Actuarial Models: The Mathematics of Insurance

Front Cover ... 1
Contents ... 10
Introduction ... 22
Chapter 0. Preliminary Facts from Probability and Interest ... 28
1 PROBABILITY AND RANDOM VARIABLES ... 28
1.1 Sample space, events, probability measure ... 28
1.2 Independence and conditional probabilities ... 29
1.3 Random variables, random vectors, and their distributions ... 31
1.3.1 Random variables ... 31
1.3.2 Random vectors ... 32
1.3.3 Cumulative distribution functions ... 35
1.3.4 Quantiles ... 37
1.3.5 Mixtures of distributions ... 38
2 EXPECTATION ... 39
2.1 De?nitions ... 39
2.2 Integration by parts and a formula for expectation ... 42
2.3 Can we encounter an in?nite expected value in models of real phenomena? ... 42
2.4 Moments of r.v.’s. Correlation ... 44
2.4.1 Variance and other moments ... 44
2.4.2 The Cauchy-Schwarz inequality ... 44
2.4.3 Covariance and correlation ... 45
2.5 Inequalities for deviations ... 46
2.6 Linear transformations of r.v.’s. Normalization ... 47
3 SOME BASIC DISTRIBUTIONS ... 48
3.1 Discrete distributions ... 48
3.1.1 The binomial distribution ... 48
3.1.2 The multinomial distribution ... 49
3.1.3 The geometric distribution ... 49
3.1.4 The negative binomial distribution ... 50
3.1.5 The Poisson distribution ... 51
3.2 Continuous distributions ... 52
3.2.1 The uniform distribution and simulation of r.v.’s ... 52
3.2.2 The exponential distribution ... 54
3.2.3 The ? (gamma)-distribution ... 55
3.2.4 The normal distribution ... 57
4 MOMENT GENERATING FUNCTIONS ... 58
4.1 Laplace transform ... 58
4.2 An example when a m.g.f. does not exist ... 60
4.3 The m.g.f.’s of basic distributions ... 60
4.3.1 The binomial distribution ... 60
4.3.2 The geometric and negative binomial distributions ... 60
4.3.3 The Poisson distribution ... 61
4.3.4 The uniform distribution ... 61
4.3.5 The exponential and gamma distributions ... 61
4.3.6 The normal distribution ... 62
4.4 The moment generating function and moments ... 62
4.5 Expansions for m.g.f.’s ... 63
4.5.1 Taylor’s expansions for m.g.f.’s ... 63
4.5.2 Cumulants ... 64
5 CONVERGENCE OF RANDOM VARIABLES AND DISTRIBUTIONS ... 65
6 LIMIT THEOREMS ... 68
6.1 The Law of Large Numbers (LLN) ... 68
6.2 The Central Limit Theorem (CLT) ... 69
7 CONDITIONAL EXPECTATIONS. CONDITIONING ... 70
7.1 Conditional expectation given a r.v. ... 70
7.1.1 The discrete case ... 70
7.1.2 The case of continuous distributions ... 72
7.2 Properties of conditional expectations ... 75
7.3 Conditioning and some useful formulas ... 77
7.3.1 A formula for variance ... 77
7.3.2 More detailed representations of the formula for total expectation ... 77
7.4 Conditional expectation given a random vector ... 79
7.4.1 General de?nitions ... 79
7.4.3 On the in?nite-dimensional case ... 81
8 ELEMENTS OF THE THEORY OF INTEREST ... 82
8.1 Compound interest ... 82
8.2 Nominal rate ... 85
8.3 Discount and annuities ... 85
8.4 Accumulated value ... 87
8.5 E?ective and nominal discount rates ... 87
9 EXERCISES ... 88
Chapter 1. Comparison of Random Variables. Preferences of Individuals ... 90
1 A GENERAL FRAMEWORK AND FIRST CRITERIA ... 90
1.1 Preference order ... 90
1.2 Several simple criteria ... 93
1.2.1 The mean-value criterion ... 93
1.2.2 Value-at-Risk (VaR) ... 93
