Actuarial Mathematics

Bowers.pdf......Page 0
Portada......Page 1
Preface......Page 2
Table of Contents......Page 3
Author\'s Biographies......Page 11
Author\'s Introductions And Guide to Study......Page 12
Introduction to Second Edition......Page 16
Guide to Study......Page 17
1.1 Introduction......Page 22
1.2 Utility Theory......Page 24
1.3 Insurance and Utility......Page 28
1.4 Elements of Insurance......Page 36
1.5 Optimal Insurance......Page 37
1.6 Notes and References......Page 39
Appendix......Page 40
Exercises......Page 41
2.1 Introduction......Page 47
2.2 Models for Individual Claim Random Variables......Page 48
2.3 Sums of Independent Random Variables......Page 54
2.4 Approximations for the Distribution of the Sum......Page 59
2.5 Applications to Insurance......Page 60
2.6 Notes and References......Page 66
Exercises......Page 67
3.1 Introduction......Page 70
3.2.2 Time-until-Death for a Person Age X......Page 71
3.2.3 Curtate-Future-Lifetimes......Page 73
3.2.4 Force of Mortality......Page 74
3.3.1 Relation of Life Table Functions to the Survival Function......Page 77
3.3.2 Life Table Example......Page 78
3.4 The Deterministic Survivorship Group......Page 85
3.5.1 Characteristics......Page 87
3.5.2 Recursion Formulas......Page 92
3.6 Assumptions for Fractional Ages......Page 93
3.7 Some Analytical Laws of Mortality......Page 96
3.8 Select and Ultimate Tables......Page 98
3.9 Notes and References......Page 102
Exercises......Page 103
4.1 Introduction......Page 111
4.2.1 Level Benefit Insurance......Page 112
4.2.2 Endowment Insurance......Page 119
4.2.3 Deferred Insurance......Page 121
4.3 Insurances Payable at the End of the Year of Death......Page 126
4.4 Relationships between Insurances Payable at the Moment of Death and the End of the Year of Death......Page 137
4.5 Differencial Equations for Insurances Payable at the Moment of Death......Page 143
Exercises......Page 144
5.1 Introduction......Page 151
5.2 Continuous Life Annuities......Page 152
5.3 Discrete Life Annuities......Page 161
5.4 Life Annuities with m- thly Payments......Page 167
5.5 Apportionable Annuities-Due and Complete Annuities-Immediate......Page 172
5.6 Notes and References......Page 175
Exercises......Page 176
6.1 Introduction......Page 185
6.2 Fully Continuous Premiums......Page 188
6.3 Fully Discrete Premiums......Page 198
6.4 True m-thly Payment Premiums......Page 206
6.5 Apportionable Premiums......Page 209
6.6 Accumulation-Type Benefits......Page 212
Exercises......Page 215
7.1 Introduction......Page 221
7.2 Fully Continuous Benefit Reserves......Page 224
7.3 Other Formulas for Fully Continuous Benefit Reserves......Page 230
7.4 Fully Discrete Benefit Reserves......Page 233
7.6 Benefit Reserves Based on True m-thly Benefit Premiums......Page 240
7.7 Benefit Reserves on an Apportionable or Discounted Continuous Basis......Page 243
Exercises......Page 244
8.1 Introduction......Page 248
8.2 Benefit Reserves for General Insurances......Page 249
8.3 Recursion Relations for Fully Discrete Benefit Reserves......Page 252
8.4 Benefit Reserves at Fractional Durations......Page 257
8.5 Allocation of the Risk to Insurance Years......Page 260
8.6 Differential Equations for Fully Continuous Benefit Reserves......Page 267
Exercises......Page 269
9.1 Introduction......Page 276
9.2 Joint Distributions of Future Lifetimes......Page 277
9.3 The Joint-Life Status......Page 282
9.4 The Last-Survivor Status......Page 289
9.5 More Probabilities and Expectations......Page 292
9.6.1 Common Shock......Page 295
9.6.2 Copulas......Page 298
9.7.1 Survival Statuses......Page 300
9.7.2 Special Two-Life Annuities......Page 305
9.7.3 Reversionary Annuities......Page 306
9.8.1 Gompertz  and Makeham Laws......Page 308
9.8.2 Uniform Distribution......Page 309
9.9 Simple Contingent Functions......Page 312
9.10 Evaluation- Simple Contingent Functions......Page 316
9.11 Notes and References......Page 318
Exercises......Page 319
10.1 Introduction......Page 327
10.2 Two Random Variables......Page 328
10.3 Random Survivorship Group......Page 336
10.4 Deterministic Survivorship Group......Page 338
10.5 Associated Single Decrement Tables......Page 339
10.5.1 Basic Relationships......Page 340
10.5.2 Central Rates of Multiple Decrement......Page 341
10.5.3 Constant Force Assumption for Multiple Decrements......Page 342
10.5.4 Uniform Distribution Assumption for Multiple Decrements......Page 343
10.5.5 Estimation Issues......Page 344
10.6 Construction of a Multiple Decrement Table......Page 348
10.7 Notes and References......Page 353
Exercises......Page 354
11.1 Introduction......Page 362
11.2 Actuarial Present Values and Their Numerical Evaluation......Page 363
11.3 Benefit Premiums and Reserves......Page 366
11.4 Withdrawal Beneft Patterns That Can Be Ignored in Evaluating Premiums and Reserves......Page 367
11.5.1 Demographic Assumptions......Page 371
11.5.2 Projecting Benefit Payment and Contribution Rates......Page 372
11.5.3 Defined - Benefit Plans......Page 375
11.5.4 Defined - Contribution Plans......Page 377
11.6.1 Disability Income Benefits......Page 379
11.6.2 Waiver-of-Premium Benefits......Page 380
