Fréchet lower bounds and maxima 53<br/>1.10.5 Mutual exclusivity and Fréchet lower bound 53<br/>1.11 Exercises 55<br/>2 Measuring Risk 59<br/>2.1 Introduction 59<br/>2.2 Risk Measures 60<br/>2.2.1 Definition 60<br/>2.2.2 Premium calculation principles 61<br/>2.2.3 Desirable properties 62<br/>2.2.4 Coherent risk measures 65<br/>2.2.5 Coherent and scenario-based risk measures 65<br/>2.2.6 Economic capital 66<br/>2.2.7 Expected risk-adjusted capital 66<br/>2.3 Value-at-Risk 67<br/>2.3.1 Definition 67<br/>2.3.2 Properties 67<br/>2.3.3 VaR-based economic capital 70<br/>2.3.4 VaR and the capital asset pricing model 71</p><p>2.4 Tail Value-at-Risk 72<br/>2.4.1 Definition 72<br/>2.4.2 Some related risk measures 72<br/>2.4.3 Properties 74<br/>2.4.4 TVaR-based economic capital 77<br/>2.5 Risk Measures Based on Expected Utility Theory 77<br/>2.5.1 Brief introduction to expected utility theory 77<br/>2.5.2 Zero-Utility Premiums 81<br/>2.5.3 Esscher risk measure 82<br/>2.6 Risk Measures Based on Distorted Expectation Theory 84<br/>2.6.1 Brief introduction to distorted expectation theory 84<br/>2.6.2 Wang risk measures 88<br/>2.6.3 Some particular cases of Wang risk measures 92<br/>2.7 Exercises 95<br/>2.8 Appendix: Convexity and Concavity 100<br/>2.8.1 Definition 100<br/>2.8.2 Equivalent conditions 100<br/>2.8.3 Properties 101<br/>2.8.4 Convex sequences 102<br/>2.8.5 Log-convex functions 102<br/>3 Comparing Risks 103<br/>3.1 Introduction 103<br/>3.2 Stochastic Order Relations 105<br/>3.2.1 Partial orders among distribution functions 105<br/>3.2.2 Desirable properties for stochastic orderings 106<br/>3.2.3 Integral stochastic orderings 106<br/>3.3 Stochastic Dominance 108<br/>3.3.1 Stochastic dominance and risk measures 108<br/>3.3.2 Stochastic dominance and choice under risk 110<br/>3.3.3 Comparing claim frequencies 113<br/>3.3.4 Some properties of stochastic dominance 114<br/>3.3.5 Stochastic dominance and notions of ageing 118<br/>3.3.6 Stochastic increasingness 120<br/>3.3.7 Ordering mixtures 121<br/>3.3.8 Ordering compound sums 121<br/>3.3.9 Sufficient conditions 122<br/>3.3.10 Conditional stochastic dominance I: Hazard rate order 123<br/>3.3.11 Conditional stochastic dominance II: Likelihood ratio order 127<br/>3.3.12 Comparing shortfalls with stochastic dominance: Dispersive order 133<br/>3.3.13 Mixed stochastic dominance: Laplace transform order 137<br/>3.3.14 Multivariate extensions 142<br/>3.4 Convex and Stop-Loss Orders 149<br/>3.4.1 Convex and stop-loss orders and stop-loss premiums 149<br/>3.4.2 Convex and stop-loss orders and choice under risk 150<br/>3.4.3 Comparing claim frequencies 154</p><p>3.4.4 Some characterizations for convex and stop-loss orders 155<br/>3.4.5 Some properties of the convex and stop-loss orders 162<br/>3.4.6 Convex ordering and notions of ageing 166<br/>3.4.7 Stochastic (increasing) convexity 167<br/>3.4.8 Ordering mixtures 169<br/>3.4.9 Ordering compound sums 169<br/>3.4.10 Risk-reshaping contracts and Lorenz order 169<br/>3.4.11 Majorization 171<br/>3.4.12 Conditional stop-loss order: Mean-excess order 173<br/>3.4.13 Comparing shortfall with the stop-loss order: Right-spread order 175<br/>3.4.14 Multivariate