Assume constant interest and benefits at the end of the

Question

1. A modified reserve system has a first-year premium of b₀vq_x a second-year premium of b₁vq_x+1, followed by a level renewal premium of β. Assume constant interest and benefits at the end of the year of death. Find an expression for β, when premiums are payable for n years.

2. Consider a policy issued at age x with benefits payable at the end of the year of death. A modified reserve system, used in Canada for tax purposes (sometimes known as the 1½ preliminary term), takes an initial premium to b₀vq_x, a premium at time 1 of β, and a level premium of γ payable after time 1, such that β is the average of b₁vq_x+1 and γ. Assume constant interest and benefits paid at the end of the year of death. If premiums are payable for life, show that

γ= [A_x+1(b_°1) + vp_x+1 A_x+2(b_°2)]/[a_x+1vp_x+1a_x+2]