Actuarial Mathematics for Life Contingent Risks (International Series on Actuarial Science)

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 16
Acknowledgements......Page 18
1.2 Background......Page 21
1.3.1 Introduction......Page 23
1.3.2 Traditional insurance contracts......Page 24
With-profit insurance......Page 25
1.3.3 Modern insurance contracts......Page 26
1.3.5 Underwriting......Page 28
1.3.6 Premiums......Page 30
1.3.7 Life annuities......Page 31
1.5 Pension benefits......Page 32
1.5.2 Defined benefit pension design......Page 33
1.7 Typical problems......Page 34
1.9 Exercises......Page 35
2.2 The future lifetime random variable......Page 37
2.3 The force of mortality......Page 41
2.4 Actuarial notation......Page 46
2.5 Mean and standard deviation of Tx......Page 49
2.6.1 Kx and ex......Page 52
2.6.2 The complete and curtate expected future lifetimes, ex and ex......Page 54
2.7 Notes and further reading......Page 55
2.8 Exercises......Page 56
Answers to selected exercises......Page 59
3.2 Life tables......Page 61
3.3.1 Uniform distribution of deaths......Page 64
3.3.2 Constant force of mortality......Page 68
3.4 National life tables......Page 69
3.5 Survival models for life insurance policyholders......Page 72
3.6 Life insurance underwriting......Page 74
3.7 Select and ultimate survival models......Page 76
3.8 Notation and formulae for select survival models......Page 78
3.9 Select life tables......Page 79
3.11 Exercises......Page 87
Answers to selected exercises......Page 92
4.2 Introduction......Page 93
4.3 Assumptions......Page 94
4.4.1 Whole life insurance: the continuous case, Ax......Page 95
4.4.2 Whole life insurance: the annual case, Ax......Page 98
4.4.3 Whole life insurance: the 1/mthly case, A(m)x......Page 99
4.4.4 Recursions......Page 101
4.4.5 Term insurance......Page 106
4.4.6 Pure endowment......Page 108
4.4.7 Endowment insurance......Page 109
4.4.8 Deferred insurance benefits......Page 111
4.5.1 Using the uniform distribution of deaths assumption......Page 113
4.5.2 Using the claims acceleration approach......Page 115
4.6 Variable insurance benefits......Page 116
4.8 Notes and further reading......Page 121
4.9 Exercises......Page 122
Answers to selected exercises......Page 126
5.2 Introduction......Page 127
5.4 Annual life annuities......Page 128
5.4.1 Whole life annuity-due......Page 129
5.4.2 Term annuity-due......Page 132
5.4.3 Whole life immediate annuity......Page 133
5.4.4 Term immediate annuity......Page 134
5.5.1 Whole life continuous annuity......Page 135
5.5.2 Term continuous annuity......Page 137
5.6.1 Introduction......Page 138
5.6.2 Life annuities payable m times a year......Page 139
5.6.3 Term annuities payable m times a year......Page 140
5.7 Comparison of annuities by payment frequency......Page 141
5.8 Deferred annuities......Page 143
5.9 Guaranteed annuities......Page 145
5.10.1 Arithmetically increasing annuities......Page 147
5.10.2 Geometrically increasing annuities......Page 149
5.11.1 Recursions......Page 150
5.11.2 Applying the UDD assumption......Page 151
5.11.3 Woolhouse's formula......Page 152
5.12 Numerical illustrations......Page 155
5.13 Functions for select lives......Page 156
5.15 Exercises......Page 157
Answers to selected exercises......Page 161
6.2 Preliminaries......Page 162
6.3 Assumptions......Page 163
6.4 The present value of future loss random variable......Page 165
6.5.1 Net premiums......Page 166
6.6 Gross premium calculation......Page 170
6.7 Profit......Page 174
6.8 The portfolio percentile premium principle......Page 182
6.9.2 Constant addition to µx......Page 185
6.9.3 Constant multiple of mortality rates......Page 187
6.10 Notes and further reading......Page 189
6.11 Exercises......Page 190
Answers to selected exercises......Page 194
7.3.1 The future loss random variable......Page 196
7.3.2 Policy values for policies with annual cash flows......Page 202
7.3.3 Recursive formulae for policy values......Page 211
7.3.4 Annual profit......Page 216
7.3.5 Asset shares......Page 220
7.4 Policy values for policies with cash flows at discreteintervals other than annually......Page 223
7.4.1 Recursions......Page 224
7.4.2 Valuation between premium dates......Page 225
7.5.1 Thiele's differential equation......Page 227
7.5.2 Numerical solution of Thiele's differential equation......Page 231
7.6 Policy alterations......Page 233
7.7 Retrospective policy value......Page 239
