Life insurance mathematics

Hans U. Gerber

Life Insurance Mathematics with exercises contributed by Samuel H. Cox

Third Edition 1997

Springer Swiss Association of Actuaries Zurich

Contents

1 The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Mathematics of Compound Interest Mathematical Bases of Life Contingencies Effective Interest Rates Nominal Interest Rates Continuous Payments Interest in Advance Perpetuities Annuities Repayment of a Debt Internal Rate of Return

1 1 2 3 4 6 9 11 13

2 The 2.1 2.2 2.3 2.4 2.5 2.6

Future Lifetime of a Life Aged x The Model The Force of Mortality Analytical Distributions of T The Curtate Future Lifetime of (a;) Life Tables Probabilities of Death for Fractions of a Year

15 16 17 18 20 21

3 Life Insurance 3.1 Introduction 3.2 Elementary Insurance Types 3.2.1 Whole Life and Term Insurance 3.2.2 Pure Endowments 3.2.3 Endowments 3.3 Insurances Payable at the Moment of Death 3.4 General Types of Life Insurance 3.5 Standard Types of Variable Life Insurance 3.6 Recursive Formulae 4

Life Annuities 4.1 Introduction 4.2 Elementary Life Annuities

23 23 23 24 25 26 27 29 31

35 35

XIV 4.3 4.4 4.5 4.6 4.7 4.8 5

7

Payments made more Frequently than Once a Year Variable Life Annuities Standard Types of Life Annuity Recursion Formulae Inequalities Payments Starting at Non-integral Ages

37 39 41 42 43 46

Net Premiums 5.1 5.2 5.3

6

Contents

5.4 5.5 5.6 5.7

Introduction An Example Elementary Forms of Insurance 5.3.1 Whole Life and Term Insurance 5.3.2 Pure Endowments 5.3.3 Endowments 5.3.4 Deferred Life Annuities Premiums Paid m Times a Year A General Type of Life Insurance Policies with Premium Refund Stochastic Interest

49 49 52 52 53 54 54 54 55 56 56

Net 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

P r e m i u m Reserves Introduction Two Examples Recursive Considerations The Survival Risk The Net Premium Reserve of a Whole Life Insurance Net Premium Reserves at Fractional Durations Allocation of the Overall Loss to Policy Years Conversion of an Insurance Technical Gain Procedure for Pure Endowments The Continuous Model

59 59 61 63 63 64 65 68 69 70 71

Multiple D e c r e m e n t s 7.1 7.2 7.3 7.4 7.5 7.6

The Model Forces of Decrement The Curtate Lifetime of (a;) A General Type of Insurance The Net Premium Reserve The Continuous Model

75 76 76 77 78 80

Contents 8

9

XV

Multiple Life Insurance 8.1 Introduction 8.2 The Joint-Life Status 8.3 Simplifications 8.4 The Last-Survivor Status 8.5 The General Symmetric Status 8.6 The Schuette-Nesbitt Formula 8.7 Asymmetric Annuities 8.8 Asymmetric Insurances The 9.1 9.2 9.3 9.4 9.5 9.6 9.7

83 83 84 85 87 89 90 91

Total Claim Amount in a Portfolio Introduction 93 The Normal Approximation 93 Exact Calculation of the Total Claim Amount Distribution . . 94 The Compound Poisson Approximation 96 Recursive Calculation of the Compound Poisson Distribution . 98 Reinsurance 100 Stop-Loss Reinsurance 101

10 Expense Loadings 10.1 Introduction 10.2 The Expense-Loaded Premium 10.3 Expense-Loaded Premium Reserves

103 104 105

11 Estimating Probabilities of Death 11.1 Problem Description 11.2 The Classical Method 11.3 Alternative Solution 11.4 The Maximum Likelihood Method 11.5 Statistical Inference 11.6 The Bayesian Approach 11.7 Multiple Causes of Decrement 11.8 Interpretation of Results

109 110 Ill 112 112 116 116 118

Appendix A. Commutation Functions A.I A.2 A.3 A.4 A.5

Introduction The Deterministic Model Life Annuities Life Insurance Net Annual Premiums and Premium Reserves

Appendix B. Simple Interest

119 119 120 121 122

125

XVI

Contents

Appendix C. Exercises CO C.I C.I.I C.1.2 C.2 C.2.1 C.2.2 C.3 C.3.1 C.3.2 C.4 C.4.1 C.4.2 C.5 C.5.1 C.5.2 C.5.3 C.6 C.6.1 C.6.2 C.7 C.7.1 C.8 C.8.1 C.8.2 C.9 C.9.1 CIO C.10.1 C.10.2 C.ll C.ll.l

Introduction Mathematics of Compound Interest: Exercises Theory Exercises Spreadsheet Exercises The Future Lifetime of a Life Aged x: Exercises Theory Exercises Spreadsheet Exercises Life Insurance Theory Exercises Spreadsheet Exercises Life Annuities Theory Exercises Spreadsheet Exercises Net Premiums Notes Theory Exercises Spreadsheet Exercises Net Premium Reserves Theory Exercises Spreadsheet Exercises Multiple Decrements: Exercises Theory Exercises Multiple Life Insurance: Exercises Theory Exercises Spreadsheet Exercises The Total Claim Amount in a Portfolio Theory Exercises Expense Loadings Theory Exercises Spreadsheet Exercises Estimating Probabilities of Death Theory Exercises

128 129 129 130 133 134 136 138 138 141 142 142 144 146 146 146 150 152 152 155 156 156 158 158 160 161 161 163 163 164 165 165

Appendix D . Solutions D.O D.I D.I.I D.I.2 D.2 D.2.1 D.2.2 D.3 D.3.1 D.3.2

Introduction Mathematics of Compound Interest Solutions to Theory Exercises Solutions to Spreadsheet Exercises The Future Lifetime of a Life Aged x Solutions to Theory Exercise Solutions to Spreadsheet Exercises Life Insurance Solutions to Theory Exercises Solution to Spreadsheet Exercises

168 169 169 170 172 172 174 175 175 178

Contents

D.4 D.4.1 D.4.2 D.5 D.5.1 D.5.2 D.6 D.6.1 D.6.2 D.7 D.7.1 D.8 D.8.1 D.8.2 D.9 D.9.1 D.10 D.10.1 D.10.2 D.ll D.ll.l

Life Annuities Solutions to Theory Exercises Solutions to Spreadsheet Exercises Net Premiums: Solutions Theory Exercises Solutions to Spreadsheet Exercises Net Premium Reserves: Solutions Theory Exercises Solutions to Spreadsheet Exercises Multiple Decrements: Solutions Theory Exercises Multiple Life Insurance: Solutions Theory Exercises Solutions to Spreadsheet Exercises The Total Claim Amount in a Portfolio Theory Exercises Expense Loadings Theory Exercises Spreadsheet Exercises Estimating Probabilities of Death Theory Exercises

XVII

179 179 181 183 183 187 189 189 190 192 192 194 194 197 198 198 200 200 201 203 203

Appendix E. Tables E.O Illustrative Life Tables E.I Commutation Columns E.2 Multiple Decrement Tables

207 209 211

References

213

Index

215