Fundamentals of Actuarial Mathematics

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

Preface xvii

Acknowledgements xxi

Notation index xxiii

Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 1

1 Introduction and motivation 3

1.1 Risk and insurance 3

1.2 Deterministic versus stochastic models 4

1.3 Finance and investments 5

1.4 Adequacy and equity 5

1.5 Reassessment 6

1.6 Conclusion 6

2 The basic deterministic model 7

2.1 Cash flows 7

2.2 An analogy with currencies 8

2.3 Discount functions 9

2.4 Calculating the discount function 11

2.5 Interest and discount rates 12

2.6 Constant interest 12

2.7 Values and actuarial equivalence 13

2.8 Vector notation 17

2.9 Regular pattern cash flows 18

2.10 Balances and reserves 20

2.11 Time shifting and the splitting identity 26

2.11 Change of discount function 27

2.12 Internal rates of return 28

2.13 Forward prices and term structure 30

2.14 Standard notation and terminology 33

2.15 Spreadsheet calculations 34

Notes and references 35

Exercises 35

3 The life table 39

3.1 Basic definitions 39

3.2 Probabilities 40

3.3 Constructing the life table from the values of qx 41

3.4 Life expectancy 42

3.5 Choice of life tables 44

3.6 Standard notation and terminology 44

3.7 A sample table 45

Notes and references 45

Exercises 45

4 Life annuities 47

4.1 Introduction 47

4.2 Calculating annuity premiums 48

4.3 The interest and survivorship discount function 50

4.4 Guaranteed payments 53

4.5 Deferred annuities with annual premiums 55

4.6 Some practical considerations 56

4.7 Standard notation and terminology 57

4.8 Spreadsheet calculations 58

Exercises 59

5 Life insurance 61

5.1 Introduction 61

5.2 Calculating life insurance premiums 61

5.3 Types of life insurance 64

5.4 Combined insurance–annuity benefits 64

5.5 Insurances viewed as annuities 69

5.6 Summary of formulas 70

5.7 A general insurance–annuity identity 70

5.8 Standard notation and terminology 72

5.9 Spreadsheet applications 74

Exercises 74

6 Insurance and annuity reserves 78

6.1 Introduction to reserves 78

6.2 The general pattern of reserves 81

6.3 Recursion 82

6.4 Detailed analysis of an insurance or annuity contract 83

6.5 Bases for reserves 87

6.6 Nonforfeiture values 88

6.7 Policies involving a return of the reserve 88

6.8 Premium difference and paid-up formulas 90

6.9 Standard notation and terminology 91

6.10 Spreadsheet applications 93

Exercises 94

7 Fractional durations 98

7.1 Introduction 98

7.2 Cash flows discounted with interest only 99

7.3 Life annuities paid

7.4 Immediate annuities 104

7.5 Approximation and computation 105

7.6 Fractional period premiums and reserves 106

7.7 Reserves at fractional durations 107

7.8 Standard notation and terminology 109

Exercises 109

8 Continuous payments 112

8.1 Introduction to continuous annuities 112

8.2 The force of discount 113

8.3 The constant interest case 114

8.4 Continuous life annuities 115

8.5 The force of mortality 118

8.6 Insurances payable at the moment of death 119

8.7 Premiums and reserves 122

8.8 The general insurance–annuity identity in the continuous case 123

8.9 Differential equations for reserves 124

8.10 Some examples of exact calculation 125

8.11 Further approximations from the life table 129

8.12 Standard actuarial notation and terminology 131

Notes and references 132

Exercises 132

9 Select mortality 137

9.1 Introduction 137

9.2 Select and ultimate tables 138

9.3 Changes in formulas 139

9.4 Projections in annuity tables 141

9.5 Further remarks 142

Exercises 142

10 Multiple-life contracts 144

10.1 Introduction 144

10.2 The joint-life status 144

10.3 Joint-life annuities and insurances 146

10.4 Last-survivor annuities and insurances 147

10.5 Moment of death insurances 149

10.6 The general two-life annuity contract 150

10.7 The general two-life insurance contract 152

10.8 Contingent insurances 153

10.9 Duration problems 156

10.10 Applications to annuity credit risk 159

10.11 Standard notation and terminology 160

10.12 Spreadsheet applications 161

Notes and references 161

Exercises 161

11 Multiple-decrement theory 166

11.1 Introduction 166

11.2 The basic model 166

11.3 Insurances 169

11.4 Determining the model from the forces of decrement 170

11.5 The analogy with joint-life statuses 171

11.6 A machine analogy 171

11.7 Associated single-decrement tables 175

Notes and references 181

Exercises 181

12 Expenses and Profits 184

12.1 Introduction 184

12.2 Effect on reserves 186

12.3 Realistic reserve and balance calculations 187

12.4 Profit measurement 189

Notes and references 196

Exercises 196

13 Specialized topics 199

13.1 Universal life 199

13.2 Variable annuities 203

