TABLE OF CONTENTS. GENERAL AND HISTORICAL PREFACE iii SIXTH EDITION PREFACE v PART ONE: REVIEW AND BACKGROUND MATERIAL

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1 TABLE OF CONTENTS GENERAL AND HISTORICAL PREFACE iii SIXTH EDITION PREFACE v PART ONE: REVIEW AND BACKGROUND MATERIAL CHAPTER ONE: REVIEW OF INTEREST THEORY Interest Measures Level Annuity Functions Annuity-Immediate Annuity-due Continuous Annuity Non-Level Annuity Functions Annuities-Immediate Annuities-due Continuous Annuities Equation of Value 13 CHAPTER TWO: REVIEW OF PROBABILITY Random Variables and Their Distributions Discrete Random Variables Continuous Random Variables Mixed Random Variables More on Moments of Random Variables Survey of Particular Discrete Distributions The Discrete Uniform Distribution The Binomial Distribution The Negative Binomial Distribution The Geometric Distribution The Poisson Distribution Survey of Particular Continuous Distributions The Continuous Uniform Distribution The Normal Distribution The Exponential Distribution The Gamma Distribution Multivariate Probability The Discrete Case The Continuous Case 3 vii

2 viii TABLE OF CONTENTS CHAPTER THREE: REVIEW OF MARKOV CHAINS Discrete-Time Markov Chains Transition Probabilities State Vector Probabilities over Multiple Steps Properties of Homogeneous Discrete-Time Markov Chains The Non-Homogeneous Discrete-Time Model Probability of Remaining in State i Application to Multi-State Models Transition Only at Fixed Time Points Continuous-Time Markov Chains Forces of Transition Formulas for ij tpx = Pr[ X ( t) = j X () = i] Payments Exercises 45 CHAPTER FOUR: CHARACTERISTICS OF INSURANCE AND PENSIONS Background and Principles Life Insurance and Annuities Types of Insurance Contracts Types of Annuity Contracts Distribution Underwriting Other Types of Insurance Pension Benefits Defined Benefit Plans Defined Contribution Plans Recent Developments in Insurance The Role of Actuaries Exercises 54 PART TWO: MODELS FOR SURVIVAL-CONTINGENT RISKS CHAPTER FIVE: SURVIVAL MODELS (CONTINUOUS PARAMETRIC CONTEXT) The Age-at-Failure Random Variable The Cumulative Distribution Function of T The Survival Distribution Function of T The Probability Density Function of T The Hazard Rate Function of T The Moments of the Age-at-Failure Random Variable T Actuarial Survival Models Examples of Parametric Survival Models The Uniform Distribution The Exponential Distribution The Gompertz Distribution The Makeham Distribution Summary of Parametric Survival Models 68

3 TABLE OF CONTENTS ix 5.3 The Time-to-Failure Random Variable The Survival Distribution Function of T x The Cumulative Distribution Function of T x The Probability Density Function of T x The Hazard Rate Function of T x Moments of the Future Lifetime Random Variable T x Discrete Time-to-Failure Random Variable Select Survival Models Multi-State Model Interpretation Written-Answer Question Examples Exercises 81 CHAPTER SIX: THE LIFE TABLE (DISCRETE TABULAR CONTEXT) Definition of the Life Table The Traditional Form of the Life Table Other Functions Derived from l x The Force of Failure The Probability Density Function of T Conditional Probabilities and Densities The Curtate Expectation of Life Summary of Concepts and Notation Multi-State Model Interpretation Methods for Non-Integral Ages Linear Form for l x+ t Exponential Form for l x+ t Hyperbolic Form for l x+ t Summary Select Life Tables Life Table Summary Written-Answer Question Examples Exercises 113 CHAPTER SEVEN: CONTINGENT PAYMENT MODELS (INSURANCE MODELS) Discrete Stochastic Models The Discrete Random Variable for Time of Failure The Present Value Random Variable Modifications of the Present Value Random Variable Applications to Life Insurance Group Deterministic Approach Continuous Stochastic Models The Continuous Random Variable for Time to Failure The Present Value Random Variable Modifications of the Present Value Random Variable Applications to Life Insurance Continuous Functions Evaluated from Parametric Survival Models Contingent Payment Models with Varying Payments 139

