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# Actuarial Mathematics for Life Contingent Risks

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

1st

### 9780521118255

0521118255

Hardcover

10/26/2009

Cambridge University Press

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## 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 T_{x} | p. 29 |

Curtate future lifetime | p. 32 |

K_{x} and e_{x} | p. 32 |

The complete and curtate expected future lifetimes, e_{x} and e_{x} | 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, A_{x} | 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}, A_{x} 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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