Actuarial Principles

Actuarial Principles

Lifetables and Mortality Models

1st Edition - October 29, 2021

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  • Author: Andrew Leung
  • Paperback ISBN: 9780323901727
  • eBook ISBN: 9780323901734

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Description

Actuarial Principles: Lifetables and Mortality Models explores the core of actuarial science: the study of mortality and other risks and applications. Including the CT4 and CT5 UK courses, but applicable to a global audience, this work lightly covers the mathematical and theoretical background of the subject to focus on real life practice. It offers a brief history of the field, why actuarial notation has become universal, and how theory can be applied to many situations. Uniquely covering both life contingency risks and survival models, the text provides numerous exercises (and their solutions), along with complete self-contained real-world assignments.

Key Features

  • Provides detailed coverage of life contingency risks and survival models
  • Presents self-contained chapters with coverage of key topics from both practitioner and theoretical viewpoints
  • Includes numerous real world exercises that are accompanied by enlightening solutions
  • Covers useful background information on how and why the subject has evolved and developed

Readership

Advanced undergraduate and graduate courses. Researchers/Professionals - qualified/certified actuaries in insurance, pensions, health

Table of Contents

  • Cover image
  • Title page
  • Table of Contents
  • Copyright
  • Chapter One: Lifetables and their applications
  • Abstract
  • 1.1. Introduction
  • Bibliography
  • Chapter Two: Lifetables and the principle of equivalence
  • Abstract
  • 2.1. Modelling mortality
  • 2.2. Lifetables
  • 2.3. Uncertainty in benefits
  • Bibliography
  • Chapter Three: Modelling mortality
  • Abstract
  • 3.1. The Australian life tables 2005–2007
  • 3.2. Mortality rates
  • 3.3. The curtate lifespan
  • 3.4. Types of lifetable
  • Chapter Four: Basic types of benefit (CT5: §1, §2, §3.4, §4)
  • Abstract
  • 4.1. The value of a lifetime insurance
  • 4.2. The value of a lifetime annuity
  • 4.3. Other types of benefits
  • 4.4. The value of an endowment insurance
  • 4.5. The value of an annuity with finite term
  • 4.6. The value of a pure endowment
  • 4.7. Value of deferred benefits
  • 4.8. Variances of insurances and annuities
  • 4.9. Increasing insurances and annuities (CT5: §6.1–6.3)
  • 4.10. An increasing annuity certain
  • 4.11. Annuities payable frequently
  • Appendix 4.A. Supplementary material
  • Chapter Five: Annuities payable frequently – a simpler approach
  • Abstract
  • Appendix 5.A. Supplementary material
  • Chapter Six: Benefits in continuous time (CT5: §1.8, 2.8)
  • Abstract
  • 6.1. Insurances payable continuously
  • 6.2. The value of a continuous lifetime insurance
  • 6.3. The value of a continuous lifetime annuity
  • Chapter Seven: Select mortality (CT5: §3.6)
  • Abstract
  • 7.1. How selection operates in practice
  • Chapter Eight: Benefits involving multiple lives (CT5: §8, §9)
  • Abstract
  • 8.1. The joint life table
  • 8.2. The curtate joint lifespan
  • 8.3. A joint lifetime insurance
  • 8.4. A joint lifetime annuity
  • Chapter Nine: The last survivor status
  • Abstract
  • 9.1. Relationship between curtate lifespans
  • 9.2. Last survivor insurances and annuities
  • Chapter Ten: Pricing and reserving in theory (CT5: §5)
  • Abstract
  • 10.1. Introduction
  • 10.2. Pricing
  • 10.3. Premium for an endowment insurance
  • 10.4. The insurer's risk
  • 10.5. Reserving
  • 10.6. Prospective reserves
  • 10.7. Retrospective reserves
  • 10.8. Equality of prospective and retrospective reserves
  • 10.9. Mortality profit/loss
  • 10.10. Death strain at risk
  • 10.11. Thiele's equation
  • Chapter Eleven: Pricing in practice (CT5: §6, §7)
