Random Variables and Their Distributions 19 2.1.1 Discrete Random Variables 20 2.1.2 Continuous Random Variables 23 2.1.3 Mixed Random Variables 25 2.1.4 More on Moments of Random Variables 25 Survey of Particular Discrete Distributions 27 2.2.1 The Discrete Uniform Distribution 27 2.2.2 The Binomial Distribution 28 2.2.3 The Negative Binomial Distribution 28 2.2.4 The Geometric Distribution 30 2.2.5 The Poisson Distribution 30 ix
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2.3
Survey of Particular Continuous Distributions 31 2.3.1 The Continuous Uniform Distribution 31 2.3.2 The Normal Distribution 32 2.3.3 The Exponential Distribution 34 2.3.4 The Gamma Distribution 35 2.4 Multivariate Probability 36 2.4.1 The Discrete Case 36 2.4.2 The Continuous Case 39 2.5 Sums of Independent Random Variables 41 2.5.1. The Moments of S 41 2.5.2 Distributions Closed Under Convolution 42 2.5.3 The Method of Convolutions 44 2.5.4 Approximating the Distribution of S 45 2.6 Compound Distributions 45 - 2.6.1 The Moments of S 46 2.6.2 The Compound Poisson Distribution 48 PART TWO: MODELS FOR SURVIVAL-CONTINGENT RISKS
The Age-at-Failure Random Variable 51 3.1.1 The Cumulative Distribution Function of X 53 3.1.2 The Survival Distribution Function of X 53 3.1.3 The Probability Density Function of X 54 3.1.4 The Hazard Rate Function of X 55 3.1.5 The Moments of the Age-at-Failure Random Variable X 57 3.1.6 Actuarial Survival Models 58 Examples of Parametric Survival Models 60 3.2.1 The Uniform Distribution 60 3.2.2 The Exponential Distribution 61 3.2.3 The Gompertz Distribution 62 3.2.4 The Makeham Distribution 62 3.2.5 The Weibull Distribution 63 3.2.6 Summary of Parametric Survival Models 63 The Time-to-Failure Random Variable 64 3.3.1 The Survival Distribution Function of Tx 65 3.3.2 The Cumulative Distribution Function of Tx 66
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XI
3.3.3 3.3.4 3.3.5
3.4 3.5 3.6 CHAPTER 4
4.1 4.2 4.3
4.4 4.5
4.6 4.7 4.8 CHAPTER 5
5.1
The Probability Density Function of Tx 67 The Hazard Rate Function of Tx 68 Moments of the Future Lifetime Random Variable Tx 68 3.3.6 The Time-to-Failure Random Variable Kx 71 The Central Rate 73 Select Survival Models 75 Exercises 76 T H E L I F E TABLE (DISCRETE TABULAR CONTEXT)
81
Definition of the Life Table 81 The Traditional Form of the Life Table 83 Other Functions Derived from lx 85 4.3.1 The Force of Failure 86 4.3.2 The Probability Density Function of X 88 4.3.3 Conditional Probabilities and Densities 90 4.3.4 The Curtate Expectation of Life 94 4.3.5 The Central Rate 96 Summary of Concepts and Notation 97 Methods for Non-Integral Ages 97 4.5.1 Linear Form for £x+t 100 4.5.2 Exponential Form for £x+t 104 4.5.3 Hyperbolic Form for £x+t 106 4.5.4 Summary 108 Select Life Tables 109 Life Table Summary 112 Exercises 113 CONTINGENT PAYMENT MODELS (INSURANCE MODELS) 121
Discrete Stochastic Models 122 5.1.1 The Discrete Random Variable for Time of Failure 122 5.1.2 The Present Value Random Variable 122 5.1.3 Modifications of the Present Value Random Variable 126 5.1.4 Applications to Life Insurance 131
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5.2 5.3
5.4 5.5
5.6 5.7 CHAPTER 6
6.1
6.2
6.3
6.4
6.5 6.6 6.7
Group Deterministic Approach 135 Continuous Stochastic Models 138 5.3.1 The Continuous Random Variable for Time to Failure 138 5.3.2 The Present Value Random Variable 139 5.3.3 Modifications of the Present Value Random Variable 141 5.3.4 Applications to Life Insurance 141 5.3.5 Continuous Functions Evaluated from Parametric Survival Models 142 Contingent Payment Models with Varying Payments 145 Continuous and m'hly Functions Approximated from the Life Table 148 5.5.1 Continuous Contingent Payment Models 148 5.5.2 m'hly Contingent Payment Models 151 Miscellaneous Examples 153 Exercises 156 CONTINGENT ANNUITY MODELS ( L I F E ANNUITIES) 161
