MODELS FOR QUANTIFYING RISK

Transcription

1 MODELS FOR QUANTIFYING RISK THIRD EDITION ROBIN J. CUNNINGHAM, FSA, PH.D. THOMAS N. HERZOG, ASA, PH.D. RICHARD L. LONDON, FSA B ACTEX PUBLICATIONS, INC. WINSTED, CONNECTICUT

2 PREFACE iii THIRD EDITION PREFACE vii PART ONE: REVIEW AND BACKGROUND MATERIAL CHAPTER 1 REVIEW OF INTEREST THEORY Interest Measures Level Annuity Functions Immediate Annuity Annuity-due Continuous Annuity Non-Level Annuity Functions Immediate Annuities Annuities-due Continuous Annuities Equation of Value 17 CHAPTER 2 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 30 ix

3 2.3 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 Sums of Independent Random Variables The Moments of S Distributions Closed Under Convolution The Method of Convolutions Approximating the Distribution of S Compound Distributions The Moments of S The Compound Poisson Distribution 48 PART TWO: MODELS FOR SURVIVAL-CONTINGENT RISKS CHAPTER 3 SURVIVAL MODELS ^ (CONTINUOUS PARAMETRIC CONTEXT) The Age-at-Failure Random Variable The Cumulative Distribution Function of X The Survival Distribution Function of X The Probability Density Function of X The Hazard Rate Function of X The Moments of the Age-at-Failure Random Variable X Actuarial Survival Models Examples of Parametric Survival Models The Uniform Distribution The Exponential Distribution The Gompertz Distribution The Makeham Distribution The Weibull Distribution Summary of Parametric Survival Models The Time-to-Failure Random Variable The Survival Distribution Function of T x The Cumulative Distribution Function of T x 66

4 XI The Probability Density Function of T x The Hazard Rate Function of T x Moments of the Future Lifetime Random Variable T x The Time-to-Failure Random Variable K x 71 The Central Rate 73 Select Survival Models 75 Exercises 76 CHAPTER 4 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 X Conditional Probabilities and Densities The Curtate Expectation of Life The Central Rate Summary of Concepts and Notation Methods for Non-Integral Ages Linear Form for x+t Exponential Form for x+t Hyperbolic Form for x+t Summary Select Life Tables Life Table Summary Exercises 113 CHAPTER 5 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 131

5 xii TABLE OF CONTENTS 5.2 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 Continuous and m' hly Functions Approximated from the Life Table Continuous Contingent Payment Models m' hly Contingent Payment Models Miscellaneous Examples Exercises 156 CHAPTER 6 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 Contingent Annuities Payable m My The Immediate Case The Due Case Random Variable Analysis Numerical Evaluation in the m thly and Continuous Cases Non-Level Payment Annuity Functions Miscellaneous Examples Exercises 203

6 CHAPTER CHAPTER 8 FUNDING PLANS FOR CONTINGENT CONTRACTS (ANNUAL PREMIUMS) 211 Annual Funding Schemes for Contingent Payment Models Discrete Contingent Payment Models Continuous Contingent Payment Models Contingent Annuity Models Non-Level Premium Contracts 218 Random Variable Analysis 219 Continuous Payment Funding Schemes Discrete Contingent Payment Models Continuous Contingent Payment Models 225 Funding Schemes with m Wy Payments 228 Funding Plans Incorporating Expenses 230 Miscellaneous Examples 233 Exercises 240 CONTINGENT CONTRACT RESERVES (BENEFIT RESERVES) Reserves for Contingent Payment Models with Annual Payment Funding Reserves by the Prospective Method Reserves by the Retrospective Method Additional Terminal Reserve Expressions Random Variable Analysis Reserve for Contingent Contracts with Immediate Payment of Claims Reserves for Contingent Annuity Models Recursive Relationships for Discrete Models with Annual Premiums Group Deterministic Approach Random Variable Analysis - Cash Basis Random Variable Analysis - Accrued Basis Reserves for Contingent Payment Models with Continuous Payment Funding Discrete Whole Life Contingent Payment Models Continuous Whole Life Contingent Payment Models Random Variable Analysis 275