extensions 178<br/>3.5 Exercises 182<br/>PART II DEPENDENCE BETWEEN RISKS 189<br/>4 Modelling Dependence 191<br/>4.1 Introduction 191<br/>4.2 Sklar’s Representation Theorem 194<br/>4.2.1 Copulas 194<br/>4.2.2 Sklar’s theorem for continuous marginals 194<br/>4.2.3 Conditional distributions derived from copulas 198<br/>4.2.4 Probability density functions associated with copulas 201<br/>4.2.5 Copulas with singular components 201<br/>4.2.6 Sklar’s representation in the general case 203<br/>4.3 Families of Bivariate Copulas 204<br/>4.3.1 Clayton’s copula 205<br/>4.3.2 Frank’s copula 205<br/>4.3.3 The normal copula 207<br/>4.3.4 The Student copula 208<br/>4.3.5 Building multivariate distributions with given marginals<br/>from copulas 210<br/>4.4 Properties of Copulas 213<br/>4.4.1 Survival copulas 213<br/>4.4.2 Dual and co-copulas 215<br/>4.4.3 Functional invariance 216<br/>4.4.4 Tail dependence 217<br/>4.5 The Archimedean Family of Copulas 218<br/>4.5.1 Definition 218<br/>4.5.2 Frailty models 219<br/>4.5.3 Probability density function associated with<br/>Archimedean copulas 220<br/>4.5.4 Properties of Archimedean copulas 221<br/>4.6 Simulation from Given Marginals and Copula 223<br/>4.6.1 General method 223<br/>4.6.2 Exploiting Sklar’s decomposition 224<br/>4.6.3 Simulation from Archimedean copulas 224</p><p>4.7 Multivariate Copulas 225<br/>4.7.1 Definition 225<br/>4.7.2 Sklar’s representation theorem 225<br/>4.7.3 Functional invariance 226<br/>4.7.4 Examples of multivariate copulas 226<br/>4.7.5 Multivariate Archimedean copulas 229<br/>4.8 Loss–Alae Modelling with Archimedean Copulas: A Case Study 231<br/>4.8.1 Losses and their associated ALAEs 231<br/>4.8.2 Presentation of the ISO data set 231<br/>4.8.3 Fitting parametric copula models to data 232<br/>4.8.4 Selecting the generator for Archimedean copula models 234<br/>4.8.5 Application to loss–ALAE modelling 238<br/>4.9 Exercises 242<br/>5 Measuring Dependence 245<br/>5.1 Introduction 245<br/>5.2 Concordance Measures 246<br/>5.2.1 Definition 246<br/>5.2.2 Pearson’s correlation coefficient 247<br/>5.2.3 Kendall’s rank correlation coefficient 253<br/>5.2.4 Spearman’s rank correlation coefficient 257<br/>5.2.5 Relationships between Kendall’s and Spearman’s rank<br/>correlation coefficients 259<br/>5.2.6 Other dependence measures 260<br/>5.2.7 Constraints on concordance measures in bivariate discrete data 262<br/>5.3 Dependence Structures 264<br/>5.3.1 Positive dependence notions 264<br/>5.3.2 Positive quadrant dependence 265<br/>5.3.3 Conditional increasingness in sequence 274<br/>5.3.4 Multivariate total positivity of order 2 276<br/>5.4 Exercises 279<br/>6 Comparing Dependence 285<br/>6.1 Introduction 285<br/>6.2 Comparing Dependence in the Bivariate Case Using the Correlation Order 287<br/>6.2.1 Definition 287<br/>6.2.2 Relationship with orthant orders 288<br/>6.2.3 Relationship with positive quadrant dependence 289<br/>6.2.4 Characterizations in terms of supermodular functions 289<br/>6.2.5 Extremal elements 290<br/>6.2.6 Relationship with convex and stop-loss orders 290<br/>6.2.7 Correlation order and copulas 292<br/>6.2.8 Correlation order and correlation coefficients 292<br/>6.2.9 