7.10 Exercises......Page 240
Answers to selected exercises......Page 248
8.2.1 The alive–dead model......Page 250
8.2.3 The permanent disability model......Page 252
8.2.4 The disability income insurance model......Page 253
8.2.5 The joint life and last survivor model......Page 254
8.3 Assumptions and notation......Page 255
8.4 Formulae for probabilities......Page 259
8.4.1 Kolmogorov's forward equations......Page 262
8.5 Numerical evaluation of probabilities......Page 263
8.6 Premiums......Page 267
8.7 Policy values and Thiele's differential equation......Page 270
8.7.1 The disability income model......Page 271
8.7.2 Thiele's differential equation – the general case......Page 275
8.8 Multiple decrement models......Page 276
8.9.1 The model and assumptions......Page 281
Notation......Page 282
8.9.3 Joint life and last survivor annuity and insurance functions......Page 284
8.9.4 An important special case: independent survival models......Page 290
8.10 Transitions at specified ages......Page 294
8.11 Notes and further reading......Page 298
8.12 Exercises......Page 299
Answers to selected exercises......Page 308
9.2 Introduction......Page 310
9.3 The salary scale function......Page 311
9.4 Setting the DC contribution......Page 314
9.5 The service table......Page 317
9.6.1 Final salary plans......Page 326
Withdrawal pension......Page 329
9.6.2 Career average earnings plans......Page 332
9.7 Funding plans......Page 334
9.9 Exercises......Page 339
Answers to selected exercises......Page 345
10.2 The yield curve......Page 346
10.3 Valuation of insurances and life annuities......Page 350
10.3.1 Replicating the cash flows of a traditional non-participating product......Page 352
10.4 Diversifiable and non-diversifiable risk......Page 354
10.4.1 Diversifiable mortality risk......Page 355
10.4.2 Non-diversifiable risk......Page 356
Comments......Page 359
10.5 Monte Carlo simulation......Page 362
10.7 Exercises......Page 368
Answers to selected exercises......Page 371
11.2.1 The net cash flows for a policy......Page 373
11.2.2 Reserves......Page 375
11.3 Profit measures......Page 378
11.4 A further example of a profit test......Page 380
11.6 Exercises......Page 389
Answers to selected exercises......Page 393
12.2 Equity-linked insurance......Page 394
12.3 Deterministic profit testing for equity-linked insurance......Page 395
12.4 Stochastic profit testing......Page 404
12.5 Stochastic pricing......Page 408
12.6.1 Reserving for policies with non-diversifiable risk......Page 410
12.6.2 Quantile reserving......Page 411
12.6.3 CTE reserving......Page 413
12.6.4 Comments on reserving......Page 414
12.8 Exercises......Page 415
Answers to selected exercises......Page 419
13.2 Introduction......Page 421
13.3 The `no arbitrage' assumption......Page 422
13.4 Options......Page 423
13.5.2 Pricing over a single time period......Page 425
13.5.3 Pricing over two time periods......Page 430
13.5.4 Summary of the binomial model option pricing technique......Page 433
13.6.1 The model......Page 434
13.6.2 The Black–Scholes–Merton option pricing formula......Page 436
13.7 Notes and further reading......Page 447
13.8 Exercises......Page 448
Answers to selected exercises......Page 450
14.2 Introduction......Page 451
14.3.1 Pricing......Page 453
14.3.2 Reserving......Page 456
14.4.1 Pricing......Page 458
14.4.2 Reserving......Page 460
14.5 Pricing methods for embedded options......Page 464
14.6 Risk management......Page 467
14.7 Emerging costs......Page 469
14.8 Notes and further reading......Page 477
14.9 Exercises......Page 478
Answers to selected exercises......Page 482
A.1.2 Uniform distribution......Page 484
A.1.3 Normal distribution......Page 485
A.1.4 Lognormal distribution......Page 486
A.3 Functions of a random variable......Page 489
A.3.2 Continuous random variables......Page 490
A.3.3 Mixed random variables......Page 491
A.4 Conditional expectation and conditional variance......Page 492
A.5 Notes and further reading......Page 493
B.1.1 The trapezium rule......Page 494
B.1.2 Repeated Simpson's rule......Page 496
B.1.3 Integrals over an infinite interval......Page 497
B.2 Woolhouse's formula......Page 498
B.3 Notes and further reading......Page 499
C.1 The inverse transform method......Page 500
C.2 Simulation from a normal distribution......Page 501
C.3 Notes and further reading......Page 502
References......Page 503
Author Index......Page 507
Index......Page 508