1.2.3 An important remark: risk measures rather than criteria ... 96
1.2.4 Tail-Value-at-Risk (TailVaR) ... 96
1.2.5 The mean-variance criterion ... 100
1.3 On coherent measures of risk ... 103
2 COMPARISON OF R.V.’S AND LIMIT THEOREMS ... 107
2.1 A simple model of insurance with many clients ... 107
2.2 St. Petersburg’s paradox ... 109
3 EXPECTED UTILITY ... 110
3.1 Expected utility maximization (EUM) ... 110
3.1.1 Utility function ... 110
3.1.2 Expected utility maximization criterion ... 111
3.1.3 Some “classical” examples of utility functions ... 114
3.2 Utility and insurance ... 116
3.3 How to determine the utility function in particular cases ... 119
3.4 Risk aversion ... 119
3.4.1 A de?nition ... 119
3.4.2 Jensen’s inequality ... 121
3.4.3 How to measure risk aversion in the EUM case ... 122
3.4.4 Proofs ... 124
3.5 A new perspective: EUM as a linear criterion ... 125
3.5.1 Preferences on distributions ... 125
3.5.2 The ?rst stochastic dominance ... 126
3.5.3 The second stochastic dominance ... 128
3.5.4 The EUM criterion ... 129
3.5.5 Linearity of the utility functional ... 131
3.5.6 An axiomatic approach ... 134
4 NON-LINEAR CRITERIA ... 136
4.1 Allais’ paradox ... 136
4.2 Weighted utility ... 137
4.3 Implicit or comparative utility ... 140
4.3.1 De?nitions and examples ... 140
4.3.2 In what sense the implicit utility criterion is linear ... 142
4.4 Rank Dependent Expected Utility ... 144
4.4.1 De?nitions and examples ... 144
4.4.2 Application to insurance ... 147
4.4.3 Further discussion and the main axiom ... 147
5 OPTIMAL PAYMENT FROM THE STANDPOINT OF AN INSURED ... 150
5.1 Arrow’s theorem ... 150
5.2 A generalization ... 153
5.3 Historical remarks regarding the whole chapter ... 154
6 EXERCISES ... 155
Chapter 2. An Individual Risk Model for a Short Period ... 162
1 THE DISTRIBUTION OF AN INDIVIDUAL PAYMENT ... 162
1.1 The distribution of the loss given that it has occurred ... 162
1.1.1 Characterization of tails ... 162
1.1.2 Some particular light-tailed distributions ... 166
1.1.3 Some particular heavy-tailed distributions ... 167
1.1.4 The asymptotic behavior of tails and moments ... 169
1.2 The distribution of the loss ... 171
1.3 The distribution of the payment and types of insurance ... 172
2 THE AGGREGATE PAYMENT ... 179
2.1 Convolutions ... 180
2.1.1 De?nition and examples ... 180
2.1.2 Some classical examples ... 183
2.1.3 An additional remark regarding convolutions: Stable distributions ... 186
2.1.4 The analogue of the binomial formula for convolutions ... 187
2.2 Moment generating functions ... 187
3 PREMIUMS AND SOLVENCY. APPROXIMATIONS FOR AGGREGATE CLAIM DISTRIBUTIONS ... 190
3.1 Premiums and normal approximation. A heuristic approach ... 190
3.1.1 Normal approximation and security loading ... 190
3.1.2 An important remark: the standard deviation principle ... 194
3.2 A rigorous estimation ... 196
3.3 The number of contracts needed to maintain a given security level ... 199
3.4 Approximations taking into account the asymmetry of S ... 201
3.4.1 The skewness coe?cient ... 202
3.4.2 The ? -approximation ... 203
3.4.3 Asymptotic expansions and Normal Power (NP) approximation ... 204
4 SOME GENERAL PREMIUM PRINCIPLES ... 206
5 EXERCISES ... 211
Chapter 3. A Collective Risk Model for a Short Period ... 220