4 x TABLE OF CONTENTS 7.5 Continuous and m thly Functions Approximated from the Life Table Continuous Contingent Payment Models m thly Contingent Payment Models Multi-State Model Representation Discrete Models Continuous Models Extension to Models with Varying Payments Written-Answer Question Examples Exercises 15 CHAPTER EIGHT: CONTINGENT ANNUITY MODELS (LIFE ANNUITIES) Whole Life Annuity Models The Immediate Case The Due Case The Continuous Case Temporary Annuity Models The Immediate Case The Due Case The Continuous Case Deferred Whole Life Annuity Models The Immediate Case The Due Case The Continuous Case Summary of Annual Payment Annuities Life Annuities Payable m thly The Immediate Case The Due Case Random Variable Analysis Numerical Evaluation in the m thly and Continuous Cases Summary of m thly Payment Annuities Non-Level Payment Annuity Functions Multi-State Model Representation Mortality Improvement Projection Written-Answer Question Examples Exercises 195 CHAPTER NINE: FUNDING PLANS FOR CONTINGENT CONTRACTS 23 (ANNUAL PREMIUMS) 9.1 Annual Funding Schemes for Contingent Payment Models Discrete Contingent Payment Models Continuous Contingent Payment Models Contingent Annuity Models Non-Level Premium Contracts Random Variable Analysis The Percentile Premium Principle Continuous Payment Funding Schemes Discrete Contingent Payment Models Continuous Contingent Payment Models Funding Schemes with m thly Payments 222

5 TABLE OF CONTENTS xi 9.6 Funding Plans Incorporating Expenses Written-Answer Question Examples Exercises 228 CHAPTER TEN: CONTINGENT CONTRACT RESERVES (NET LEVEL PREMIUM RESERVES) NLP Reserves for Contingent Payment Models with Annual Payment Funding NLP Reserves by the Prospective Method NLP Reserves by the Retrospective Method Additional NLP Terminal Reserve Expressions Random Variable Analysis NLP Reserves for Contingent Contracts with Immediate Payment of Claims NLP Reserves for Life Annuity Models Recursive Relationships for Discrete Models with Annual Premiums NLP Reserves for Contingent Payment Models with Continuous Funding Discrete Whole Life Contingent Payment Models Continuous Whole Life Contingent Payment Models Approximate Values of Fully Continuous Reserves Random Variable Analysis NLP Reserves for Contingent Payment Models with m thl Payment Funding Multi-State Model Representation Gain and Loss Analysis Contingent Insurance Contracts Contingent Annuity Contracts Written-Answer Question Examples Exercises 263 CHAPTER ELEVEN: CONTINGENT CONTRACT RESERVES (RESERVES AS FINANCIAL LIABILITIES) Modified Reserves Reserve Modification in General Full Preliminary Term Modified Reserves Deficiency Reserves Negative Reserves Net Premium Reserves at Fractional Durations Generalization to Non-Level Benefits and Non-Level Net Premiums Discrete Models Continuous Models Incorporation of Expenses Gain and Loss Analysis Written-Answer Question Examples Exercises 287 CHAPTER TWELVE: MODELS DEPENDENT ON MULTIPLE SURVIVALS (MULTI-LIFE MODELS) The Joint-Life Model The Time-to-Failure Random Variable for a Joint-Life Status The Survival Distribution Function of T xy The Cumulative Distribution Function of T xy 292