13.3 Pension plans 204

Exercises 207

Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL 209

14 Survival distributions and failure times 211

14.1 Introduction to survival distributions 211

14.2 The discrete case 212

14.3 The continuous case 213

14.4 Examples 215

14.5 Shifted distributions 216

14.6 The standard approximation 217

14.7 The stochastic life table 219

14.8 Life expectancy in the stochastic model 220

14.9 Stochastic interest rates 221

Notes and references 222

Exercises 222

15 The stochastic approach to insurance and annuities 224

15.1 Introduction 224

15.2 The stochastic approach to insurance benefits 225

15.3 The stochastic approach to annuity benefits 229

15.4 Deferred contracts 233

15.5 The stochastic approach to reserves 233

15.6 The stochastic approach to premiums 235

15.7 The variance of rL 241

15.8 Standard notation and terminology 243

Notes and references 244

Exercises 244

16 Simplifications under level benefit contracts 248

16.1 Introduction 248

16.2 Variance calculations in the continuous case 248

16.3 Variance calculations in the discrete case 250

16.4 Exact distributions 252

16.5 Some non-level benefit examples 254

Exercises 256

17 The minimum failure time 259

17.1 Introduction 259

17.2 Joint distributions 259

17.3 The distribution of T 261

17.4 The joint distribution of (T,J) 261

17.5 Other problems 270

17.6 The common shock model 271

17.7 Copulas 273

Notes and references 276

Exercises 276

Part III ADVANCED STOCHASTIC MODELS 279

18 An introduction to stochastic processes 281

18.1 Introduction 281

18.2 Markov chains 283

18.3 Martingales 286

18.4 Finite-state Markov chains 287

18.5 Introduction to continuous time processes 293

18.6 Poisson processes 293

18.7 Brownian motion 295

Notes and references 299

Exercises 300

19 Multi-state models 304

19.1 Introduction 304

19.2 The discrete-time model 305

19.3 The continuous-time model 311

19.4 Recursion and differential equations for multi-state reserves 324

19.5 Profit testing in multi-state models 327

19.6 Semi-Markov models 328

Notes and references 328

Exercises 329

20 Introduction to the Mathematics of Financial Markets 333

20.1 Introduction 333

20.2 Modelling prices in financial markets 333

20.3 Arbitrage 334

20.4 Option contracts 337

20.5 Option prices in the one-period binomial model 339

20.6 The multi-period binomial model 342

20.7 American options 346

20.8 A general financial market 348

20.9 Arbitrage-free condition 351

20.10 Existence and uniqueness of risk neutral measures 353

20.11 Completeness of markets 358

20.12 The Black–Scholes–Merton formula 361

20.13 Bond markets 364

Notes and references 372

Exercises 373

Part IV RISK THEORY 375

21 Compound distributions 377

21.1 Introduction 377

21.2 The mean and variance of S 379

21.3 Generating functions 380

21.4 Exact distribution of S 381

21.5 Choosing a frequency distribution 381

21.6 Choosing a severity distribution 383

21.7 Handling the point mass at 0 384

21.8 Counting claims of a particular type 385

21.9 The sum of two compound Poisson distributions 387

21.10 Deductibles and other modifications 388

21.11 A recursion formula for S 393

Notes and references 398

Exercises 398

22 Risk assessment 403

22.1 Introduction 403

22.2 Utility theory 403

22.3 Convex and concave functions: Jensen’s inequality 406

22.4 A general comparison method 408

22.5 Risk measures for capital adequacy 412

Notes and references 417

Exercises 417

23 Ruin models 420

23.1 Introduction 420

23.2 A functional equation approach 422

23.3 The martingale approach to ruin theory 424

23.4 Distribution of the deficit at ruin 433

23.5 Recursion formulas 434

23.6 The compound Poisson surplus process 438

23.7 The maximal aggregate loss 441

Notes and references 445

Exercises 445

24 Credibility theory 449

24.1 Introductory material 449

24.2 Conditional expectation and variance with respect to another random variable 453

24.3 General framework for Bayesian credibility 457

24.4 Classical examples 459

24.5 Approximations 462

24.6 Conditions for exactness 465

24.7 Estimation 469

Notes and References 473

Exercises 473

Appendix A review of probability theory 477

A.1 Sample spaces and probability measures 477

A.2 Conditioning and independence 479

A.3 Random variables 479

A.4 Distributions 480

A.5 Expectations and moments 481

A.6 Expectation in terms of the distribution function 482

A.7 Joint distributions 483

A.8 Conditioning and independence for random variables 485

A.9 Moment generating functions 486

A.10 Probability generating functions 487

A.11 Some standard distributions 489

A.12 Convolution 495

A.13 Mixtures 499

Answers to exercises 501

References 517

Index 523