Basic Life Insurance Mathematics

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-prot 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 denitions 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 dierential 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 Dierential 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 ele- mentary methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.3 The general Markov multistate policy . . . . . . . . . . . . . . . 111 8.4 Dierential equations for statewise distributions . . . . . . . . . . 112 8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9 Reserves 119 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.2 General denitions of reserves and statement of some relation- ships between them . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.3 Description of payment streams appearing in life and pension insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 9.4 The Markov chain model . . . . . . . . . . . . . . . . . . . . . . . 126 9.5 Reserves in the Markov chain model . . . . . . . . . . . . . . . . 131 9.6 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10 Safety loadings and bonus 145 10.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . 145 10.2 First and second order bases . . . . . . . . . . . . . . . . . . . . . 146 10.3 The technical surplus and how it emerges . . . . . . . . . . . . . 147 10.4 Dividends and bonus . . . . . . . . . . . . . . . . . . . . . . . . . 149 10.5 Bonus prognoses . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 10.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10.7 Including expenses . . . . . . . . . . . . . . . . . . . . . . . . . . 161 CONTENTS 10.8 Discussions 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 11 Statistical inference in the Markov chain model 167 11.1 Estimating a mortality law from fully observed life lengths . . . . 167 11.2 Parametric inference in the Markov model . . . . . . . . . . . . . 172 11.3 Condence regions . . . . . . . . . . . . . . . . . . . . . . . . . . 176 11.4 More on simultaneous condence intervals . . . . . . . . . . . . . 177 11.5 Piecewise constant intensities . . . . . . . . . . . . . . . . . . . . 179 11.6 Impact of the censoring scheme . . . . . . . . . . . . . . . . . . . 183 12 Heterogeneity models 185 12.1 The notion of heterogeneity { a two-stage model . . . . . . . . . 185 12.2 The proportional hazard model . . . . . . . . . . . . . . . . . . . 187 13 Group life insurance 190 13.1 Basic characteristics of group insurance . . . . . . . . . . . . . . 190 13.2 A proportional hazard model for complete individual policy and claim records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 13.3 Experience rated net premiums . . . . . . . . . . . . . . . . . . . 194 13.4 The uctuation reserve . . . . . . . . . . . . . . . . . . . . . . . . 195 13.5 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . 197 14 Hattendor and Thiele 198 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 14.2 The general Hattendor theorem . . . . . . . . . . . . . . . . . . 199 14.3 Application to life insurance . . . . . . . . . . . . . . . . . . . . . 201 14.4 Excerpts from martingale theory . . . . . . . . . . . . . . . . . . 205 15 Financial mathematics in insurance 212 15.1 Finance in insurance . . . . . . . . . . . . . . . . . . . . . . . . . 212 15.2 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 15.3 A Markov chain nancial market - Introduction . . . . . . . . . . 218 15.4 The Markov chain market . . . . . . . . . . . . . . . . . . . . . . 219 15.5 Arbitrage-pricing of derivatives in a complete market . . . . . . . 226 15.6 Numerical procedures . . . . . . . . . . . . . . . . . . . . . . . . 229 15.7 Risk minimization in incomplete markets . . . . . . . . . . . . . 229 15.8 Trading with bonds: How much can be hedged? . . . . . . . . . . 232 15.9 The Vandermonde matrix in nance . . . . . . . . . . . . . . . . 235 15.10Two properties of the Vandermonde matrix . . . . . . . . . . . . 236 15.11Applications to nance . . . . . . . . . . . . . . . . . . . . . . . . 237 15.12Martingale methods . . . . . . . . . . . . . . . . . . . . . . . . . 240 A Calculus 4 B Indicator functions 9 C Distribution of the number of occurring events 12 CONTENTS 4 D Asymptotic results from statistics 15 E The G82M mortality table 17 F Exercises 1 G Solutions to exercises 1 ... - tailieumienphi.vn