Actuarial Theory for Dependent Risks: Measures, Orders and Models

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 ageing 166

3.4.7 Stochastic (increasing) convexity 167

3.4.8 Ordering mixtures 169

3.4.9 Ordering compound sums 169

3.4.10 Risk-reshaping contracts and Lorenz order 169

3.4.11 Majorization 171

3.4.12 Conditional stop-loss order: Mean-excess order 173

3.4.13 Comparing shortfall with the stop-loss order: Right-spread order 175

3.4.14 Multivariate extensions 178

3.5 Exercises 182

PART II DEPENDENCE BETWEEN RISKS 189

4 Modelling Dependence 191

4.1 Introduction 191

4.2 Sklar’s Representation Theorem 194

4.2.1 Copulas 194

4.2.2 Sklar’s theorem for continuous marginals 194

4.2.3 Conditional distributions derived from copulas 198

4.2.4 Probability density functions associated with copulas 201

4.2.5 Copulas with singular components 201

4.2.6 Sklar’s representation in the general case 203

4.3 Families of Bivariate Copulas 204

4.3.1 Clayton’s copula 205

4.3.2 Frank’s copula 205

4.3.3 The normal copula 207

4.3.4 The Student copula 208

4.3.5 Building multivariate distributions with given marginal from copulas 210

4.4 Properties of Copulas 213

4.4.1 Survival copulas 213

4.4.2 Dual and co-copulas 215

4.4.3 Functional invariance 216

4.4.4 Tail dependence 217

4.5 The Archimedean Family of Copulas 218

4.5.1 Definition 218

4.5.2 Frailty models 219

4.5.3 Probability density function associated with Archimedean copulas 220

4.5.4 Properties of Archimedean copulas 221

4.6 Simulation from Given Marginals and Copula 223

4.6.1 General method 223

4.6.2 Exploiting Sklar’s decomposition 224

4.6.3 Simulation from Archimedean copulas 224

4.7 Multivariate Copulas 225

4.7.1 Definition 225

4.7.2 Sklar’s representation theorem 225

4.7.3 Functional invariance 226

4.7.4 Examples of multivariate copulas 226

4.7.5 Multivariate Archimedean copulas 229

4.8 Loss–Alae Modelling with Archimedean Copulas: A Case Study 231

4.8.1 Losses and their associated ALAEs 231

4.8.2 Presentation of the ISO data set 231

4.8.3 Fitting parametric copula models to data 232

4.8.4 Selecting the generator for Archimedean copula models 234

4.8.5 Application to loss–ALAE modelling 238

4.9 Exercises 242

5 Measuring Dependence 245

5.1 Introduction 245

5.2 Concordance Measures 246

5.2.1 Definition 246

5.2.2 Pearson’s correlation coefficient 247

5.2.3 Kendall’s rank correlation coefficient 253

5.2.4 Spearman’s rank correlation coefficient 257

5.2.5 Relationships between Kendall’s and Spearman’s rank correlation coefficients 259

5.2.6 Other dependence measures 260

5.2.7 Constraints on concordance measures in bivariate discrete data 262

5.3 Dependence Structures 264

5.3.1 Positive dependence notions 264

5.3.2 Positive quadrant dependence 265

5.3.3 Conditional increasingness in sequence 274

5.3.4 Multivariate total positivity of order 2 276

5.4 Exercises 279

6 Comparing Dependence 285

6.1 Introduction 285

6.2 Comparing Dependence in the Bivariate Case Using the Correlation Order 287

6.2.1 Definition 287

6.2.2 Relationship with orthant orders 288

6.2.3 Relationship with positive quadrant dependence 289

6.2.4 Characterizations in terms of supermodular functions 289

6.2.5 Extremal elements 290

6.2.6 Relationship with convex and stop-loss orders 290

6.2.7 Correlation order and copulas 292

6.2.8 Correlation order and correlation coefficients 292

6.2.9 Ordering Archimedean copulas 292

6.2.10 Ordering compound sums 293

6.2.11 Correlation order and diversification benefit 294

6.3 Comparing Dependence in the Multivariate Case Using the Supermodular Order 295

6.3.1 Definition 295

6.3.2 Smooth supermodular functions 296

6.3.3 Restriction to distributions with identical marginals 296

6.3.4 A companion order: The symmetric supermodular order 297

6.3.5 Relationships between supermodular-type orders 297

6.3.6 Supermodular order and dependence measures 297

6.3.7 Extremal dependence structures in the supermodular sense 298

6.3.8 Supermodular, stop-loss and convex orders 298

6.3.9 Ordering compound sums 299

6.3.10 Ordering random vectors with common values 300

6.3.11 Stochastic analysis of duplicates in life insurance portfolios 302

6.4 Positive Orthant Dependence Order 304

6.4.1 Definition 304

6.4.2 Positive orthant dependence order and correlation coefficients 304

6.5 Exercises 305

PART III APPLICATIONS TO INSURANCE MATHEMATICS 309

7 Dependence in Credibility Models Based on Generalized Linear Models 311

7.1 Introduction 311

7.2 Poisson Credibility Models for Claim Frequencies 312

7.2.1 Poisson static credibility model 312

7.2.2 Poisson dynamic credibility models 315

