Collective annuity estimation of joint-life status

Consider the case of collective life insurance for which a useful abstraction is a status. Let m individuals of ages x_m conclude an insurance contract. In accordance with the notation of actuarial mathematics, let the random variable X be lifetime, T(xk) = X - xk be residual lifetime of the k-th individual. We put in line to the set of m numbers T(x_:),...,T(x_m) the status U, which has its own lifetime T (U) . The joint-life status is denoted U := x_: :...: x_m and is considered to be destroyed if at least one of the individuals has died, i.e. T (U) = mm (T (x,),..., T (xm)). It is clear that P{T(U) > t} = P{mm(T(x,),...,T(xm)) > t} = P{T(x,) > t,...,T(xm) > t} = S^ (t)••• S^ (t), where S_x (t) = -(-) is the survival function of the random variable T(x). It is for this status, we define the present value of continu- S ( x) ous time life annuity. By analogy with the case of individual insurance, the annuity is expressed in terms of the net premium: j(( ^.xm ), x,..:xm 5 where 5 is the rate of interest, the net premium A _{: :}x is expressed by the formula 4,„x_m =Jef_x,...:x_m (t)dt, 0 in which the distribution density of the joint-life status is defined as m U, _m (t) = X Sx_i(t )-fx, (t )• Sx_m (t). i=1 Here f_x (t) is the distribution density of the random variable T(x). The paper deals with the problem of finding numerical values for functionals of annuities for the join-life status of two persons for a number of parameterized distributions of actuarial mathematics. Also, the corresponding estimates of annuities for the join-life status of two persons are found.

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