Actuarial Mathematics for Life Contingent Risks 3rd Edition by David C. M. Dickson, ISBN-13: 978-1108478083

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Actuarial Mathematics for Life Contingent Risks 3rd Edition by David C. M. Dickson, ISBN-13: 978-1108478083

[PDF eBook eTextbook]

• Publisher: ‎ Cambridge University Press; 3rd edition (January 30, 2020)
• Language: ‎ English
• 782 pages
• ISBN-10: ‎ 1108478085
• ISBN-13: ‎ 978-1108478083

The substantially updated third edition of the popular Actuarial Mathematics for Life Contingent Risks is suitable for advanced undergraduate and graduate students of actuarial science, for trainee actuaries preparing for professional actuarial examinations, and for life insurance practitioners who wish to increase or update their technical knowledge. The authors provide intuitive explanations alongside mathematical theory, equipping readers to understand the material in sufficient depth to apply it in real-world situations and to adapt their results in a changing insurance environment. Topics include modern actuarial paradigms, such as multiple state models, cash-flow projection methods and option theory, all of which are required for managing the increasingly complex range of contemporary long-term insurance products. Numerous exam-style questions allow readers to prepare for traditional professional actuarial exams, and extensive use of Excel ensures that readers are ready for modern, Excel-based exams and for the actuarial work environment.

Table of Contents:

