A Reading of the Theory of Life Contingency Models: A ...

1 A Reading of the Theory of Life Contingency Models: A Preparation for Exam MLC/3L. Marcel B. Finan Arkansas Tech University c All Rights Reserved Preliminary Draft 2. To My Daughter Nadia Contents Preface 11. Prerequisite Material 13. Brief Review of Interest Theory 15. 1 The Basics of Interest Theory .. 15. 2 Equations of Value and Time Diagrams .. 26. 3 Level Annuities .. 28. Level Annuity-Immediate .. 28. Level Annuity-Due .. 30. Level Continuous Annuity .. 32. 4 Varying Annuities .. 34. Varying Annuity-Immediate .. 34. Varying Annuity-Due .. 38. Continuous Varying Annuities .. 42. Continuously Payable Varying Annuities .. 45. 5 Annuity Values on Any Date: Deferred Annuity .. 46. A Brief Review of Probability Theory 51. 6 Basic Definitions of Probability .. 51. 7 Classification of Random Variables .. 57. 8 Discrete Random Variables .. 59. 9 Continuous Random Variables .. 69.

2 10 Raw and Central Moments .. 77. 11 Median, Mode, Percentiles, and Quantiles .. 82. 12 Mixed Distributions .. 85. 13 A List of Commonly Encountered Discrete .. 88. Discrete Uniform Distribution .. 88. 3. 4 CONTENTS. The Binomial Distribution .. 90. The Negative Binomial Distribution .. 91. The Geometric Distribution .. 92. The Poisson Distribution .. 93. 14 A List of Commonly Encountered Continuous .. 94. Continuous Uniform Distribution .. 94. Normal and Standard Normal Distributions .. 96. Exponential Distribution .. 102. Gamma Distribution .. 104. 15 Bivariate Random Variables .. 106. Joint CDFs .. 106. Bivariate Distributions: The Discrete Case .. 108. Bivariate Distributions: The Continuous Case .. 110. Independent Random Variables .. 113. Conditional Distributions: The Discrete Case .. 116. Conditional Distributions: The Continuous Case .. 119. The Expected Value of g(X, Y ).

3 123. Conditional Expectation .. 127. 16 Sums of Independent Random Variables .. 132. Moments of S .. 132. Distributions Closed Under Convolution .. 133. Distribution of S : Convolutions .. 135. Estimating the Distribution of S : The Central Limit Theorem .. 138. 17 Compound Probability Distributions .. 140. Mean and Variance of S .. 140. Moment Generating Function of S .. 142. Actuarial Survival Models 143. 18 Age-At-Death Random Variable .. 144. The Cumulative Distribution Function of X .. 144. The Survival Distribution Function of X .. 147. The Probability Density Function of X .. 150. Force of Mortality of X .. 154. The Mean and Variance of X .. 159. 19 Selected Parametric Survival Models .. 163. The Uniform or De Moivre's Model .. 163. The Exponential Model .. 167. CONTENTS 5. The Gompertz Model .. 171. The Modified Gompertz Model: The Makeham's Model . 174. The Weibull Model.

4 177. 20 Time-Until-Death Random Variable .. 180. The Survival Function of T (x) .. 180. The Cumulative Distribution Function of T (x) .. 185. Probability Density Function of T (x) .. 189. Force of Mortality of T (x) .. 193. Mean and Variance of T (x) .. 198. Curtate-Future-Lifetime .. 202. 21 Central Death Rates .. 208. The Life Table Format 213. 22 The Basic Life Table .. 214. 23 Mortality Functions in Life Table Notation .. 220. Force of Mortality Function .. 220. The Probability Density Function of X .. 224. Mean and Variance of X .. 226. Conditional Probabilities .. 230. Mean and Variance of T (x) .. 233. Temporary Complete Life Expectancy .. 236. The Curtate Expectation of Life .. 240. The n Lx Notation .. 244. Central Death Rate .. 247. 24 Fractional Age Assumptions .. 251. Linear Interpolation: Uniform Distribution of Deaths (UDD) .. 251. Constant Force of Mortality Assumption: Exponential Interpolation.

5 259. Harmonic (Balducci) Assumption .. 264. 25 Select-and-Ultimate Mortality Tables .. 270. Life Insurance: Contingent Payment Models 277. 26 Insurances Payable at the Moment of Death .. 279. Level Benefit Whole Life Insurance .. 279. Finite Term Insurance Payable at the Moment of Death . 287. Endowments .. 291. Pure Endowments .. 291. 6 CONTENTS. Endowment Insurance .. 295. Deferred Life Insurance .. 298. 27 Insurances Payable at the End of the Year of Death .. 303. 28 Recursion Relations for Life Insurance .. 312. 29 Variable Insurance Benefit .. 318. Non-level Payments: A Simple Example .. 318. Increasing or Decreasing Insurances Payable at the Mo- ment of Death .. 322. Increasing and Decreasing Insurances Payable at the End of Year of Death .. 327. 30 Expressing APV's of Continuous Models in Terms of Discrete Ones .. 331. thly 31 m Contingent Payments .. 337. 32 Applications of Life Insurance.