Whole Life Annuity Models 162 6.1.1 The Immediate Case 163 6.1.2 The Due Case 169 6.1.3 The Continuous Case 171 Temporary Annuity Models 174 6.2.1 The Immediate Case 174 6.2.2 The Due Case 179 6.2.3 The Continuous Case 182 Deferred Whole Life Annuity Models 185 6.3.1 The Immediate Case 185 6.3.2 The Due Case 187 6.3.3 The Continuous Case 188 Contingent Annuities Payable mMy 191 6.4.1 The Immediate Case 191 6.4.2 The Due Case 192 6.4.3 Random Variable Analysis 193 6.4.4 Numerical Evaluation in the mthly and Continuous Cases 195 Non-Level Payment Annuity Functions 197 Miscellaneous Examples 198 Exercises 203
TABLE OF CONTENTS
CHAPTER 7
7.1
7.2 7.3
7.4 .7.5 7.6 7.7 CHAPTER 8
8.1
8.2
8.3
FUNDING PLANS FOR CONTINGENT CONTRACTS (ANNUAL PREMIUMS) 211
Reserves for Contingent Payment Models with Annual Payment Funding 247 8.1.1 Reserves by the Prospective Method 247 8.1.2 Reserves by the Retrospective Method 250 8.1.3 Additional Terminal Reserve Expressions 253 8.1.4 Random Variable Analysis 255 8.1.5 Reserve for Contingent Contracts with Immediate Payment of Claims 257 8.1.6 Reserves for Contingent Annuity Models 258 Recursive Relationships for Discrete Models with Annual Premiums 259 8.2.1 Group Deterministic Approach 259 8.2.2 Random Variable Analysis - Cash Basis 263 8.2.3 Random Variable Analysis - Accrued Basis 266 Reserves for Contingent Payment Models with Continuous Payment Funding 270 8.3.1 Discrete Whole Life Contingent Payment Models 270 8.3.2 Continuous Whole Life Contingent Payment Models 271 8.3.3 Random Variable Analysis 275
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8.4 8.5 8.6 8.7
8.8 8.9 CHAPTER 9
9.1
9.2
9.3 9.4
9.5
9.6 9.7
Reserves for Contingent Payment Models with m'hly Payment Funding 276 Incorporation of Expenses 279 Reserves at Fractional Durations 280 Generalization to Non-Level Benefits and Premiums 282 8.7.1 Discrete Models 282 8.7.2 Continuous Models 286 Miscellaneous Examples 288 Exercises 292 MODELS DEPENDENT ON MULTIPLE SURVIVALS (MULTI-LIFE MODELS) 299
The Joint-Life Model 299 s 9.1.1 The Time-to-Failure Random Variable for a Joint-Life Status 300 9.1.2 Survival Distribution Function of Txy 300 9.1.3 Cumulative Distribution Function of T^ 300 9.1.4 Probability Density Function of Txy 302 9.1.5 Hazard Rate Function of T^ 303 9.1.6 Conditional Probabilities 303 9.1.7 Moments of Txv 305 The Last-Survivor Model 306 9.2.1 The Time-to-Failure Random Variable for a Last-Survivor Status 306 9.2.2 Functions of the Random Variable Tw 307 9.2.3 Relationships Between Txy and T^ 310 Contingent Probability Functions 311 Contingent Contracts Involving Multi-Life Statuses 314 9.4.1 Contingent Payment Models 314 9.4.2 Contingent Annuity Models 316 9.4.3 Annual Premiums and Reserves 317 9.4.4 Reversionary Annuities 319 9.4.5 Contingent Insurance Functions 321 General Random Variable Analysis 322 9.5.1 Marginal Distributions of Tx and Ty 322 9.5.2 TheCovarianceofr x and7; 323 9.5.3 Other Joint Functions of Tx and Ty 325 9.5.4 Joint and Last-Survivor Status Functions 328 Common Shock - A Model for Lifetime Dependency 330 Exercises 333
TABLE OF CONTENTS
CHAPTER 10
10.1
10.2 10.3 10.4
10.5 10.6 10.7
10.8
MULTIPLE CONTINGENCIES WITH APPLICATIONS (MULTIPLE-DECREMENT MODELS) 339
