Fundamentals Of Actuarial Mathematics, Third Edition.pdf
Provides a comprehensive coverage of both the deterministic and stochastic models of life contingencies, risk theory, credibility theory, multi-state models, and an introduction to modern mathematical finance. New edition restructures the material to fit into modern computational methods and provides several spreadsheet examples throughout. Covers the syllabus for the Institute of Actuaries subject CT5, Contingencies Includes new chapters covering stochastic investments returns, universal life insurance. Elements of option pricing and the Black-Scholes formula will be introduced.
Preface Notation Index 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 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 Constant interest 2.7 Values and actuarial equivalence 2.8 Vector Notation 2.9 Regular pattern cashflows 2.10 Balances and reserves 2.11 Time shifting and the splitting identity 2.12 Internal rates of return 2.13 Standard notation and terminology 2.14 Spreadsheet calculations 2.15 Notes and references Exercises Spreadsheet exercises 3 The life table 3.1 Basic definitions 3.2 Probabilities 3.3 Constructing the life table from the values of qx 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 Exercises 4 Life annuities 4.1 Introduction 4.2 Calculating annuity premiums 4.3 The interest and survivorship discount function 4.3.1 The basic definition 4.3.2 Relations between yx for various values of x 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 Exercises 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 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 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 Exercises 7 Fractional durations 7.1 Introduction 7.2 Cashflows discounted with interest only 7.3 Life annuities paid mthly 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 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 calculation 8.11 Further approximations from the life table 8.12 Standard actuarial notation and terminology 8.13 Notes and references Exercises 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 Exercises 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 10.13 Notes and references 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 Exercises 12 Expenses and Profits 12.1 Introduction 12.2 Effect on reserves 12.3 Realistic reserve and balance calculations 12.4 Profit Measurement 12.5 Notes and references Exercises 13 Specialized topics 13.1 Universal Life 13.2 Variable annuities 13.3 Pension Plans Exercises 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 14.10 Notes and references Exercises 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 rL 15.8 Standard notation and terminology 15.9 Notes and references Exercises 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 Exercises 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 17.8 Notes and references Exercises 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 18.8 Notes and references Exercises 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 19.7 Notes and references Exercises 20 Introduction to the Mathematics of Financial Markets 20.1 Introduction 20.2 Modeling 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 20.14 Notes and references Exercises 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 21.12 Notes and references Exercises 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 22.6 Notes and References Exercises 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 23.8 Notes and references Exercises 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 Exercises Answers to exercises 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.11.1 The Binomial A.11.2 The Poisson A.11.3 The negative binomial and geometric distributions A.11.4 The Continuous uniform A.11.5 The normal distribution A.11.6 The gamma and exponential distributions A.11.7 The lognormal distribution A.11.8 The Pareto distribution A.12 Convolution A.12.1 The discrete case A.12.2 The continuous case A.12.3 Notation and remarks A.13 Mixtures References Index