6 341. Contingent Annuity Models 347. 33 Continuous Whole Life Annuities .. 348. 34 Continuous Temporary Life Annuities .. 356. 35 Continuous Deferred Life Annuities .. 361. 36 The Certain-and-Life Annuity .. 364. 37 Discrete Life Annuities .. 366. Whole Life Annuity Due .. 366. Temporary Life Annuity-Due .. 374. Discrete Deferred Life Annuity-Due .. 380. Discrete Certain and Life Annuity-Due .. 383. Life Annuity-Immediate .. 386. 38 Life Annuities with mthly Payments .. 392. 39 Non-Level Payments Annuities .. 401. The Discrete Case .. 401. The Continuous Case .. 405. Calculating Benefit Premiums 409. 40 Fully Continuous Premiums .. 410. Continuous Whole Life Policies .. 410. n year Term Policies .. 417. Continuous n year Endowment Insurance .. 421. Continuous n year Pure Endowment .. 425. Continuous n year Deferred Insurance .. 428. CONTENTS 7. Continuous n year Deferred Whole Life Annuity.

7 430. 41 Fully Discrete Benefit Premiums .. 434. Fully Discrete Whole Life Insurance .. 434. Fully Discrete n year Term .. 439. Fully Discrete n year Pure Endowment .. 443. Fully Discrete n year Endowment Insurance .. 446. Fully Discrete n year Deferred Insurance .. 450. Fully Discrete n year Deferred Annuity-Due .. 453. 42 Benefit Premiums for Semicontinuous Models .. 456. Semicontinuous Whole Life Insurance .. 456. Semicontinuous n year Term Insurance .. 460. Semicontinuous n year Endowment Insurance .. 463. Semicontinuous n year Deferred Insurance .. 466. thly 43 m Benefit Premiums .. 469. mthly Payments with Benefit Paid at Moment of Death . 469. mthly Payments with Benefit Paid at End of Year of Death474. 44 Non-Level Benefit/Premium Payments and the Equivalence Prin- ciple .. 478. 45 Percentile Premium Principle .. 488. Benefit Reserves 495. 46 Fully Continuous Benefit Reserves.

8 497. Fully Continuous Whole Life .. 497. Reserves by the Prospective Method .. 497. Other Special Formulas for the Prospective Re- serve .. 503. Retrospective Reserve Formula .. 507. Fully Continuous n year Term .. 510. Fully Continuous n year Endowment Insurance .. 514. Fully Continuous n year Pure Endowment .. 518. n year Deferred Whole Life Annuity .. 520. 47 Fully Discrete Benefit Reserves .. 523. Fully Discrete Whole Life Insurance .. 523. Fully Discrete n year Term Insurance .. 530. Fully Discrete n year Endowment .. 533. Fully n year Deferred Whole Life Annuity .. 537. 48 Semicontinuous Reserves .. 539. 49 Reserves Based on True mthly Premiums .. 544. 8 CONTENTS. Reserves for Contracts with Nonlevel Benefits and Premiums 549. 50 Reserves for Fully Discrete General Insurances .. 550. 51 Reserves for Fully Continuous General Insurances .. 557. 52 Recursive Formulas for Fully Discrete Benefit Reserves.

9 560. 53 Miscellaneous Examples .. 569. 54 Benefit Reserves at Fractional Durations .. 573. 55 Calculation of Variances of Loss Random Variables: The Hat- tendorf's Theorem .. 578. Multiple Life Models 585. 56 The Joint-Life Status Model .. 586. The Joint Survival Function of T (xy) .. 586. The Joint CDF/PDF of T (xy) .. 589. The Force of Mortality of T (xy) .. 592. Mean and Variance of T (xy) .. 594. 57 The Last-Survivor Status Model .. 597. 58 Relationships Between T (xy) and T (xy) .. 606. 59 Contingent Probability Functions .. 609. 60 Contingent Policies for Multiple Lives .. 615. 61 Special Two-life Annuities: Reversionary Annuities .. 624. 62 Dependent Future Lifetimes Model: The Common Shock .. 629. 63 Joint Distributions of Future Lifetimes .. 635. Multiple Decrement Models 639. 64 The Continuous Case .. 640. 65 Associated Single Decrement Models .. 647. 66 Discrete Multiple-Decrement Models.

10 652. 67 Uniform Distribution of Decrements .. 661. 68 Valuation of Multiple Decrement Benefits .. 669. 69 Valuation of Multiple Decrement Premiums and Reserves .. 677. Incorporating Expenses in Insurance Models 685. 70 Expense-Augmented Premiums .. 686. 71 Types of Expenses .. 695. 72 The Mathematics of Asset Share .. 702. CONTENTS 9. Multiple-State Transition Models 709. 73 Introduction to Markov Chains Process .. 710. 74 Longer Term Transition Probabilities .. 715. 75 Valuation of Cash Flows .. 721. Cash Flows Upon Transitions .. 721. Cash Flows while in State .. 724. Benefit Premiums and Reserves .. 726. Probability Models: Poisson Processes 733. 76 The Poisson Process .. 734. 77 Interarrival and Waiting Time Distributions .. 741. 78 Superposition and Decomposition of Poisson Process .. 747. 79 Non-Homogeneous Poisson Process .. 758. 80 Compound Poisson Process .. 764. 81 Conditional Poisson Processes.