Download Life Insurance Theory: Actuarial Perspectives by F. Etienne De Vylder PDF

By F. Etienne De Vylder

ISBN-10: 144195189X

ISBN-13: 9781441951892

ISBN-10: 1475726163

ISBN-13: 9781475726169

This publication isn't the same as all different books on lifestyles assurance via a minimum of one of many following features 1-4. 1. The therapy of existence insurances at 3 diversified degrees: time-capital, current price and value point. We name time-capital any distribution of a capital over the years: (*) is the time-capital with quantities Cl, ~, ... , C at moments Tl, T , ..• , T resp. N 2 N for example, permit (x) be a existence at immediate zero with destiny lifetime X. Then the entire oO oO existence assurance 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 worth ofA 00 and ä 00 is x x zero zero 2 A = ~ and ä = 1 + v + v + ... + v'X resp. x x the associated fee (or top class) of a time-capital is the expectancy of its current price. specifically, 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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The price ofthis insurance is AX<~). Let us now assume that the interest rate i' equals the interest rate i used by the insurer. ,k at h. ],tk , tk) Its price is (36) Let us verify that the latter price equals Ä x ( ~). e. ,k=O ifk is large enough (then no convergence problems do occur). ,l~oo Inqx a", V . In practice, the time-capital (35) is some c1assical annuity-certain in most cases. 6. Variable interest rates. The notations are completed with the indication 0" in brackets, if present values are calculated in the financial model with instantaneous interest rate function 0".

N-l) is partitioned in r periods [s+k+v/r,s+k+(v+ 1)/r] (v=O,I, ... ,r-l), lIr lIr lIr lIr 1----1----1---1---1 s+k s+k+l 34 Life Insurance Theory and the amOlmt 1/r is attached at each of these periods. It is attached at the end of the period in case of the partitioned annuity-immediate sjmax(r)oo and at the beginning ofthe period in case ofthe partitioned annuity-due sjmä(r)oo. The amount attached at an instant is paid by the insurer if x is alive at that instant. Hence, slna/)oo := (1/r) LI~V::illr (1xts+v/r, s+v/r), slnä}r) 00 := (1/r) Losv::illr-l (1xts+v/r, s+v/r) The price of the partitioned annuities is slnax(r) = (1/r) LI~v::illr s+v/rEx, (12) slnä/) = (1/r) Lo~V::illr-1 s+v/rEx.

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. (r-l)/(2r). (56) By (52) and by a summation byparts, Dx n(Dä)x = Lo~-l (n-k) DX+k = - Lo~-l (n-k) ANX+k = -[(n-k)NX+k1on - Lo~-l NX+k+l = nNx - (~ - ~)NX+k+l = nNx - Sx+l + Sx+n+l.

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