Actuarial Mathematics for Life Contingent Risks

by David C. M. Dickson , Mary R. Hardy , Howard R. Waters

• ISBN13: 9780521118255
• ISBN10: 0521118255
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-10-26
• Publisher: Cambridge University Press

Summary

How can actuaries equip themselves for the products and risk structures of the future? Using the powerful framework of multiple state models, three leaders in actuarial science give a modern perspective on life contingencies, and develop and demonstrate a theory that can be adapted to changing products and technologies. The book begins traditionally, covering actuarial models and theory, and emphasizing practical applications using computational techniques. The authors then develop a more contemporary outlook, introducing multiple state models, emerging cash flows and embedded options. Using spreadsheet-style software, the book presents large-scale, realistic examples. Over 150 exercises and solutions teach skills in simulation and projection through computational practice. Balancing rigor with intuition, and emphasizing applications, this text is ideal for university courses, but also for individuals preparing for professional actuarial exams and qualified actuaries wishing to freshen up their skills.

Table of Contents

Preface	p. xiv
Introduction to life insurance	p. 1
Summary	p. 1
Background	p. 1
Life insurance and annuity contracts	p. 3
Introduction	p. 3
Traditional insurance contracts	p. 4
Modern insurance contracts	p. 6
Distribution methods	p. 8
Underwriting	p. 8
Premiums	p. 10
Life annuities	p. 11
Other insurance contracts	p. 12
Pension benefits	p. 12
Defined benefit and defined contribution pensions	p. 13
Defined benefit pension design	p. 13
Mutual and proprietary insurers	p. 14
Typical problems	p. 14
Notes and further reading	p. 15
Exercises	p. 15
Survival models	p. 17
Summary	p. 17
The future lifetime random variable	p. 17
The force of mortality	p. 21
Actuarial notation	p. 26
Mean and standard deviation of Tx	p. 29
Curtate future lifetime	p. 32
Kx and ex	p. 32
The complete and curtate expected future lifetimes, ex and ex	p. 34
Notes and further reading	p. 35
Exercises	p. 36
Life tables and selection	p. 41
Summary	p. 41
Life tables	p. 41
Fractional age assumptions	p. 44
Uniform distribution of deaths	p. 44
Constant force of mortality	p. 48
National life tables	p. 49
Survival models for life insurance policyholders	p. 52
Life insurance underwriting	p. 54
Select and ultimate survival models	p. 56
Notation and formulae for select survival models	p. 58
Select life tables	p. 59
Notes and further reading	p. 67
Exercises	p. 67
Insurance benefits	p. 73
Summary	p. 73
Introduction	p. 73
Assumptions	p. 74
Valuation of insurance benefits	p. 75
Whole life insurance: the continuous case, &Abar;x	p. 75
Whole life insurance: the annual case, Ax	p. 78
Whole life insurance: the 1 /mthly case, A(m)x	p. 79
Recursions	p. 81
Term insurance	p. 86
Pure endowment	p. 88
Endowment insurance	p. 89
Deferred insurance benefits	p. 91
Relating &Abar;x, Ax and A(m)x	p. 93
Using the uniform distribution of deaths assumption	p. 93
Using the claims acceleration approach	p. 95
Variable insurance benefits	p. 96
Functions for select lives	p. 101
Notes and further reading	p. 101
Exercises	p. 102
Annuities	p. 107
Summary	p. 107
Introduction	p. 107
Review of annuities-certain	p. 108
Annual life annuities	p. 108
Whole life annuity-due	p. 109
Term annuity-due	p. 112
Whole life immediate annuity	p. 113
Term immediate annuity	p. 114
Annuities payable continuously	p. 115
Whole life continuous annuity	p. 115
Term continuous annuity	p. 117
Annuities payable m times per year	p. 118
Introduction	p. 118
Life annuities payable m times a year	p. 119
Term annuities payable m times a year	p. 120
Comparison of annuities by payment frequency	p. 121
Deferred annuities	p. 123
Guaranteed annuities	p. 125
Increasing annuities	p. 127
Arithmetically increasing annuities	p. 127
Geometrically increasing annuities	p. 129
Evaluating annuity functions	p. 130
Recursions	p. 130
Applying the UDD assumption	p. 131
Woolhouse's formula	p. 132
Numerical illustrations	p. 135
Functions for select lives	p. 136
Notes and further reading	p. 137
Exercises	p. 137
Premium calculation	p. 142
Summary	p. 142
Preliminaries	p. 142
Assumptions	p. 143
The present value of future loss random variable	p. 145
The equivalence principle	p. 146
Net premiums	p. 146
Gross premium calculation	p. 150
Profit	p. 154
The portfolio percentile premium principle	p. 162
Extra risks	p. 165
Age rating	p. 165
Constant addition to ¿x	p. 165
Constant multiple of mortality rates	p. 167
Notes and further reading	p. 169
Exercises	p. 170
Policy values	p. 176
Summary	p. 176
Assumptions	p. 176
Policies with annual cash flows	p. 176
