Actuarial Mathematics & Life-Table Statistics

Actuarial Mathematics & Life-Table Statistics -...

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Actuarial Mathematics and Life-Table Statistics Eric V. Slud Mathematics Department University of Maryland, College Park c 2001
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c 2001 Eric V. Slud Statistics Program Mathematics Department University of Maryland College Park, MD 20742
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Contents 0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Basics of Probability & Interest 1 1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Theory of Interest . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Variable Interest Rates . . . . . . . . . . . . . . . . . . 10 1.2.2 Continuous-time Payment Streams . . . . . . . . . . . 15 1.3 Exercise Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Useful Formulas from Chapter 1 . . . . . . . . . . . . . . . . . 21 2 Interest & Force of Mortality 23 2.1 More on Theory of Interest . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Annuities & Actuarial Notation . . . . . . . . . . . . . 24 2.1.2 Loan Amortization & Mortgage Refinancing . . . . . . 29 2.1.3 Illustration on Mortgage Refinancing . . . . . . . . . . 30 2.1.4 Computational illustration in Splus . . . . . . . . . . . 32 2.1.5 Coupon & Zero-coupon Bonds . . . . . . . . . . . . . . 35 2.2 Force of Mortality & Analytical Models . . . . . . . . . . . . . 37 i
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ii CONTENTS 2.2.1 Comparison of Forces of Mortality . . . . . . . . . . . . 45 2.3 Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5 Useful Formulas from Chapter 2 . . . . . . . . . . . . . . . . . 58 3 Probability & Life Tables 61 3.1 Interpreting Force of Mortality . . . . . . . . . . . . . . . . . . 61 3.2 Interpolation Between Integer Ages . . . . . . . . . . . . . . . 62 3.3 Binomial Variables & Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 66 3.3.1 Exact Probabilities, Bounds & Approximations . . . . 71 3.4 Simulation of Life Table Data . . . . . . . . . . . . . . . . . . 74 3.4.1 Expectation for Discrete Random Variables . . . . . . 76 3.4.2 Rules for Manipulating Expectations . . . . . . . . . . 78 3.5 Some Special Integrals . . . . . . . . . . . . . . . . . . . . . . 81 3.6 Exercise Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.7 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.8 Useful Formulas from Chapter 3 . . . . . . . . . . . . . . . . . 93 4 Expected Present Values of Payments 95 4.1 Expected Payment Values . . . . . . . . . . . . . . . . . . . . 96 4.1.1 Types of Insurance & Life Annuity Contracts . . . . . 96 4.1.2 Formal Relations among Net Single Premiums . . . . . 102 4.1.3 Formulas for Net Single Premiums . . . . . . . . . . . 103 4.1.4 Expected Present Values for m = 1 . . . . . . . . . . . 104 4.2 Continuous Contracts & Residual Life . . . . . . . . . . . . . 106
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CONTENTS iii 4.2.1 Numerical Calculations of Life Expectancies . . . . . . 111 4.3 Exercise Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.5 Useful Formulas from Chapter 4 . . . . . . . . . . . . . . . . . 121 5 Premium Calculation 123 5.1 m-Payment Net Single Premiums . . . . . . . . . . . . . . . . 124 5.1.1 Dependence Between Integer & Fractional Ages at Death124 5.1.2 Net Single Premium Formulas — Case (i) . . . . . . . 126 5.1.3 Net Single Premium Formulas — Case (ii) . . . . . . . 129 5.2 Approximate Formulas via Case(i) . . . . . . . . . . . . . . . . 132 5.3 Net Level Premiums . . . . . . . . . . . . . . . . . . . . . . . 134 5.4 Benefits Involving Fractional Premiums . . . . . . . . . . . . . 136 5.5 Exercise Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.6 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.7 Useful Formulas from Chapter 5 . . . . . . . . . . . . . . . . . 145 6 Commutation & Reserves 147 6.1 Idea of Commutation Functions . . . . . . . . . . . . . . . . . 147 6.1.1 Variable-benefit Commutation Formulas . . . . . . . . 150 6.1.2 Secular Trends in Mortality . . . . . . . . . . . . . . . 152 6.2 Reserve & Cash Value of a Single Policy . . . . . . . . . . . . 153 6.2.1 Retrospective Formulas & Identities . . . . . . . . . . . 155 6.2.2 Relating Insurance & Endowment Reserves . . . . . . . 158 6.2.3 Reserves under Constant Force of Mortality . . . . . . 158 6.2.4 Reserves under Increasing Force of Mortality . . . . . . 160
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iv CONTENTS 6.2.5 Recursive Calculation of Reserves . . . . . . . . . . . . 162 6.2.6 Paid-Up Insurance . . . . . . . . . . . . . . . . . . . . 163 6.3 Select Mortality Tables & Insurance . . . . . . . . . . . . . . . 164 6.4 Exercise Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.5 Illustration of Commutation Columns . . . . . . . . . . . . . . 168 6.6 Examples on Paid-up Insurance . . . . . . . . . . . . . . . . . 169 6.7 Useful formulas from Chapter 6 . . . . . . . . . . . . . . . . . 171 7 Population Theory 161 7.1 Population Functions & Indicator Notation . . . . . . . . . . . 161 7.1.1 Expectation & Variance of Residual Life . . . . . . . . 164 7.2 Stationary-Population Concepts . . . . . . . . . . . . . . . . . 167 7.3 Estimation Using Life-Table Data . . . . . . . . . . . . . . . . 170 7.4 Nonstationary Population Dynamics . . . . . . . . . . . . . . 174 7.4.1 Appendix: Large-time Limit of λ ( t, x ) . . . . . . . . . 176 7.5 Exercise Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.6
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