F. Etienne De Vylder 
Life Insurance Theory [PDF ebook] 
Actuarial Perspectives

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 o O o O life insurance A is the time-capital (I, X). The whole life annuity a is the x x time-capital (1, 0) + (1, 1) + (1, 2) + … + (I, ‘X), where ‘X is the integer part of X. The present value at 0 of time-capital (*) is the random variable T1 T TN Cl V + ~ v , + … + CNV . (**) In particular, the present value of A 00 and a 00 is x x 0 0 2 A = ~ and a = 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 of A 00 and ax 00 is x 2 A = E(~) and a = E(I + v + v + … + v’X) resp.