Fundamentals of Actuarial Mathematics

, by S. David Promislow (York University, Toronto, Canada)

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ISBN: 9780470016893 | 0470016892
Cover: Hardcover
Copyright: 2/1/2006

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Actuarial work is the application of mathematics and statistics to the analysis of financial problems in life insurance, pensions, general insurance and investments. This unique introduction to the topic employs both a deterministic and stochastic treatment of the subject. It combines interest theory and life contingencies in a unified manner as well as covering basic risk theory. Fundamentals of Actuarial Mathematics presents the concepts in an original, accessible style, assuming a minimal formal background. * Provides a complete review of necessary probability theory. * Covers the Society of Actuaries' syllabus on Actuarial Models. * Orders the topics specifically to facilitate learning, beginning with the simplest case of the deterministic discrete model, and then moving to the more complicated stochastic, continuous models. * Employs modern calculation and computing techniques, such as spreadsheets. * Contains a variety of exercises, both computational and theoretical. * Supported by a website featuring exercises and further examples. * Written by a highly respected academic with over 35 years teaching experience. This book will be invaluable to senior undergraduate and graduate students, as well as actuarial professionals working in the life insurance or pension fields. Applied mathematicians and economists will also benefit greatly from the clear presentation and numerous examples.

Preface.

PART 1 - THE DETERMINISTIC MODEL.

1 Introduction and Motivation.

1.1 Risk and insurance.

1.2 Deterministic versus stochastic models.

1.3 Finance and investments.

1.4 Adequacy and equity.

1.5 Reassessment.

1.6 Conclusion.

2 A general deterministic model.

2.1 Cashflows.

2.2 An analogy with currencies.

2.3 Discount functions.

2.4 Calculating the discount function.

2.5 Interest and discount rates.

2.6 The constant interest case.

2.7 Values and actuarial equivalence.

2.8 The case of equal cashflows.

2.9 Balances and reserves.

2.10 Time shifting and the splitting identity.

*2.11 Changing the discount function.

* 2.12 Internal rate of return.

2.13 Standard notation and terminology.

2.14 Spreadsheet applications.

2.15 Notes and references.

2.16 Exercises.

3 The life table.

3.1 Basic definitions.

3.2 Probabilities.

3.3 Constructing the life table from ox.

3.4 Life expectancy.

3.5 Choice of life tables.

3.6 Standard notation and terminology.

3.7 A sample table.

3.8 Notes and references.

3.9 Exercises.

4 Life annuities.

4.1 Introduction to life annuities.

4.2 Calculating annuity premiums.

4.3 The interest and survivorship discount function.

4.4 Guaranteed payments.

4.5 Deferred annuities with annual premiums.

4.6 Some practical aspects of annuities.

4.7 Standard notation and terminology.

4.8 Spreadsheet applications.

4.9 Exercises.

5 Life insurance.

5.1 Introduction to life insurance.

5.2 Calculating life insurance premiums.

5.3 Types of life insurance.

5.4 Combined benefits.

5.5 Insurances viewed as annuities.

5.6 Summary of formulas.

5.7 A general insurance-annuity identify.

5.8 Standard notation and terminology.

5.9 Spreadsheet applications.

5.10 Exercises.

6 Insurance and annuity reserves.

6.1 Introduction to reserves.

6.2 The general pattern of reserves.

6.3 Recursion.

6.4 Detailed analysis of an insurance or annuity contact.

6.5 Bases for reserves.

6.6 Nonforfwiture values.

6.7 Policies involving a return of the reserve.

6.8 Premium difference and paid up formulas.

6.9 Standard notation and terminology.

6.10 Spreadsheet applications.

6.11 Exercises.

7 Fractional durations.

7.1 Introduction.

7.2 Cashflows discounted at interest only.

7.3 Life annuities paid m-thly.

7.4 Immediate annuities.

7.5 Approximation and computation.

*7.6 Fractional period premiums and reserves.

