Textbook - Actuarial Mathematics for Life Contingent Risks...

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Unformatted text preview: Actuarial Mathematics for Life Contingent Risks David C M Dickson, Mary R Hardy, Howard R Waters September 5, 2008 2 Contents 1 Introduction to Life Insurance 15 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Life and annuity contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.2 Traditional insurance contracts . . . . . . . . . . . . . . . . . . . . 19 1.3.3 Modern insurance contracts . . . . . . . . . . . . . . . . . . . . . . 22 1.3.4 Distribution methods . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.5 Underwriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.6 Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3.7 Life annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Other insurance contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Pension benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.1 Defined benefit and defined contribution pensions . . . . . . . . . . 30 1.5.2 Defined benefits pension design . . . . . . . . . . . . . . . . . . . . 31 1.6 Typical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 4 CONTENTS 1.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Survival Models 35 2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 The future lifetime random variable . . . . . . . . . . . . . . . . . . . . . . 35 2.3 The force of mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Actuarial Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Mean and Standard Deviation of T x . . . . . . . . . . . . . . . . . . . . . . 50 2.6 Curtate future lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6.1 K x and e x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.6.2 The complete and curtate expected future lifetimes, e x and ◦ e x . . . 56 2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 Life Tables and Selection 65 3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Life tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Fractional age assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3.1 Uniform distribution of deaths . . . . . . . . . . . . . . . . . . . . . 69 3.3.2 Constant force of mortality . . . . . . . . . . . . . . . . . . . . . . 74 3.4 National life tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76National life tables ....
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Textbook - Actuarial Mathematics for Life Contingent Risks...

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