Fundamentals of Actuarial Mathematics, 3rd Edition
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- Published: 26 December 2014
- ISBN: 9781118782460
- Author(s): S. David Promislow
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- Provides a comprehensive coverage of both the deterministic and stochastic models of life contingencies, risk theory, credibility theory, multi-state models, and an introduction to modern mathematical ﬁnance.
- New edition restructures the material to ﬁt into modern computational methods and provides several spreadsheet examples throughout.
- Covers the syllabus for the Institute of Actuaries subject CT5, Contingencies
- Includes new chapters covering stochastic investments returns, universal life insurance. Elements of option pricing and the Black-Scholes formula will be introduced.
Preface xvii
Acknowledgements xxi
Notation index xxiii
Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 1
1 Introduction and motivation 3
1.1 Risk and insurance 3
1.2 Deterministic versus stochastic models 4
1.3 Finance and investments 5
1.4 Adequacy and equity 5
1.5 Reassessment 6
1.6 Conclusion 6
2 The basic deterministic model 7
2.1 Cash flows 7
2.2 An analogy with currencies 8
2.3 Discount functions 9
2.4 Calculating the discount function 11
2.5 Interest and discount rates 12
2.6 Constant interest 12
2.7 Values and actuarial equivalence 13
2.8 Vector notation 17
2.9 Regular pattern cash flows 18
2.10 Balances and reserves 20
2.11 Time shifting and the splitting identity 26
2.11 Change of discount function 27
2.12 Internal rates of return 28
2.13 Forward prices and term structure 30
2.14 Standard notation and terminology 33
2.15 Spreadsheet calculations 34
Notes and references 35
Exercises 35
3 The life table 39
3.1 Basic definitions 39
3.2 Probabilities 40
3.3 Constructing the life table from the values of q_{x} 41
3.4 Life expectancy 42
3.5 Choice of life tables 44
3.6 Standard notation and terminology 44
3.7 A sample table 45
Notes and references 45
Exercises 45
4 Life annuities 47
4.1 Introduction 47
4.2 Calculating annuity premiums 48
4.3 The interest and survivorship discount function 50
4.4 Guaranteed payments 53
4.5 Deferred annuities with annual premiums 55
4.6 Some practical considerations 56
4.7 Standard notation and terminology 57
4.8 Spreadsheet calculations 58
Exercises 59
5 Life insurance 61
5.1 Introduction 61
5.2 Calculating life insurance premiums 61
5.3 Types of life insurance 64
5.4 Combined insurance–annuity benefits 64
5.5 Insurances viewed as annuities 69
5.6 Summary of formulas 70
5.7 A general insurance–annuity identity 70
5.8 Standard notation and terminology 72
5.9 Spreadsheet applications 74
Exercises 74
6 Insurance and annuity reserves 78
6.1 Introduction to reserves 78
6.2 The general pattern of reserves 81
6.3 Recursion 82
6.4 Detailed analysis of an insurance or annuity contract 83
6.5 Bases for reserves 87
6.6 Nonforfeiture values 88
6.7 Policies involving a return of the reserve 88
6.8 Premium difference and paid-up formulas 90
6.9 Standard notation and terminology 91
6.10 Spreadsheet applications 93
Exercises 94
7 Fractional durations 98
7.1 Introduction 98
7.2 Cash flows discounted with interest only 99
7.3 Life annuities paid
7.4 Immediate annuities 104
7.5 Approximation and computation 105
7.6 Fractional period premiums and reserves 106
7.7 Reserves at fractional durations 107
7.8 Standard notation and terminology 109
Exercises 109
8 Continuous payments 112
8.1 Introduction to continuous annuities 112
8.2 The force of discount 113
8.3 The constant interest case 114
8.4 Continuous life annuities 115
8.5 The force of mortality 118
8.6 Insurances payable at the moment of death 119
8.7 Premiums and reserves 122
8.8 The general insurance–annuity identity in the continuous case 123
8.9 Differential equations for reserves 124
8.10 Some examples of exact calculation 125
8.11 Further approximations from the life table 129
8.12 Standard actuarial notation and terminology 131
Notes and references 132
Exercises 132
9 Select mortality 137
9.1 Introduction 137
9.2 Select and ultimate tables 138
9.3 Changes in formulas 139
9.4 Projections in annuity tables 141
9.5 Further remarks 142
Exercises 142
10 Multiple-life contracts 144
10.1 Introduction 144
10.2 The joint-life status 144
10.3 Joint-life annuities and insurances 146
10.4 Last-survivor annuities and insurances 147
10.5 Moment of death insurances 149
10.6 The general two-life annuity contract 150
10.7 The general two-life insurance contract 152
10.8 Contingent insurances 153
10.9 Duration problems 156
10.10 Applications to annuity credit risk 159
10.11 Standard notation and terminology 160
10.12 Spreadsheet applications 161
Notes and references 161
Exercises 161
11 Multiple-decrement theory 166
11.1 Introduction 166
11.2 The basic model 166
11.3 Insurances 169
11.4 Determining the model from the forces of decrement 170
11.5 The analogy with joint-life statuses 171
11.6 A machine analogy 171
11.7 Associated single-decrement tables 175
