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Fundamentals of Actuarial Mathematics, 3rd Edition

Book 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.

  • Table of Contents

    1. Preface
    2. Acknowledgements
    3. About the companion website
    4. Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL
      1. 1 Introduction and motivation
        1. 1.1 Risk and insurance
        2. 1.2 Deterministic versus stochastic models
        3. 1.3 Finance and investments
        4. 1.4 Adequacy and equity
        5. 1.5 Reassessment
        6. 1.6 Conclusion
      2. 2 The basic deterministic model
        1. 2.1 Cash flows
        2. 2.2 An analogy with currencies
        3. 2.3 Discount functions
        4. 2.4 Calculating the discount function
        5. 2.5 Interest and discount rates
        6. 2.6 Constant interest
        7. 2.7 Values and actuarial equivalence
        8. 2.8 Vector notation
        9. 2.9 Regular pattern cash flows
        10. 2.10 Balances and reserves
        11. 2.11 Time shifting and the splitting identity
        12. *2.11 Change of discount function
        13. 2.12 Internal rates of return
        14. *2.13 Forward prices and term structure
        15. 2.14 Standard notation and terminology
        16. 2.15 Spreadsheet calculations
        17. Notes and references
        18. Exercises
      3. 3 The life table
        1. 3.1 Basic definitions
        2. 3.2 Probabilities
        3. 3.3 Constructing the life table from the values of <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">q<sub>x</sub></i>
        4. 3.4 Life expectancy
        5. 3.5 Choice of life tables
        6. 3.6 Standard notation and terminology
        7. 3.7 A sample table
        8. Notes and references
        9. Exercises
      4. 4 Life annuities
        1. 4.1 Introduction
        2. 4.2 Calculating annuity premiums
        3. 4.3 The interest and survivorship discount function
        4. 4.4 Guaranteed payments
        5. 4.5 Deferred annuities with annual premiums
        6. 4.6 Some practical considerations
        7. 4.7 Standard notation and terminology
        8. 4.8 Spreadsheet calculations
        9. Exercises
      5. 5 Life insurance
        1. 5.1 Introduction
        2. 5.2 Calculating life insurance premiums
        3. 5.3 Types of life insurance
        4. 5.4 Combined insurance–annuity benefits
        5. 5.5 Insurances viewed as annuities
        6. 5.6 Summary of formulas
        7. 5.7 A general insurance–annuity identity
        8. 5.8 Standard notation and terminology
        9. 5.9 Spreadsheet applications
        10. Exercises
      6. 6 Insurance and annuity reserves
        1. 6.1 Introduction to reserves
        2. 6.2 The general pattern of reserves
        3. 6.3 Recursion
        4. 6.4 Detailed analysis of an insurance or annuity contract
        5. 6.5 Bases for reserves
        6. 6.6 Nonforfeiture values
        7. 6.7 Policies involving a return of the reserve
        8. 6.8 Premium difference and paid-up formulas
        9. 6.9 Standard notation and terminology
        10. 6.10 Spreadsheet applications
        11. Exercises
      7. 7 Fractional durations
        1. 7.1 Introduction
        2. 7.2 Cash flows discounted with interest only
        3. 7.3 Life annuities paid <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">m</i>thlythly
        4. 7.4 Immediate annuities
        5. 7.5 Approximation and computation
        6. *7.6 Fractional period premiums and reserves
        7. 7.7 Reserves at fractional durations
        8. 7.8 Standard notation and terminology
        9. Exercises
      8. 8 Continuous payments
        1. 8.1 Introduction to continuous annuities
        2. 8.2 The force of discount
        3. 8.3 The constant interest case
        4. 8.4 Continuous life annuities
        5. 8.5 The force of mortality
        6. 8.6 Insurances payable at the moment of death
        7. 8.7 Premiums and reserves
        8. 8.8 The general insurance–annuity identity in the continuous case
        9. 8.9 Differential equations for reserves
        10. 8.10 Some examples of exact calculation
        11. 8.11 Further approximations from the life table
