# Life Insurance Theory: Actuarial Perspectives

Springer Science & Business Media, Aug 31, 1997 - Business & Economics - 184 pages
This book is different from all other books on Life Insurance by at least one of the following characteristics 1-4. 1. The treatment of life insurances at three different levels: time-capital, present value and price level. We call time-capital any distribution of a capital over time: (*) is the time-capital with amounts Cl, ~, ... , C at moments Tl, T , ..• , T resp. N 2 N For instance, let (x) be a life at instant 0 with future lifetime X. Then the whole oO oO life insurance A is the time-capital (I,X). The whole life annuity ä is the x x time-capital (1,0) + (1,1) + (1,2) + ... + (I,'X), where 'X is the integer part ofX. The present value at 0 of time-capital (*) is the random variable T1 T TN Cl V + ~ v , + ... + CNV . (**) In particular, the present value ofA 00 and ä 00 is x x 0 0 2 A = ~ and ä = 1 + v + v + ... + v'X resp. x x The price (or premium) of a time-capital is the expectation of its present value. In particular, the price ofA 00 and äx 00 is x 2 A = E(~) and ä = E(I + v + v + ... + v'X) resp.

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### Contents

 Financial Models 1 13 Variable interest rates 4 14 Deterministic timecapitals 7 15 Stochastic timecapitals 8 16 Annuitiescertain 9 17 Stochastic interests 10 Mortality Models 11 23 Force of mortality 12
 Ruin Probability of a Life Insurance Company 83 102 Profit of a contract 84 104 Probability of ruin in a closed portfolio 85 105 Solvency parameter of a portfolio 86 108 Probability of ruin in an open portfolio 87 1010 Open portfolio with exponential growth 88 1012 Evaluation of variances General methodology 89 1013 Deferred life capital 90

 24 Decease in the middle of the year 13 25 Expected future lifetime 14 26 Analytic life tables 15 27 Restricted life tables 16 29 Commutation functions 17 Construction of Life Tables 19 32 National tables 20 33 Private tables 21 34 Analytic leastsquares graduation 22 36 Determination of initial parameters in the Makeham case 24 Basic Concepts of Life Insurance Mathematics 25 42 Contracts 26 44 Validity level of relations 27 46 Null events 28 Life Annuities One Life 29 52 Constant life annuities 30 53 Partitioned life annuities 33 54 General variable life annuities 35 55 Classical variable life annuities 37 56 Annuities on status x 40 57 Variable interest rates 41 Life Insurances One Life 43 62 General variable life insurances 46 63 Classical variable life insurances 47 64 Endowments 48 65 Insurance of a remaining debt at death 49 66 Variable interest rates 50 Relations Between Life Annuities and Life Insurances One Life 51 72 Constant annuities and insurances Present value level 52 73 Variable annuities and insurances General discrete case 53 74 Variable annuities and insurances General continuous case 54 75 Classical variable annuities and insurances 55 Decomposition of TimeCapitols One Life 57 82 The decomposition formula 59 83 Evaluation of a reserve at a noninteger instant 60 84 Fourets Formula 62 86 Insurances payable in the middle of the year of death 64 Life Insurance Contracts One Life 65 92 Reserves of a contract 66 93 Practical constraints on contracts 68 94 Contracts with partitioned premiums 69 general endowment insurance 71 97 Positive reserves analytic proofs 72 98 Variation of prices with interest rate i 74 99 Variation of reserves with interest rate i 75 910 Variation of reserves with time t 78 912 Expense loadings 80
 1014 General life insurance 91 1016 Variance of reserves 94 Insurance on a Status Several Lives 95 112 Probabilities on a status 97 113 Deferred capitals on a status 100 114 Life annuities on a status 101 115 Life insurances on a status 102 116 Alternative notations 103 Decomposition of TimeCapitals Several Lives 105 122 Timecapitals vanishing at first decease 106 123 The decomposition formula 107 125 Evaluation of a reserve at a noninteger instant 108 126 Fourets formula 109 Life Insurance Contracts Several Lives 111 133 Practical constraints on contracts 112 134 Contracts with partitioned premiums 113 Multiple Decrement Models 115 142 Other graphs 118 143 Events and probabilities on a graph 119 144 Annuities on states of a graph 123 145 Transition capitals on a graph 125 146 Transition Theorem for timecapitals Price level 126 147 Illustration in the case of graph Grxy 128 Variances Several Lives 133 152 Evaluation of variances General methodology 134 153 Deferred life capitals 136 156 Variance of reserves 138 Population groups on a Graph 139 162 Open graph model 140 163 Estimation of instantaneous transition rates 141 164 Estimations in a graph with two states 142 165 Estimations in a graph with three states 145 166 Estimations in a graph with four states 148 167 Evaluation of state probabilities 152 168 State probabilities in a graph with two states 153 1610 State probabilities in a graph with four states 154 1611 Mortality estimations 155 SUMMATION BY PARTS 157 LINEAR INTERPOLATIONS 159 PROBABILITY THEORY 163 A DIFFERENTIAL EQUATION 169 INVERSION OF A POWER SERIES 171 SUMMARY OF FORMULAS 173 References 175 Notation Index 177 Subject Index 181 Copyright