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Basic Life Insurance Mathematics

Although experience with enforcing mandates for health insurance is limited, some lessons can be drawn from other types of enforcement. National compliance rates for certain other mandates—such as provisions requiring that workers receive at least the minimum wage, children be vaccinated, drivers purchase auto insurance and wear seat belts, and people pay taxes—range from 63 percent to 85 percent, and the rates appear to improve as enforcement intensifies (Ashenfelter and Smith, 1979; Davis and Gaglia, 2005; Insurance Research Council, 2006; IRS, 2007; NHTSA, 2008). The data from those studies of compliance can provide some insight about compliance overall. The rates. | Basic Life Insurance Mathematics Ragnar Norberg Version September 2002 Contents 1 Introduction 5 1.1 Banking versus insurance . 5 1.2 Mortality. 7 1.3 Banking. 9 1.4 Insurance. 11 1.5 With-profit contracts Surplus and bonus. 14 1.6 Unit-linked insurance. 16 1.7 Issues for further study. 17 2 Payment streams and interest 19 2.1 Basic definitions and relationships . 19 2.2 Application to loans. 25 3 Mortality 28 3.1 Aggregate mortality. 28 3.2 Some standard mortality laws. 33 3.3 Actuarial notation. 35 3.4 Select mortality. 37 4 Insurance of a single life 39 4.1 Some standard forms of insurance . 39 4.2 The principle of equivalence. 43 4.3 Prospective reserves. 45 4.4 Thiele s differential equation. 52 4.5 Probability distributions. 56 4.6 The stochastic process point of view . 57 5 Expenses 59 5.1 A single life insurance policy . 59 5.2 The general multi-state policy. 62 6 Multi-life insurances 63 6.1 Insurances depending on the number of survivors. 63 1 CONTENTS 2 7 Markov chains in life insurance 67 7.1 The insurance policy as a stochastic process. 67 7.2 The time-continuous Markov chain. 68 7.3 Applications. 73 7.4 Selection phenomena. 77 7.5 The standard multi-state contract . 79 7.6 Select mortality revisited . 86 7.7 Higher order moments of present values. 89 7.8 A Markov chain interest model . 94 7.8.1 The Markov model. 94 7.8.2 Differential equations for moments of present values . 95 7.8.3 Complement on Markov chains. 98 7.9 Dependent lives.100 7.9.1 Introduction .100 7.9.2 Notions of positive dependence.101 7.9.3 Dependencies between present values.103 7.9.4 A Markov chain model for two lives .103 7.10 Conditional Markov chains .106 7.10.1 Retrospective fertility analysis .106 8 Probability distributions of present values 109 8.1 Introduction.109 8.2 Calculation of probability distributions of present values by elementary methods . 110 8.3 The general Markov multistate policy .111 8.4 Differential equations for statewise distributions.112 8.5 .