1 THREE BASIC PROPOSITIONS ... 220
2 COUNTING OR FREQUENCY DISTRIBUTIONS ... 222
2.1 The Poisson distribution and theorem ... 222
2.1.1 A heuristic approximation ... 222
2.1.2 The accuracy of the Poisson approximation ... 226
2.2 Some other “counting” distributions ... 228
2.2.1 The mixed Poisson distribution ... 228
2.2.2 Compound mixing ... 233
3 THE DISTRIBUTION OF THE AGGREGATE CLAIM ... 236
3.1 The case of a homogeneous group ... 236
3.1.1 The convolution method ... 236
3.1.2 The case where N has a Poisson distribution ... 240
3.1.3 The m.g.f. method ... 243
3.2 The case of several homogeneous groups ... 244
3.2.1 The probability of coming from a particular group ... 245
3.2.2 A general scheme and reduction to one group ... 246
4 PREMIUMS AND SOLVENCY. NORMAL APPROXIMATION ... 249
4.1 Limit theorems ... 249
4.1.1 The Poisson case ... 249
4.1.2 The general case ... 250
4.2 Estimation of premiums ... 254
4.3 The accuracy of normal approximation ... 256
4.4 Proof of Theorem 12 ... 257
5 EXERCISES ... 259
Chapter 4. Random Processes and their Applications I ... 264
1 A GENERAL FRAMEWORK AND TYPICAL SITUATIONS ... 264
1.1 Preliminaries ... 264
1.2 Processes with independent increments ... 266
1.2.1 The simplest counting process ... 266
1.2.2 Brownian motion ... 266
1.3 Markov processes ... 269
2 POISSON AND OTHER COUNTING PROCESSES ... 271
2.1 The homogeneous Poisson process ... 271
2.2 The non-homogeneous Poisson process ... 275
2.2.1 A model and examples ... 275
2.2.2 Another perspective: In?nitesimal approach ... 278
2.2.3 Proof of Proposition 1 ... 279
2.3 The Cox process ... 281
3 COMPOUND PROCESSES ... 282
4 MARKOV CHAINS. CASH FLOWS IN THE MARKOV ENVIRONMENT ... 284
4.1 Preliminaries ... 284
4.2 Variables de?ned on a Markov chain. Cash ?ows ... 290
4.2.1 Variables de?ned on states ... 290
4.2.2 Mean discounted payments ... 292
4.2.3 The case of absorbing states ... 294
4.2.4 Variables de?ned on transitions ... 296
4.2.5 What to do if the chain is not homogeneous ... 297
4.3 The ?rst step analysis. An in?nite horizon ... 298
4.3.1 Mean discounted payments in the case of in?nite time horizon ... 299
4.3.2 The ?rst step approach to random walk problems ... 300
4.4 Limiting probabilities and stationary distributions ... 305
4.5 The ergodicity property and classi?cation of states ... 310
4.5.1 Classes of states ... 310
4.5.2 The recurrence property ... 311
4.5.3 Recurrence and travel times ... 314
4.5.4 Recurrence and ergodicity ... 315
5 EXERCISES ... 317
Chapter 5. Random Processes and their Applications II ... 324
1 BROWNIAN MOTION AND ITS GENERALIZATIONS ... 324
1.1 More on properties of the standard Brownian motion ... 324
1.1.1 Non-di?erentiability of trajectories ... 324
1.1.2 Brownian motion as an approximation. The Donsker–Prokhorov invariance principle ... 325
1.1.3 The distribution of wt, hitting times, and the maximum value of Brownian motion ... 326
1.2 The Brownian motion with drift ... 329
1.2.1 Modeling of the surplus process. What a Brownian motion with drift approximates in this case ... 329
1.2.2 A reduction to the standard Brownian motion ... 331
1.3 Geometric Brownian motion ... 332
2 MARTINGALES ... 333
2.1 Two formulas of a general nature ... 333
2.2 Martingales: General properties and examples ... 334