7.7 Reserves at a fractional duration.

7.8 Notes and references.

7.9 Exercises.

8 Continuous payments.

8.1 Introduction to continuous annuities.

8.2 The force of discount.

8.3 The constant interest case.

8.4 Continuous life annuities.

8.5 The force of mortality.

8.6 Insurances payable at the moment of death.

8.7 Premiums and reserves.

8.8 The general insurance-annuity identity in the continuous case.

8.9 Differential equations for reserves.

8.10 Some examples of exact calculations.

8.11 Standard notations and terminology,.

8.12 Notes and references.

8.13 Exercises.

9 Select mortality.

9.1 Introduction.

9.2 Select and ultimate tables.

9.3 Changes in formulas.

9.4 Further remarks.

9.5 Exercises.

10 Multiple-life contracts.

10.1 Introduction.

10.2 The joint-life status.

10.3 The joint-life annuities and insurances.

10.4 Last-survivor annuities and insurances.

10.5 Moment of death insurances.

10.6 The general two-life annuity.

10.7 The general two-life insurance contract.

10.8 Contingent insurance.

10.9 Standard notation and terminology.

10.10 Spreadsheet applications.

10.11 Notes and references.

10.12 Exercises.

11 Multiple-decrement theory.

11.1 Introduction.

11.2 The basic model.

11.3 Insurances.

11.4 Determining the model from the forces of decrement.

11.5 The analogy with joint-life statuses.

11.6 A machine analogy.

11.7 Associated single-decrement tables.

11.8 Notes and references.

11.9 Exercises.

12 Expenses.

12.1 Introduction.

12.2 Effect on reserves.

12.3 Realistic reserve and balance calculations.

12.4 Notes and references.

12.5 Exercises.

PART 11: THE STOCHASTIC MODEL.

13 Survival distributions and failure times.

13.1 Introduction to survival distributions.

13.2 The discrete case.

13.3 The continuous case.

13.4 Examples.

13.5 Shifted distributions.

13.6 The standard approximation.

13.7 The stochastic life table.

13.8 Life expectancy in the stochastic model.

13.9 Notes and references.

13.10 Exercises.

14 The stochastic approach to life insurance and annuities.

14.1 Introduction.

14.2 The stochastic approach to life insurance.

14.3 The stochastic approach to life annuities.

14.4 Deferred contracts.

14.5 The stochastic approach to reserves.

14.6 The stochastic approach to premiums.

14.7 The variance of rL.].

14.8 Standard notation and terminology.

14.9 Notes and references.

14.10 Exercises.

15 Simplifications under constant benefit contracts.

15.1 Introduction.

15.2 Variances in the continuous case.

15.3 Variances in the discrete case.

15.4 Exact distributions.

15.5 Some examples with nonconstant benefits.

15.6 Exercises.

16 The minimum failure time.

16.1 Introduction.

16.2 Joint distribution.

16.3 The distribution of T.

16.4 The joint distribution of (T,J).

16.5 Approximations.

16.6 Other problems.

16.7 The common shock model.

16.8 Copulas.

16.9 Notes and references.

16.10 Exercises.

PART 111: RISK THEORY.

17 Compound distributions.

17.1 Introduction.

17.2 The mean and variance of S.

17.3 Generating functions.

17.4 Exact distribution of S.

17.5 Choosing a frequency distribution.

17.6 Choosing a severity distribution.

17.7 Handling the point mass at 0.

17.8 Counting claims of a particular type.

17.9 The sum of two compound Poisson distributions.

17.10 Deductibles and other modifications.

17.11 A recursion formula for S.

17.12 Notes and references.

17.13 Exercises.

18 An introduction to stochastic processes.

18.1 Introduction.

18.2 Markov Chains.

18.3 Examples.

18.4 Martingales.

18.5 Finite state Markov chains.

18.6 Multi-state insurances and annuities.

18.7 Notes and references.

18.8 Exercises.

19 Poisson processes.

19.1 Introduction.

19.2 Definition of a Poisson process.

19.3 Waiting times.

19.4 Some properties of the Poisson process.

19.5 Non-homogeneous Poisson process.

19.6 Compound Poisson process.

19.7 Notes and references.

19.8 Exercises.

20 Ruin Models.

20.1 Introduction.

20.2 A functional equation approach.

20.3 The martingale approach to ruin theory.

20.4 Distribution of the deficit at ruin.

20.5 Recursion formulas.

20.6 The compound Poisson surplus process.

20.7 The maximal aggregate loss.

20.8 Notes and references.

20.9 Exercises.

Appendix A.

A.1 Introduction.

A.2 Sample spaces and probability measures.

A.3 Conditioning and independence.

A.4 Random variables.

A.5 Distributions.

A.6 Expectations and moments.

A.7 Expectation in terms of the distribution function.

A.8 Joint distributions.

A.9 Conditioning and independence for random variables.

A.10 Convolution.

A.11 Moment generating functions.

A.12 Probability generating functions.

A.13 Mixtures.

Answers to Exercises.

Index.

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Fundamentals of Actuarial Mathematics

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