1. A modified reserve system has a first-year premium of b0vqx a second-year premium of b1vqx+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 b0vqx, a premium at time 1 of β, and a level premium of γ payable after time 1, such that β is the average of b1vqx+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= [Ax+1(b°1) + vpx+1 Ax+2(b°2)]/[ax+1vpx+1ax+2]