Fundamentals of Actuarial Mathematics

S. David Promislow is the author of Fundamentals of Actuarial Mathematics, 3rd Edition, published by Wiley.

• Preface xviiAcknowledgements xxiNotation index xxiiiPart I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 11 Introduction and motivation 31.1 Risk and insurance 31.2 Deterministic versus stochastic models 41.3 Finance and investments 51.4 Adequacy and equity 51.5 Reassessment 61.6 Conclusion 62 The basic deterministic model 72.1 Cash flows 72.2 An analogy with currencies 82.3 Discount functions 92.4 Calculating the discount function 112.5 Interest and discount rates 122.6 Constant interest 122.7 Values and actuarial equivalence 132.8 Vector notation 172.9 Regular pattern cash flows 182.10 Balances and reserves 202.11 Time shifting and the splitting identity 262.11 Change of discount function 272.12 Internal rates of return 282.13 Forward prices and term structure 302.14 Standard notation and terminology 332.15 Spreadsheet calculations 34Notes and references 35Exercises 353 The life table 393.1 Basic definitions 393.2 Probabilities 403.3 Constructing the life table from the values of qx 413.4 Life expectancy 423.5 Choice of life tables 443.6 Standard notation and terminology 443.7 A sample table 45Notes and references 45Exercises 454 Life annuities 474.1 Introduction 474.2 Calculating annuity premiums 484.3 The interest and survivorship discount function 504.4 Guaranteed payments 534.5 Deferred annuities with annual premiums 554.6 Some practical considerations 564.7 Standard notation and terminology 574.8 Spreadsheet calculations 58Exercises 595 Life insurance 615.1 Introduction 615.2 Calculating life insurance premiums 615.3 Types of life insurance 645.4 Combined insurance–annuity benefits 645.5 Insurances viewed as annuities 695.6 Summary of formulas 705.7 A general insurance–annuity identity 705.8 Standard notation and terminology 725.9 Spreadsheet applications 74Exercises 746 Insurance and annuity reserves 786.1 Introduction to reserves 786.2 The general pattern of reserves 816.3 Recursion 826.4 Detailed analysis of an insurance or annuity contract 836.5 Bases for reserves 876.6 Nonforfeiture values 886.7 Policies involving a return of the reserve 886.8 Premium difference and paid-up formulas 906.9 Standard notation and terminology 916.10 Spreadsheet applications 93Exercises 947 Fractional durations 987.1 Introduction 987.2 Cash flows discounted with interest only 997.3 Life annuities paid7.4 Immediate annuities 1047.5 Approximation and computation 1057.6 Fractional period premiums and reserves 1067.7 Reserves at fractional durations 1077.8 Standard notation and terminology 109Exercises 1098 Continuous payments 1128.1 Introduction to continuous annuities 1128.2 The force of discount 1138.3 The constant interest case 1148.4 Continuous life annuities 1158.5 The force of mortality 1188.6 Insurances payable at the moment of death 1198.7 Premiums and reserves 1228.8 The general insurance–annuity identity in the continuous case 1238.9 Differential equations for reserves 1248.10 Some examples of exact calculation 1258.11 Further approximations from the life table 1298.12 Standard actuarial notation and terminology 131Notes and references 132Exercises 1329 Select mortality 1379.1 Introduction 1379.2 Select and ultimate tables 1389.3 Changes in formulas 1399.4 Projections in annuity tables 1419.5 Further remarks 142Exercises 14210 Multiple-life contracts 14410.1 Introduction 14410.2 The joint-life status 14410.3 Joint-life annuities and insurances 14610.4 Last-survivor annuities and insurances 14710.5 Moment of death insurances 14910.6 The general two-life annuity contract 15010.7 The general two-life insurance contract 15210.8 Contingent insurances 15310.9 Duration problems 15610.10 Applications to annuity credit risk 15910.11 Standard notation and terminology 16010.12 Spreadsheet applications 161Notes and references 161Exercises 16111 Multiple-decrement theory 16611.1 Introduction 16611.2 The basic model 16611.3 Insurances 16911.4 Determining the model from the forces of decrement 17011.5 The analogy with joint-life statuses 17111.6 A machine analogy 17111.7 Associated single-decrement tables 175Notes and references 181Exercises 18112 Expenses and Profits 18412.1 Introduction 18412.2 Effect on reserves 18612.3 Realistic reserve and balance calculations 18712.4 Profit measurement 189Notes and references 196Exercises 19613 Specialized topics 19913.1 Universal life 19913.2 