  • Abstract
  • 11.1. How an insurance company works
  • 11.2. Practical issues
  • 11.3. Types of expense
  • 11.4. Expense recovery
  • 11.5. Premium assessment
  • 11.6. Bonus loadings
  • 11.7. The with-profit gross premium
  • Chapter Twelve: Multiple decrement models (CT5: §10.2, §13)
  • Abstract
  • 12.1. Multiple decrement tables
  • 12.2. Dependent and independent rates of decrement
  • Chapter Thirteen: Defined benefit superannuation (CT5: §14)
  • Abstract
  • 13.1. Trusts
  • 13.2. Defined benefits
  • 13.3. The salary scale
  • 13.4. Benefit rules
  • 13.5. An example of defined benefits
  • 13.6. Accrued defined benefits
  • 13.7. The value of accrued defined benefits
  • 13.8. Future contributions
  • 13.9. Value of future service benefits
  • 13.10. Future contributions
  • 13.11. A more complex defined benefit fund
  • 13.12. Final average salary
  • 13.13. Accrued benefits
  • 13.14. Value of accrued benefits
  • 13.15. Value of future service benefits
  • Chapter Fourteen: Life insurance modelling (CT5: §10; §11)
  • Abstract
  • 14.1. Profit testing
  • 14.2. A model of an insurer
  • 14.3. The profit vector
  • 14.4. The profit signature
  • Chapter Fifteen: Profit signature
  • Abstract
  • 15.1. Profit signature
  • 15.2. The return on transfers
  • 15.3. Unit linked insurances
  • 15.4. Testing unit linked insurances
  • 15.5. Reserves under profit testing
  • 15.6. Zeroisation
  • Chapter Sixteen: Multiple decrements and multiple states (CT5: §10)
  • Abstract
  • 16.1. Multiple decrements and multiple states in continuous time
  • 16.2. Multiple states
  • 16.3. Transition intensities
  • 16.4. Kolmogorov forward equations
  • 16.5. A sickness example
  • 16.6. A sickness example with no recovery
  • 16.7. Using transition probabilities
  • Chapter Seventeen: More complex benefits on multiple lives (CT5: §9)
  • Abstract
  • 17.1. More complex contingent insurances
  • 17.2. Contingent insurances as double summations
  • 17.3. Contingent insurances dependent on term
  • 17.4. Even more complex contingent insurances
  • 17.5. Contingent annuities
  • Chapter Eighteen: Risk factors in mortality
  • Abstract
  • 18.1. Life tables
  • 18.2. Mortality factors
  • 18.3. Selection
  • 18.4. How selection operates in practice
  • Chapter Nineteen: Types of life tables
  • Abstract
  • 19.1. Life table construction
  • 19.2. Construction of a life table
  • 19.3. Exposures to death
  • 19.4. Assessing mortality rates in practice
  • 19.5. Mortality comparisons
  • 19.6. Comparing mortality of two different populations
  • 19.7. Indirect standardisation
  • Chapter Twenty: Constructing a life table
  • Abstract
  • 20.1. Lifetime distribution
  • 20.2. Censoring
  • 20.3. Censoring examples
  • 20.4. Information from censoring
  • 20.5. Censoring and truncation
  • Chapter Twenty-One: Maximum likelihood estimation
  • Abstract
  • 21.1. Background
  • 21.2. Likelihood function
  • 21.3. Maximising the log likelihood
  • 21.4. Second-order conditions
  • 21.5. Conditions for a local maximum
  • 21.6. General conditions for a local maximum
  • 21.7. Properties of ML estimators
  • Chapter Twenty-Two: Binomial and Poisson models (CT4: §10)
  • Abstract
  • 22.1. Estimating q
  • 22.2. The MLE for q
  • 22.3. Different observation periods
  • 22.4. Likelihood function
  • 22.5. The Balducci assumption
  • 22.6. The Balducci estimator
  • 22.7. Initial and central risk exposures
  • Chapter Twenty-Three: The Poisson model
  • Abstract
  • 23.1. Properties of the Poisson model
  • 23.2. The MLE for the Poisson model
  • 23.3. Choosing between the binomial and Poisson models
  • Chapter Twenty-Four: The Kaplan–Meier estimator (CT4: §8)
  • Abstract
  • 24.1. Mortality investigation with censoring
  • 24.2. Notation for the investigation
  • 24.3. The likelihood function
  • 24.4. Kaplan–Meier estimator
  • 24.5. Application of the KM method
  • 24.6. Nelson–Aalen estimator
