Fundamentals of Actuarial Mathematics. 3rd Edition

Transcription

1 Brochure More information from Fundamentals of Actuarial Mathematics. 3rd Edition Description: - 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?nance. - New edition restructures the material to?t 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. Contents: Preface xvii Acknowledgements xxi Notation index xxiii Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 1 1 Introduction and motivation Risk and insurance Deterministic versus stochastic models Finance and investments Adequacy and equity Reassessment Conclusion 6 2 The basic deterministic model Cash flows An analogy with currencies Discount functions Calculating the discount function Interest and discount rates Constant interest Values and actuarial equivalence Vector notation Regular pattern cash flows Balances and reserves Time shifting and the splitting identity 26

2 2.11 Change of discount function Internal rates of return Forward prices and term structure Standard notation and terminology Spreadsheet calculations 34 Notes and references 35 Exercises 35 3 The life table Basic definitions Probabilities Constructing the life table from the values of qx Life expectancy Choice of life tables Standard notation and terminology A sample table 45 Notes and references 45 Exercises 45 4 Life annuities Introduction Calculating annuity premiums The interest and survivorship discount function Guaranteed payments Deferred annuities with annual premiums Some practical considerations Standard notation and terminology Spreadsheet calculations 58 Exercises 59 5 Life insurance Introduction Calculating life insurance premiums Types of life insurance Combined insurance annuity benefits 64

3 5.5 Insurances viewed as annuities Summary of formulas A general insurance annuity identity Standard notation and terminology Spreadsheet applications 74 Exercises 74 6 Insurance and annuity reserves Introduction to reserves The general pattern of reserves Recursion Detailed analysis of an insurance or annuity contract Bases for reserves Nonforfeiture values Policies involving a return of the reserve Premium difference and paid-up formulas Standard notation and terminology Spreadsheet applications 93 Exercises 94 7 Fractional durations Introduction Cash flows discounted with interest only Life annuities paid 7.4 Immediate annuities Approximation and computation Fractional period premiums and reserves Reserves at fractional durations Standard notation and terminology 109 Exercises Continuous payments Introduction to continuous annuities The force of discount 113

4 8.3 The constant interest case Continuous life annuities The force of mortality Insurances payable at the moment of death Premiums and reserves The general insurance annuity identity in the continuous case Differential equations for reserves Some examples of exact calculation Further approximations from the life table Standard actuarial notation and terminology 131 Notes and references 132 Exercises Select mortality Introduction Select and ultimate tables Changes in formulas Projections in annuity tables Further remarks 142 Exercises Multiple-life contracts Introduction The joint-life status Joint-life annuities and insurances Last-survivor annuities and insurances Moment of death insurances The general two-life annuity contract The general two-life insurance contract Contingent insurances Duration problems Applications to annuity credit risk Standard notation and terminology Spreadsheet applications 161

5 Notes and references 161 Exercises Multiple-decrement theory Introduction The basic model Insurances Determining the model from the forces of decrement The analogy with joint-life statuses A machine analogy Associated single-decrement tables 175 Notes and references 181 Exercises Expenses and Profits Introduction Effect on reserves Realistic reserve and balance calculations Profit measurement 189 Notes and references 196 Exercises Specialized topics Universal life Variable annuities Pension plans 204 Exercises 207 Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL Survival distributions and failure times Introduction to survival distributions The discrete case The continuous case Examples Shifted distributions 216

6 14.6 The standard approximation The stochastic life table Life expectancy in the stochastic model Stochastic interest rates 221 Notes and references 222 Exercises The stochastic approach to insurance and annuities Introduction The stochastic approach to insurance benefits The stochastic approach to annuity benefits Deferred contracts The stochastic approach to reserves The stochastic approach to premiums The variance of rl Standard notation and terminology 243 Notes and references 244 Exercises Simplifications under level benefit contracts Introduction Variance calculations in the continuous case Variance calculations in the discrete case Exact distributions Some non-level benefit examples 254 Exercises The minimum failure time Introduction Joint distributions The distribution of T The joint distribution of (T,J) Other problems The common shock model Copulas 273

7 Notes and references 276 Exercises 276 Part III ADVANCED STOCHASTIC MODELS An introduction to stochastic processes Introduction Markov chains Martingales Finite-state Markov chains Introduction to continuous time processes Poisson processes Brownian motion 295 Notes and references 299 Exercises Multi-state models Introduction The discrete-time model The continuous-time model Recursion and differential equations for multi-state reserves Profit testing in multi-state models Semi-Markov models 328 Notes and references 328 Exercises Introduction to the Mathematics of Financial Markets Introduction Modelling prices in financial markets Arbitrage Option contracts Option prices in the one-period binomial model The multi-period binomial model American options A general financial market 348

8 20.9 Arbitrage-free condition Existence and uniqueness of risk neutral measures Completeness of markets The Black Scholes Merton formula Bond markets 364 Notes and references 372 Exercises 373 Part IV RISK THEORY Compound distributions Introduction The mean and variance of S Generating functions Exact distribution of S Choosing a frequency distribution Choosing a severity distribution Handling the point mass at Counting claims of a particular type The sum of two compound Poisson distributions Deductibles and other modifications A recursion formula for S 393 Notes and references 398 Exercises Risk assessment Introduction Utility theory Convex and concave functions: Jensen s inequality A general comparison method Risk measures for capital adequacy 412 Notes and references 417 Exercises Ruin models Introduction 420

9 23.2 A functional equation approach The martingale approach to ruin theory Distribution of the deficit at ruin Recursion formulas The compound Poisson surplus process The maximal aggregate loss 441 Notes and references 445 Exercises Credibility theory Introductory material Conditional expectation and variance with respect to another random variable General framework for Bayesian credibility Classical examples Approximations Conditions for exactness 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

10 A.13 Mixtures 499 Answers to exercises 501 References 517 Index 000 Ordering: Order Online - Order by Fax - using the form below Order by Post - print the order form below and send to Research and Markets, Guinness Centre, Taylors Lane, Dublin 8, Ireland.

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