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Preface xiv
1 Introduction to life insurance 1
1.1 Summary 1
1.2 Background 1
1.3 Life insurance and annuity contracts 3
1.3.1 Introduction 3
1.3.2 Traditional insurance contracts 4
1.3.3 Modern insurance contracts 6
1.3.4 Distribution methods 8
1.3.5 Underwriting 8
1.3.6 Premiums 10
1.3.7 Life annuities 11
1.4 Other insurance contracts 12
1.5 Pension benefits 12
1.5.1 Defined benefit and defined contribution pensions 13
1.5.2 Defined benefit pension design 13
1.6 Mutual and proprietary insurers 14
1.7 Typical problems 14
1.8 Notes and further reading 15
1.9 Exercises 15
2 Survival models 17
2.1 Summary 17
2.2 The future lifetime random variable 17
2.3 The force of mortality 21
2.4 Actuarial notation 26
2.5 Mean and standard deviation of Tx 29
2.6 Curtate future lifetime 32
2.6.1 Kx and ex 32
2.6.2 The complete and curtate expected future lifetimes, ex and ex 34
2.7 Notes and further reading 35
2.8 Exercises 36
3 Life tables and selection 41
3.1 Summary 41
3.2 Life tables 41
3.3 Fractional age assumptions 44
3.3.1 Uniform distribution of deaths 44
3.3.2 Constant force of mortality 48
3.4 National life tables 49
3.5 Survival models for life insurance policyholders 52
3.6 Life insurance underwriting 54
3.7 Select and ultimate survival models 56
3.8 Notation and formulae for select survival models 58
3.9 Select life tables 59
3.10 Notes and further reading 67
3.11 Exercises 67
4 Insurance benefits 73
4.1 Summary 73
4.2 Introduction 73
4.3 Assumptions 74
4.4 Valuation of insurance benefits 75
4.4.1 Whole life insurance: the continuouscase, &Abar;x 75
4.4.2 Whole life insurance: the annual case, Ax 78
4.4.3 Whole life insurance: the 1 /mthly case, A(m)x 79
4.4.4 Recursions 81
4.4.5 Term insurance 86
4.4.6 Pure endowment 88
4.4.7 Endowment insurance 89
4.4.8 Deferred insurance benefits 91
4.5 Relating &Abar;x, Ax and A(m)x 93
4.5.1 Using the uniform distribution of deaths assumption 93
4.5.2 Using the claims acceleration approach 95
4.6 Variable insurance benefits 96
4.7 Functions for select lives 101
4.8 Notes and further reading 101
4.9 Exercises 102
5 Annuities 107
5.1 Summary 107
5.2 Introduction 107
5.3 Review of annuities-certain 108
5.4 Annual life annuities 108
5.4.1 Whole life annuity-due 109
5.4.2 Term annuity-due 112
5.4.3 Whole life immediate annuity 113
5.4.4 Term immediate annuity 114
5.5 Annuities payable continuously 115
5.5.1 Whole life continuous annuity 115
5.5.2 Term continuous annuity 117
5.6 Annuities payable m times per year 118
5.6.1 Introduction 118
5.6.2 Life annuities payable m times a year 119
5.6.3 Term annuities payable m times a year 120
5.7 Comparison of annuities by payment frequency 121
5.8 Deferred annuities 123
5.9 Guaranteed annuities 125
5.10 Increasing annuities 127
5.10.1 Arithmetically increasing annuities 127
5.10.2 Geometrically increasing annuities 129
5.11 Evaluating annuity functions 130
5.11.1 Recursions 130
5.11.2 Applying the UDD assumption 131
5.11.3 Woolhouse's formula 132
5.12 Numerical illustrations 135
5.13 Functions for select lives 136
5.14 Notes and further reading 137
5.15 Exercises 137
6 Premium calculation 142
6.1 Summary 142
6.2 Preliminaries 142
6.3 Assumptions 143
6.4 The present value of future loss random variable 145
6.5 The equivalence principle 146
6.5.1 Net premiums 146
6.6 Gross premium calculation 150
6.7 Profit 154
6.8 The portfolio percentile premium principle 162
6.9 Extra risks 165
6.9.1 Age rating 165
6.9.2 Constant addition to μx 165
6.9.3 Constant multiple of mortality rates 167
6.10 Notes and further reading 169
6.11 Exercises 170
7 Policy values 176
7.1 Summary 176
7.2 Assumptions 176
7.3 Policies with annual cash flows 176
7.3.1 The future loss random variable 176
7.3.2 Policy values for policies with annual cash flows 182
7.3.3 Recursive formulae for policy values 191
7.3.4 Annual profit 196
7.3.5 Asset shares 200
7.4 Policy values for policies with cash flows at discrete intervals other than annually 203
7.4.1 Recursions 204
7.4.2 Valuation between premium dates 205
7.5 Policy values with continuous cash flows 207
7.5.1 Thiele's differential equation 207
7.5.2 Numerical solution of Thiele's differential equation 211
7.6 Policy alterations 213
7.7 Retrospective policy value 219
7.8 Negative policy values 220
7.9 Notes and further reading 220
7.10 Exercises 220
8 Multiple state models 230
