<p><strong>Actuarial Mathematics and Life-Table Statistics<br/>Eric V. Slud<br/>Mathematics Department<br/>University of Maryland, College Park<br/> 2001</strong></p><p><strong>Contents<br/>0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi<br/>1 Basics of Probability & Interest 1<br/>1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br/>1.2 Theory of Interest . . . . . . . . . . . . . . . . . . . . . . . . . 7<br/>1.2.1 Variable Interest Rates . . . . . . . . . . . . . . . . . . 10<br/>1.2.2 Continuous-time Payment Streams . . . . . . . . . . . 15<br/>1.3 Exercise Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16<br/>1.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 18<br/>1.5 Useful Formulas from Chapter 1 . . . . . . . . . . . . . . . . . 21<br/>2 Interest & Force of Mortality 23<br/>2.1 More on Theory of Interest . . . . . . . . . . . . . . . . . . . . 23<br/>2.1.1 Annuities & Actuarial Notation . . . . . . . . . . . . . 24<br/>2.1.2 Loan Amortization & Mortgage Re¯nancing . . . . . . 29<br/>2.1.3 Illustration on Mortgage Re¯nancing . . . . . . . . . . 30<br/>2.1.4 Computational illustration in Splus . . . . . . . . . . . 32<br/>2.1.5 Coupon & Zero-coupon Bonds . . . . . . . . . . . . . . 35<br/>2.2 Force of Mortality & Analytical Models . . . . . . . . . . . . . 37<br/>i<br/>ii CONTENTS<br/>2.2.1 Comparison of Forces of Mortality . . . . . . . . . . . . 45<br/>2.3 Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51<br/>2.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 54<br/>2.5 Useful Formulas from Chapter 2 . . . . . . . . . . . . . . . . . 58<br/>3 Probability & Life Tables 61<br/>3.1 Interpreting Force of Mortality . . . . . . . . . . . . . . . . . . 61<br/>3.2 Interpolation Between Integer Ages . . . . . . . . . . . . . . . 62<br/>3.3 Binomial Variables &<br/>Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 66<br/>3.3.1 Exact Probabilities, Bounds & Approximations . . . . 71<br/>3.4 Simulation of Life Table Data . . . . . . . . . . . . . . . . . . 74<br/>3.4.1 Expectation for Discrete Random Variables . . . . . . 76<br/>3.4.2 Rules for Manipulating Expectations . . . . . . . . . . 78<br/>3.5 Some Special Integrals . . . . . . . . . . . . . . . . . . . . . . 81<br/>3.6 Exercise Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 84<br/>3.7 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 87<br/>3.8 Useful Formulas from Chapter 3 . . . . . . . . . . . . . . . . . 93<br/>4 Expected Present Values of Payments 95<br/>4.1 Expected Payment Values . . . . . . . . . . . . . . . . . . . . 96<br/>4.1.1 Types of Insurance & Life Annuity Contracts . . . . . 96<br/>4.1.2 Formal Relations among Net Single Premiums . . . . . 102<br/>4.1.3 Formulas for Net Single Premiums . . . . . . . . . . . 103<br/>4.1.4 Expected Present Values for m = 1 . . . . . . . . . . . 104<br/>4.2 Continuous Contracts & Residual Life . . . . . . . . . . . . . 106<br/>CONTENTS iii<br/>4.2.1 Numerical Calculations of Life Expectancies . . . . . . 111<br/>4.3 Exercise Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113<br/>4.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 118<br/>4.5 Useful Formulas from Chapter 4 . . . . . . . . . . . . . . . . . 121<br/>5 Premium Calculation 123<br/>5.1 m-Payment Net Single Premiums . . . . . . . . . . . . . . . . 124<br/>5.1.1 Dependence Between Integer & Fractional Ages at Death124<br/>5.1.2 Net Single Premium Formulas | Case (i) . . . . . . . 126<br/>5.1.3 Net Single Premium Formulas | Case (ii) . . . . . . . 129<br/>5.2 Approximate Formulas via Case(i) . . . . . . . . . . . . . . . . 132<br/>5.3 Net Level Premiums . . . . . . . . . . . . . . . . . . . . . . . 134<br/>5.4 Bene¯ts Involving Fractional Premiums . . . . . . . . . . . . . 136<br/>5.5 Exercise Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138<br/>5.