2.3 Martingale transform ... 340
2.4 Optional stopping time and some applications ... 341
2.4.1 De?nitions and examples ... 341
2.4.2 Wald’s identity ... 344
2.4.3 The ruin probability for the simple random walk ... 346
2.4.4 The ruin probability for the Brownian motion with drift ... 347
2.4.5 The distribution of the ruin time in the case of Brownian motion ... 349
2.4.6 The hitting time for the Brownian motion with drift ... 350
2.5 Generalizations ... 351
2.5.1 The martingale property in the case of random stopping time ... 351
2.5.2 A reduction to the standard Brownian motion in the case of ran-dom time ... 352
2.5.3 The distribution of the ruin time in the case of Brownian motion: another approach ... 353
2.5.4 Proof of Theorem 12 ... 354
2.5.5 Veri?cation of Condition 3 of Theorem 6 ... 355
3 EXERCISES ... 356
Chapter 6. Global Characteristics of the Surplus Process ... 360
Chapter 7. Survival Distributions ... 412
1 THE PROBABILITY DISTRIBUTION OF LIFETIME ... 412
1.1 Survival functions and force of mortality ... 412
1.2 The time-until-death for a person of a given age ... 417
1.3 Curtate-future-lifetime ... 421
1.4 Survivorship groups ... 422
1.5 Life tables and interpolation ... 423
1.5.1 Life tables ... 423
1.5.2 Interpolation for fractional ages ... 428
1.6 Analytical laws of mortality ... 430
2 A MULTIPLE DECREMENT MODEL ... 432
2.1 A single life ... 432
2.2 Another view: net probabilities of decrement ... 436
2.3 Survivorship group ... 440
2.4 Proof of Proposition 1 ... 441
3 MULTIPLE LIFE MODELS ... 442
3.1 The joint distribution ... 443
3.2 The lifetime of statuses ... 445
3.3 A model of dependency: conditional independence ... 449
3.3.1 A de?nition and the ?rst example ... 450
3.3.2 The common shock model ... 451
4 EXERCISES ... 453
Chapter 8. Life Insurance Models ... 458
1 A GENERAL MODEL ... 458
1.1 The present value of a future payment ... 458
1.2 The present value of payments for a portfolio of many policies ... 461
2 SOME PARTICULAR TYPES OF CONTRACTS ... 464
2.1 Whole life insurance ... 464
2.1.1 The continuous time case (bene?ts payable at the moment of death) ... 464
2.1.2 The discrete time case (bene?ts payable at the end of the year of death) ... 464
2.1.3 A relation between Ax and Ax ... 467
2.1.4 The case of bene?ts payable at the end of the m-thly period ... 468
2.2 Deferred whole life insurance ... 470
2.2.1 The continuous time case ... 470
2.2.2 The discrete time case ... 471
2.3 Term insurance ... 471
2.3.1 Continuous time ... 471
2.3.2 Discrete time ... 473
2.4 Endowments ... 474
2.4.1 Pure endowment ... 474
2.4.2 Endowment ... 475
3 VARYING BENEFITS ... 477
3.1 Certain payments ... 477
3.2 Random payments ... 481
4 MULTIPLE DECREMENT AND MULTIPLE LIFE MODELS ... 482
4.1 Multiple decrements ... 482
4.2 Multiple life insurance ... 485
5 ON THE ACTUARIAL NOTATION ... 488
6 EXERCISES ... 488
Chapter 9. Annuity Models ... 494
1 TWO APPROACHES TO THE EVALUATION OF ANNUITIES ... 494
1.1 Continuous annuities ... 494
1.2 Discrete annuities ... 496
2 LEVEL ANNUITIES. A CONNECTION WITH INSURANCE ... 499
2.1 Certain annuities ... 499
2.2 Random annuities ... 500
3 SOME PARTICULAR TYPES OF LEVEL ANNUITIES ... 501
3.1 Whole life annuities ... 501
3.2 Temporary annuities ... 504