The future loss random variable	p. 176
Policy values for policies with annual cash flows	p. 182
Recursive formulae for policy values	p. 191
Annual profit	p. 196
Asset shares	p. 200
Policy values for policies with cash flows at discrete intervals other than annually	p. 203
Recursions	p. 204
Valuation between premium dates	p. 205
Policy values with continuous cash flows	p. 207
Thiele's differential equation	p. 207
Numerical solution of Thiele's differential equation	p. 211
Policy alterations	p. 213
Retrospective policy value	p. 219
Negative policy values	p. 220
Notes and further reading	p. 220
Exercises	p. 220
Multiple state models	p. 230
Summary	p. 230
Examples of multiple state models	p. 230
The alive-dead model	p. 230
Term insurance with increased benefit on accidental death	p. 232
The permanent disability model	p. 232
The disability income insurance model	p. 233
The joint life and last survivor model	p. 234
Assumptions and notation	p. 235
Formulae for probabilities	p. 239
Kolmogorov's forward equations	p. 242
Numerical evaluation of probabilities	p. 243
Premiums	p. 247
Policy values and Thiele's differential equation	p. 250
The disability income model	p. 251
Thiele's differential equation - the general case	p. 255
Multiple decrement models	p. 256
Joint life and last survivor benefits	p. 261
The model and assumptions	p. 261
Joint life and last survivor probabilities	p. 262
Joint life and last survivor annuity and insurance functions	p. 264
An important special case: independent survival models	p. 270
Transitions at specified ages	p. 274
Notes and further reading	p. 278
Exercises	p. 279
Pension mathematics	p. 290
Summary	p. 290
Introduction	p. 290
The salary scale function	p. 291
Setting the DC contribution	p. 294
The service table	p. 297
Valuation of benefits	p. 306
Final salary plans	p. 306
Career average earnings plans	p. 312
Funding plans	p. 314
Notes and further reading	p. 319
Exercises	p. 319
Interest rate risk	p. 326
Summary	p. 326
The yield curve	p. 326
Valuation of insurances and life annuities	p. 330
Replicating the cash flows of a traditional non-participating product	p. 332
Diversifiable and non-diversifiable risk	p. 334
Diversifiable mortality risk	p. 335
Non-diversifiable risk	p. 336
Monte Carlo simulation	p. 342
Notes and further reading	p. 348
Exercised	p. 348
Emerging costs for traditional life insurance	p. 353
Summary	p. 353
Profit testing for traditional life insurance	p. 353
The net cash flows for a policy	p. 353
Reserves	p. 355
Profit measures	p. 358
A further example of a profit test	p. 360
Notes and further reading	p. 369
Exercises	p. 369
Emerging costs for equity-linked insurance	p. 374
Summary	p. 374
Equity-linked insurance	p. 374
Deterministic profit testing for equity-linked insurance	p. 375
Stochastic profit testing	p. 384
Stochastic pricing	p. 388
Stochastic reserving	p. 390
Reserving for policies with non-diversifiable risk	p. 390
Quantile reserving	p. 391
CTE reserving	p. 393
Comments on reserving	p. 394
Notes and further reading	p. 395
Exercises	p. 395
Option pricing	p. 401
Summary	p. 401
Introduction	p. 401
The'no arbitrageÆassumption	p. 402
Options	p. 403
The binomial option pricing model	p. 405
Assumptions	p. 405
Pricing over a single time period	p. 405
Pricing over two time periods	p. 410
Summary of the binomial model option pricing technique	p. 413
The Black-Scholes-Merton model	p. 414
The model	p. 414
The Black-Scholes-Merton option pricing formula	p. 416
Notes and further reading	p. 427
Exercises	p. 428
Embedded options	p. 431
Summary	p. 431
Introduction	p. 431
Guaranteed minimum maturity benefit	p. 433
Pricing	p. 433
Reserving	p. 436
Guaranteed minimum death benefit	p. 438
Pricing	p. 438
Reserving	p. 440
Pricing methods for embedded options	p. 444
Risk management	p. 447
Emerging costs	p. 449
Notes and further reading	p. 457
Exercises	p. 458
Probability theory	p. 464
Probability distributions	p. 464
Binomial distribution	p. 464
Uniform distribution	p. 464
Normal distribution	p. 465
Lognormal distribution	p. 466
The central limit theorem	p. 469
Functions of a random variable	p. 469
Discrete random variables	p. 470
Continuous random variables	p. 470
Mixed random variables	p. 471
Conditional expectation and conditional variance	p. 472
Notes and further reading	p. 473
Numerical techniques	p. 474
Numerical integration	p. 474
The trapezium rule	p. 474
Repeated Simpson's rule	p. 476
Integrals over an infinite interval	p. 477
Woolhouse's formula	p. 478
Notes and further reading	p. 479
Simulation	p. 480
The inverse transform method	p. 480
Simulation from a normal distribution	p. 481
The Box-Muller method	p. 482
The polar method	p. 482
Notes and further reading	p. 482
References	p. 483
Author index	p. 487
Index	p. 488
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