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Basic Life Insurance Mathematics

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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-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 ele- mentary methods . 110 8.3 The general Markov multistate policy . 111 8.4 Differential equations for statewise distributions . 112 8.5 Applications . 116 9 Reserves 119 9.1 Introduction . 119 9.2 General definitions 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 3 10.8 Discussions . 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 Confidence regions . 176 11.4 More on simultaneous confidence 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 fluctuation reserve . 195 13.5 Estimation of parameters . 197 14 Hattendorff and Thiele 198 14.1 Introduction . 198 14.2 The general Hattendorff 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 financial 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 finance . 235 15.10Two properties of the Vandermonde matrix . 236 15.11Applications to finance . 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 Chapter 1 Introduction 1.1 Banking versus insurance A. The bank savings contract. Upon celebrating his 55th anniversary Mr. (55) (let us call him so) decides to invest money to secure himself economically in his old age. The first idea that occurs to him is to deposit a capital of S0 = 1 (e.g. one hundred thousand pounds) on a savings account today and draw the entire amount with earned compound interest in 15 years, on his 70th birthday. The account bears interest at rate i = 0:045 (4:5%) per year. In one year the capital will increase to S1 = S0 +S0 i = S0(1+i), in two years it will increase to 2 S = S1 + S1 i = S0(1 + i) , and so on until in 15 years it will have accumulated to 15 15 S15 = S0 (1 + i) = 1:045 = 1:935 : (1.1) This simple calculation takes no account of the fact that (55) will die sooner or later, maybe sooner than 15 years. Suppose he has no heirs (or he dislikes the ones he has) so that in the event of death before 70 he would consider his savings waisted. Checking population statistics he learns that about 75% of those who are 55 will survive to 70. Thus, the relevant prospects of the contract are: { with probability 0.75 (55) survives to 70 and will then possess S15; { with probability 0.25 (55) dies before 70 and loses the capital. In this perspective the expected amount at (55)'s disposal after 15 years is 0:75 S15 : (1.2) B. A small scale mutual fund. Having thought things over, (55) seeks to make the following mutual arrangement with (55)∗ and (55)∗∗, who are also 55 years old and are in exactly the same situation as (55). Each of the three deposits S0 = 1 on the savings account, and those who survive to 70, if any, will then share the total accumulated capital 3 S15 equally. The prospects of this scheme are given in Table 1.1, where + and − signify survival and death, respectively, L70 is the number of survivors at age 70, and 5 CHAPTER 1. INTRODUCTION 6 Table 1.1: Possible outcomes of a savings scheme with three participants. (55) (55)∗ (55)∗∗ L70 3 S15=L70 Probability + + + 3 S15 0:75 · 0:75 · 0:75 = 0:422 + + − 2 1:5 S15 0:75 · 0:75 · 0:25 = 0:141 + − + 2 1:5 S15 0:75 · 0:25 · 0:75 = 0:141 + − − 1 3S15 0:75 · 0:25 · 0:25 = 0:047 − + + 2 1:5 S15 0:25 · 0:75 · 0:75 = 0:141 − + − 1 3 S15 0:25 · 0:75 · 0:25 = 0:047 − − + 1 3 S15 0:25 · 0:25 · 0:75 = 0:047 − − − 0 undefined 0:25 · 0:25 · 0:25 = 0:016 3 S15=L70 is the amount at disposal per survivor (undefined if L70 = 0). There are now the following possibilities: { with probability 0.422 (55) survives to 70 together with (55)∗ and (55)∗∗ and will then possess S15; { with probability 2 · 0:141 = 0:282 (55) survives to 70 together with one more survivor and will then possess 1:5 S15; { with probability 0.047 (55) survives to 70 while both (55)∗ and (55)∗∗ die (may they rest in peace) and he will cash the total savings 3S15; { with probability 0.25 (55) dies before 70 and will get nothing. This scheme is superior to the one described in Paragraph A, with separate individual savings contracts: If (55) survives to 70, which is the only scenario of interest to him, he will cash no less than the amount S15 he would cash under the individual scheme, and it is likely that he will get more. As compared with (1.2), the expected amount at (55)'s disposal after 15 years is now 0:422 · S15 + 0:282 · 1:5 · S15 + 0:047 · 3 S15 = 0:985 S15 : The point is that under the present scheme the savings of those who die are bequeathed to the survivors. Thus the total savings are retained for the group so that nothing is left to others unless the unlikely thing happens that the whole group goes extinct within the term of the contract. This is essentially the kind of solidarity that unites the members of a pension fund. From the point of view of the group as a whole, the probability that all three participants will die before 70 is only 0.016, which should be compared to the probability 0:25 that (55) will die and lose everything under the individual savings program. C. A large scale mutual scheme. Inspired by the success of the mutual fund idea already on the small scale of three participants, (55) starts to play with the idea of extending it to a large number of participants. Let us assume that a total number of L55 persons, who are in exactly the same situation as (55), agree to join a scheme similar to the one described for the three.
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