Models for quantifying risk

Cover......Page 1
Models for Quantifying Risk, 2nd ed.......Page 2
Copyright......Page 3
Preface......Page 4
Table of Contents......Page 8
Part I: Review and Background Material......Page 18
1.1 Interest Measures......Page 20
1.2.1 Immediate Annuity......Page 24
1.2.2 Annuity-Due......Page 25
1.2.3 Continuous Annuity......Page 26
1.3.1 Immediate Annuities......Page 28
1.3.2 Annuities-Due......Page 30
1.3.3 Continuous Annuities......Page 32
1.4 Equation of Value......Page 34
2.1 Random Variables and Their Distributions......Page 36
2.1.1 Discrete Random Variables......Page 37
2.1.2 Continuous Random Variables......Page 40
2.1.4 More on Moments of Random Variables......Page 42
2.2.1 The Discrete Uniform Distribution......Page 44
2.2.3 The Negative Binomial Distribution......Page 45
2.2.5 The Poisson Distribution......Page 47
2.3.1 The Continuous Uniform Distribution......Page 48
2.3.2 The Normal Distribution......Page 49
2.3.3 The Exponential Distribution......Page 51
2.3.4 The Gamma Distribution......Page 52
2.4.1 The Discrete Case......Page 53
2.4.2 The Continuous Case......Page 56
2.5.1 The Moments of S......Page 58
2.5.2 Distributions Closed Under Convolution......Page 59
2.5.3 The Method of Convolutions......Page 61
2.6 Compound Distributions......Page 62
2.6.1 The Moments of S......Page 63
2.6.2 The Compound Poisson Distribution......Page 65
Part II: Models for Survival-Contingent Risks......Page 66
3.1 The Age-at-Failure Random Variable......Page 68
3.1.2 The Survival Distribution Function of X......Page 70
3.1.3 The Probability Density Function of X......Page 71
3.1.4 The Hazard Rate Function of X......Page 72
3.1.5 The Moments of the Age-at-Failure Random Variable X......Page 74
3.1.6 Actuarial Survival Models......Page 75
3.2.1 The Uniform Distribution......Page 77
3.2.2 The Exponential Distribution......Page 78
3.2.4 The Makeham Distribution......Page 79
3.2.6 Summary of Parametric Survival Models......Page 80
3.3 The Time-to-Failure Random Variable......Page 81
3.3.1 The Survival Distribution Function of T[sub(x)]......Page 82
3.3.2 The Cumulative Distribution Function of T[sub(x)]......Page 83
3.3.3 The Probability Density Function of T[sub(x)]......Page 84
3.3.5 Moments of the Future Lifetime Random Variable T[sub(x)]......Page 85
3.3.6 The Time-to-Failure Random Variable K[sub(x)]......Page 88
3.4 The Central Rate......Page 90
3.5 Select Survival Models......Page 92
3.6 Exercises......Page 93
4.1 Definition of the Life Table......Page 98
4.2 The Traditional Form of the Life Table......Page 100
4.3 Other Functions Derived from ℓ[sub(x)]......Page 102
4.3.1 The Force of Failure......Page 103
4.3.2 The Probability Density Function of X......Page 105
4.3.3 Conditional Probabilities and Densities......Page 107
4.3.4 The Curtate Expectation of Life......Page 111
4.3.5 The Central Rate......Page 113
4.5 Methods for Non-Integral Ages......Page 114
4.5.1 Linear Form for ℓ[sub(x+t)]......Page 117
4.5.2 Exponential Form for ℓ[sub(x+t)]......Page 121
4.5.3 Hyperbolic Form for ℓ[sub(x+t)]106......Page 0
4.5.4 Summary......Page 125
4.6 Select Life Tables......Page 126
4.7 Exercises......Page 129
Chapter 5. Contingent Payment Models (Insurance Models)......Page 138
5.1.2 The Present Value Random Variable......Page 139
5.1.3 Modifications of the Present Value Random Variable......Page 143
5.1.4 Applications to Life Insurance......Page 148
5.2 Group Deterministic Approach......Page 152
5.3.1 The Continuous Random Variable for Time to Failure......Page 155
5.3.2 The Present Value Random Variable......Page 156
5.3.4 Applications to Life Insurance......Page 158
5.3.5 Continuous Functions Evaluated from Parametric Survival Models......Page 159
5.4 Contingent Payment Models with Varying Payments......Page 162
5.5.1 Continuous Contingent Payment Models......Page 165
5.5.2 m[sup(thly)] Contingent Payment Models......Page 168
5.6 Miscellaneous Examples......Page 170
5.7 Exercises......Page 173
Chapter 6. Contingent Annuity Models (Life Annuities)......Page 178
6.1 Whole Life Annuity Models......Page 179
6.1.1 The Immediate Case......Page 180
6.1.2 The Due Case......Page 186
6.1.3 The Continuous Case......Page 188
