By F. Etienne De Vylder

This publication isn't the same as all different books on existence assurance by way of not less than one of many following features 1-4. 1. The therapy of existence insurances at 3 various degrees: time-capital, current price and value point. We name time-capital any distribution of a capital through the years: (*) is the time-capital with quantities Cl, ~, ... , C at moments Tl, T , ..• , T resp. N 2 N for example, allow (x) be a existence at quick zero with destiny lifetime X. Then the complete oO oO existence coverage A is the time-capital (I,X). the complete existence annuity ä is the x x time-capital (1,0) + (1,1) + (1,2) + ... + (I,'X), the place 'X is the integer half ofX. the current price at zero of time-capital (*) is the random variable T1 T TN Cl V + ~ v , + ... + CNV . (**) specifically, the current price ofA 00 and ä 00 is x x zero zero 2 A = ~ and ä = 1 + v + v + ... + v'X resp. x x the cost (or top rate) of a time-capital is the expectancy of its current price. particularly, the fee ofA 00 and äx 00 is x 2 A = E(~) and ä = E(I + v + v + ... + v'X) resp.

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Then x and y denote at the same time the persons and their ages at the origin of time. The future lifetime of x and y is the random variable X and Y resp. Hence, x dies at instant X at age x+X and y dies at instant Y at age y+Y. The life tables for x and y are I; and ITJ resp. They may be different (although the same 1is used). The lives x and y are independent if the random variables X and Y are stochastically independent. The independence assumption is not very realistic in some cases: the future lifetimes X and Y of a married couple are certainly not independent (for instance, because they use the same car simulteneously).

1. Life insurance models The conjooction of a financial model and a mortality model is a Iife wurance model The c1assical Iife wurance model is the conjooction of the classical financial model (with constant interest rate i) and the classical mortality model (with survival probabilities resulting from a continuous life table I;). The classical life insurance model is adopted if nothing else is stated explicitly. Some developments have direct generalizations in case of variable interest rates: it is enough to replace the discoWlt factor v" by v" everywhere.

N-l n n+l n+2... (43) instants and the capital-function of the annuities (Ia)x°o r,(r)] is defined on [O,oo[ as follows 1 2 n n+l n+2 ... ~' r--r--'I-I---r--r--r--r-- o 1 2 n-l n n+1 n+2... (r-l)/(2r). By (45) and (40) (47) 1 (48) (49) and by(41) c. n(Da)XOO := olnax(Ct)OO, n(Dä)xOO := Olnäx(Ct)OO, n(Da)/)oo := olnax(r)(Ctto, n(Dä)x(r)oo := olnä/)(Ct)OO, where Ct is defined on [O,n] as follows: n n-l n-2 ... 1 t-;-t- ----n~l~stants ~ 1 3t- (51) Then n(Dä)x = Lo~-l (n-k) kEx = n + Ll~-l (n-k) kEx, (52) n(Da)x = Ll~ (n-k+ 1) kEx = Ll~-l (n-k+ 1) kEx + ,Ä, (53) n(Dä)x - n(Da)x = n - Ll~-l kEx - ,Ä = n - nax.