11.6.3 Benefit Premiums and Reserves......Page 381
11.7 Notes and References......Page 382
Exercises......Page 383
12.1 Introduction......Page 388
12.2 The Distribution of Aggregate Claims......Page 389
12.3.1 The Distribution of N......Page 393
12.3.2 The Individual Claim amount Distribution......Page 398
12.4 Properties of Certain Compound Distributions......Page 399
12.5 Approximations to the Distribution of Aggregate Claims......Page 406
12.6 Notes and References......Page 411
Appendix......Page 412
Exercises......Page 414
13.1 Introduction......Page 419
13.2 A Discrete Time Model......Page 421
13.3 A Continuous Time Model......Page 426
13.4 Ruin Probabilities and the Claim Amount Distribution......Page 429
13.5 The First Surplus below the Initial Level......Page 435
13.6 The Maximal Aggregate Loss......Page 437
13.7 Notes and References......Page 443
Appendix......Page 445
Exercises......Page 450
14.1 Introduction......Page 455
14.2 Claim Amount Distributions......Page 456
14.3 Approximating the Individual Model......Page 461
14.4 Stop-Loss Reinsurance......Page 465
14.5 Analysis of Reinsurance Using Ruin Theory......Page 471
14.6 Notes and References......Page 479
Appendix......Page 480
Exercises......Page 481
15.1 Introduction......Page 485
15.2.1 Premiums and Reserves......Page 486
15.2.2 Accounting......Page 489
15.3.1 Premiums and Reserves......Page 492
15.4 Types of Expenses......Page 496
15.5 Algebraic Foundations of Accounting: Single Decrement Model......Page 501
15.6.1 Recursion Relations......Page 505
15.6.2 Accounting......Page 507
15.7 Expenses, Reserves, and General Insurances......Page 509
15.8 Notes and References......Page 511
Exercises......Page 512
16.2 Cash Values......Page 518
16.3.1 Paid-up Insurance......Page 521
16.3.2 Extended Term......Page 523
16.3.3 Automatic Premium Loan......Page 525
16.4 Premiums and Economic Considerations......Page 526
16.4.2 Fund Objective......Page 527
16.4.3 Rate of Return Objective......Page 528
16.4.4 Risk-Based Objectives......Page 530
16.5 Experience Adjustments......Page 531
16.6 Modified Reserve Methods......Page 534
16.7 Full Preliminary Term......Page 538
16.8 Modified Preliminary Term......Page 540
16.9.1 Valuation......Page 542
16.9.2 Cash Values......Page 545
16.10 Notes and References......Page 547
Exercises......Page 548
17.2 Special Types of Annuity Benefits......Page 554
17.3 Family Income Insurances......Page 556
17.4.1 Variable Annuity......Page 558
17.4.3 Fixed Premium Variable Life Insurance......Page 560
17.4.4 Paid-up Insurance Increments......Page 561
17.5.1 Flexible Plan Illustration......Page 562
17.5.2 An Alternative Design......Page 565
17.6 Accelerated Benefits......Page 566
17.6.1 Lump Sum Benefits......Page 567
17.6.2 Income Benefits......Page 569
Exercises......Page 570
18.1 Introduction......Page 574
18.2 More General Statuses......Page 575
18.3 Compound Statuses......Page 581
18.4 Contingent Probabilities and Insurances......Page 583
18.5 Compound Contingent Functions......Page 585
18.6 More Reversionary Annuities......Page 589
18.7 Benefit Premiums and Reserves......Page 592
18.8 Notes and References......Page 594
Appendix......Page 596
Exercises......Page 597
19.2 The Lexis Diagram......Page 604
19.3 A Continuous Model......Page 606
19.4 Stationary and Stable Populations......Page 612
19.5 Acturarial Applications......Page 614
19.6 Population Dynamics......Page 618
Exercises......Page 622
20.1 Introduction......Page 626
20.2 The Model......Page 627
20.3 Terminal Funding......Page 628
20.4.2 Benefit Payment Rate......Page 630
20.4.3 The Allocation Equation......Page 631
20.5 Accrual of Actuarial Liability......Page 633
20.6.1 Acturarial Present Value of Future Benefits......Page 634
20.6.2 Normal Cost Rate......Page 635
20.6.3 Acturarial Accrued Liability......Page 637
20.6.4 Actuarial Present Value of Future Normal Costs......Page 638
20.7 Individual Acturarial Cost Methods......Page 641
20.8 Group Acturarial Cost Methods......Page 643
20.9 Basic Functions for Active and Retired Members Combined......Page 647
Exercises......Page 649
21.1 Introduction......Page 654
21.1.2 Notation and Preliminaries......Page 656
21.2.1 Deterministic Scenarios......Page 657
21.2.2 Random Scenarios: Deterministic Interest Rates......Page 660
21.3 Independent Interest Rates......Page 662
21.4.1 Moving Average Model......Page 668
21.4.2 Implementation......Page 673
21.5.1 Information in Prices and Maturities......Page 674
21.5.2 Stochastic Models......Page 678
21.6.1 Immunization......Page 682
21.6.2 General Stochastic Model......Page 684
21.7 Notes and References......Page 685
Exercises......Page 686
Appendix 1 Normal Distribution Table......Page 692
Appendix 2A Illustrative Life Table......Page 694
Appendix 2B Illustrative Service Table......Page 703
Appendix 3 Symbol Index......Page 705
Appendix 4 General Rules for Symbols of Actuarial Functions......Page 710
Appendix 5 Some Mathematical Formulas Useful in Actuarial Mathematics......Page 715
Appendix 6 Bibliography......Page 719
Appendix 7 Answers to Exercises......Page 732
Index......Page 762