Hans U. Gerber

Life InsuranceMathematicswith exercises contributed by Samuel H. Cox

Third Edition 1997

Springer

Swiss Association of Actuaries Zurich

Contents

1 The Mathematics of Compound Interest1.1 Mathematical Bases of Life Contingencies 11.2 Effective Interest Rates 11.3 Nominal Interest Rates 21.4 Continuous Payments 31.5 Interest in Advance 41.6 Perpetuities 61.7 Annuities 91.8 Repayment of a Debt 111.9 Internal Rate of Return 13

2 The Future Lifetime of a Life Aged x

2.1 The Model 152.2 The Force of Mortality 162.3 Analytical Distributions of T 172.4 The Curtate Future Lifetime of (a;) 182.5 Life Tables 202.6 Probabilities of Death for Fractions of a Year 21

3 Life Insurance3.1 Introduction 233.2 Elementary Insurance Types 23

3.2.1 Whole Life and Term Insurance 233.2.2 Pure Endowments 243.2.3 Endowments 25

3.3 Insurances Payable at the Moment of Death 263.4 General Types of Life Insurance 273.5 Standard Types of Variable Life Insurance 293.6 Recursive Formulae 31

4 Life Annuities4.1 Introduction 354.2 Elementary Life Annuities 35

XIV Contents

4.3 Payments made more Frequently than Once a Year 374.4 Variable Life Annuities 394.5 Standard Types of Life Annuity 414.6 Recursion Formulae 424.7 Inequalities 434.8 Payments Starting at Non-integral Ages 46

5 Net P r emiums

5.1 Introduction 495.2 An Example 495.3 Elementary Forms of Insurance 52

5.3.1 Whole Life and Term Insurance 525.3.2 Pure Endowments 535.3.3 Endowments 545.3.4 Deferred Life Annuities 54

5.4 Premiums Paid m Times a Year 545.5 A General Type of Life Insurance 555.6 Policies with Premium Refund 565.7 Stochastic Interest 56

6 Net P r e m i u m Reserves

6.1 Introduction 596.2 Two Examples 596.3 Recursive Considerations 616.4 The Survival Risk 636.5 The Net Premium Reserve of a Whole Life Insurance 636.6 Net Premium Reserves at Fractional Durations 646.7 Allocation of the Overall Loss to Policy Years 656.8 Conversion of an Insurance 686.9 Technical Gain 696.10 Procedure for Pure Endowments 706.11 The Continuous Model 71

7 Mult iple Decrements

7.1 The Model 757.2 Forces of Decrement 767.3 The Curtate Lifetime of (a;) 767.4 A General Type of Insurance 777.5 The Net Premium Reserve 787.6 The Continuous Model 80

Contents XV

8 Multiple Life Insurance8.1 Introduction 838.2 The Joint-Life Status 838.3 Simplifications 848.4 The Last-Survivor Status 858.5 The General Symmetric Status 878.6 The Schuette-Nesbitt Formula 898.7 Asymmetric Annuities 908.8 Asymmetric Insurances 91

9 The Total Claim Amount in a Portfolio9.1 Introduction 939.2 The Normal Approximation 939.3 Exact Calculation of the Total Claim Amount Distribution . . 949.4 The Compound Poisson Approximation 969.5 Recursive Calculation of the Compound Poisson Distribution . 989.6 Reinsurance 1009.7 Stop-Loss Reinsurance 101

10 Expense Loadings10.1 Introduction 10310.2 The Expense-Loaded Premium 10410.3 Expense-Loaded Premium Reserves 105

11 Estimating Probabilities of Death11.1 Problem Description 10911.2 The Classical Method 11011.3 Alternative Solution I l l11.4 The Maximum Likelihood Method 11211.5 Statistical Inference 11211.6 The Bayesian Approach 11611.7 Multiple Causes of Decrement 11611.8 Interpretation of Results 118

Appendix A. Commutation FunctionsA.I Introduction 119A.2 The Deterministic Model 119A.3 Life Annuities 120A.4 Life Insurance 121A.5 Net Annual Premiums and Premium Reserves 122

Appendix B. Simple Interest 125

XVI Contents

Appendix C. Exercises

CO Introduction 128C.I Mathematics of Compound Interest: Exercises 129C.I.I Theory Exercises 129C.1.2 Spreadsheet Exercises 130C.2 The Future Lifetime of a Life Aged x: Exercises 133C.2.1 Theory Exercises 134C.2.2 Spreadsheet Exercises 136C.3 Life Insurance 138C.3.1 Theory Exercises 138C.3.2 Spreadsheet Exercises 141C.4 Life Annuities 142C.4.1 Theory Exercises 142C.4.2 Spreadsheet Exercises 144C.5 Net Premiums 146C.5.1 Notes 146C.5.2 Theory Exercises 146C.5.3 Spreadsheet Exercises 150C.6 Net Premium Reserves 152C.6.1 Theory Exercises 152C.6.2 Spreadsheet Exercises 155C.7 Multiple Decrements: Exercises 156C.7.1 Theory Exercises 156C.8 Multiple Life Insurance: Exercises 158C.8.1 Theory Exercises 158C.8.2 Spreadsheet Exercises 160C.9 The Total Claim Amount in a Portfolio 161C.9.1 Theory Exercises 161CIO Expense Loadings 163C.10.1 Theory Exercises 163C.10.2 Spreadsheet Exercises 164C.ll Estimating Probabilities of Death 165C.ll.l Theory Exercises 165

Appendix D. Solutions

D.O Introduction 168D.I Mathematics of Compound Interest 169D.I.I Solutions to Theory Exercises 169D.I.2 Solutions to Spreadsheet Exercises 170D.2 The Future Lifetime of a Life Aged x 172D.2.1 Solutions to Theory Exercise 172D.2.2 Solutions to Spreadsheet Exercises 174D.3 Life Insurance 175D.3.1 Solutions to Theory Exercises 175D.3.2 Solution to Spreadsheet Exercises 178

Contents XVII

D.4 Life Annuities 179D.4.1 Solutions to Theory Exercises 179D.4.2 Solutions to Spreadsheet Exercises 181D.5 Net Premiums: Solutions 183D.5.1 Theory Exercises 183D.5.2 Solutions to Spreadsheet Exercises 187D.6 Net Premium Reserves: Solutions 189D.6.1 Theory Exercises 189D.6.2 Solutions to Spreadsheet Exercises 190D.7 Multiple Decrements: Solutions 192D.7.1 Theory Exercises 192D.8 Multiple Life Insurance: Solutions 194D.8.1 Theory Exercises 194D.8.2 Solutions to Spreadsheet Exercises 197D.9 The Total Claim Amount in a Portfolio 198D.9.1 Theory Exercises 198D.10 Expense Loadings 200D.10.1 Theory Exercises 200D.10.2 Spreadsheet Exercises 201D.ll Estimating Probabilities of Death 203D.ll.l Theory Exercises 203

Appendix E. TablesE.O Illustrative Life Tables 207E.I Commutation Columns 209E.2 Multiple Decrement Tables 211

References 213

Index 215