3.3 Deferred annuities ... 507
3.4 Certain and life annuities ... 510
4 MORE ON VARYING PAYMENTS ... 512
5 ANNUITIES WITH m-thly PAYMENTS ... 514
6 MULTIPLE DECREMENT AND MULTIPLE LIFE MODELS ... 516
6.1 Multiple decrement ... 516
6.2 Multiple life annuities ... 519
7 EXERCISES ... 521
Chapter 10. Premiums and Reserves ... 526
1 PREMIUM ANNUITIES ... 526
1.1 General principles ... 526
1.2 Bene?t premiums: The case of a single risk ... 528
1.2.1 Net rate ... 528
1.2.2 The case where “Y is consistent with Z” ... 532
1.2.3 Variances ... 533
1.2.4 Premiums paid m times a year ... 535
1.2.5 Combinations of insurances ... 536
1.3 Accumulated values ... 537
1.4 Percentile premiums ... 538
1.4.1 The case of a single risk ... 538
1.4.2 The case of many risks. Normal approximation ... 540
1.5 Exponential premiums ... 543
2 RESERVES ... 544
2.1 De?nitions and preliminary remarks ... 544
2.2 Examples of direct calculations ... 545
2.3 Formulas for some standard types of insurance ... 547
2.4 Recursive relations ... 548
3 EXERCISES ... 551
Chapter 11. Pension Plans ... 554
1 VALUATION OF INDIVIDUAL PENSION PLANS ... 554
1.1 DB plans ... 555
1.1.1 The APV of future bene?ts ... 555
1.1.2 More examples of the bene?t rate function B(x,h,y) ... 557
1.2 DC plans ... 559
1.2.1 Calculations at the time of retirement ... 559
1.2.2 Calculations at the time of entering a plan ... 561
2 PENSION FUNDING. COST METHODS ... 562
2.1 A dynamic DB fund model ... 562
2.1.1 Modeling enrollment ?ow and pension payments ... 562
2.1.2 Normal cost ... 563
2.1.3 The bene?t payment rate and the APV of future bene?ts ... 564
2.2 More on cost methods ... 566
2.2.1 The unit-credit method ... 566
2.2.2 The entry-age-normal method ... 570
3 EXERCISES ... 571
Chapter 12. Risk Exchange: Reinsurance and Coinsurance ... 574
1 REINSURANCE FROM THE STANDPOINT OF A CEDENT ... 574
1.1 Some optimization considerations ... 574
1.1.1 Expected utility maximization ... 575
1.1.2 Variance as a measure of risk ... 577
1.2 Proportional reinsurance: Adding a new contract to an existing portfolio ... 579
1.2.1 The case of a ?xed security loading coe?cient ... 579
1.2.2 The case of the standard deviation premium principle ... 582
1.3 Long-term insurance: Ruin probability as a criterion ... 584
1.3.1 An example with proportional reinsurance ... 584
1.3.2 An example with excess-of-loss insurance ... 586
2 RISK EXCHANGE AND RECIPROCITY OF COMPANIES ... 587
2.1 A general framework and some examples ... 587
2.2 Two more examples with expected utility maximization ... 595
2.3 The case of the mean-variance criterion ... 599
2.3.1 Minimization of variances ... 599
2.3.2 The exchange of portfolios ... 602
3 REINSURANCE MARKET ... 607
3.1 A model of the exchange market of random assets ... 607
3.2 An example concerning reinsurance ... 610
4 EXERCISES ... 613
Appendix ... 616
1 SUMMARY TABLES FOR BASIC DISTRIBUTIONS ... 616
2 TABLES FOR THE STANDARD NORMAL DISTRIBUTION ... 618
3 ILLUSTRATIVE LIFE TABLE ... 619
4 SOME FACTS FROM CALCULU ... 622
4.1 The “little o and big O” notation ... 622
4.1.1 Little o ... 622
4.1.2 Big O ... 623
4.2 Taylor expansions ... 624
4.2.1 A general expansion ... 624
4.2.2 Some particular expansions ... 625
4.3 Concavity ... 626
References ... 628
Answers to Exercises ... 638