Variable annuities 20313.3 Pension plans 204Exercises 207Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL 20914 Survival distributions and failure times 21114.1 Introduction to survival distributions 21114.2 The discrete case 21214.3 The continuous case 21314.4 Examples 21514.5 Shifted distributions 21614.6 The standard approximation 21714.7 The stochastic life table 21914.8 Life expectancy in the stochastic model 22014.9 Stochastic interest rates 221Notes and references 222Exercises 22215 The stochastic approach to insurance and annuities 22415.1 Introduction 22415.2 The stochastic approach to insurance benefits 22515.3 The stochastic approach to annuity benefits 22915.4 Deferred contracts 23315.5 The stochastic approach to reserves 23315.6 The stochastic approach to premiums 23515.7 The variance of rL 24115.8 Standard notation and terminology 243Notes and references 244Exercises 24416 Simplifications under level benefit contracts 24816.1 Introduction 24816.2 Variance calculations in the continuous case 24816.3 Variance calculations in the discrete case 25016.4 Exact distributions 25216.5 Some non-level benefit examples 254Exercises 25617 The minimum failure time 25917.1 Introduction 25917.2 Joint distributions 25917.3 The distribution of T 26117.4 The joint distribution of (T,J) 26117.5 Other problems 27017.6 The common shock model 27117.7 Copulas 273Notes and references 276Exercises 276Part III ADVANCED STOCHASTIC MODELS 27918 An introduction to stochastic processes 28118.1 Introduction 28118.2 Markov chains 28318.3 Martingales 28618.4 Finite-state Markov chains 28718.5 Introduction to continuous time processes 29318.6 Poisson processes 29318.7 Brownian motion 295Notes and references 299Exercises 30019 Multi-state models 30419.1 Introduction 30419.2 The discrete-time model 30519.3 The continuous-time model 31119.4 Recursion and differential equations for multi-state reserves 32419.5 Profit testing in multi-state models 32719.6 Semi-Markov models 328Notes and references 328Exercises 32920 Introduction to the Mathematics of Financial Markets 33320.1 Introduction 33320.2 Modelling prices in financial markets 33320.3 Arbitrage 33420.4 Option contracts 33720.5 Option prices in the one-period binomial model 33920.6 The multi-period binomial model 34220.7 American options 34620.8 A general financial market 34820.9 Arbitrage-free condition 35120.10 Existence and uniqueness of risk neutral measures 35320.11 Completeness of markets 35820.12 The Black–Scholes–Merton formula 36120.13 Bond markets 364Notes and references 372Exercises 373Part IV RISK THEORY 37521 Compound distributions 37721.1 Introduction 37721.2 The mean and variance of S 37921.3 Generating functions 38021.4 Exact distribution of S 38121.5 Choosing a frequency distribution 38121.6 Choosing a severity distribution 38321.7 Handling the point mass at 0 38421.8 Counting claims of a particular type 38521.9 The sum of two compound Poisson distributions 38721.10 Deductibles and other modifications 38821.11 A recursion formula for S 393Notes and references 398Exercises 39822 Risk assessment 40322.1 Introduction 40322.2 Utility theory 40322.3 Convex and concave functions: Jensen’s inequality 40622.4 A general comparison method 40822.5 Risk measures for capital adequacy 412Notes and references 417Exercises 41723 Ruin models 42023.1 Introduction 42023.2 A functional equation approach 42223.3 The martingale approach to ruin theory 42423.4 Distribution of the deficit at ruin 43323.5 Recursion formulas 43423.6 The compound Poisson surplus process 43823.7 The maximal aggregate loss 441Notes and references 445Exercises 44524 Credibility theory 44924.1 Introductory material 44924.2 Conditional expectation and variance with respect to another random variable 45324.3 General framework for Bayesian credibility 45724.4 Classical examples 45924.5 Approximations 46224.6 Conditions for exactness 46524.7 Estimation 469Notes and References 473Exercises 473Appendix A review of probability theory 477A.1 Sample spaces and probability measures 477A.2 Conditioning and independence 479A.3 Random variables 479A.4 Distributions 480A.5 Expectations and moments 481A.6 Expectation in terms of the distribution function 482A.7 Joint distributions 483A.8 Conditioning and independence for random variables 485A.9 Moment generating functions 486A.10 Probability generating functions 487A.11 Some standard distributions 489A.12 Convolution 495A.13 Mixtures 499Answers to exercises 501References 517Index 523