In Woolhouse approximations, ax:n equals which of the following relationships?

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Multiple Choice

In Woolhouse approximations, ax:n equals which of the following relationships?

Woolhouse approximations give a simple way to estimate the present value of a finite-term life annuity using the whole-life value, life expectancy, and the annuity at a higher age. Let a_x be the present value of a life annuity payable for life at age x, a_{x+n} the same for age x+n, and E_x the expected remaining lifetime at age x. The idea is to split the whole-life annuity into two parts: payments during the first n years (the finite-term part) and payments after year n (the tail). The tail is approximated by n times the expected remaining years, E_x, times the perpetuity value at age x+n, a_{x+n}. Therefore the whole-life value satisfies a_x ≈ a_x:n + n E_x a_{x+n}, which rearranges to a_x:n ≈ a_x − n E_x a_{x+n}. The relation shown is the standard Woolhouse approximation.

Woolhouse approximations give a simple way to estimate the present value of a finite-term life annuity using the whole-life value, life expectancy, and the annuity at a higher age. Let a_x be the present value of a life annuity payable for life at age x, a_{x+n} the same for age x+n, and E_x the expected remaining lifetime at age x. The idea is to split the whole-life annuity into two parts: payments during the first n years (the finite-term part) and payments after year n (the tail). The tail is approximated by n times the expected remaining years, E_x, times the perpetuity value at age x+n, a_{x+n}. Therefore the whole-life value satisfies a_x ≈ a_x:n + n E_x a_{x+n}, which rearranges to a_x:n ≈ a_x − n E_x a_{x+n}. The relation shown is the standard Woolhouse approximation.

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