  • Chapter Twenty-Five: The Cox regression model (CT4: §9)
  • Abstract
  • 25.1. The Cox model
  • 25.2. Proportional hazards
  • 25.3. Partial likelihood
  • 25.4. Example of Cox model
  • 25.5. Breslow's approximation
  • 25.6. Example of Breslow's approximation
  • 25.7. Applying the Cox model
  • Chapter Twenty-Six: Graduation
  • Abstract
  • 26.1. Some issues with raw estimates
  • 26.2. Definition of ‘graduation’
  • 26.3. Aim of the graduation process
  • 26.4. Do mortality rates progress smoothly?
  • Chapter Twenty-Seven: Graduation techniques (CT4: §12)
  • Abstract
  • 27.1. Pros and cons of graphical techniques
  • 27.2. Mathematical graduation methods
  • 27.3. Makeham and Gompertz
  • 27.4. Example for estimating the Gompertz curve
  • 27.5. Perk's curve
  • 27.6. Barnett's formula
  • 27.7. Heligman–Pollard
  • Chapter Twenty-Eight: Methods of estimation of the parameters
  • Abstract
  • 28.1. Maximum likelihood (MLE)
  • 28.2. Least squares (OLS/WLS)
  • 28.3. Minimum chi-square method (MCSM)
  • 28.4. Which estimation method to use?
  • 28.5. Pros and cons of the mathematical approach
  • 28.6. Standard table graduation
  • 28.7. Variations of the standard table approach
  • 28.8. Summary of the standard table approach
  • Chapter Twenty-Nine: Assessing a graduation (CT4: §13)
  • Abstract
  • 29.1. Testing for smoothness
  • 29.2. Testing for fidelity
  • 29.3. Standardised mortality differences zx
  • 29.4. The χ2 test for goodness-of-fit
  • 29.5. The χ2 test for normality
  • 29.6. The sign test for overall bias
  • 29.7. Example of χ2 test and sign tests
  • 29.8. The cumulative test for goodness-of-fit
  • 29.9. Tests based on ordering
  • 29.10. The grouped signs test
  • 29.11. Example of the grouped signs test
  • 29.12. The serial correlation test
  • Chapter Thirty: Summary: estimation and graduation
  • Abstract
  • 30.1. Modelling
  • 30.2. Graduation
  • Chapter Thirty-One: Experience rating and Markov processes
  • Abstract
  • 31.1. The risk premium
  • 31.2. Exposures
  • 31.3. Exposures and asymmetry in information
  • 31.4. Experience rating
  • 31.5. NCB schemes
  • 31.6. Modelling NCB schemes
  • 31.7. Equilibrium distribution
  • 31.8. Equilibrium distribution
  • 31.9. Exercises
  • Appendix A: International actuarial notation
  • Discrete time
  • Continuous time
  • AMC00 – life table and benefit values
  • Appendix B: Useful mathematical techniques
  • B.1. Eigenvalues and eigenvectors
  • B.2. Diagonalisation
  • B.3. The benefits of diagonalisation
  • B.4. Nondiagonisable matrices
  • Appendix C: Exercises for Section 1.9 – defined benefits
  • C.1. Assignment
  • C.2. Assignment
  • C.3. Assignment
  • C.4. Assignment
  • C.5. Supplementary material
  • Appendix D: Sample exams
  • D.1. Sample Exam 1
  • D.2. Sample Exam 2
  • D.3. Sample Exam 3
  • D.4. Sample Exam 4
  • Bibliography
  • Bibliography
  • Index

Product details

  • No. of pages: 262
  • Language: English
  • Copyright: © Academic Press 2022
  • Published: October 29, 2021
  • Imprint: Academic Press
  • Paperback ISBN: 9780323901727
  • eBook ISBN: 9780323901734

About the Author

Andrew Leung

Dr Andrew Leung is a fellow of the actuarial professions in both the UK and Australia. He also has post graduate qualifications in pure mathematics and mathematical economics. He has worked for international actuarial consulting firms, in banking, insurance and investment. For more than 10 years he led the professional investment courses for the Australian profession and was the editor of the Australian Actuarial Journal. More recently, he established the actuarial course at Monash University, Melbourne, on which his book is based.

Affiliations and Expertise

Senior Consultant and Research Associate, Towers Perrin and Russell Investments; Senior Lecturer and program founder in Actuarial Studies, Monash University, Australia

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