## Life Insurance Theory: Actuarial PerspectivesThis 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

175 | |

177 | |

181 | |

### Other editions - View all

### Common terms and phrases

alive approximations basic random variable capital-function Chapter classical ﬁnancial model closed portfolio commutation functions consider constant continuous function contract vanishes decomposition formula deferred life capital deﬁnitions denote discount factor Dxtn equals equation equivalence principle estimator evaluation expected number Figure ﬁrst decease ﬁxed force of mortality fumishes future lifetime graph with four graph with retums Hence indicator function instant instantaneous interest rate insurance contracts insurer pays integer interval s,s+t iterative relation joint-life last member Let Q linear interpolation lives x notations number of individuals number of jumps observe open portfolio partitioned power series present value level probability of ruin proﬁt Proof radius of convergence replaced resp s+k,s+k+1 satisﬁes simple status solvency parameter staircase stochastic time-capital tables Theorem true interest rate valid vanishes at ﬁrst Variable interest rates variance Variation of reserves whole life insurance Zosksn-1