Ordering Archimedean copulas 292<br/>6.2.10 Ordering compound sums 293<br/>6.2.11 Correlation order and diversification benefit 294</p><p>6.3 Comparing Dependence in the Multivariate Case Using<br/>the Supermodular Order 295<br/>6.3.1 Definition 295<br/>6.3.2 Smooth supermodular functions 296<br/>6.3.3 Restriction to distributions with identical marginals 296<br/>6.3.4 A companion order: The symmetric supermodular order 297<br/>6.3.5 Relationships between supermodular-type orders 297<br/>6.3.6 Supermodular order and dependence measures 297<br/>6.3.7 Extremal dependence structures in the supermodular sense 298<br/>6.3.8 Supermodular, stop-loss and convex orders 298<br/>6.3.9 Ordering compound sums 299<br/>6.3.10 Ordering random vectors with common values 300<br/>6.3.11 Stochastic analysis of duplicates in life insurance portfolios 302<br/>6.4 Positive Orthant Dependence Order 304<br/>6.4.1 Definition 304<br/>6.4.2 Positive orthant dependence order and correlation coefficients 304<br/>6.5 Exercises 305<br/>PART III APPLICATIONS TO INSURANCE MATHEMATICS 309<br/>7 Dependence in Credibility Models Based on Generalized Linear Models 311<br/>7.1 Introduction 311<br/>7.2 Poisson Credibility Models for Claim Frequencies 312<br/>7.2.1 Poisson static credibility model 312<br/>7.2.2 Poisson dynamic credibility models 315<br/>7.2.3 Association 316<br/>7.2.4 Dependence by mixture and common mixture models 320<br/>7.2.5 Dependence in the Poisson static credibility model 323<br/>7.2.6 Dependence in the Poisson dynamic credibility models 325<br/>7.3 More Results for the Static Credibility Model 329<br/>7.3.1 Generalized linear models and generalized additive models 329<br/>7.3.2 Some examples of interest to actuaries 330<br/>7.3.3 Credibility theory and generalized linear mixed models 331<br/>7.3.4 Exhaustive summary of past claims 332<br/>7.3.5 A posteriori distribution of the random effects 333<br/>7.3.6 Predictive distributions 334<br/>7.3.7 Linear credibility premium 334<br/>7.4 More Results for the Dynamic Credibility Models 339<br/>7.4.1 Dynamic credibility models and generalized linear mixed models 339<br/>7.4.2 Dependence in GLMM-based credibility models 340<br/>7.4.3 A posteriori distribution of the random effects 341<br/>7.4.4 Supermodular comparisons 342<br/>7.4.5 Predictive distributions 343<br/>7.5 On the Dependence Induced by Bonus–Malus Scales 344<br/>7.5.1 Experience rating in motor insurance 344<br/>7.5.2 Markov models for bonus–malus system scales 344<br/>7.5.3 Positive dependence in bonus–malus scales 345</p><p>7.6 Credibility Theory and Time Series for Non-Normal Data 346<br/>7.6.1 The classical actuarial point of view 346<br/>7.6.2 Time series models built from copulas 346<br/>7.6.3 Markov models for random effects 348<br/>7.6.4 Dependence induced by autoregressive copula models<br/>in dynamic frequency credibility models 349<br/>7.7 Exercises 350<br/>8 Stochastic Bounds on Functions of Dependent Risks 355<br/>8.1 Introduction 355<br/>8.2 Comparing Risks With Fixed Dependence Structure 357<br/>8.2.1 The problem 357<br/>8.2.2 Ordering random vectors with fixed dependence structure with<br/>stochastic dominance 358<br/>8.2.3 