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Contents
Flag-signs in the margins designate a route: either Route 1 (which is entirely contained in
Route 2), or Route 2 (which is entirely contained in Route 3), or Route 3; see the Preface
and Introduction for a more detailed description of these routes. A new flag-sign is posted
only at the moment of a route change.
Introduction 1 1,2,3
Chapter 0. Preliminary Facts from Probability and Interest 7
1 Probability and Random Variables . . . . . . . . . . . . . . . . . . . . . . 7
2 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Some Basic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Convergence of Random Variables and Distributions . . . . . . . . . . . . 44
6 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7 Conditional Expectations. Conditioning . . . . . . . . . . . . . . . . . . . 49
8 Elements of the Theory of Interest . . . . . . . . . . . . . . . . . . . . . . 61
9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter 1. Comparison of Random Variables. Preferences of Individuals 69
2 Comparison of R.V.’s and Limit Theorems . . . . . . . . . . . . . . . . . . 86
3 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Non-Linear Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Optimal Payment from the Standpoint of an Insured . . . . . . . . . . . . . 129
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Chapter 2. An Individual Risk Model for a Short Period 141
1 The Distribution of an Individual Payment . . . . . . . . . . . . . . . . . . 141
2 The Aggregate Payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3 Premiums and Solvency. Approximations for Aggregate Claim Distributions
4 Some General Premium Principles . . . . . . . . . . . . . . . . . . . . . . 185
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
Chapter 3. A Collective Risk Model for a Short Period 199
1 Three Basic Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
2 Counting or Frequency Distributions . . . . . . . . . . . . . . . . . . . . . 201
3 The Distribution of the Aggregate Claim . . . . . . . . . . . . . . . . . . . 215
4 Premiums and Solvency. Normal Approximation . . . . . . . . . . . . . . 228
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Chapter 4. Random Processes and their Applications I 243
2 Poisson and Other Counting Processes . . . . . . . . . . . . . . . . . . . . 250
3 Compound Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 1,2,3
4 Markov Chains. Cash Flows in the Markov Environment . . . . . . . . . . 263
5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Chapter 5. Random Processes and their Applications II 303
2,3
1 Brownian Motion and Its Generalizations . . . . . . . . . . . . . . . . . . 303
2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Contents xv
Chapter 6. Global Characteristics of the Surplus Process 339 1,2,3 1 A General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
2 Ruin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
3 Criteria Connected with Paying Dividends . . . . . . . . . . . . . . . . . . 380
4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Chapter 7. Survival Distributions 391 1,2,3
1 The Probability Distribution of Lifetime . . . . . . . . . . . . . . . . . . . 391
2 A Multiple Decrement Model . . . . . . . . . . . . . . . . . . . . . . . . . 411
3 Multiple Life Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Chapter 8. Life Insurance Models 437
1,2,3
1 A General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
2 Some Particular Types of Contracts . . . . . . . . . . . . . . . . . . . . . . 443
3 Varying Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
4 Multiple Decrement and Multiple Life Models . . . . . . . . . . . . . . . . 461
5 On the Actuarial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
1,2,3
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467Contents xvii
Chapter 9. Annuity Models 473
1 Two Approaches to the Evaluation of Annuities . . . . . . . . . . . . . . . 473
2 Level Annuities. A Connection with Insurance . . . . . . . . . . . . . . . . 478
3 Some Particular Types of Level Annuities . . . . . . . . . . . . . . . . . . 480
4 More on Varying Payments . . . . . . . . . . . . . . . . . . . . . . . . . . 491
6 Multiple Decrement and Multiple Life Models . . . . . . . . . . . . . . . . 495
7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Chapter 10. Premiums and Reserves 505 1,2,3 1 Premium Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
1.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . . 505
2 Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
Chapter 11. Pensions Plans 533
1 Valuation of Individual Pension Plans . . . . . . . . . . . . . . . . . . . . 533
2 Pension Funding. Cost Methods . . . . . . . . . . . . . . . . . . . . . . . 541
3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
Chapter 12. Risk Exchange: Reinsurance and Coinsurance 553
1 Reinsurance from the Standpoint of a Cedent . . . . . . . . . . . . . . . . 553
2 Risk Exchange and Reciprocity of Companies . . . . . . . . . . . . . . . . 566
3
3 Reinsurance Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
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