8.1 Summary 230
8.2 Examples of multiple state models 230
8.2.1 The alive-dead model 230
8.2.2 Term insurance with increased benefit on accidental death 232
8.2.3 The permanent disability model 232
8.2.4 The disability income insurance model 233
8.2.5 The joint life and last survivor model 234
8.3 Assumptions and notation 235
8.4 Formulae for probabilities 239
8.4.1 Kolmogorov's forward equations 242
8.5 Numerical evaluation of probabilities 243
8.6 Premiums 247
8.7 Policy values and Thiele's differential equation 250
8.7.1 The disability income model 251
8.7.2 Thiele's differential equation - the general case 255
8.8 Multiple decrement models 256
8.9 Joint life and last survivor benefits 261
8.9.1 The model and assumptions 261
8.9.2 Joint life and last survivor probabilities 262
8.9.3 Joint life and last survivor annuity and insurance functions 264
8.9.4 An important special case: independent survival models 270
8.10 Transitions at specified ages 274
8.11 Notes and further reading 278
8.12 Exercises 279
9 Pension mathematics 290
9.1 Summary 290
9.2 Introduction 290
9.3 The salary scale function 291
9.4 Setting the DC contribution 294
9.5 The service table 297
9.6 Valuation of benefits 306
9.6.1 Final salary plans 306
9.6.2 Career average earnings plans 312
9.7 Funding plans 314
9.8 Notes and further reading 319
9.9 Exercises 319
10 Interest rate risk 326
10.1 Summary 326
10.2 The yield curve 326
10.3 Valuation of insurances and life annuities 330
Replicating the cash flows of a traditional non-participating product 332
10.4 Diversifiable and non-diversifiable risk 334
10.4.1 Diversifiable mortality risk 335
10.4.2 Non-diversifiable risk 336
10.5 Monte Carlo simulation 342
10.6 Notes and further reading 348
10.7 Exercised 348
11 Emerging costs for traditional life insurance 353
11.1 Summary 353
11.2 Profit testing for traditional life insurance 353
11.2.1 The net cash flows for a policy 353
11.2.2 Reserves 355
11.3 Profit measures 358
11.4 A further example of a profit test 360
11.5 Notes and further reading 369
11.6 Exercises 369
12 Emerging costs for equity-linked insurance 374
12.1 Summary 374
12.2 Equity-linked insurance 374
12.3 Deterministic profit testing for equity-linked insurance 375
12.4 Stochastic profit testing 384
12.5 Stochastic pricing 388
12.6 Stochastic reserving 390
12.6.1 Reserving for policies with non-diversifiable risk 390
12.6.2 Quantile reserving 391
12.6.3 CTE reserving 393
12.6.4 Comments on reserving 394
12.7 Notes and further reading 395
12.8 Exercises 395
13 Option pricing 401
13.1 Summary 401
13.2 Introduction 401
13.3 The'no arbitrage'assumption 402
13.4 Options 403
13.5 The binomial option pricing model 405
13.5.1 Assumptions 405
13.5.2 Pricing over a single time period 405
13.5.3 Pricing over two time periods 410
13.5.4 Summary of the binomial model option pricing technique 413
13.6 The Black-Scholes-Merton model 414
13.6.1 The model 414
13.6.2 The Black-Scholes-Merton option pricing formula 416
13.7 Notes and further reading 427
13.8 Exercises 428
14 Embedded options 431
14.1 Summary 431
14.2 Introduction 431
14.3 Guaranteed minimum maturity benefit 433
14.3.1 Pricing 433
14.3.2 Reserving 436
14.4 Guaranteed minimum death benefit 438
14.4.1 Pricing 438
14.4.2 Reserving 440
14.5 Pricing methods for embedded options 444
14.6 Risk management 447
14.7 Emerging costs 449
14.8 Notes and further reading 457
14.9 Exercises 458
A Probability theory 464
A.1 Probability distributions 464
A.1.l Binomial distribution 464
A.1.2 Uniform distribution 464
A.1.3 Normal distribution 465
A.1.4 Lognormal distribution 466
A.2 The central limit theorem 469
A.3 Functions of a random variable 469
A.3.1 Discrete random variables 470
A.3.2 Continuous random variables 470
A.3.3 Mixed random variables 471
A.4 Conditional expectation and conditional variance 472
A.5 Notes and further reading 473
B Numerical techniques 474
B.1 Numerical integration 474
B.1.1 The trapezium rule 474
B.1.2 Repeated Simpson's rule 476
B.1.3 Integrals over an infinite interval 477
B.2 Woolhouse's formula 478
B.3 Notes and further reading 479
C Simulation 480
C.1 The inverse transform method 480
C.2 Simulation from a normal distribution 481
C.2.1 The Box-Muller method 482
C.2.2 The polar method 482
C.3 Notes and further reading 482
References 483
Author index 487
Index 488
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