        12. 8.12 Standard actuarial notation and terminology
        13. Notes and references
        14. Exercises
      9. 9 Select mortality
        1. 9.1 Introduction
        2. 9.2 Select and ultimate tables
        3. 9.3 Changes in formulas
        4. 9.4 Projections in annuity tables
        5. 9.5 Further remarks
        6. Exercises
      10. 10 Multiple-life contracts
        1. 10.1 Introduction
        2. 10.2 The joint-life status
        3. 10.3 Joint-life annuities and insurances
        4. 10.4 Last-survivor annuities and insurances
        5. 10.5 Moment of death insurances
        6. 10.6 The general two-life annuity contract
        7. 10.7 The general two-life insurance contract
        8. 10.8 Contingent insurances
        9. 10.9 Duration problems
        10. *10.10 Applications to annuity credit risk
        11. 10.11 Standard notation and terminology
        12. 10.12 Spreadsheet applications
        13. Notes and references
        14. Exercises
      11. 11 Multiple-decrement theory
        1. 11.1 Introduction
        2. 11.2 The basic model
        3. 11.3 Insurances
        4. 11.4 Determining the model from the forces of decrement
        5. 11.5 The analogy with joint-life statuses
        6. 11.6 A machine analogy
        7. 11.7 Associated single-decrement tables
        8. Notes and references
        9. Exercises
      12. 12 Expenses and profits
        1. 12.1 Introduction
        2. 12.2 Effect on reserves
        3. 12.3 Realistic reserve and balance calculations
        4. 12.4 Profit measurement
        5. Notes and references
        6. Exercises
      13. *13 Specialized topics
        1. 13.1 Universal life
        2. 13.2 Variable annuities
        3. 13.3 Pension plans
        4. Exercises
    5. Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL
      1. 14 Survival distributions and failure times
        1. 14.1 Introduction to survival distributions
        2. 14.2 The discrete case
        3. 14.3 The continuous case
        4. 14.4 Examples
        5. 14.5 Shifted distributions
        6. 14.6 The standard approximation
        7. 14.7 The stochastic life table
        8. 14.8 Life expectancy in the stochastic model
        9. 14.9 Stochastic interest rates
        10. Notes and references
        11. Exercises
      2. 15 The stochastic approach to insurance and annuities
        1. 15.1 Introduction
        2. 15.2 The stochastic approach to insurance benefits
        3. 15.3 The stochastic approach to annuity benefits
        4. *15.4 Deferred contracts
        5. 15.5 The stochastic approach to reserves
        6. 15.6 The stochastic approach to premiums
        7. 15.7 The variance of <sub xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg"><i>r</i></sub>&#160; <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">L</i>
        8. 15.8 Standard notation and terminology
        9. Notes and references
        10. Exercises
      3. 16 Simplifications under level benefit contracts
        1. 16.1 Introduction
        2. 16.2 Variance calculations in the continuous case
        3. 16.3 Variance calculations in the discrete case
        4. 16.4 Exact distributions
        5. 16.5 Some non-level benefit examples
        6. Exercises
      4. 17 The minimum failure time
        1. 17.1 Introduction
        2. 17.2 Joint distributions
        3. 17.3 The distribution of <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">T</i>
        4. 17.4 The joint distribution of (<i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">T</i>, , <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">J</i>))
        5. 17.5 Other problems
        6. 17.6 The common shock model
        7. 17.7 Copulas
        8. Notes and references
        9. Exercises
    6. Part III ADVANCED STOCHASTIC MODELS
      1. 18 An introduction to stochastic processes
        1. 18.1 Introduction
        2. 18.2 Markov chains
        3. 18.3 Martingales
        4. 18.4 Finite-state Markov chains
        5. 18.5 Introduction to continuous time processes
        6. 18.6 Poisson processes
        7. 18.7 Brownian motion
        8. Notes and references