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 142<br/>5.7 Useful Formulas from Chapter 5 . . . . . . . . . . . . . . . . . 145<br/>6 Commutation & Reserves 147<br/>6.1 Idea of Commutation Functions . . . . . . . . . . . . . . . . . 147<br/>6.1.1 Variable-bene¯t Commutation Formulas . . . . . . . . 150<br/>6.1.2 Secular Trends in Mortality . . . . . . . . . . . . . . . 152<br/>6.2 Reserve & Cash Value of a Single Policy . . . . . . . . . . . . 153<br/>6.2.1 Retrospective Formulas & Identities . . . . . . . . . . . 155<br/>6.2.2 Relating Insurance & Endowment Reserves . . . . . . . 158<br/>6.2.3 Reserves under Constant Force of Mortality . . . . . . 158<br/>6.2.4 Reserves under Increasing Force of Mortality . . . . . . 160<br/>iv CONTENTS<br/>6.2.5 Recursive Calculation of Reserves . . . . . . . . . . . . 162<br/>6.2.6 Paid-Up Insurance . . . . . . . . . . . . . . . . . . . . 163<br/>6.3 Select Mortality Tables & Insurance . . . . . . . . . . . . . . . 164<br/>6.4 Exercise Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 166<br/>6.5 Illustration of Commutation Columns . . . . . . . . . . . . . . 168<br/>6.6 Examples on Paid-up Insurance . . . . . . . . . . . . . . . . . 169<br/>6.7 Useful formulas from Chapter 6 . . . . . . . . . . . . . . . . . 171<br/>7 Population Theory 161<br/>7.1 Population Functions & Indicator Notation . . . . . . . . . . . 161<br/>7.1.1 Expectation & Variance of Residual Life . . . . . . . . 164<br/>7.2 Stationary-Population Concepts . . . . . . . . . . . . . . . . . 167<br/>7.3 Estimation Using Life-Table Data . . . . . . . . . . . . . . . . 170<br/>7.4 Nonstationary Population Dynamics . . . . . . . . . . . . . . 174<br/>7.4.1 Appendix: Large-time Limit of ¸(t; x) . . . . . . . . . 176<br/>7.5 Exercise Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 178<br/>7.6 Population Word Problems . . . . . . . . . . . . . . . . . . . . 179<br/>8 Estimation from Life-Table Data 185<br/>8.1 General Life-Table Data . . . . . . . . . . . . . . . . . . . . . 186<br/>8.2 ML Estimation for Exponential Data . . . . . . . . . . . . . . 188<br/>8.3 MLE for Age Speci¯c Force of Mortality . . . . . . . . . . . . 191<br/>8.3.1 Extension to Random Entry & Censoring Times . . . . 193<br/>8.4 Kaplan-Meier Survival Function Estimator . . . . . . . . . . . 194<br/>8.5 Exercise Set 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 195<br/>8.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 195<br/>CONTENTS v<br/>9 Risk Models & Select Mortality 197<br/>9.1 Proportional Hazard Models . . . . . . . . . . . . . . . . . . . 198<br/>9.2 Excess Risk Models . . . . . . . . . . . . . . . . . . . . . . . . 201<br/>9.3 Select Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . 202<br/>9.4 Exercise Set 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 204<br/>9.5 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 204<br/>10 Multiple Decrement Models 205<br/>10.1 Multiple Decrement Tables . . . . . . . . . . . . . . . . . . . . 206<br/>10.2 Death-Rate Estimators . . . . . . . . . . . . . . . . . . . . . . 209<br/>10.2.1 Deaths Uniform within Year of Age . . . . . . . . . . . 209<br/>10.2.2 Force of Mortality Constant within Year of Age . . . . 210<br/>10.2.3 Cause-Speci¯c Death Rate Estimators . . . . . . . . . 210<br/>10.3 Single-Decrement Tables and Net Hazards of Mortality . . . . 212<br/>10.4 Cause-Speci¯c Life Insurance Premiums . . . . . . . . . . . . 213<br/>10.5 Exercise Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 213<br/>10.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 214<br/>11 Central Limit Theorem & Portfolio Risks 215<br/>13 Bibliography 217<br/>Solutions & Hints 219</strong></p><p> <br/> <br/> <br/></p><p><strong></strong> </p>

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