Half-title

Series information

Title page

Copyright information

Dedication

Contents

Preface to the third edition

1 Introduction to life and long-term health insurance

1.1 Summary

1.2 Background

1.3 Traditional life insurance contracts

1.3.1 Introduction

1.3.2 Term insurance

1.3.3 Whole life insurance

1.3.4 Endowment insurance

1.3.5 Options and variations on traditional insurance

1.4 Modern insurance contracts

1.4.1 Why innovate?

1.4.2 Universal life insurance

1.4.3 Unitized with-profit

1.4.4 Equity-linked insurance

1.5 Marketing, pricing and issuing life insurance

1.5.1 Insurance distribution methods

1.5.2 Underwriting

1.5.3 Premiums

1.6 Life annuities

1.7 Long-term coverages in health insurance

1.7.1 Disability income insurance

1.7.2 Long-term care insurance

1.7.3 Critical illness insurance

1.7.4 Chronic illness insurance

1.8 Mutual and proprietary insurers

1.9 Other life contingent contracts

1.9.1 Continuing care retirement communities

1.9.2 Structured settlements

1.10 Pensions

1.10.1 Defined Benefit pensions

1.10.2 Defined Contribution

1.11 Typical problems

1.12 Notes and further reading

1.13 Exercises

2 Survival models

2.1 Summary

2.2 The future lifetime random variable

2.3 The force of mortality

2.3.1 Mortality laws

2.4 Actuarial notation

2.5 Mean and standard deviation of T[sub(x)]

2.6 Curtate future lifetime

2.6.1 K[sub(x)] and e[sub(x)]

2.6.2 Comparing [circ(e) sub(x)] and e[sub(x)]

2.7 Notes and further reading

2.8 Exercises

3 Life tables and selection

3.1 Summary

3.2 Life tables

3.3 Fractional age assumptions

3.3.1 Uniform distribution of deaths

3.3.2 Constant force of mortality

3.4 National life tables

3.5 Survival models for life insurance policyholders

3.6 Life insurance underwriting

3.7 Select and ultimate survival models

3.8 Notation and formulae for select survival models

3.9 Select life tables

3.10 Some comments on heterogeneity in mortality

3.11 Mortality improvement modelling

3.12 Mortality improvement scales

3.12.1 Single-factor mortality improvement scales

3.12.2 Two-factor mortality improvement scales

3.12.3 Cubic spline mortality improvement scales

3.13 Notes and further reading

3.14 Exercises

4 Insurance benefits

4.1 Summary

4.2 Introduction

4.3 Assumptions

4.4 Valuation of insurance benefits

4.4.1 Whole life insurance: the continuous case, [bar(A)sub(x)]

4.4.2 Whole life insurance: the annual case, [bar(A)sub(x)]

4.4.3 Whole life insurance: the 1/mthly case, A[sup((m))][sub(x)]

4.4.4 Recursions

4.4.5 Term insurance

4.4.6 Pure endowment

4.4.7 Endowment insurance

4.4.8 Deferred insurance

4.5 Relating [bar(A)sub(x)], A[sub(x)] and A[sup((m))][sub(x)]

4.5.1 Using the uniform distribution of deaths assumption

4.5.2 Using the claims acceleration approach

4.6 Variable insurance benefits

4.7 Functions for select lives

4.8 Notes and further reading

4.9 Exercises

5 Annuities

5.1 Summary

5.2 Introduction

5.3 Review of annuities-certain

5.4 Annual life annuities

5.4.1 Whole life annuity-due

5.4.2 Term annuity-due

5.4.3 Immediate life annuities

5.5 Annuities payable continuously

5.6 Annuities payable 1/mthly

5.7 Comparison of annuities by payment frequency

5.8 Deferred annuities

5.9 Guaranteed annuities

5.10 Increasing annuities

5.10.1 Arithmetically increasing annuities

5.10.2 Geometrically increasing annuities

5.11 Evaluating annuity functions

5.11.1 Recursions

5.11.2 Applying the UDD assumption

5.11.3 Woolhouse’s formula

5.12 Numerical illustrations

5.13 Functions for select lives

5.14 Notes and further reading

5.15 Exercises

6 Premium calculation

6.1 Summary

6.2 Preliminaries

6.2.1 Assumptions

6.3 The loss at issue random variable

6.4 The equivalence principle premium

6.4.1 Net premiums

6.4.2 Gross premiums

6.5 Profit

6.6 The portfolio percentile premium principle

6.7 Extra risks

6.7.1 Age rating

6.7.2 Constant addition to μ[sub(x)]

6.7.3 Constant multiple of mortality rates

6.8 Notes and further reading

6.9 Exercises

7 Policy values

7.1 Summary

7.2 Policies with annual cash flows

7.2.1 The future loss random variable

7.2.2 Policy values for policies with annual cash flows

7.2.3 Recursive formulae for policy values

7.2.4 Analysis of surplus

7.2.5 Asset shares

7.3 Policy values for policies with cash flows at 1/mthly intervals

7.3.1 Recursions with 1/mthly cash flows

7.3.2 Valuation between premium dates

7.4 Policy values with continuous cash flows

7.4.1 Thiele’s differential equation

7.4.2 Numerical solution of Thiele’s differential equation

7.5 Policy alterations

7.6 Retrospective policy values

7.6.1 Prospective and retrospective valuation

7.6.2 Defining the retrospective net premium policy value