Discrete Multiple-Decrement Models 339 10.1.1 The Multiple-Decrement Table 341 10.1.2 Random Variable Analysis 344 Theory of Competing Risks 346 Continuous Multiple-Decrement Models 347 Uniform Distribution of Decrements 352 10.4.1 Uniform Distribution in the Multiple-Decrement Context 352 10.4.2 Uniform Distribution in the Associated SingleDecrement Tables 354 Actuarial Present Value 357 Asset Shares 363 Multi-State Models 366 10.7.1 The Homogeneous Process 366 10.7.2 The Nonhomogeneous Process 372 Exercises 374
PART T H R E E : M O D E L S FOR NON-SURVIVAL-CONTINGENT RISKS CHAPTER 11
11.1
11.2
11.3
XV
CLAIM FREQUENCY M O D E L S 383
Section 2.2 (Discrete Distributions) Revisited 383 11.1.1 The Binomial Distribution 383 11.1.2 The Poisson Distribution 384 11.1.3 The Negative Binomial Distribution 389 11.1.4 The Geometric Distribution 393 11.1.5 Summary of the Recursive Relationships 393 11.1.6 Probability Generating Functions 394 Creating Additional Counting Distributions 397 11.2.1 Compound Frequency Models 397 11.2.2 Mixture Frequency Models 402 11.2.3 Truncation or Modification at Zero 405 Counting Processes 409 11.3.1 Properties of Counting Processes 410 11.3.2 The Poisson Counting Process 411
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11.4 CHAPTER 12
12.1
11.3.3 Further Properties of the Poisson Counting Process 412 11.3.4 Poisson Mixture Processes 415 11.3.5 The Nonstationary Poisson Counting Process 415 Exercises 418 CLAIM SEVERITY MODELS 425
Fundamental Continuous Distributions 426 12.1.1 The Normal and Exponential Distributions 426 12.1.2 The Pareto Distribution 426 12.1.3 Analytic Measures of Tail Weight 429 12.2 Generating New Distributions 431 12.2.1 Summation 431 , 12.2.2 Scalar Multiplication 431 12.2.3 Power Operations 434 12.2.4 Exponentiation and the Lognormal Distribution 437 12.2.5 Summary of Severity Distributions 439 12.2.6 Mixtures of Distributions 442 12.2.7 Spliced Distributions 448 12.2.8 Limiting Distributions 450 12.3 Modifications of the Loss Random Variable 451 12.3.1 Deductibles 452 12.3.2 Policy Limits 454 12.3.3 Relationships between Deductibles and Policy Limits 457 12.3.4 Coinsurance Factors 460 12.3.5 The Effect of Inflation 461 12.3.6 Effect of Coverage Deductibles on Frequency Models 463 12.4 Tail Weight Revisited; Risk Measures 470 12.4.1 The Mean Excess Loss Function 471 12.4.2 Conditional Tail Expectation 474 12.4.3 Value at Risk 475 12.4.4 Distortion Risk Measures 476 12.4.5 Risk Measures Using Discrete Distributions 479 12.4.6 Other Risk Measures 481 12.5 Empirical Loss Distributions 481 12.6 Exercises 485
TABLE OF CONTENTS
CHAPTER 13
13.1 13.2
13.3
13.4
13.5
13.6 CHAPTER 14
14.1
14.2 14.3
14.4
14.5 14.6
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MODELS FOR AGGREGATE PAYMENTS
495
Individual Risk versus Collective Risk 495 Selection of Frequency and Severity Distributions 499 13.2.1 Frequency 499 13.2.2 Severity 500 13.2.3 Frequency-Severity Interaction 501 More on the Collective Risk Model 502 13.3.1 Convolutions of the Probability Function ofX 502 13.3.2 Convolutions of the CDF ofX 508 13.3.3 Continuous Severity Distributions 512 13.3.4 A Final Thought Regarding Convolutions 519 Effect of Coverage Modifications 520 13.4.1 Modifications Applied to Individual Losses 521 13.4.2 Modifications Applied to the Aggregate Loss (Stop-Loss Reinsurance) 523 Infinitely Divisible Distributions 528 13.5.1 Definition of Infinite Divisibility 528 13.5.2 The Poisson Distribution 529 13.5.3 The Negative Binomial Distribution 529 Exercises 529 PROCESS MODELS
537
The Compound Poisson Process 537 14.1.1 Moments of the Compound Poisson Process 538 14.1.2 Other Properties of the Compound Poisson Process 539 The Surplus Process Model 540 The Probability of Ruin 543 14.3.1 The Adjustment Coefficient 544 14.3.2 The Probability of Ruin 547 The Distribution of Surplus Deficit 550 14.4.1 The Event of U{t)