        9. Exercises
      2. 19 Multi-state models
        1. 19.1 Introduction
        2. 19.2 The discrete-time model
        3. 19.3 The continuous-time model
        4. 19.4 Recursion and differential equations for multi-state reserves
        5. 19.5 Profit testing in multi-state models
        6. 19.6 Semi-Markov models
        7. Notes and references
        8. Exercises
      3. 20 Introduction to the Mathematics of Financial Markets
        1. 20.1 Introduction
        2. 20.2 Modelling prices in financial markets
        3. 20.3 Arbitrage
        4. 20.4 Option contracts
        5. 20.5 Option prices in the one-period binomial model
        6. 20.6 The multi-period binomial model
        7. 20.7 American options
        8. 20.8 A general financial market
        9. 20.9 Arbitrage-free condition
        10. 20.10 Existence and uniqueness of risk-neutral measures
        11. 20.11 Completeness of markets
        12. 20.12 The Black–Scholes–Merton formula
        13. 20.13 Bond markets
        14. Notes and references
        15. Exercises
    7. Part IV RISK THEORY
      1. 21 Compound distributions
        1. 21.1 Introduction
        2. 21.2 The mean and variance of <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">S</i>
        3. 21.3 Generating functions
        4. 21.4 Exact distribution of <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">S</i>
        5. 21.5 Choosing a frequency distribution
        6. 21.6 Choosing a severity distribution
        7. 21.7 Handling the point mass at 0
        8. 21.8 Counting claims of a particular type
        9. 21.9 The sum of two compound Poisson distributions
        10. 21.10 Deductibles and other modifications
        11. 21.11 A recursion formula for <i xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:svg="http://www.w3.org/2000/svg">S</i>
        12. Notes and references
        13. Exercises
      2. 22 Risk assessment
        1. 22.1 Introduction
        2. 22.2 Utility theory
        3. 22.3 Convex and concave functions: Jensen’s inequality
        4. 22.4 A general comparison method
        5. 22.5 Risk measures for capital adequacy
        6. Notes and references
        7. Exercises
      3. 23 Ruin models:
        1. 23.1 Introduction
        2. 23.2 A functional equation approach
        3. 23.3 The martingale approach to ruin theory
        4. 23.4 Distribution of the deficit at ruin
        5. 23.5 Recursion formulas
        6. 23.6 The compound Poisson surplus process
        7. 23.7 The maximal aggregate loss
        8. Notes and references
        9. Exercises
      4. 24 Credibility theory:
        1. 24.1 Introductory material
        2. 24.2 Conditional expectation and variance with respect to another random variable
        3. 24.3 General framework for Bayesian credibility
        4. 24.4 Classical examples
        5. 24.5 Approximations
        6. 24.6 Conditions for exactness
        7. 24.7 Estimation
        8. Notes and References
        9. Exercises
    8. Answers to exercises
      1. Chapter 2
      2. Chapter 3
      3. Chapter 4
      4. Chapter 5
      5. Chapter 6
      6. Chapter 7
      7. Chapter 8
      8. Chapter 9
      9. Chapter 10
      10. Chapter 11
      11. Chapter 12
      12. Chapter 13
      13. Chapter 14
      14. Chapter 15
      15. Chapter 16
      16. Chapter 17
      17. Chapter 18
      18. Chapter 19
      19. Chapter 20
      20. Chapter 21
      21. Chapter 22
      22. Chapter 23
      23. Chapter 24
    9. Appendix A review of probability theory
      1. A.1 Sample spaces and probability measures
      2. A.2 Conditioning and independence
      3. A.3 Random variables
      4. A.4 Distributions
      5. A.5 Expectations and moments
      6. A.6 Expectation in terms of the distribution function
      7. A.7 Joint distributions
      8. A.8 Conditioning and independence for random variables
      9. A.9 Moment generating functions
      10. A.10 Probability generating functions
      11. A.11 Some standard distributions
      12. A.12 Convolution
      13. A.13 Mixtures
    10. References
    11. Notation index
    12. Index
    13. End User License Agreement