Yanyun Zhu. Actuarial Model: Life Insurance & Annuity. Series in Actuarial Science. Volume I. ir* International Press.

Transcription

1 Yanyun Zhu Actuarial Model: Life Insurance & Annuity Series in Actuarial Science Volume I ir* International Press

2 Contents Preface v 1 Interest and Annuity-Certain Introduction Interest Simple Interest Compound Interest Interest Convertible m-thly Force of Interest Relationship among Interest Rates The Accumulation Factor The Discount Factor Annuities-Certain Annual Annuities-Certain Continuous Annuities-Certain m-thly Annuities-Certain Accumulated Values of Annuities-Certain at Time n Summary Exercise 17 2 Individual Future Lifetime Introduction A Newborn's Future Lifetime X Future Lifetime of (x) Relationship Between Probability Functions of X and T(x) Curtate-Future-Lifetime of (x) Conditional Average Death Time Central Force of Mortality Life Table Aggregate Life Table Select-and-Ultimate Life Table Summary Exercise 52 3 Life Insurance Introduction Continuous Life Insurance Level Life Insurance 60 i

3 ii CONTENTS A General Continuous Life Insurance Discrete Life Insurance Level Life Insurance A General Discrete Life Insurance Commutation Functions m-thly Life Insurance Endowment Insurance Summary Exercise 85 4 Life Annuities Introduction Continuous Life Annuities Level Life Annuities Varying Continuous Life Annuities Annual Life Annuities Level Annual Life Annuities Varying Annual Life Annuities Commutation Functions Special Life Annuities m-thly Life Annuities n-year-certain-and-life Annuities Apportionable Annuities-Due Complete Annuities-Immediate Summary Exercise Insurance Premiums Introduction Insurance Pricing Principles The Three Pricing Principles Single Benefit Premiums Benefit Premiums Fully Continuous Benefit Premiums Fully Discrete Benefit Premiums m-thly Benefit Premiums Apportionable Benefit Premiums Gross Insurance Premiums Classification of Expenses Gross Premiums Under the Equivalence Principle Summary Exercises Insurance Reserves Introduction Insurance Reserve Principles The Prospective Loss Random Variable The Three Common Principles Insurance Benefit Reserves Benefit Reserves for Fully Continuous Life Insurance 178

4 CONTENTS iii Benefit Reserves for Fully Discrete Life Insurance Benefit Reserves with the Retrospective Method Recursive Formula between Discrete Benefit Reserves Benefit Reserves for Special Life Insurance Benefit Reserves for m-thly Life Insurance Benefit Reserves for Mixed Life Insurance Benefit Reserves with Apportionable Premiums Gross Insurance Reserves Summary Excercise Joint-Life Functions Introduction Joint Distributions of Future Lifetimes The Joint-Life Status Last-Survivor Status (xy) Relationship among T(x),T(y), Txy, and Tm Contingent Probabilities Dependent Models Common Shock Model Frank's Copula Life Insurance on Two Individuals Life Insurance on (xy) and (xy) Contingent Life Insurance Life Annuities on Two Individuals Life Annuities on (xy) and (xy) Reversionary Annuities Summary Exercise Multiple-Decrement Model Introduction A Double-Decrement Model Future Lifetimes of Two Risks Probabilities of Decrement A General m-decrement Model Probabilities of Decrement Central Rates from a Multiple-Decrement Table Constructing a Multiple-Decrement Table Discretionary Life Insurance Benefit Premiums for Discretionary Life Insurance Benefit Reserves for Discretionary Life Insurance Asset Share Summary Exercise 305 Appendix 1 Standard Normal Table 321 Appendix 2A Illustrative Life Table with i =

5 iv CONTENTS Appendix 2B Illustrative Service Table with i = Appendix 2C Interest Rate Function at i = Appendix 3 Probability Theorem and Random Variables 328 Appendix 4 Interest Rate and Annuity-Certain 331 Bibliography 332 Symbol Index 334 Index 339