Actuarial Mathematics for Life Contingent Risks

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

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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

Prefacep. xiv
Introduction to life insurancep. 1
Summaryp. 1
Backgroundp. 1
Life insurance and annuity contractsp. 3
Introductionp. 3
Traditional insurance contractsp. 4
Modern insurance contractsp. 6
Distribution methodsp. 8
Underwritingp. 8
Premiumsp. 10
Life annuitiesp. 11
Other insurance contractsp. 12
Pension benefitsp. 12
Defined benefit and defined contribution pensionsp. 13
Defined benefit pension designp. 13
Mutual and proprietary insurersp. 14
Typical problemsp. 14
Notes and further readingp. 15
Exercisesp. 15
Survival modelsp. 17
Summaryp. 17
The future lifetime random variablep. 17
The force of mortalityp. 21
Actuarial notationp. 26
Mean and standard deviation of Txp. 29
Curtate future lifetimep. 32
Kx and exp. 32
The complete and curtate expected future lifetimes, ex and exp. 34
Notes and further readingp. 35
Exercisesp. 36
Life tables and selectionp. 41
Summaryp. 41
Life tablesp. 41
Fractional age assumptionsp. 44
Uniform distribution of deathsp. 44
Constant force of mortalityp. 48
National life tablesp. 49
Survival models for life insurance policyholdersp. 52
Life insurance underwritingp. 54
Select and ultimate survival modelsp. 56
Notation and formulae for select survival modelsp. 58
Select life tablesp. 59
Notes and further readingp. 67
Exercisesp. 67
Insurance benefitsp. 73
Summaryp. 73
Introductionp. 73
Assumptionsp. 74
Valuation of insurance benefitsp. 75
Whole life insurance: the continuous case, &Abar;xp. 75
Whole life insurance: the annual case, Axp. 78
Whole life insurance: the 1 /mthly case, A(m)xp. 79
Recursionsp. 81
Term insurancep. 86
Pure endowmentp. 88
Endowment insurancep. 89
Deferred insurance benefitsp. 91
Relating &Abar;x, Ax and A(m)xp. 93
Using the uniform distribution of deaths assumptionp. 93
Using the claims acceleration approachp. 95
Variable insurance benefitsp. 96
Functions for select livesp. 101
Notes and further readingp. 101
Exercisesp. 102
Annuitiesp. 107
Summaryp. 107
Introductionp. 107
Review of annuities-certainp. 108
Annual life annuitiesp. 108
Whole life annuity-duep. 109
Term annuity-duep. 112
Whole life immediate annuityp. 113
Term immediate annuityp. 114
Annuities payable continuouslyp. 115
Whole life continuous annuityp. 115
Term continuous annuityp. 117
Annuities payable m times per yearp. 118
Introductionp. 118
Life annuities payable m times a yearp. 119
Term annuities payable m times a yearp. 120
Comparison of annuities by payment frequencyp. 121
Deferred annuitiesp. 123
Guaranteed annuitiesp. 125
Increasing annuitiesp. 127
Arithmetically increasing annuitiesp. 127
Geometrically increasing annuitiesp. 129
Evaluating annuity functionsp. 130
Recursionsp. 130
Applying the UDD assumptionp. 131
Woolhouse's formulap. 132
Numerical illustrationsp. 135
Functions for select livesp. 136
Notes and further readingp. 137
Exercisesp. 137
Premium calculationp. 142
Summaryp. 142
Preliminariesp. 142
Assumptionsp. 143
The present value of future loss random variablep. 145
The equivalence principlep. 146
Net premiumsp. 146
Gross premium calculationp. 150
Profitp. 154
The portfolio percentile premium principlep. 162
Extra risksp. 165
Age ratingp. 165
Constant addition to żxp. 165
Constant multiple of mortality ratesp. 167
Notes and further readingp. 169
Exercisesp. 170
Policy valuesp. 176
Summaryp. 176
Assumptionsp. 176
Policies with annual cash flowsp. 176
The future loss random variablep. 176
Policy values for policies with annual cash flowsp. 182
Recursive formulae for policy valuesp. 191
Annual profitp. 196
Asset sharesp. 200
Policy values for policies with cash flows at discrete intervals other than annuallyp. 203
Recursionsp. 204
Valuation between premium datesp. 205
Policy values with continuous cash flowsp. 207
Thiele's differential equationp. 207
Numerical solution of Thiele's differential equationp. 211
Policy alterationsp. 213
Retrospective policy valuep. 219
Negative policy valuesp. 220
