# Fundamentals of Actuarial Mathematics, 3ed

ISBN: 9788126574872

552 pages

Exclusively distributed by Shri Adhya Educational Books

## Description

The book is an introduction to actuarial mathematics and features new topics including an introduction to time diagrams and credibility theory, multi-state models and continuous time models. More complex types of contingent insurances will be featured and will include cases which involve durations or benefit amounts contingent on the time of the first death and will include applications on credit risk in life annuities. This new edition also includes additional material which is on the current syllabus of the Society of Actuaries examination on Life Contingencies. Furthermore a new chapter dealing with stochastic investment returns is also featured introducing the reader to various new ideas including the elements of option pricing.

Preface

Acknowledgements

Notation index

Part I The Deterministic Life Contingencies 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 The basic deterministic model

2.1 Cash flows

2.2 An analogy with currencies

2.3 Discount functions

2.4 Calculating the discount function

2.5 Interest and discount rates

2.6 Constant interest

2.7 Values and actuarial equivalence

2.8 Vector notation

2.9 Regular pattern cash flows

2.10 Balances and reserves

2.11 Time shifting and the splitting identity

2.11 Change of discount function

2.12 Internal rates of return

2.13 Forward prices and term structure

2.14 Standard notation and terminology

2.15 Spreadsheet calculations

3 The life table

3.1 Basic definitions

3.2 Probabilities

3.3 Constructing the life table from the values of q_{x}

3.4 Life expectancy

3.5 Choice of life tables

3.6 Standard notation and terminology

3.7 A sample table

4 Life annuities

4.1 Introduction

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 considerations

4.7 Standard notation and terminology

4.8 Spreadsheet calculations

5 Life insurance

5.1 Introduction

5.2 Calculating life insurance premiums

5.3 Types of life insurance

5.4 Combined insurance--annuity benefits

5.5 Insurances viewed as annuities

5.6 Summary of formulas

5.7 A general insurance--annuity identity

5.8 Standard notation and terminology

5.9 Spreadsheet applications

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 contract

6.5 Bases for reserves

6.6 Nonforfeiture 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

7 Fractional durations

7.1 Introduction

7.2 Cash flows discounted with interest only

7.3 Life annuities paid

7.4 Immediate annuities

7.5 Approximation and computation

7.6 Fractional period premiums and reserves

7.7 Reserves at fractional durations

7.8 Standard notation and terminology

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 calculation

8.11 Further approximations from the life table

8.12 Standard actuarial notation and terminology

9 Select mortality

9.1 Introduction

9.2 Select and ultimate tables

9.3 Changes in formulas

9.4 Projections in annuity tables

9.5 Further remarks

10 Multiple-life contracts

10.1 Introduction

10.2 The joint-life status

10.3 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 contract

10.7 The general two-life insurance contract

10.8 Contingent insurances

10.9 Duration problems

10.10 Applications to annuity credit risk

10.11 Standard notation and terminology

10.12 Spreadsheet applications

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

12 Expenses and Profits

12.1 Introduction

12.2 Effect on reserves

12.3 Realistic reserve and balance calculations

12.4 Profit measurement

13 Specialized topics

13.1 Universal life

13.2 Variable annuities

13.3 Pension plans

Part II The Stochastic Life Contingencies Model

14 Survival distributions and failure times

14.1 Introduction to survival distributions

14.2 The discrete case

14.3 The continuous case

14.4 Examples

14.5 Shifted distributions

14.6 The standard approximation

14.7 The stochastic life table

14.8 Life expectancy in the stochastic model

14.9 Stochastic interest rates

15 The stochastic approach to insurance and annuities

15.1 Introduction

15.2 The stochastic approach to insurance benefits

15.3 The stochastic approach to annuity benefits

15.4 Deferred contracts

15.5 The stochastic approach to reserves

15.6 The stochastic approach to premiums

15.7 The variance of _{r}L

15.8 Standard notation and terminology

16 Simplifications under level benefit contracts

16.1 Introduction

16.2 Variance calculations in the continuous case

16.3 Variance calculations in the discrete case

16.4 Exact distributions

16.5 Some non-level benefit examples

17 The minimum failure time

17.1 Introduction

17.2 Joint distributions

17.3 The distribution of T

17.4 The joint distribution of (T,J)

17.5 Other problems

17.6 The common shock model

17.7 Copulas

Part III Advanced Stochastic Models

18 An introduction to stochastic processes

18.1 Introduction

18.2 Markov chains

18.3 Martingales

18.4 Finite-state Markov chains

18.5 Introduction to continuous time processes

18.6 Poisson processes

18.7 Brownian motion

19 Multi-state models

19.1 Introduction

19.2 The discrete-time model

19.3 The continuous-time model

19.4 Recursion and differential equations for multi-state reserves

19.5 Profit testing in multi-state models

19.6 Semi-Markov models

20 Introduction to the Mathematics of Financial Markets

20.1 Introduction

20.2 Modelling prices in financial markets

20.3 Arbitrage

20.4 Option contracts

20.5 Option prices in the one-period binomial model

20.6 The multi-period binomial model

20.7 American options

20.8 A general financial market

20.9 Arbitrage-free condition

20.10 Existence and uniqueness of risk neutral measures

20.11 Completeness of markets

20.12 The Black--Scholes--Merton formula

20.13 Bond markets

Part IV Risk Theory

21 Compound distributions

21.1 Introduction

21.2 The mean and variance of S

21.3 Generating functions

21.4 Exact distribution of S

21.5 Choosing a frequency distribution

21.6 Choosing a severity distribution

21.7 Handling the point mass at 0

21.8 Counting claims of a particular type

21.9 The sum of two compound Poisson distributions

21.10 Deductibles and other modifications

21.11 A recursion formula for S

22 Risk assessment

22.1 Introduction

22.2 Utility theory

22.3 Convex and concave functions: Jensen's inequality

22.4 A general comparison method

22.5 Risk measures for capital adequacy

23 Ruin models

23.1 Introduction

23.2 A functional equation approach

23.3 The martingale approach to ruin theory

23.4 Distribution of the deficit at ruin

23.5 Recursion formulas

23.6 The compound Poisson surplus process

23.7 The maximal aggregate loss

24 Credibility theory

24.1 Introductory material

24.2 Conditional expectation and variance with respect to another random variable

24.3 General framework for Bayesian credibility

24.4 Classical examples

24.5 Approximations

24.6 Conditions for exactness

24.7 Estimation

Appendix A review of probability theory

A.1 Sample spaces and probability measures

A.2 Conditioning and independence

A.3 Random variables

A.4 Distributions

A.5 Expectations and moments

A.6 Expectation in terms of the distribution function

A.7 Joint distributions

A.8 Conditioning and independence for random variables

A.9 Moment generating functions

A.10 Probability generating functions

A.11 Some standard distributions

A.12 Convolution

A.13 Mixtures

Answers to exercises

References

Index