Risk and insurance

Preface......Page 6
Contents......Page 9
  Special Typeface......Page 12
  Standard Distributions......Page 13
  Abbreviations......Page 14
 1 Actuarial Versus Financial Pricing......Page 15
 2 Utility......Page 19
 3 Premium Rules......Page 26
 4 Reinsurance......Page 28
  5.1 The Poisson Distribution as a Binomial Limit......Page 30
  5.2 The Poisson Process......Page 31
  5.3 Superposition and Thinning......Page 34
 1 Bayes and Empirical Bayes......Page 38
  1.1 The Bayes Premium......Page 41
 2 Exponential Families and Conjugate Priors......Page 43
 3 Credibility Premiums......Page 46
  3.1 The Simple Credibility Model......Page 47
  3.2 The General Credibility Model......Page 50
  3.3 The Bühlmann Model......Page 51
  3.4 The Bühlmann–Straub Model......Page 52
  3.5 Quadratic Loss......Page 54
  4.1 Introduction......Page 56
  4.2 Loimaranta Efficiency......Page 58
  4.3 Bayes Premiums......Page 60
 1 Introduction......Page 63
  2.1 Definition of Subexponentiality and Sufficient Conditions......Page 68
  2.2 Further Mathematical Properties......Page 71
  3.1 The Cumulant Function......Page 76
  3.2 The Legendre–Fenchel Transform......Page 77
  3.3 Exponential Families and Change of Measure......Page 78
  3.4 The Chernoff Bound and the Saddlepoint Approximation......Page 79
 4 Tails of Sums of Light-Tailed Random Variables......Page 81
 5 Aggregate Claims and Compound Sums: Generalities......Page 85
  5.1 Poisson Compounding......Page 86
 6 Panjer's Recursion......Page 88
  7.1 Heavy Tails: The Subexponential Approximation......Page 91
  7.2 Light Tails: The Saddlepoint Approximation......Page 92
  7.3 The NP Approximation......Page 93
 1 The Cramér–Lundberg Model......Page 96
 2 First Results: Martingale Techniques......Page 99
 3 Ladder Heights. Heavy Tails......Page 101
 4 Proof of the Cramér–Lundberg Approximation......Page 104
 5 Finite Time Ruin Probabilities......Page 107
  5.1 Finite Horizon Ruin with Light Tails......Page 108
  5.2 Finite Horizon Ruin with Heavy Tails......Page 109
 6 Markov Regime Switching......Page 112
  6.1 The Averaged Model......Page 113
  6.2 The Matrix m.g.f.......Page 114
  6.3 Cramér–Lundberg Theory......Page 116
 7 Level-Dependent Premiums......Page 117
 8 The Diffusion Approximation......Page 120
 1 The Contract Payments and the Probability Model......Page 123
  2.1 Mortality Modelling......Page 124
  2.2 The Survival Model......Page 127
  2.3 The Disability Model......Page 130
  2.4 The Spouse Model......Page 133
 3 Valuation of the Payments......Page 135
  4.1 The Survival Model......Page 144
  4.2 The Disability Model......Page 146
Chapter VI: Financial Mathematics in Life Insurance......Page 150
 1 Background and Simple Claims......Page 151
 2 Payment Streams......Page 160
 3 Unit-Link Insurance......Page 164
 4 With-Profit Insurance and the Dynamics of the Surplus......Page 174
 5 Cash Dividends and Market Reserve......Page 180
 6 The Pure Case of Cash Dividends......Page 182
 7 Bonus Payments and Market Reserve......Page 184
 8 The Pure Case of Bonus Payments......Page 190
 9 Comparison of Products......Page 192
 1 Duration-Dependent Intensities and Payments......Page 197
 2 Reserve-Dependent Payments and Intensities......Page 201
 3 Bonds and Forward Interest Rates......Page 207
 4 Survival Probabilities and Forward Mortality Rates......Page 210
 5 Dependent Interest and Mortality Rates......Page 216
 6 Stochastic Interest and Mortality Rate Models......Page 219
 7 Reserves Revisited......Page 227
 8 Incidental Policy Holder Behavior......Page 229
 9 Rational Policy Holder Behavior......Page 237
 10 Higher Order Moments. Hattendorf's Theorem......Page 247
 1 Stochastic Ordering of Risks......Page 254
  2.1 Preliminaries on Stop-Loss Transforms and Convexity......Page 256
  2.2 Convex Ordering......Page 258
  2.3 Increasing Convex Ordering......Page 264
 3 Closure Properties of Orderings......Page 266
  4.1 Underinsurance and Coinsurance......Page 268
  4.2 Utility and Reinsurance......Page 270
  4.3 Local Versus Global Reinsurance......Page 271