6 xii TABLE OF CONTENTS The Probability Density Function of T xy The Hazard Rate Function of T xy Conditional Probabilities Moments of T xy The Last-Survivor Model The Time-to-Failure Random Variable for a Last-Survivor Status Functions of the Random Variable T xy Relationships Between T xy and T xy Contingent Probability Functions Contingent Contracts Involving Multi-Life Statuses Contingent Payment Models Contingent Annuity Models Annual Premiums and Reserves Reversionary Annuities Contingent Insurance Functions Multi-State Model Representation The General Model Notation Annuity Contracts Insurance Contracts Solving the Kolmogorov Forward Equation Thiele s Equation in the Multi-Life Model General Random Variable Analysis Marginal Distributions of T x and T y The Covariance of T x and T y Other Joint Functions of T x and T y Joint and Last-Survivor Status Functions Common Shock A Model for Lifetime Dependency Written-Answer Question Examples Exercises 329 CHAPTER THIRTEEN: MULTIPLE-DECREMENT MODELS (THEORY) Discrete Multiple-Decrement Models The Multiple-Decrement Table Random Variable Analysis Theory of Competing Risks Continuous Multiple-Decrement Models Uniform Distribution of Decrements Uniform Distribution in the Multiple-Decrement Context Uniform Distribution in the Associated Single-Decrement Tables Constant Forces of Decrement Written-Answer Question Examples Exercises 356 CHAPTER FOURTEEN: MULTIPLE-DECREMENT MODELS (APPLICATIONS) Actuarial Present Value Asset Shares 365

7 TABLE OF CONTENTS xiii 14.3 Non-Forfeiture Options Cash Values Reduced Paid-Up Insurance Extended Term Insurance Multi-State Model Representation, with Illustrations The General Multiple-Decrement Model The Total and Permanent Disability Model Disability Model Allowing for Recovery Continuing Care Retirement Communities Thiele s Differential Equation in the Multiple-Decrement Case Defined Benefit Pension Plans Normal Retirement Benefits Early Retirement Benefits Withdrawal and Other Benefits Funding and Reserving Gain and Loss Analysis Written-Answer Question Examples Exercises 4 PART THREE: SPECIALIZED TOPICS CHAPTER FIFTEEN: MODELS WITH VARIABLE INTEREST RATES Actuarial Present Values Using Variable Interest Rates Deterministic Interest Rate Scenarios Spot Interest Rates and the Term Structure of Interest Rates Forward Interest Rates Transferring the Interest Rate Risk Exercises 422 CHAPTER SIXTEEN: UNIVERSAL LIFE INSURANCE Definitions and Basic Mechanics Universal Life with Variable Death Benefit (Type B) Universal Life with Fixed Death Benefit (Type A) Corridor Factors Surrender Benefits Policy Loan Provisions Variations on the Basic Form Variable Universal Life (VUL) Insurance Secondary Guarantees Indexed Universal Life Insurance Pricing Considerations Mortality Lapse Expenses Investment Income Pricing for Secondary Guarantees 44

8 xiv TABLE OF CONTENTS 16.4 Reserving Considerations Basic Universal Life Variable Universal Life Indexed Universal Life Contracts with Secondary Guarantees Exercises 448 CHAPTER SEVENTEEN: PROFIT ANALYSIS Definitions of Basic Concepts Pre-Contract Expenses The Profit Vector The Profit Signature Net Present Value Internal Rate of Return Profit Margin Discounted Payback Period A Comprehensive Example Commentary on the Comprehensive Example Uses of Profit Analysis Premium Determination Reserve Determination Cash Management Profit Emergence Complete Financial Evaluation Using Profit Analysis to Determine Reserves Profit Distribution Participating Insurance Actual vs. Expected Profit Gain and Loss Distributable Surplus (Profit) Forms of Distribution Cash Premium Reduction Terminal Bonuses Purchase of Additional Insurance Distribution to Terminating Policyholders Exercises 473 APPENDIX A COMPUTATION OF ACTUARIAL FUNCTIONS 479 APPENDIX B DERIVATION OF THE KOLMOGOROV FORWARD EQUATION 493 APPENDIX C THE MATHEMATICS OF RISK DIVERSIFICATION 495 ANSWERS TO THE EXERCISES 497 BIBLIOGRAPHY 517 INDEX 519

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