Actuarial Principles

Lifetables and Mortality Models

1st Edition - October 29, 2021
Imprint: Academic Press
Author: Andrew Leung
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 9 0 1 7 2 - 7
eBook ISBN:
9 7 8 - 0 - 3 2 3 - 9 0 1 7 3 - 4

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… Read more

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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.

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

Life Sciences

Physical Sciences & Engineering

Social Sciences & Humanities

Health