Life Tables and Life Insurance and Annuity Contracts
Life-table mechanics
This domain tests assurance benefits, annuity benefits, premiums and reserves using life tables. From l_x, d_x=l_x-l_{x+1}, p_x=l_{x+1}/l_x and q_x=d_x/l_x. Deferred mortality is m|n q_x=(l_{x+m}-l_{x+m+n})/l_x. A whole-life assurance paying 1 at the end of the year of death has A_x=sum v^{k+1} k p_x q_{x+k}. A whole-life annuity-due paying 1 while alive has a-due_x=sum v^k k p_x. For level annual net premiums P, the equivalence principle gives P=A_x/a-due_x for whole-life assurance.
Premiums and reserves
Gross premiums also cover expenses and profit; net premiums cover benefits only. A prospective reserve at duration t is EPV of future benefits minus EPV of future net premiums. For whole-life assurance, t V_x=A_{x+t}-P a-due_{x+t}. Reserves increase with duration for whole-life contracts because the probability of claim rises and fewer premiums remain.
Worked example
For a whole-life assurance of 1 on a life aged x, assume A_x=0.25 and a-due_x=15. The level annual net premium is P=0.25/15=0.016667 per unit of sum assured, or 16.67 per 1,000. After one year, if A_{x+1}=0.27 and a-due_{x+1}=14.5, the prospective reserve is 0.27-0.016667(14.5)=0.02833 per unit, about 28.33 per 1,000.
Traps
Using an annuity-immediate when premiums are payable in advance; paying the death benefit at the start of the year; confusing sum assured with premium; and calculating a reserve without updating age and remaining term. Select mortality can also make A_{[x]+t} differ from A_{x+t}.