6.2.1 The Immediate Case......Page 191
6.2.2 The Due Case......Page 196
6.2.3 The Continuous Case......Page 199
6.3.1 The Immediate Case......Page 202
6.3.2 The Due Case......Page 204
6.3.3 The Continuous Case......Page 205
6.4.1 The Immediate Case......Page 208
6.4.2 The Due Case......Page 209
6.4.3 Random Variable Analysis......Page 210
6.4.4 Numerical Evaluation in the m[sup(thly)] and Continuous Cases......Page 212
6.5 Non-Level Payment Annuity Functions......Page 214
6.6 Miscellaneous Examples......Page 215
6.7 Exercises......Page 220
Chapter 7. Funding Plans for Contingent Contracts (Annual Premiums)......Page 228
7.1.1 Discrete Contingent Payment Models......Page 229
7.1.2 Continuous Contingent Payment Models......Page 234
7.1.4 Non-Level Premium Contracts......Page 235
7.2 Random Variable Analysis......Page 236
7.3.1 Discrete Contingent Payment Models......Page 241
7.3.2 Continuous Contingent Payment Models......Page 242
7.4 Funding Schemes with m[sup(thly)] Payments......Page 245
7.5 Funding Plans Incorporating Expenses......Page 247
7.6 Miscellaneous Examples......Page 250
7.7 Exercises......Page 257
Chapter 8. Contingent Contract Reserves (Benefit Reserves)......Page 262
8.1.1 Reserves by the Prospective Method......Page 264
8.1.2 Reserves by the Retrospective Method......Page 267
8.1.3 Additional Terminal Reserve Expressions......Page 270
8.1.4 Random Variable Analysis......Page 272
8.1.5 Reserve for Contingent Contracts with Immediate Payment of Claims......Page 274
8.1.6 Reserves for Contingent Annuity Models......Page 275
8.2.1 Group Deterministic Approach......Page 276
8.2.2 Random Variable Analysis — Cash Basis......Page 280
8.2.3 Random Variable Analysis — Accrued Basis......Page 283
8.3.1 Discrete Whole Life Contingent Payment Models......Page 287
8.3.2 Continuous Whole Life Contingent Payment Models......Page 288
8.3.3 Random Variable Analysis......Page 290
8.4 Reserves for Contingent Payment Models with m[sup(thly)] Payment Funding......Page 291
8.5 Incorporation of Expenses......Page 294
8.6 Reserves at Fractional Durations......Page 295
8.7.1 Discrete Models......Page 298
8.7.2 Continuous Models......Page 301
8.8 Miscellaneous Examples......Page 303
8.9 Exercises......Page 307
9.1 The Joint-Life Model......Page 314
9.1.2 Survival Distribution Function of T[sub(xy)]......Page 315
9.1.3 Cumulative Distribution Function of T[sub(xy)]......Page 316
9.1.4 Probability Density Function of T[sub(xy)]......Page 317
9.1.6 Conditional Probabilities......Page 318
9.1.7 Moments of T[sub(xy)]......Page 320
9.2.1 The Time-to-Failure Random Variable for a Last-Survivor Status......Page 321
9.2.2 Functions of the Random Variable T[sub(x̅y̅)]......Page 322
9.2.3 Relationships Between T[sub(xy)] and T[sub(x̅y̅)]......Page 325
9.3 Contingent Probability Functions......Page 326
9.4.1 Contingent Payment Models......Page 329
9.4.2 Contingent Annuity Models......Page 331
9.4.3 Annual Premiums and Reserves......Page 332
9.4.4 Reversionary Annuities......Page 334
9.4.5 Contingent Insurance Functions......Page 336
9.5.1 Marginal Distributions of T[sub(x)] and T[sub(y)]......Page 337
9.5.2 The Covariance of T[sub(x)] and T[sub(y)]......Page 338
9.5.3 Other Joint Functions of T[sub(y)] and T[sub(y)]......Page 340
9.5.4 Joint and Last-Survivor Status Functions......Page 343
9.6 Common Shock — A Model for Lifetime Dependency......Page 345
9.7 Exercises......Page 348
10.1 Discrete Multiple-Decrement Models......Page 354
10.1.1 The Multiple-Decrement Table......Page 356
10.1.2 Random Variable Analysis......Page 359
10.2 Theory of Competing Risks......Page 361
10.3 Continuous Multiple-Decrement Models......Page 362
10.4 Uniform Distribution of Decrements......Page 366
10.4.1 Uniform Distribution in the Multiple-Decrement Context......Page 367
10.4.2 Uniform Distribution in the Associated Single-Decrement Tables......Page 369
10.5 Actuarial Present Value......Page 372
10.6 Asset Shares......Page 378
10.7.1 The Homogeneous Process......Page 381
10.7.2 The Nonhomogeneous Process......Page 388
10.8 Exercises......Page 389
Part III: Models for Non-Survival-Contingent Risks......Page 396
11.1.1 The Binomial Distribution......Page 398