  4.4 Optimal Compensation Functions......Page 272
 5 Applications to Ruin Theory......Page 275
 6 Maximizing the Adjustment Coefficient......Page 276
 1 Introduction......Page 281
 2 Elementary Examples and Considerations......Page 282
  2.1 The Exceedance Time......Page 285
 3 Convergence Results......Page 286
  3.1 MDA(Gumbel)......Page 288
  3.3 MDA(Weibull)......Page 291
  3.4 Further Remarks and Examples......Page 292
 4 Proof of the Fisher–Tippett Theorem......Page 296
 5 Records......Page 298
Chapter X: Dependence and Further Topics in Risk Management......Page 304
  1.1 General Theory......Page 305
  1.2 Value-at-Risk and Expected Shortfall......Page 307
  1.3 Further Remarks......Page 310
  1.4 Quotes on Risk Diversification......Page 311
 2 The Fréchet–Höffding Bounds. Comonotonicity......Page 313
  2.1 Applications of Comonotonicity......Page 316
 3 Special Dependence Structures......Page 318
  3.2 Normal Mixtures......Page 319
  3.3 Spherical and Elliptical Distributions......Page 320
  3.4 Multivariate Regular Variation......Page 321
 4 Copulas......Page 325
  4.1 Archimedean Copulas......Page 326
  4.2 Extreme Value Copulas......Page 328
  4.4 Further Examples......Page 329
  5.1 Linear (Pearson) Correlation......Page 332
  5.2 Kendall's Tau and Spearman's Rho......Page 334
  5.3 Empirical Versions......Page 335
  6.1 Association......Page 337
  6.2 Tail Dependence......Page 338
 7 Tails of Sums of Dependent Risks......Page 341
  7.1 Bounds......Page 342
  7.2 A Copula Representation......Page 343
  7.3 Asymptotics......Page 344
 8 Dependence Orderings......Page 347
  8.1 Orthant Dependencies and Orderings......Page 348
  8.2 The Supermodular Ordering......Page 351
  8.3 The Multivariate Normal Distribution......Page 355
 1 Introduction......Page 359
  2.1 General Theory......Page 363
  2.2 Reinsurance Control......Page 366
  2.3 Investment Control......Page 368
  2.4 Premium Control......Page 369
 3 The Hamilton–Jacobi–Bellman Equation......Page 370
 4 Optimal Dividends......Page 374
 5 Control Problems for the Cramér–Lundberg Model......Page 377
  5.1 The Generator and the HJB Equation......Page 378
  5.2 Ruin Probability Minimization......Page 379
  5.3 Optimal Investment......Page 383
 6 Examples Involving Game Theory......Page 385
  6.1 An Optimality Property of Stop-Loss Reinsurance......Page 387
  6.2 Premiums Based on Customer Preferences......Page 388
Chapter XII: Stochastic Control in Life Insurance......Page 390
 1 The Diffusion Approximation......Page 391
 2 Finite-State Markov Process Linear Regulation......Page 397
 3 The Consumption-Investment Problem......Page 403
 4 Uncertain Lifetime......Page 411
 5 The Consumption-Investment-Insurance Problem......Page 416
 6 The Multi-State Problem......Page 420
  6.1 The Survival Model......Page 428
 7 The Pension Fund's Problem......Page 429
  7.1 The Survival Model......Page 433
 1 Claims Reserving......Page 436
 2 Multivariate Extreme Value Theory......Page 442
 3 Statistical Methods for Tails and Extreme Values......Page 445
  3.2 The Hill Estimator......Page 446
  3.4 Block Maxima......Page 448
  4.1 The Large Deviations Principle. One-Dimensional Cramér......Page 449
  4.2 Schilder and Mogulski......Page 452
  4.3 Implementation and Examples......Page 453
  4.4 LD Results for SDEs......Page 457
 5 Gaussian Maxima......Page 460
  5.1 Borell's Inequality......Page 462
  5.2 Slepian's Inequality......Page 465
 A.1 Integral Formulas......Page 467
  A.2.1 The One-Dimensional Case......Page 469
  A.2.2 Systems of ODEs......Page 470
 A.3 Inhomogeneous Markov Processes......Page 473
  A.3.1 Feed-Forward Systems......Page 476
 A.4 Itô's Formula......Page 477
 A.5 Diffusion First Passage Probabilities......Page 479
  A.6.1 Least Squares......Page 480
  A.6.2 Conditional Expectations......Page 481
  A.7.2 Mill's Ratio......Page 483
  A.7.3 A PDE for the Multivariate Normal Density......Page 484
 A.8 Generalized Inverses......Page 486
 A.9 The Distributional Transform......Page 489
 A.10 Types of Distributions......Page 490
 A.11 Transforms......Page 492
References......Page 494
Index......Page 501