Estimation of the actuarial present value of the whole continuous life annuity

Consider the estimation problem of the present value for the whole continuous life annuity 1 - A _x Cx (5) = x 5 where A _x is a net premium (mathematical expectation of the present value of an insured unitary sum for the whole life insurance at age x), 5 is a force of interest. As an estimate a _x (5) from a random lifetime sample X ₁.. .X _N , we take the statistics ( ₅ N _х e Z e '/ (X _t > x) i =1 x = 7 5 1 - - N Z I (X, > x) i =1 where I(A) is an indicator of a set A. The main part of the asymptotic mean square error of the estimate (5) is found, and the asymptotic normality of (5) is proved. The simulations were carried out for the de Moivre model with the limiting age 100 years. In this case, _, _я, 5(ю - x) - 1 + e ^^ - (5) = --г-, 5>- x) the present value of the whole continuous life annuity for a person at the age x = 45 years, by 5 = 0.09531 and monthly payments in the size of 1000 rubles, is equal to 12000• - ₄₅(0.09531) = 12000• 8,501 = 102012 rub. The simulations show that the empirical mean square errors of life annuity estimates decrease when the sample size N increases.

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