7.7 Negative policy values

7.8 Deferred acquisition expenses and modified net premium reserves

7.8.1 Full Preliminary Term reserve

7.9 Other reserves

7.10 Notes and further reading

7.11 Exercises

8 Multiple state models

8.1 Summary

8.2 Examples of multiple state models

8.2.1 The alive–dead model

8.2.2 Term insurance with increased benefit on accidental death

8.2.3 The permanent disability model

8.2.4 The sickness–death model

8.3 Assumptions and notation

8.4 Formulae for probabilities

8.4.1 Kolmogorov’s forward equations

8.5 Numerical evaluation of probabilities

8.6 State-dependent insurance and annuity functions

8.6.1 State-dependent annuities

8.7 Premiums

8.8 Policy values

8.8.1 Recursions for state-dependent policy values

8.8.2 General recursion for h-yearly cash flows

8.8.3 Thiele’s differential equation

8.9 Applications of multiple state models in long-term health and disability insurance

8.9.1 Disability income insurance

8.9.2 Long-term care

8.9.3 Critical illness insurance

8.9.4 Continuing care retirement communities

8.9.5 Structured settlements

8.10 Markov multiple state models in discrete time

8.10.1 The Chapman–Kolmogorov equations

8.10.2 Transition matrices

8.11 Notes and further reading

8.12 Exercises

9 Multiple decrement models

9.1 Summary

9.2 Examples of multiple decrement models

9.3 Actuarial functions for multiple decrement models

9.4 Multiple decrement tables

9.4.1 Fractional age assumptions for decrements

9.5 Constructing a multiple decrement table

9.5.1 Deriving independent rates from dependent rates

9.5.2 Deriving dependent rates from independent rates

9.6 Comments on multiple decrement notation

9.7 Transitions at exact ages

9.8 Exercises

10 Joint life and last survivor benefits

10.1 Summary

10.2 Joint life and last survivor benefits

10.3 Joint life notation

10.4 Independent future lifetimes

10.5 A multiple state model for independent future lifetimes

10.6 A model with dependent future lifetimes

10.7 The common shock model

10.8 Notes and further reading

10.9 Exercises

11 Pension mathematics

11.1 Summary

11.2 Introduction

11.3 The salary scale function

11.4 Setting the contribution for a DC plan

11.5 The service table

11.6 Valuation of final salary plans

11.6.1 Accrued benefits

11.6.2 A general formula for the EPV of the projected accrued age retirement pension

11.6.3 Withdrawal benefits

11.6.4 Valuing the current accrued benefit

11.7 Valuing career average earnings plans

11.8 Funding the benefits

11.9 Projected Unit Credit funding

11.9.1 The normal contribution formula using PUC funding

11.10 Traditional Unit Credit funding

11.10.1 The normal contribution formula using TUC funding

11.11 Comparing PUC and TUC funding methods

11.12 Retiree health benefits

11.12.1 Introduction

11.12.2 Valuing retiree health benefits

11.12.3 Funding retiree health benefits

11.13 Notes and further reading

11.14 Exercises

12 Yield curves and non-diversifiable risk

12.1 Summary

12.2 The yield curve

12.3 Valuation of insurances and life annuities

12.3.1 Replicating the cash flows of a traditional non-participating product

12.4 Diversifiable and non-diversifiable risk

12.4.1 Diversifiable mortality risk

12.4.2 Non-diversifiable risk

12.5 Monte Carlo simulation

12.6 Notes and further reading

12.7 Exercises

13 Emerging costs for traditional life insurance

13.1 Summary

13.2 Introduction

13.3 Profit testing a term insurance policy

13.3.1 Time step

13.3.2 Profit test basis

13.3.3 Incorporating reserves

13.3.4 Profit signature

13.4 Profit testing principles

13.4.1 Assumptions

13.4.2 The profit vector

13.4.3 The profit signature

13.4.4 The net present value

13.4.5 Notes on the profit testing method

13.5 Profit measures

13.6 Using the profit test to calculate the premium

13.7 Using the profit test to calculate reserves

13.8 Profit testing for participating insurance

13.9 Profit testing for multiple state-dependent insurance

13.10 Notes

13.11 Exercises

14 Universal life insurance

14.1 Summary

14.2 Introduction

14.3 Universal life insurance

14.3.1 Introduction

14.3.2 Key design features

14.3.3 Projecting account values

14.3.4 Profit testing Universal life policies

14.3.5 Universal life Type B profit test

14.3.6 Universal life Type A profit test

14.3.7 No lapse guarantees

14.3.8 Comments on UL profit testing

14.4 Notes and further reading

14.5 Exercises

15 Emerging costs for equity-linked insurance

15.1 Summary

15.2 Equity-linked insurance

15.3 Deterministic profit testing for equity-linked insurance

15.4 Stochastic profit testing

15.5 Stochastic pricing

15.6 Stochastic reserving

15.6.1 Reserving for policies with non-diversifiable risk

15.6.2 Quantile reserving