7.2.3 Association 316

7.2.4 Dependence by mixture and common mixture models 320

7.2.5 Dependence in the Poisson static credibility model 323

7.2.6 Dependence in the Poisson dynamic credibility models 325

7.3 More Results for the Static Credibility Model 329

7.3.1 Generalized linear models and generalized additive models 329

7.3.2 Some examples of interest to actuaries 330

7.3.3 Credibility theory and generalized linear mixed models 331

7.3.4 Exhaustive summary of past claims 332

7.3.5 A posteriori distribution of the random effects 333

7.3.6 Predictive distributions 334

7.3.7 Linear credibility premium 334

7.4 More Results for the Dynamic Credibility Models 339

7.4.1 Dynamic credibility models and generalized linear mixed models 339

7.4.2 Dependence in GLMM-based credibility models 340

7.4.3 A posteriori distribution of the random effects 341

7.4.4 Supermodular comparisons 342

7.4.5 Predictive distributions 343

7.5 On the Dependence Induced by Bonus–Malus Scales 344

7.5.1 Experience rating in motor insurance 344

7.5.2 Markov models for bonus–malus system scales 344

7.5.3 Positive dependence in bonus–malus scales 345

7.6 Credibility Theory and Time Series for Non-Normal Data 346

7.6.1 The classical actuarial point of view 346

7.6.2 Time series models built from copulas 346

7.6.3 Markov models for random effects 348

7.6.4 Dependence induced by autoregressive copula models in dynamic frequency credibility models 349

7.7 Exercises 350

8 Stochastic Bounds on Functions of Dependent Risks 355

8.1 Introduction 355

8.2 Comparing Risks With Fixed Dependence Structure 357

8.2.1 The problem 357

8.2.2 Ordering random vectors with fixed dependence structure with stochastic dominance 358

8.2.3 Ordering random vectors with fixed dependence structure with convex order 358

8.3 Stop-Loss Bounds on Functions of Dependent Risks 360

8.3.1 Known marginals 360

8.3.2 Unknown marginals 360

8.4 Stochastic Bounds on Functions of Dependent Risks 363

8.4.1 Stochastic bounds on the sum of two risks 363

8.4.2 Stochastic bounds on the sum of several risks 365

8.4.3 Improvement of the bounds on sums of risks under positive dependence 367

8.4.4 Stochastic bounds on functions of two risks 368

8.4.5 Improvements of the bounds on functions of risks under positive quadrant dependence 370

8.4.6 Stochastic bounds on functions of several risks 370

8.4.7 Improvement of the bounds on functions of risks under positive orthant dependence 371

8.4.8 The case of partially specified marginals 372

8.5 Some Financial Applications 375

8.5.1 Stochastic bounds on present values 375

8.5.2 Stochastic annuities 376

8.5.3 Life insurance 379

8.6 Exercises 382

9 Integral Orderings and Probability Metrics 385

9.1 Introduction 385

9.2 Integral Stochastic Orderings 386

9.2.1 Definition 386

9.2.2 Properties 386

9.3 Integral Probability Metrics 388

9.3.1 Probability metrics 388

9.3.2 Simple probability metrics 389

9.3.3 Integral probability metrics 389

9.3.4 Ideal metrics 390

9.3.5 Minimal metric 392

9.3.6 Integral orders and metrics 392

9.4 Total-Variation Distance 393

9.4.1 Definition 393

9.4.2 Total-variation distance and integral metrics 394

9.4.3 Comonotonicity and total-variation distance 395

9.4.4 Maximal coupling and total-variation distance 396

9.5 Kolmogorov Distance 396

9.5.1 Definition 396

9.5.2 Stochastic dominance, Kolmogorov and total-variation distances 397

9.5.3 Kolmogorov distance under single crossing condition for probability density functions 397

9.6 Wasserstein Distance 398

9.6.1 Definition 398

9.6.2 Properties 399

9.6.3 Comonotonicity and Wasserstein distance 400

9.7 Stop-Loss Distance 401

9.7.1 Definition 401

9.7.2 Stop-loss order, stop-loss and Wasserstein distances 401

9.7.3 Computation of the stop-loss distance under stochastic dominance or dangerousness order 401

9.8 Integrated Stop-Loss Distance 403

9.8.1 Definition 403

9.8.2 Properties 403

9.8.3 Integrated stop-loss distance and positive quadrant dependence 405

9.8.4 Integrated stop-loss distance and cumulative dependence 405

9.9 Distance Between the Individual and Collective Models in Risk Theory 407

9.9.1 Individual model 407

9.9.2 Collective model 407

9.9.3 Distance between compound sums 408

9.9.4 Distance between the individual and collective models 410

9.9.5 Quasi-homogeneous portfolios 412

9.9.6 Correlated risks in the individual model 414

9.10 Compound Poisson Approximation for a Portfolio of Dependent Risks 414

9.10.1 Poisson approximation 414

9.10.2 Dependence in the quasi-homogeneous individual model 418

9.11 Exercises 421

References 423

Index 439