Notes and references 181
Exercises 181
12 Expenses and Profits 184
12.1 Introduction 184
12.2 Effect on reserves 186
12.3 Realistic reserve and balance calculations 187
12.4 Profit measurement 189
Notes and references 196
Exercises 196
13 Specialized topics 199
13.1 Universal life 199
13.2 Variable annuities 203
13.3 Pension plans 204
Exercises 207
Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL 209
14 Survival distributions and failure times 211
14.1 Introduction to survival distributions 211
14.2 The discrete case 212
14.3 The continuous case 213
14.4 Examples 215
14.5 Shifted distributions 216
14.6 The standard approximation 217
14.7 The stochastic life table 219
14.8 Life expectancy in the stochastic model 220
14.9 Stochastic interest rates 221
Notes and references 222
Exercises 222
15 The stochastic approach to insurance and annuities 224
15.1 Introduction 224
15.2 The stochastic approach to insurance benefits 225
15.3 The stochastic approach to annuity benefits 229
15.4 Deferred contracts 233
15.5 The stochastic approach to reserves 233
15.6 The stochastic approach to premiums 235
15.7 The variance of _{r}L 241
15.8 Standard notation and terminology 243
Notes and references 244
Exercises 244
16 Simplifications under level benefit contracts 248
16.1 Introduction 248
16.2 Variance calculations in the continuous case 248
16.3 Variance calculations in the discrete case 250
16.4 Exact distributions 252
16.5 Some non-level benefit examples 254
Exercises 256
17 The minimum failure time 259
17.1 Introduction 259
17.2 Joint distributions 259
17.3 The distribution of T 261
17.4 The joint distribution of (T,J) 261
17.5 Other problems 270
17.6 The common shock model 271
17.7 Copulas 273
Notes and references 276
Exercises 276
Part III ADVANCED STOCHASTIC MODELS 279
18 An introduction to stochastic processes 281
18.1 Introduction 281
18.2 Markov chains 283
18.3 Martingales 286
18.4 Finite-state Markov chains 287
18.5 Introduction to continuous time processes 293
18.6 Poisson processes 293
18.7 Brownian motion 295
Notes and references 299
Exercises 300
19 Multi-state models 304
19.1 Introduction 304
19.2 The discrete-time model 305
19.3 The continuous-time model 311
19.4 Recursion and differential equations for multi-state reserves 324
19.5 Profit testing in multi-state models 327
19.6 Semi-Markov models 328
Notes and references 328
Exercises 329
20 Introduction to the Mathematics of Financial Markets 333
20.1 Introduction 333
20.2 Modelling prices in financial markets 333
20.3 Arbitrage 334
20.4 Option contracts 337
20.5 Option prices in the one-period binomial model 339
20.6 The multi-period binomial model 342
20.7 American options 346
20.8 A general financial market 348
20.9 Arbitrage-free condition 351
20.10 Existence and uniqueness of risk neutral measures 353
20.11 Completeness of markets 358
20.12 The Black–Scholes–Merton formula 361
20.13 Bond markets 364
Notes and references 372
Exercises 373
Part IV RISK THEORY 375
21 Compound distributions 377
21.1 Introduction 377
21.2 The mean and variance of S 379
21.3 Generating functions 380
21.4 Exact distribution of S 381
21.5 Choosing a frequency distribution 381
21.6 Choosing a severity distribution 383
21.7 Handling the point mass at 0 384
21.8 Counting claims of a particular type 385
21.9 The sum of two compound Poisson distributions 387
21.10 Deductibles and other modifications 388
21.11 A recursion formula for S 393
Notes and references 398
Exercises 398
22 Risk assessment 403
22.1 Introduction 403
22.2 Utility theory 403
22.3 Convex and concave functions: Jensen’s inequality 406
22.4 A general comparison method 408
22.5 Risk measures for capital adequacy 412
Notes and references 417
Exercises 417
23 Ruin models 420
23.1 Introduction 420
23.2 A functional equation approach 422
23.3 The martingale approach to ruin theory 424
23.4 Distribution of the deficit at ruin 433
23.5 Recursion formulas 434
23.6 The compound Poisson surplus process 438
23.7 The maximal aggregate loss 441
Notes and references 445
Exercises 445
24 Credibility theory 449
24.1 Introductory material 449
24.2 Conditional expectation and variance with respect to another random variable 453
24.3 General framework for Bayesian credibility 457
24.4 Classical examples 459
24.5 Approximations 462
24.6 Conditions for exactness 465
24.7 Estimation 469
Notes and References 473
Exercises 473
Appendix A review of probability theory 477
A.1 Sample spaces and probability measures 477
A.2 Conditioning and independence 479
A.3 Random variables 479
A.4 Distributions 480
A.5 Expectations and moments 481
A.6 Expectation in terms of the distribution function 482
A.7 Joint distributions 483
A.8 Conditioning and independence for random variables 485
A.9 Moment generating functions 486
A.10 Probability generating functions 487
A.11 Some standard distributions 489
A.12 Convolution 495
A.13 Mixtures 499
Answers to exercises 501
References 517
Index 523
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