15.6.3 CTE reserving

15.6.4 Comments on reserving

15.7 Notes and further reading

15.8 Exercises

16 Option pricing

16.1 Summary

16.2 Introduction

16.3 The ‘no arbitrage’ assumption

16.4 Options

16.5 The binomial option pricing model

16.5.1 Assumptions

16.5.2 Pricing over a single time period

16.5.3 Pricing over two time periods

16.5.4 Summary of the binomial model option pricing technique

16.6 The Black–Scholes–Merton model

16.6.1 The model

16.6.2 The Black–Scholes–Merton option pricing formula

16.7 Notes and further reading

16.8 Exercises

17 Embedded options

17.1 Summary

17.2 Introduction

17.3 Guaranteed minimum maturity benefit

17.3.1 Pricing

17.3.2 Reserving

17.4 Guaranteed minimum death benefit

17.4.1 Pricing

17.4.2 Reserving

17.5 Funding methods for embedded options

17.6 Risk management

17.7 Profit testing

17.8 Notes and further reading

17.9 Exercises

18 Estimating survival models

18.1 Summary

18.2 Introduction

18.3 Actuarial lifetime data

18.3.1 Left truncation

18.3.2 Right censoring

18.4 Non-parametric survival function estimation

18.4.1 The empirical distribution for seriatim data

18.4.2 The empirical distribution for grouped data

18.4.3 The Kaplan–Meier estimate

18.4.4 The Nelson–Aalen estimator

18.5 The alive–dead model

18.5.1 Notes on the alive–dead model

18.6 Estimation of transition intensities in multiple state models

18.7 Comments

18.8 Notes and further reading

18.9 Exercises

19 Stochastic longevity models

19.1 Summary

19.2 Introduction

19.3 The Lee–Carter model

19.4 The Cairns–Blake–Dowd models

19.4.1 The original CBD model

19.4.2 Actuarial applications of stochastic longevity models

19.4.3 Notes on stochastic longevity models

19.5 Notes and further reading

19.6 Exercises

Appendix A Probability and statistics

A.1 Probability distributions

A.1.1 Binomial distribution

A.1.2 Uniform distribution

A.1.3 Normal distribution

A.1.4 Lognormal distribution

A.2 The central limit theorem

A.3 Functions of a random variable

A.3.1 Discrete random variables

A.3.2 Continuous random variables

A.3.3 Mixed random variables

A.4 Conditional expectation and conditional variance

A.5 Maximum likelihood estimation

A.5.1 The likelihood function

A.5.2 Finding the maximum likelihood estimates

A.5.3 Properties of maximum likelihood estimates

A.6 Notes and further reading

Appendix B Numerical techniques

B.1 Numerical integration

B.1.1 The trapezium rule

B.1.2 Repeated Simpson’s rule

B.1.3 Integrals over an infinite interval

B.2 Woolhouse’s formula

B.3 Notes and further reading

Appendix C Monte Carlo simulation

C.1 The inverse transform method

C.2 Simulation from a normal distribution

C.2.1 The Box–Muller method

C.2.2 The polar method

C.3 Notes and further reading

Appendix D Tables

D.1 The Standard Select and Ultimate Life Tables

D.2 Joint life functions

D.3 Standard Sickness–Death tables

D.4 Pension plan service table

References

Index

David C. M. Dickson holds a Ph.D. in Actuarial Science from Heriot-Watt University, Edinburgh, and is a Fellow of the Institute and Faculty of Actuaries and the Institute of Actuaries of Australia. He lectured for seven years at Heriot-Watt University before moving to the University of Melbourne in 1993. In 2000 Dickson was appointed to the Chair in Actuarial Studies in Melbourne. He was Head of the Department of Economics from 2016 to 2018. He has twice been awarded the H.M. Jackson Prize of the Institute of Actuaries of Australia, most recently for his book Insurance Risk and Ruin (2016).

Mary R. Hardy is Professor of Actuarial Science at the University of Waterloo, Ontario. She earned her Ph.D. in Actuarial Science from Heriot-Watt University, Edinburgh,where she lectured for eleven years before moving to the University of Waterloo in 1997. She is a Fellow of the Institute and Faculty of Actuaries, and of the Society of Actuaries. In 2007 she was awarded the Chartered Enterprise Risk Analyst designation of the Society of Actuaries, through their thought-leader recognition program. In 2013 she was awarded the Finlaison Medal of the Institute and Faculty of Actuaries for services to the actuarial profession, in research, teaching and governance.

Howard R. Waters is Professor Emeritus at the Department of Actuarial Mathematics and Statistics at Heriot-Watt University, Edinburgh. He holds a D.Phil. in mathematics from the University of Oxford, and worked as a consulting actuary for several years before joining Heriot-Watt University. He is a Fellow of the Institute and Faculty of Actuaries. He was awarded the Finlaison medal of the Institute of Actuaries in 2006 for services to actuarial research and education.

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