1 Interest and Annuity-Certain 1
1.1 Introduction 1
1.2 Interest 1
1.2.1 Simple Interest 1
1.2.2 Compound Interest 2
1.2.3 Interest Convertible m-thly 3
1.2.4 Force of Interest 4
1.2.5 Relationship among Interest Rates 5
1.2.6 The Accumulation Factor 5
1.2.7 The Discount Factor 6
1.3 Annuities-Certain 6
1.3.1 Annual Annuities-Certain 6
1.3.2 Continuous Annuities-Certain 10
1.3.3 m-thly Annuities-Certain 12
1.3.4 Accumulated Values of Annuities-Certain at Time n 14
1.4 Summary 14
1.5 Exercise 17
2 Individual Future Lifetime 21
2.1 Introduction 21
2.2 A Newborn's Future Lifetime X 21
2.3 Future Lifetime of(x) 26
2.3.1 Relationship Between Probability Functions of X and T(x) 28
2.3.2 Curtate-Future-Lifetime of(x) 32
2.3.3 Conditional Average Death Time 33
2.3.4 Central Force of Mortality 34
2.4 Life Table 34
2.4.1 Aggregate Life Table 35
2.4.2 Select-and-Ultimate Life Table 42
2.5 Summary 49
2.6 Exercise 52
3 Life Insurance 59
3.1 Introduction 59
3.2 Continuous Life Insurance 59
3.2.1 Level Life Insurance 60
3.2.2 A General Continuous Life Insurance 65
3.3 Discrete Life Insurance 68
3.3.1 Level Life Insurance 68
3.3.2 A General Discrete Life Insurance 71
3.3.3 Commutation Functions 73
3.4 m-thly Life Insurance 77
3.5 Endowment Insurance 80
3.6 Summary 82
3.7 Exercise 85
4 Life Annuities 93
4.1 Introduction 93
4.2 Continuous Life Annuities 94
4.2.1 Level Life Annuities 94
4.2.2 Varying Continuous Life Annuities 99
4.3 Annual Life Annuities 101
4.3.1 Level Annual Life Annuities 101
4.3.2 Varying Annual Life Annuities 106
4.3.3 Commutation Functions 107
4.4 Special Life Annuities 112
4.4.1 m-thly Life Annuities 112
4.4.2 n-Year-Certain-and-Life Annuities 118
4.4.3 Apportionable Annuities-Due 119
4.4.4 Complete Annuities-Immediate 120
4.5 Summary 121
4.6 Exercise 127
5 Insurance Premiums 135
5.1 Introduction 135
5.2 Insurance Pricing Principles 135
5.2.1 The Three Pricing Principles 137
5.2.2 Single Benefit Premiums 140
5.3 Benefit Premiums 141
5.3.1 Fully Continuous Benefit Premiums 142
5.3.2 Fully Discrete Benefit Premiums 144
5.3.3 m-thly Benefit Premiums 148
5.3.4 Apportionable Benefit Premiums 150
5.4 Gross Insurance Premiums 152
5.4.1 Classification of Expenses 152
5.4.2 Gross Premiums Under the Equivalence Principle 153
5.5 Summary 155
5.6 Exercises 160
6 Insurance Reserves 173
6.1 Introduction 173
6.2 Insurance Reserve Principles 173
6.2.1 The Prcspective Loss Random Variable 174
6.2.2 The Three Common Principles 175
6.3 Insurance Benefit Reserves 177
6.3.1 Benefit Reserves for Fully Continuous Life Insurance 178
6.3.2 Benefit Reserves for Fully Discrete Life Insurance 183
6.3.3 Benefit Reserves with the Retrospective Method 187
6.3.4 Recursive Formula between Discrete Benefit Reserves 190
6.4 Benefit Reserves for Special Life Insurance 197
6.4.1 Benefit Reserves for m-thly Life Insurance 197
6.4.2 Benefit Reserves for Mixed Life Insurance 199
6.4.3 Benefit Reserves with Apportionable Premiums 200
6.4.4 Gross Insurance Reserves 202
6.5 Summary 203
6.6 Excercise 207
7 Joint-Life Functions 223
7.1 Introduction 223
7.2 Joint Distributions of Future Lifetimes 223
7.2.1 The Joint-Life Status 226
7.2.2 Last-Survivor Status(？) 230
7.3 Relationship among T(x),T(y),Txy,and T？ 233
7.4 Contingent Probabilities 236
7.5 Dependent Models 240
7.5.1 Common Shock Model 240
7.5.2 Frank's Copula 243
7.6 Life Insurance on Two Individuals 245
7.6.1 Life Insurance on(xy) and (？) 245
7.6.2 Contingent Life Insurance 249
7.7 Life Annuities on Two Individuals 250
7.7.1 Life Annuities on(xy) and (？) 250
7.7.2 Reversionary Annuities 252
7.8 Summary 254
7.9 Exercise 257
8 Multiple-Decrement Model 271
8.1 Introduction 271
8.2 A Double-Decrement Model 271
8.2.1 Future Lifetimes of Two Risks 271
8.2.2 Probabilities of Decrement 276
8.3 A General m-Decrement Model 282
8.3.1 Probabilities of Decrement 282
8.3.2 Central Rates from a Multiple-Decrement Table 287
8.3.3 Constructing a Multiple-Decrement Table 289
8.4 Discretionary Life Insurance 294
8.4.1 Benefit Premiums for Discretionary Life Insurance 296
8.4.2 Benefit Reserves for Discretionary Life Insurance 297
8.4.3 Asset Share 298
8.5 Summary 302
8.6 Exercise 305
Appendix 1 Standard Normal Table 321
Appendix 2A Illustrative Life Table with i=0.06 322
Appendix 2B Illustrative Service Table with i=0.06 326
Appendix 2C Interest Rate Function at i=0.06 327
Appendix 3 Probability Theorem and Random Variables 328
Appendix 4 Interest Rate and Annuity-Certain 331
Bibliography 333
Symbol Index 334
Index 339