Ordering random vectors with fixed dependence structure with<br/>convex order 358<br/>8.3 Stop-Loss Bounds on Functions of Dependent Risks 360<br/>8.3.1 Known marginals 360<br/>8.3.2 Unknown marginals 360<br/>8.4 Stochastic Bounds on Functions of Dependent Risks 363<br/>8.4.1 Stochastic bounds on the sum of two risks 363<br/>8.4.2 Stochastic bounds on the sum of several risks 365<br/>8.4.3 Improvement of the bounds on sums of risks under positive<br/>dependence 367<br/>8.4.4 Stochastic bounds on functions of two risks 368<br/>8.4.5 Improvements of the bounds on functions of risks under positive<br/>quadrant dependence 370<br/>8.4.6 Stochastic bounds on functions of several risks 370<br/>8.4.7 Improvement of the bounds on functions of risks under positive<br/>orthant dependence 371<br/>8.4.8 The case of partially specified marginals 372<br/>8.5 Some Financial Applications 375<br/>8.5.1 Stochastic bounds on present values 375<br/>8.5.2 Stochastic annuities 376<br/>8.5.3 Life insurance 379<br/>8.6 Exercises 382<br/>9 Integral Orderings and Probability Metrics 385<br/>9.1 Introduction 385<br/>9.2 Integral Stochastic Orderings 386<br/>9.2.1 Definition 386<br/>9.2.2 Properties 386<br/>9.3 Integral Probability Metrics 388<br/>9.3.1 Probability metrics 388<br/>9.3.2 Simple probability metrics 389<br/>9.3.3 Integral probability metrics 389<br/>9.3.4 Ideal metrics 390<br/>9.3.5 Minimal metric 392<br/>9.3.6 Integral orders and metrics 392<br/>9.4 Total-Variation Distance 393<br/>9.4.1 Definition 393<br/>9.4.2 Total-variation distance and integral metrics 394<br/>9.4.3 Comonotonicity and total-variation distance 395<br/>9.4.4 Maximal coupling and total-variation distance 396<br/>9.5 Kolmogorov Distance 396<br/>9.5.1 Definition 396<br/>9.5.2 Stochastic dominance, Kolmogorov and total-variation distances 397<br/>9.5.3 Kolmogorov distance under single crossing condition<br/>for probability density functions 397<br/>9.6 Wasserstein Distance 398<br/>9.6.1 Definition 398<br/>9.6.2 Properties 399<br/>9.6.3 Comonotonicity and Wasserstein distance 400<br/>9.7 Stop-Loss Distance 401<br/>9.7.1 Definition 401<br/>9.7.2 Stop-loss order, stop-loss and Wasserstein distances 401<br/>9.7.3 Computation of the stop-loss distance under stochastic<br/>dominance or dangerousness order 401<br/>9.8 Integrated Stop-Loss Distance 403<br/>9.8.1 Definition 403<br/>9.8.2 Properties 403<br/>9.8.3 Integrated stop-loss distance and positive quadrant dependence 405<br/>9.8.4 Integrated stop-loss distance and cumulative dependence 405<br/>9.9 Distance Between the Individual and Collective Models in Risk Theory 407<br/>9.9.1 Individual model 407<br/>9.9.2 Collective model 407<br/>9.9.3 Distance between compound sums 408<br/>9.9.4 Distance between the individual and collective models 410<br/>9.9.5 Quasi-homogeneous portfolios 412<br/>9.9.6 Correlated risks in the individual model 414<br/>9.10 Compound Poisson Approximation for a Portfolio of Dependent Risks 414<br/>9.10.1 Poisson approximation 414<br/>9.10.2 Dependence in the quasi-homogeneous individual model 418<br/>9.11 Exercises 421<br/>References 423<br/>Index 439</p><p><br/> <br/></p>

[此贴子已经被作者于2008-10-20 8:51:54编辑过]