11.1.2 The Poisson Distribution......Page 399
11.1.3 The Negative Binomial Distribution......Page 404
11.1.5 Summary of the Recursive Relationships......Page 408
11.2 Creation of Additional Counting Distributions......Page 409
11.2.1 Compound Frequency Models......Page 410
11.2.2 Mixture Frequency Models......Page 415
11.3.1 Properties of Counting Processes......Page 418
11.3.3 Further Properties of the Poisson Counting Process......Page 420
11.3.4 Poisson Mixture Processes......Page 423
11.3.5 The Nonstationary Poisson Counting Process......Page 424
11.4 Exercises......Page 426
Chapter 12. Claim Severity Models......Page 430
12.1.2 The Pareto Distribution......Page 431
12.2.2 Scalar Multiplication......Page 435
12.2.3 Power Operations......Page 437
12.2.4 Exponentiation......Page 440
12.2.5 Mixtures of Distributions......Page 442
12.2.6 Spliced Distributions......Page 445
12.3.1 Deductibles......Page 448
12.3.2 Policy Limits......Page 450
12.3.3 Relationships between Deductibles and Policy Limits......Page 452
12.3.4 Coinsurance Factors......Page 454
12.3.5 The Effect of Inflation......Page 455
12.4 Empirical Loss Distributions......Page 458
12.5 Exercises......Page 461
13.1 Individual Risk versus Collective Risk......Page 466
13.2.1 Frequency......Page 470
13.2.2 Severity......Page 471
13.2.3 Frequency–Severity Interaction......Page 472
13.3.1 Convolutions......Page 473
13.3.2 Stop-Loss Reinsurance......Page 478
13.3.3 The Compound Poisson Model......Page 480
13.4.2 The Poisson Distribution......Page 485
13.5 Exercises......Page 486
14.1 The Compound Poisson Process......Page 492
14.1.1 Moments of the Compound Poisson Process......Page 493
14.1.2 Other Properties of the Compound Poisson Process......Page 494
14.2 The Surplus Process Model......Page 495
14.3 The Probability of Ruin......Page 498
14.3.1 The Adjustment Coefficient......Page 499
14.3.2 The Probability of Ruin......Page 502
14.4.1 The Event of U(t) < u......Page 505
14.4.2 The Cumulative Loss of Surplus......Page 508
14.5 Probability of Ruin in Finite Time......Page 512
14.6 Exercises......Page 513
A.1 Markov Chains......Page 518
A.2 Probabilities for Multiple Steps......Page 523
A.3 Classification of States......Page 524
A.4.1 The Fundamental Matrix Q......Page 529
A.4.2 Probability of Visit to State j......Page 532
A.4.3 Probability of Absorption into State j......Page 535
A.4.4 Proof of the Footnote 4 Result......Page 536
A.4.5 Proof of the Footnote 5 Result......Page 537
A.5 Nonhomogeneous Markov Chains......Page 538
Appendix B. Review of Stochastic Simulation......Page 540
B.2 Multiplicative Congruential Random Number Generators......Page 541
B.3 The Inversion Method for Generating Simulated Outputs from Continuous Distributions......Page 543
B.3.2 The Pareto Distribution......Page 545
B.4 The Table Look-Up Method for Discrete Probability Distributions......Page 547
B.4.3 The Binomial Distribution......Page 548
B.4.4 The Negative Binomial Distribution......Page 550
B.4.5 The Poisson Distribution......Page 552
B.5 The Polar Method for Generating Simulated Normal Distribution Values......Page 555
C.1 Estimating the Probability of Ruin (Chapter 14)......Page 558
C.2 Estimating Aggregate Loss Distributions (Chapter 13)......Page 562
C.3 Estimating Multiple Decrement Results (Chapter 10)......Page 567
C.4 Estimating the Number of Months in Disability Status — A Markov Chain Approach (Chapter 10 and Appendix A)......Page 569
C.5 Miscellaneous Simulation Examples......Page 571
D.1 Downloading the Basic Life Table from the ACTEX Web Site......Page 578
D.2.1 Review of Formulas......Page 579
D.2.2 Using Microsoft Excel to Do the Above Calculations......Page 580
D.3 Calculating A[sub(x)] Using Visual Basic Macros......Page 582
D.4 Calculating [sub(n)]E[sub(x)] Using Microsoft Excel......Page 584
D.5 Calculating Second Moments and Variances......Page 586
D.6 Calculating Joint-Life Status Functions......Page 587
Chapter 3......Page 594
Chapter 4......Page 596
Chapter 5......Page 598
Chapter 7......Page 600
Chapter 8......Page 601
Chapter 9......Page 604
Chapter 10......Page 605
Chapter 11......Page 608
Chapter 13......Page 609
Chapter 14......Page 611
Bibliography......Page 614
Index......Page 616