S. David Promislow

Fundamentals of Actuarial Mathematics (eBook, PDF)

• Format: PDF

Produktbeschreibung

* 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.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 552
• Erscheinungstermin: 27. Oktober 2014
• Englisch
• ISBN-13: 9781118782491
• Artikelnr.: 41821727

Autorenporträt

S. David Promislow is the author of Fundamentals of Actuarial Mathematics, 3rd Edition, published by Wiley.

Inhaltsangabe

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 qx 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 rL 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