7 xiv TABLE OF CONTENTS 8.4 Reserves for Contingent Payment Models with m' hly Payment Funding Incorporation of Expenses Reserves at Fractional Durations Generalization to Non-Level Benefits and Premiums Discrete Models Continuous Models Miscellaneous Examples Exercises 292 CHAPTER 9 MODELS DEPENDENT ON MULTIPLE SURVIVALS (MULTI-LIFE MODELS) The Joint-Life Model 299 s The Time-to-Failure Random Variable for a Joint-Life Status Survival Distribution Function of T xy Cumulative Distribution Function of T^ Probability Density Function of T xy Hazard Rate Function of T^ Conditional Probabilities Moments of T xv The Last-Survivor Model The Time-to-Failure Random Variable for a Last-Survivor Status Functions of the Random Variable T w Relationships Between T xy and T^ Contingent Probability Functions Contingent Contracts Involving Multi-Life Statuses Contingent Payment Models Contingent Annuity Models Annual Premiums and Reserves Reversionary Annuities Contingent Insurance Functions General Random Variable Analysis Marginal Distributions of T x and T y TheCovarianceofr x and7; Other Joint Functions of T x and T y Joint and Last-Survivor Status Functions Common Shock - A Model for Lifetime Dependency Exercises 333

8 XV CHAPTER 10 MULTIPLE CONTINGENCIES WITH APPLICATIONS (MULTIPLE-DECREMENT MODELS) 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 Actuarial Present Value Asset Shares Multi-State Models The Homogeneous Process The Nonhomogeneous Process Exercises 374 PART THREE: MODELS FOR NON-SURVIVAL-CONTINGENT RISKS CHAPTER 11 CLAIM FREQUENCY MODELS Section 2.2 (Discrete Distributions) Revisited The Binomial Distribution The Poisson Distribution The Negative Binomial Distribution The Geometric Distribution Summary of the Recursive Relationships Probability Generating Functions Creating Additional Counting Distributions Compound Frequency Models Mixture Frequency Models Truncation or Modification at Zero Counting Processes Properties of Counting Processes The Poisson Counting Process 411

9 xvi TABLE OF CONTENTS Further Properties of the Poisson Counting Process Poisson Mixture Processes The Nonstationary Poisson Counting Process Exercises 418 CHAPTER 12 CLAIM SEVERITY MODELS Fundamental Continuous Distributions The Normal and Exponential Distributions The Pareto Distribution Analytic Measures of Tail Weight Generating New Distributions Summation 431, Scalar Multiplication Power Operations Exponentiation and the Lognormal Distribution Summary of Severity Distributions Mixtures of Distributions Spliced Distributions Limiting Distributions Modifications of the Loss Random Variable Deductibles Policy Limits Relationships between Deductibles and Policy Limits Coinsurance Factors The Effect of Inflation Effect of Coverage Deductibles on Frequency Models Tail Weight Revisited; Risk Measures The Mean Excess Loss Function Conditional Tail Expectation Value at Risk Distortion Risk Measures Risk Measures Using Discrete Distributions Other Risk Measures Empirical Loss Distributions Exercises 485

10 xvu CHAPTER 13 MODELS FOR AGGREGATE PAYMENTS Individual Risk versus Collective Risk Selection of Frequency and Severity Distributions Frequency Severity Frequency-Severity Interaction More on the Collective Risk Model Convolutions of the Probability Function ofx Convolutions of the CDF ofx Continuous Severity Distributions A Final Thought Regarding Convolutions Effect of Coverage Modifications Modifications Applied to Individual Losses Modifications Applied to the Aggregate Loss (Stop-Loss Reinsurance) Infinitely Divisible Distributions Definition of Infinite Divisibility The Poisson Distribution The Negative Binomial Distribution Exercises 529 CHAPTER 14 PROCESS MODELS The Compound Poisson Process Moments of the Compound Poisson Process Other Properties of the Compound Poisson Process The Surplus Process Model The Probability of Ruin The Adjustment Coefficient The Probability of Ruin The Distribution of Surplus Deficit The Event of U{t) <u The Cumulative Loss of Surplus Probability of Ruin in Finite Time Exercises 558

11 xviii TABLE OF CONTENTS APPENDIX A REVIEW OF MARKOV CHAINS 565 APPENDIX B REVIEW OF STOCHASTIC SIMULATION 587 APPENDIX C EVALUATION BY SIMULATION 605 APPENDIX D USING MICROSOFT EXCEL AND VISUAL BASIC MACROS TO COMPUTE ACTUARIAL FUNCTIONS 625 APPENDIX E REVIEW OF THE INCOMPLETE GAMMA FUNCTION 641 ANSWERS TO TEXT EXERCISES 649 BIBLIOGRAPHY 671 INDEX 673