Notes and further readingp. 220
Exercisesp. 220
Multiple state modelsp. 230
Summaryp. 230
Examples of multiple state modelsp. 230
The alive-dead modelp. 230
Term insurance with increased benefit on accidental deathp. 232
The permanent disability modelp. 232
The disability income insurance modelp. 233
The joint life and last survivor modelp. 234
Assumptions and notationp. 235
Formulae for probabilitiesp. 239
Kolmogorov's forward equationsp. 242
Numerical evaluation of probabilitiesp. 243
Premiumsp. 247
Policy values and Thiele's differential equationp. 250
The disability income modelp. 251
Thiele's differential equation - the general casep. 255
Multiple decrement modelsp. 256
Joint life and last survivor benefitsp. 261
The model and assumptionsp. 261
Joint life and last survivor probabilitiesp. 262
Joint life and last survivor annuity and insurance functionsp. 264
An important special case: independent survival modelsp. 270
Transitions at specified agesp. 274
Notes and further readingp. 278
Exercisesp. 279
Pension mathematicsp. 290
Summaryp. 290
Introductionp. 290
The salary scale functionp. 291
Setting the DC contributionp. 294
The service tablep. 297
Valuation of benefitsp. 306
Final salary plansp. 306
Career average earnings plansp. 312
Funding plansp. 314
Notes and further readingp. 319
Exercisesp. 319
Interest rate riskp. 326
Summaryp. 326
The yield curvep. 326
Valuation of insurances and life annuitiesp. 330
Replicating the cash flows of a traditional non-participating productp. 332
Diversifiable and non-diversifiable riskp. 334
Diversifiable mortality riskp. 335
Non-diversifiable riskp. 336
Monte Carlo simulationp. 342
Notes and further readingp. 348
Exercisedp. 348
Emerging costs for traditional life insurancep. 353
Summaryp. 353
Profit testing for traditional life insurancep. 353
The net cash flows for a policyp. 353
Reservesp. 355
Profit measuresp. 358
A further example of a profit testp. 360
Notes and further readingp. 369
Exercisesp. 369
Emerging costs for equity-linked insurancep. 374
Summaryp. 374
Equity-linked insurancep. 374
Deterministic profit testing for equity-linked insurancep. 375
Stochastic profit testingp. 384
Stochastic pricingp. 388
Stochastic reservingp. 390
Reserving for policies with non-diversifiable riskp. 390
Quantile reservingp. 391
CTE reservingp. 393
Comments on reservingp. 394
Notes and further readingp. 395
Exercisesp. 395
Option pricingp. 401
Summaryp. 401
Introductionp. 401
The'no arbitrageĈassumptionp. 402
Optionsp. 403
The binomial option pricing modelp. 405
Assumptionsp. 405
Pricing over a single time periodp. 405
Pricing over two time periodsp. 410
Summary of the binomial model option pricing techniquep. 413
The Black-Scholes-Merton modelp. 414
The modelp. 414
The Black-Scholes-Merton option pricing formulap. 416
Notes and further readingp. 427
Exercisesp. 428
Embedded optionsp. 431
Summaryp. 431
Introductionp. 431
Guaranteed minimum maturity benefitp. 433
Pricingp. 433
Reservingp. 436
Guaranteed minimum death benefitp. 438
Pricingp. 438
Reservingp. 440
Pricing methods for embedded optionsp. 444
Risk managementp. 447
Emerging costsp. 449
Notes and further readingp. 457
Exercisesp. 458
Probability theoryp. 464
Probability distributionsp. 464
Binomial distributionp. 464
Uniform distributionp. 464
Normal distributionp. 465
Lognormal distributionp. 466
The central limit theoremp. 469
Functions of a random variablep. 469
Discrete random variablesp. 470
Continuous random variablesp. 470
Mixed random variablesp. 471
Conditional expectation and conditional variancep. 472
Notes and further readingp. 473
Numerical techniquesp. 474
Numerical integrationp. 474
The trapezium rulep. 474
Repeated Simpson's rulep. 476
Integrals over an infinite intervalp. 477
Woolhouse's formulap. 478
Notes and further readingp. 479
Simulationp. 480
The inverse transform methodp. 480
Simulation from a normal distributionp. 481
The Box-Muller methodp. 482
The polar methodp. 482
Notes and further readingp. 482
Referencesp. 483
Author indexp. 487
Indexp. 488
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