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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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